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An introduction to gravitation teory

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Instituto de F
sica Te
orica
Universidade Estadual Paulista
An Introduction to
GRAVITATION THEORY
R. Aldrovandi and J. G. Pereira
2001
A Preliminary Note
These notes are intended for a one-semester post-graduate course. Addressed to future researchers in a Centre mainly devoted to Field Theory,
they avoid the ex cathedra style frequently assumed by teachers of the subject. Mainly, General Relativity is not presented as a nished theory.
Emphasis is laid on the basic tenets and on comparison of gravitation
with the other fundamental interactions of Nature. Thus, a little more space
than would be expected in such a short text is devoted to the equivalence
principle.
The equivalence principle leads to universality, a distinguishing feature of
the gravitational eld. The other fundamental interactions of Nature | the
electromagnetic, the weak and the strong interactions, which are described
in terms of gauge theories | are not universal.
Actually, universality does not preclude gravitation from presenting a
gauge description. In fact, as we are going to see, in addition to the General
Relativity description, gravitation can be given a presentation in terms of a
gauge theory. In the speci c case of the teleparallel equivalent of General
Relativity, it turns out to be a gauge theory for the translation group. In
this case, instead of a curvature, the gravitational eld is attributed to a torsion. Accordingly, instead of the metric structure of the pseudo{riemannian
spacetime, the underlying spacetime structure will be another, the so-called
teleparallel structure. A spacetime, once endowed with such a structure, is
called a Weitzenbock spacetime. Of course, other spacetime{rooted gauge
theories can be constructed, and models which are much more general than
General Relativity can be obtained. These theories, however, will not be
considered in these notes.
The text is divided into two parts. The rst is a rather standard account of
General Relativity. The second is devoted to a presentation of the teleparallel
approach to the same theory.
i
Contents
I General Relativity
1
1 Introduction
3
1.1 General Concepts . . . . . . . . . . . . .
1.2 Some Basic Notions . . . . . . . . . . . .
1.3 The Equivalence Principle . . . . . . . .
1.3.1 Inertial Forces . . . . . . . . . . .
1.3.2 The Wake of Non-Trivial Metric .
1.3.3 Towards Geometry . . . . . . . .
2 Geometry
2.1 Di erential Geometry . . . . . .
2.1.1 Spaces . . . . . . . . . .
2.1.2 Vector and Tensor Fields
2.1.3 Di erential Forms . . . .
2.1.4 Metrics . . . . . . . . .
2.2 Pseudo-Riemannian Metric . . .
2.3 The Notion of Connection . . .
2.4 The Levi{Civita Connection . .
2.5 Curvature Tensor . . . . . . . .
2.6 Bianchi Identities . . . . . . . .
2.6.1 Examples . . . . . . . .
3 Dynamics
3.1
3.2
3.3
3.4
3.5
3.6
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Geodesics . . . . . . . . . . . . . .
The Minimal Coupling Prescription
Einstein's Field Equations . . . . .
Action of the Gravitational Field .
Non-Relativistic Limit . . . . . . .
About Time, and Space . . . . . .
3.6.1 Time Recovered . . . . . . .
ii
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. 3
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20
20
22
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43
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57
59
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67
67
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81
84
87
90
90
3.6.2 Space . . . . . . . . . .
3.7 Equivalence, Once Again . . . .
3.8 More About Curves . . . . . . .
3.8.1 Geodesic Deviation . . .
3.8.2 General Observers . . .
3.8.3 Transversality . . . . . .
3.8.4 Fundamental Observers .
3.9 An Aside: Hamilton-Jacobi . .
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4.1 Transformations . . . . . . . . . . .
4.2 Small Scale Solutions . . . . . . . .
4.2.1 The Schwarzschild Solution
4.3 Large Scale Solutions . . . . . . . .
4.3.1 The Friedmann Solutions . .
4.3.2 Thermal History . . . . . .
4.3.3 de Sitter Solutions . . . . .
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4 Solutions
5 Tetrad Fields
5.1 Tetrads . . . . . . . . . . . . . . .
5.2 Linear Connections . . . . . . . . .
5.2.1 Linear Transformations . . .
5.2.2 Orthogonal Transformations
5.2.3 Connections, Revisited . . .
5.2.4 Back to Equivalence . . . .
5.2.5 Two Gates into Gravitation
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6 Gravitational Interaction of the Fundamental Fields
6.1 Minimal Coupling Prescription . . . .
6.2 General Relativity Spin Connection . .
6.3 Application to the Fundamental Fields
6.3.1 Scalar Field . . . . . . . . . . .
6.3.2 Dirac Spinor Field . . . . . . .
6.3.3 Electromagnetic Field . . . . .
7.1
7.2
7.3
7.4
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Global Noether Theorem . . . . . . . . . .
Energy{Momentum as Source of Curvature
Energy{Momentum Conservation . . . . .
Examples . . . . . . . . . . . . . . . . . .
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7 General Relativity with Matter Fields
iii
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92
95
97
97
98
100
101
104
112
112
116
116
133
133
145
152
161
161
166
166
168
170
174
179
181
181
182
184
184
185
186
190
190
191
193
195
7.4.1 Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . 195
7.4.2 Dirac Spinor Field . . . . . . . . . . . . . . . . . . . . 196
7.4.3 Electromagnetic Field . . . . . . . . . . . . . . . . . . 197
II Teleparallel Gravity
199
8 Teleparallel Equivalent of General Relativity
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
Preliminary Considerations . . . . . . . . .
General Concepts . . . . . . . . . . . . . .
Gauge Transformations . . . . . . . . . . .
Spacetime Geometric Structures . . . . . .
Lagrangian and Field Equations . . . . . .
Equivalence with General Relativity . . . .
Energy{Momentum Density of Gravitation
Bianchi Identities . . . . . . . . . . . . . .
9 Equation of Motion of Spinless Particles
9.1
9.2
9.3
9.4
9.5
Gravitational Lorentz Force .
Torsion: Force Equation . . .
Curvature: Geodesic Equation
Weak Equivalence Principle .
Equivalence of the Actions . .
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10 Teleparallel Interaction of the Fundamental Fields
10.1 Teleparallel Spin Connection . . . . . .
10.2 Teleparallel Coupling Prescription . . .
10.3 Application to the Fundamental Fields
10.3.1 Scalar Field . . . . . . . . . . .
10.3.2 Dirac Spinor Field . . . . . . .
10.3.3 Electromagnetic Field . . . . .
11 Teleparallel Gravity with Matter Fields
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11.1 Energy-Momentum as Source of Torsion
11.2 Energy{Momentum Conservation . . . .
11.3 Examples . . . . . . . . . . . . . . . . .
11.3.1 Scalar Field . . . . . . . . . . . .
11.3.2 Dirac Spinor Field . . . . . . . .
11.3.3 Electromagnetic Field . . . . . .
iv
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Bibliography
242
v
Part I
General Relativity
1
Chapter 1
Introduction
1.1 General Concepts
1.1
All elementary particles feel gravitation the same. More speci cally,
particles with di erent masses experience a di erent gravitational force, but
in such a way that all of them acquire the same acceleration and, given the
same initial conditions, follow the same path. Such universality of response
is the most fundamental characteristic of the gravitational interaction. It is a
unique property, peculiar to gravitation: no other basic interaction of Nature
has it.
Due to universality, the gravitational interaction admits a description
which makes no use of the concept of force. In this description, instead of
acting through a force, the presence of a gravitational eld is represented
by a deformation of the spacetime structure. This deformation, however,
preserves the pseudo-riemannian character of the at Minkowski spacetime,
the non-deformed spacetime that represents absence of gravitation. In other
words, the presence of a gravitational eld is supposed to produce curvature,
but no other kind of spacetime deformation.
A free particle in at space follows a straight line, that is, a curve keeping
a constant direction. A geodesic is a curve keeping a constant direction on
a curved space. As the only e ect of the gravitational interaction is to bend
spacetime so as to endow it with curvature, a particle submitted exclusively
to gravity will follow a geodesic of the deformed spacetime.
x
3
This is the approach of Einstein's General Relativity, according to which
the gravitational interaction is described by a geometrization of spacetime.
It is important to remark that only an interaction presenting the property of
universality can be described by such a geometrization.
1.2 Some Basic Notions
1.2
Before going further, let us recall some notions taken from classical
physics, put in a language loose enough to make them valid both in the
relativistic and the non-relativistic cases.
x
frame a reference frame is a coordinate system for space positions, to which
a clock is bound.
inertia a reference frame such that free (unsubmitted to any forces) mo-
tion takes place with constant velocity is an inertialk frame; in classical
physics, the force law in an inertial frame is m dvdt = F k ; in Special
Relativity, the force law in an inertial frame is
m
d a
U = F a;
ds
(1.1)
p
where U is the four-velocity U = ( ; v=c), with = 1= 1 v 2=c2 (as
U is dimensionless, F above has not the mechanical dimension of a force
| only F c2 has). Incidentally, we are stuck to cartesian coordinates to
discuss accelerations: the second time derivative of a coordinate is an
acceleration only if that coordinate is cartesian.
transitivity a reference frame moving with constant velocity with respect
to an inertial frame is also an inertial frame;
relativity all the
laws of nature are the same in all inertial frames; or,
alternatively, the equations describing them are invariant under the
transformations (of space coordinates and time) taking one inertial
frame into the other; or still, the equations describing the laws of Nature
in terms of space coordinates and time keep their forms in di erent
inertial frames; this \principle" can be seen as an experimental fact; in
non-relativistic classical physics, the transformations referred to belong
to the Galilei group; in Special Relativity, to the Poincare group
4
causality in non-relativistic classical physics the interactions are given by
the potential energy, which usually depends only on the space coordinates; forces on a given particle, caused by all the others, depend
only on their position at a given instant; a change in position changes
the force instantaneously; this instantaneous propagation e ect | or
action at a distance | is a typicallly classical, non-relativistic feature;
it violates special-relativistic causality, which says that no e ect can
propagate faster than the velocity of light
elds there have been tentatives to preserve action at a distance in a rel-
ativistic context, but a simpler way to consider interactions while respecting Special Relativity is of common use in eld theory: interactions are mediated by a eld, which has a well-de ned behaviour under
transformations; disturbances propagate with nite velocities.
1.3 The Equivalence Principle
Equivalence is a guiding principle, which inspired Einstein in his construction
of General Relativity. It is rmly rooted on experience.
In its most usual form, the Principle includes three sub{principles: the
weak, the strong and that which is called \Einstein's equivalence principle".
We shall come back and forth to them along these notes. Let us shortly list
them with a few comments.
The weak equivalence principle: universality of free fall, or inertial
mass = gravitational mass.
x
1.3
In a gravitational eld, all pointlike structureless particles follow one same path; that path is xed once given (i) an initial
position x(t0) and (ii) the correspondent velocity x_ (t0 ).
This leads to a force equation which is a second order ordinary di erential
equation. No characteristic of any special particle, no particular property
appears in the equation. Gravitation is consequently universal. Being universal, it can be seen as a property of space itself. It determines geometrical
Those interested in the experimental status will nd a recent appraisal in C. M. Will,
The Confrontation between General Relativity and Experiment, arXiv:gr-qc/0103036 12
Mar 2001. Theoretical issues are discussed by B. Mashhoon, Measurement Theory and
General Relativity, gr-qc/0003014, and Relativity and Nonlocality, gr-qc/0011013 v2.
5
properties which are common to all particles. The weak equivalence principle goes back to Galileo. It raises to the status of fundamental principle a
deep experimental fact: the equality of inertial and gravitational masses of
all bodies.
The strong
equivalence principle: (Einstein's lift) says that
Gravitation can be made to vanish locally through an appropriate choice of frame.
It requires that, for any and every particle and at each point x0 , there exists
a frame in which x = 0.
Einstein's equivalence principle requires, besides the weak principle,
the local validity of Poincare invariance | that is, of Special Relativity. This
invariance is, in Minkowski space, summed up in the Lorentz metric. The
requirement suggests that the above deformation caused by gravitation is a
change in that metric.
In its complete form, the equivalence principle
1. provides an operational de nition of the gravitational interaction;
2. geometrizes it;
3. xes the equation of motion of the test particles.
x
1.4 Use has been made above of some unde
x
1.5 Now, forces equally felt by all bodies were known since long. They are
ned concepts, such as \path",
and \local". A more precise formulation requires more mathematics, and will
be left to later sections. We shall, for example, rephrase the Principle as a
prescription saying how an expression valid in Special Relativity changes in
the presence of a gravitational eld. What changes is the notion of derivative,
and that change requires the concept of connection. The prescription (of
\minimal coupling") will be seen after that notion is introduced.
the inertial forces, whose name comes from their turning up in non-inertial
frames. Examples on Earth (not an inertial system !) are the centrifugal
6
force and the Coriolis force. We shall begin by recalling what such forces
are in Classical Mechanics, in particular how they appear related to changes
of coordinates. We shall then show how a metric appears in an non-inertial
frame, and how that metric changes the law of force in a very special way.
1.3.1 Inertial Forces
x 1.6 In a frame attached to Earth (that is, rotating with a certain angular
velocity !), a body of mass m moving with velocity X_ on which an external
force Fext acts will actually experience a \strange" total force. Let us recall
in rough brushstrokes how that happens.
A simpli ed model for the motion of a particle in a system attached to
Earth is taken from the classical formalism of rigid body motion.y It is as
follows:
Start with an inertial cartesian system, the space system (\inertial" means
| we insist | devoid of proper acceleration). A point particle will
have coordinates fxi g, collectively written as a column vector x = (xi ).
Under the action of a force f , its velocity and acceleration will be, with
respect to that system, x_ and x . If the particle has mass m, the force
will be f = m x .
Consider now another coordinate system (the body system) which rotates
around the origin of the rst. The point particle will have coordinates
X in this system. The relation between the coordinates will be given
by a rotation matrix R,
X = R x:
The forces acting on the particle in both systems are related by the same
relation,
F = R f:
y The standard approach is given in H. Goldstein,Classical Mechanics, Addison{Wesley,
Reading, Mass., 1982. A modern description can be found in J. L. McCauley,Classical
Mechanics, Cambridge University Press, Cambridge, 1997.
7
Now comes the crucial point: as Earth is rotating with respect to the space
system, a di erent rotation is necessary at each time to pass from that
system to the body system; this is to say that the rotation matrix R
is time-dependent. In consequence, the velocity and the acceleration
seen from Earth's system are given by
X_ = R_ x + R x_
X = R x + 2R_ x_ + R x :
(1.2)
Introduce the matrix ! = R 1 R_ . It is an antisymmetric 3 3 matrix,
consequently equivalent to a vector. This vector, with components
! k = 21 k ij ! ij
(1.3)
(which is the same as ! ij = ijk !k ), is Earth's angular velocity seen
from the space system. ! is, thus, a matrix version of the angular
velocity. It will correspond, in the body system, to
= R!R
1
=
R_ R 1 :
Just in case, ijk is the 3-dimensional Kronecker symbol in 3dimensional space: 123 = 1; any odd exchange of indices changes the sign; ijk = 0
if there are repeated indices. Indices are raised and lowered with the Kronecker
delta Æij , de ned by Æii = 1 and Æij = 0 if i 6= j . In consequence, ijk = ijk =
ijk , etc. The usual vector product has components given by (v u)i = (v ^ u)i
= ijk uj vk . An antisymmetric matrix like !, acting on a vector will give !ij vj =
ijk!k vj = (! v)i .
Comment 1.1
A few relations turn out without much ado: 2 = R! 2 R 1 , _ = R!R
_
and
!_ !! = R 1 R ;
or
R = R [!_
!! ] :
Substitutions put then Eq. (1.2) into the form
X + 2 X_ + [ _ +
8
2]
X = R x
1
The above relationship between 3 3 matrices and vectors takes matrix
action on vectors into vector products: ! x = ! x, etc. Transcribing
into vector products and multiplying by the mass, the above equation
acquires its standard form,
=
mX
|
m {z
X}
centrifugal
2m{z X_} | m {z_ X} + Fext :
Coriolis
fluctuation
|
We have indicated the usual names of the contributions. A few words
on each of them
uctuation force: can be neglected for Earth,
very nearly constant.
whose angular velocity is
centrifugal force: opposite to Earth's attraction, it is already taken into
account by any balance (you are fatter than you think, your mass is
larger than suggested by your your weight by a few grams ! the ratio
is 3=1000 at the equator).
Coriolis force: responsible for trade winds, rivers' one-sided over
ows, assymmetric wear of rails by trains, and the e ect shown by the Foucault
pendulum.
1.7
Inertial forces have once been called \ cticious", because they disappear when seen from an inertial system at rest. We have met them when
we started from such a frame and transformed to coordinates attached to
Earth. We have listed the measurable e ects to emphasize that they are
actually very real forces, though frame-dependent.
x
1.8
The remarkable fact is that each body feels them the same. Think of
the examples given for the Coriolis force: air, water and iron feel them, and
in the same way. Inertial forces are \universal", just like gravitation. This
has led Einstein to his formidable stroke of genius, to conceive gravitation as
an inertial force.
x
1.9
Nevertheless, if gravitation were an inertial e ect, it should be obtained by changing to a non-inertial frame. And here comes a problem. In
x
9
Classical Mechanics, time is a parameter, external to the coordinate system.
In Special Relativity, with Minkowski's invention of spacetime, time underwent a violent conceptual change: no more a parameter, it became the fourth
coordinate (in our notation, the zeroth one).
Classical non-inertial frames are obtained from inertial frames by transformations which depend on time. Relativistic non-inertial frames should be
obtained by transformations which depend on spacetime. Time{dependent
coordinate changes ought to be special cases of more general transformations, dependent on all the spacetime coordinates. In order to be put into
a position closer to inertial forces, and concomitantly respect Special Relativity, gravitation should be related to the dependence of frames on all the
coordinates.
1.10
Universality of inertial forces has been the rst hint towards General
Relativity. A second ingredient is the notion of eld. The concept allows the
best approach to interactions coherent with Special Relativity. All known
forces are mediated by elds on spacetime. Now, if gravitation is to be
represented by a eld, it should, by the considerations above, be a universal
eld, equally felt by every particle. It should change spacetime itself. And,
of all the elds present in a space the metric | the rst fundamental form,
as it is also called | seemed to be the basic one. The simplest way to
change spacetime would be to change its metric. Furthermore, the metric
does change when looked at from a non-inertial frame.
x
1.11 The Lorentz metric of Special Relativity is rather trivial.
There
is a coordinate system (the cartesian system) in which the line element of
Minkowski space takes the form
x
ds2 = ab dxadxb = dx0 dx0
= c2 dt2
dx1dx1
dx2 dy 2
dx2 dx2
dz 2 :
dx3dx3
(1.4)
Take two points P and Q in Minkowski spacetime, and consider the integral
Z
Z
Q
P
ds =
Qp
ab dxa dxb :
P
10
Lorentz
metric
Its value depends on the path chosen. It is actually a functional on the space
of paths between P and Q,
S[
P Q] =
Z
PQ
ds:
(1.5)
An extremal of this functional would be a curve such that ÆS [ ] =
= 0. Now,
Æds2 = 2 ds Æds = 2 ab dxa Ædxb;
so that
R
Æds
dxa
Ædxb :
ds
Thus, commuting d and Æ and integrating by parts,
Æds = ab
ÆS [ ] =
Z Q
P
Z Q
dxa dÆxb
ds =
ab
ds ds
Z Q
P
ab
d dxa b
Æx ds
ds ds
d a b
U Æx ds:
ds
P
The variations Æxb are arbitrary. If we want to have ÆS [ ] = 0, the integrand
must vanish. Thus, an extremal of S [ ] will satisfy
=
ab
d a
U = 0:
(1.6)
ds
This is the equation of a straight line, the force law (1.1) when F a = 0.
The solution of this di erential equation is xed once initial conditions are
given. We learn here that a vanishing acceleration is related to an extremal
of S [ P Q ].
x
1.12 Let us see through an example what happens when a force is present.
For that it is better to notice beforehand that, when considering elds, it is
in general the action which is extremal. Simple dimensional analysis shows
that, in order to have a real physical action, we must take
S=
mc
11
Z
ds
(1.7)
instead of the \length". Consider the case of a charged test particle. The
coupling of a particle of charge e to an electromagnetic potential A is given
by Aa j a = e AaU a , so that the action along a curve is
Z
Z
e
e
a
Sem [ ] =
AaU ds =
Aa dxa :
c
c
The variation is
Z
Z
Z
Z
e
e
e
e
a
a
a
ÆAadx
Aa dÆx =
ÆAadx +
dAb Æxb
ÆSem [ ] =
c
c
c
c
Z
Z
Z
e
e
e
dxa
b
a
b
a
=
@b AaÆx dx +
@a AbÆx dx =
[@b Aa @a Ab]Æxb ds
c
c
c
ds
Z
e
FbaU a Æxb ds :
=
c
Combining the two pieces, the variation of the total action
S = mc
is
ÆS =
Z Q
P
ds
Z Q
The extremal satis es
P
d
ab mc U a
ds
e
c
Z Q
P
Aa dxa
(1.8)
e
F U a Æxbds:
c ba
d a e a b
U = F bU;
(1.9)
ds
c
which is the Lorentz force law and has the form of the general case (1.1).
mc
1.3.2 The Wake of Non-Trivial Metric
Let us see now | in another example | that the metric changes when
viewed from a non-inertial system. This fact suggests that, if gravitation is
to be related to non-inertial systems, a gravitational eld is to be related to
a non-trivial metric.
x
1.13 Consider a rotating disc (details can be seen in Mller's bookz), seen
as a system performing a uniform rotation with angular velocity ! on the x,
y plane:
x = r cos( + !t) ; y = r sin( + !t) ; Z = z ;
z C. Mller, The Theory of Relativity, Oxford at Clarendon Press, Oxford, 1966, mainly
in x8.9.
12
X = R cos ; Y = R sin :
This is the same as
x = X cos !t Y sin !t ; y = Y cos !t + X sin !t :
As there is no contraction along the radius (the motion being orthogonal
to it), R = r. Both systems coincide at t = 0. How will an observer on the
disk see the line element
ds2 = c2 dt2
dx2 dy 2
dz 2 ?
It is immediate that
dx = dr cos( + !t) r sin( + !t)[d + !dt]
dy = dr sin( + !t) + r cos( + !t)[d + !dt]
dx2 = dr2 cos2 ( + !t) + r2 sin2 ( + !t)[d + !dt]2
2rdr cos( + !t) sin( + !t)[d + !dt] ;
dy 2 = dr2 sin2( + !t) + r2 cos2 ( + !t)[d + !dt]2
+2rdr sin( + !t) cos( + !t)[d + !dt]
) dx2 + dy 2 = dR2 + R2 (d2 + ! 2 dt2 + 2!ddt):
It follows from
that
dX 2 + dY 2 = dR2 + R2 d2 ;
dx2 + dy 2 = dX 2 + dY 2 + R2 ! 2 dt2 + 2!R2 ddt:
A simple check shows that
XdY
Y dX = R2 d;
so that
dx2 + dy 2 = dX 2 + dY 2 + R2 ! 2 dt2 + 2!XdY dt 2!Y dXdt:
Thus,
ds2 = (1
! 2 R2 2 2
) c dt
c2
dX 2
dY 2
13
2!XdY dt + 2!Y dXdt
dZ 2 :
In the moving system, with coordinates (x0; x1; x2 ; x3) = (ct; X; Y; Z = z )
the metric will be
ds2 = g dX dX ;
where the only non-vanishing components of the modi ed metric g are:
g11 = g22 = g33 = 1; g01 = g10 = !Y=c; g02 = g20 = !X=c;
! 2 R2
g00 = 1
:
c2
This is better visualized as the matrix
1
0
2 2
!X=c 0
1 ! cR2 !Y=c
B !Y=c
1
0
0 C
C
(1.10)
g = (g ) = B
C :
B
@
!X=c 0
1
0 A
0
0
0
1
We can go one
step further an de ne the body time coordinate T to be
p
such that dT = 1 ! 2 R2 =c2 dt, that is,
T=
p
1
! 2 R2 =c2 t :
This is physically appealing, as the last expression is the same as T =
1 v 2=c2 t, the time-contraction of Special Relativity, if we take into account the fact that a point with coordinates (R; ) will have squared velocity
v 2 = ! 2 R2 . We see that, anyhow, the body coordinate system can be used
only for points satisfying !R < c. With body coordinates (x0 ; x1; x2; x3 ) =
(cT; X; Y; Z ), the line element becomes
dT
ds2 = c2 dT 2 dX 2 dY 2 dZ 2 + 2! [Y dX XdY ] p
:
1 ! 2 R2 =c2
(1.11)
Time, as measured by the accelerated frame, di ers from that measured in
the inertial frame. And, anyhow, the metric has changed. This is the point
we wanted to make: when we change to a non-inertial system the metric
undergoes a signi cant transformation, even in Special Relativity.
p
= !R=c. Matrix (1.10) and its inverse are
0
Y
X
0
X 0 1
1
R
R
1 2 RY
R
2
2
Y
XY
2
Y
Y
1
g = (g ) = @ R X 1 0 0 A ; g 1 = (g ) = B
@ R X R22XY 2 XR2 2
0
1 0
R
2
R
R2 1
0 0 0
1
0
0R
0
Comment 1.2
Put
14
0 1
0 C
A :
0
1
1.3.3 Towards Geometry
x 1.14 We have said that the only e
ect of a gravitational eld is to bend
spacetime, so that straight lines become geodesics. Now, there are two quite
distinct de nitions of a straight line, which coincide on at spaces but not
on spaces endowed with more sophisticated geometries. A straight line going
from a point P to a point Q is
1. among all the lines linking P to Q, that with the shortest length;
2. among all the lines linking P to Q, that which keeps the same direction
all along.
There is a clear problem with the rst de nition: length presupposes a
metric | a real, positive-de nite metric. The Lorentz metric does not de ne
lengths, but pseudo-lengths. There is always a \zero-length" path between
R
any two points in Minkowski space. In Minkowski space, ds is actually
maximal for a straight line. Curved lines, or broken ones, give a smaller
pseudo-length. We have introduced a minus sign in Eq.(1.7) in order to
conform to the current notion of \minimal action".
The second de nition can be carried over to spacetime of any kind, but
at a price. Keeping the same direction means \keeping the tangent velocity
vector constant". The derivative of that vector along the line should vanish.
Now, derivatives of vectors on non- at spaces require an extra concept, that
of connection | which, will, anyhow, turn up when the rst de nition is
used. We shall consequently feel forced to talk a lot about connections in
what follows.
x
1.15
Consider an arbitrary metric g , de ning the interval by
ds2 = g dx dx :
What happens now to the integral of Eq.(1.7) with a point-dependent metric?
Consider again a charged test particle, but now in the presence of a non-trivial
metric. We shall retrace the steps leading to the Lorentz force law, with the
action
Z
Z
e
S = mc
ds
A dx ;
(1.12)
c
PQ
PQ
15
general
metric
but now with ds =
p
g dx dx .
1. take rst the variation
Æds2 = 2dsÆds = Æ [g dx dx ] = dx dx Æg + 2g dx Ædx
dx dx
ds ds
) Æds = 21
@ g Æxds + g dxds Ædx
ds ds
We have conveniently divided and multiplied by ds.
2. we now insert this in the rst piece of the action and integrate by parts
the last term, getting
ÆS = mc
Z
d (g dx ) Æx ds
ds ds
1 dx dx
2 ds ds @ g
PQ
Z
e
c
PQ
[ÆAdx + A dÆx ]:
(1.13)
3. the derivative dsd (g dxds ) is
d
dx
dx d
d
d
(g
)=
g + g U = U U @ g + g U ds
ds
ds ds
ds
ds
d
d
= g U + U U @g = g U + 21 U U [@ g + @ g ]:
ds
ds
4. collecting terms in the metric sector, and integrating by parts in the
electromagnetic sector,
ÆS = mc
Z
PQ
d
g U ds
Z
e
c
mc
Z
PQ
g
PQ
d U
ds
e
c
1
2
U U (@
g
[@ AÆx dx
U U 12
Z
PQ
16
@ g ) Æx ds
Æx @ A dx ] =
g (@
[@ AÆx dx
+ @ g
g + @ g
Æx @ A dx ]:
(1.14)
@g )
Æx ds
(1.15)
5. we meet here an important character of all metric theories. The expression between curly brackets is the Christo el symbol, which will be
Æ
indicated by the notation :
Æ
= 21 g (@ g + @ g
@ g ) :
Christo el
symbol
(1.16)
6. after arranging the terms, we get
ÆS =
Z
PQ
mc g
d Æ U + U U
ds
e
(@ A @ A )U Æx ds:
c
(1.17)
7. the variations Æx , except at the xed endpoints, is quite arbitrary. To
have ÆS = 0, the integrand must vanish. Which gives, after contracting
with g ,
d Æ e
mc
U + U U = F U :
ds
c
(1.18)
8. this is the Lorentz law of force in the presence of a non-trivial metric.
We see that what appears as acceleration is now
d Æ U + U U :
(1.19)
ds
The Christo el symbol is a non-tensorial quantity, a connection. We
shall see later that a reference frame can be always chosen in which it
vanishes at a point. The law of force
A =
d Æ mc
U + U U = F ds
(1.20)
will, in that frame, reduce to that holding for a trivial metric, Eq. (1.1).
9. in the absence of forces, the resulting expression,
d Æ U + U U = 0;
(1.21)
ds
is the geodesic equation, de ning the \straightest" possible line on a
space in which the metric is non-trivial.
17
geodesic
equation
An accelerated frame creates the illusion of a force. Suppose a point P is
\at rest". It may represent a vessel in space, far from any other body. An astronaut in
the spacecraft can use gyros and accelerometers to check its state of motion. It will never
be able to say that it is actually at rest, only that it has some constant velocity. Its own
reference frame will be inertial. Assume another craft approaches at a velocity which is
constant relative to P , and observes P . It will measure the distance from P , see that the
velocity x_ is constant. That observer will also be inertial.
Suppose now that the second vessel accelerates towards P . It will then see x 6= 0, and
will interpret this result in the normal way: there is a force pulling P . That force is clearly
an illusion: it would have opposite sign if the accelerated observer moved away from P .
No force acts on P , the force is due to the observer's own acceleration. It comes from the
observer, not from P .
Comment 1.3
Curvature creates the illusion of a force. Two old travellers (say, Herodotus and Pausanias) move northwards on Earth, starting from the equator. Suppose they
somehow communicate, and have a means to evaluate their relative distance. They will
notice that that distance decreases with their progress until, near the pole, they will see
it dwindle to nothing. Suppose further they have ancient notions, and think the Earth is
at. How would they explain it ? They would think there were some force, some attractive
force between them. And what is the real explanation ? It is simply that Earth's surface
is a curved space. The force is an illusion, born from the atness prejudice.
Comment 1.4
x
1.16
x
1.17 Suppose that one single elementary particle did not feel gravitation.
Gravitation is very weak. To present time, no gravitational bending
in the trajectory of an elementary particle has been experimentally observed.
Only large agglomerates of fermions have been seen to experience it. Nevertheless, an e ect on the phase of the wave-function has been detected, for
neutrons and atoms.x
That would be enough to change all the picture. The underlying spacetime
would remain Minkowski's, and the metric responsible for gravitation would
be a eld g on that, by itself at, spacetime.
x The so-called \COW experiment" with neutrons is described in R. Colella, A. W.
Overhauser and S. A. Werner, Phys. Rev. Lett. 34, 1472 (1975). See also U. Bonse
and T. Wroblevski, Phys. Rev. Lett. 51, 1401 (1983). Experiments with atoms are
reviewed in C. J. Borde, Matter wave interferometers: a synthetic approach, and in B.
Young, M. Kasevich and S. Chu, Precision atom interferometry with light pulses, in Atom
Interferometry, P. R. Bergman (editor) (Academic Press, San Diego, 1997).
18
Spacetime is a geometric construct. Gravitation should change the geometry of spacetime. This comes from what has been said above: coordinates,
metric, connection and frames are part of the di erential-geometrical structure of spacetime. We shall need to examine that structure. The next chapter
is devoted to the main aspects of di erential geometry.
19
Chapter 2
Geometry
The basic equations of Physics are di erential equations. Now, not every
space accepts di erentials and derivatives. Every time a derivative is written
in some space, a lot of underlying structure is assumed, taken for granted. It
is supposed that that space is a di erential (or smooth) manifold. We shall
give in what follows a short survey of the steps leading to that concept. That
will include many other notions taken for granted, as that of \coordinate",
\parameter", \curve", \continuous", and the very idea of space.
2.1 Di erential Geometry
Physicists work with sets of numbers, provided by experiments, which they
must somehow organize. They make { always implicitly { a large number
of assumptions when conceiving and preparing their experiments and a few
more when interpreting them. For example, they suppose that the use of
coordinates is justi ed: every time they have to face a continuum set of
values, it is through coordinates that they distinguish two points from each
other. Now, not every kind of point-set accepts coordinates. Those which do
accept coordinates are speci cally structured sets called manifolds. Roughly
speaking, manifolds are sets on which, at least around each point, everything
looks usual, that is, looks Euclidean.
2.1
Let us recall that a distance function is a function d taking any pair
(p; q ) of points of a set X into the real line R and satisfying the following four
conditions : (i) d(p; q ) 0 for all pairs (p; q ); (ii) d(p; q ) = 0 if and only if
p = q ; (iii) d(p; q ) = d(q; p) for all pairs (p; q ); (iv) d(p; r) + d(r; q ) d(p; q )
for any three points p, q , r. It is thus a mapping d: X X ! R+. A space on
x
20
distance
function
which a distance function is de ned is a metric space. For historical reasons,
a distance function is here (and frequently) called a metric, though it would
be better to separate the two concepts (see in section 2.1.4, page 43, how a
positive-de nite metric, which is a tensor eld, can de ne a distance).
2.2
The Euclidean spaces are the basic spaces we shall start with. The 3dimensional space E3 consists of the set R3 of ordered triples of real numbers p
= (p1 ; p2 ; p3 ), q = (q 1; q 2; q 3), etc, endowed with the distance function d(p; q )
P3
i q i )21=2. A r-ball around p is the set of points q such that
=
(
p
i=1
d(p; q ) < r, for r a positive number. These open balls de ne a topology, that
is, a family of subsets of E3 leading to a well-de ned concept of continuity.
It was thought for much time that a topology was necessarily an o spring
of a distance function. The modern concept, presented below (x2.7), is more
abstract and does without a distance function.
Non-relativistic elds live on space E3 or, if we prefer, on the spacetime
E3 E1, with the extra E1 accounting for time. In non-relativistic physics
space and time are independent of each other, and this is encoded in the
direct{product character: there is one distance function for space, another
for time. In relativistic theories, space and time are blended together in an
inseparable way, constituting a real spacetime. The notion of spacetime was
introduced by Minkowski in Special Relativity. Minkowski spacetime, to be
described later, is the paradigm of every other spacetime.
For the n-dimensional Euclidean space En , the point set is the set Rn of
ordered n-uples p = (p1 ; p2 ; :::; pn) of real numbers and the topology is the
P
ball-topology of the distance function d(p, q) = [ ni=1 (pi q i)2 ]1=2. En is the
basic, initially assumed space, as even di erential manifolds will be presently
de ned so as to generalize it. The introduction of coordinates on a general
space S will require that S \resemble" some En around each point.
x
x
2.3 When we say \around each point", mathematicians say \locally". For
example, manifolds are \locally Euclidean" sets. But not every set of points
can resemble, even locally, an Euclidean space. In order to do so, a point set
must have very special properties. To begin with, it must have a topology. A
set with such an underlying structure is a \topological space". Manifolds are
21
euclidean
spaces
topological spaces with some particular properties which make them locally
Euclidean spaces.
The procedure then runs as follows:
it is supposed that we know everything on usual Analysis, that is,
Analysis on Euclidean spaces. Structures are then progressively
added up to the point at which it becomes possible to transfer
notions from the Euclidean to general spaces. This is, as a rule,
only possible locally, in a neighborhood around each point.
2.4
We shall later on represent physical systems by elds. Such elds are
present somewhere in space and time, which are put together in a uni ed
spacetime. We should say what we mean by that. But there is more. Fields
are idealized objects, which we represent mathematically as members of some
other spaces. We talk about vectors, matrices, functions, etc. There will be
spaces of vectors, of matrices, of functions. And still more: we operate with
these elds. We add and multiply them, sometimes integrate them, or take
their derivatives. Each one of these operations requires, in order to have
a meaning, that the objects they act upon belong to spaces with speci c
properties.
x
2.1.1 Spaces
x 2.5 Thus, rst task, it will be necessary to say what we understand by
\spaces" in general. Mathematicians have built up a systematic theory of
spaces, which describes and classi es them in a progressive order of complexity. This theory uses two primitive notions - sets and functions from one
set to another. The elements belonging to a space may be vectors, matrices,
functions, other sets, etc, but the standard language calls simply \points"
the members of a generic space.
A space S is an organized set of points, a point set plus a structure.
This structure is a division of S , a convenient family of subsets. Di erent
purposes require di erent kinds of subset families. For example, in order to
arrive at a well-de ned notion of integration, a measure space is necessary,
which demands a special type of sub-division called \ -algebra". To make of
22
general
notion
S a topological space, we decompose it in another peculiar way. The latter
will be our main interest because most spaces used in Physics are, to start
with, topological spaces.
x
2.6
x
2.7
That this is so is not evident at every moment. The customary approach is just the contrary. The physicist will implant the object he needs
without asking beforehand about the possibilities of the underlying space.
He can do that because Physics is an experimental science. He is justied in introducing an object if he obtains results con rmed by experiment.
A well-succeeded experiment brings forth evidence favoring all the assumptions made, explicit or not. Summing up: the additional objects (say, elds)
de ned on a certain space (say, spacetime) may serve to probe into the underlying structure of that space.
them.
Topological spaces are, thus, the primary spaces. Let us begin with
Given a point set S , a topology is a family T of subsets of S
to which belong: (a) the whole set S and the empty set ;; (b)
T
the intersection k Uk of any nite sub-family of members Uk of
S
T ; (c) the union k Uk of any sub-family ( nite or in nite) of
members.
A topological space (S , T ) is a set of points S on which a
topology T is de ned.
The members of the family T are, by de nition, the open sets of (S , T ).
Notice that a topological space is indicated by the pair (S , T ). There are, in
general, many di erent possible topologies on a given point set S , and each
one will make of S a di erent topological space. Two extreme topologies
are always possible on any S . The discrete space is the topological space
(S , P (S )), with the power set P (S ) | the set of all subsets of S | as the
topology. For each point p, the set fpg containing only p is open. The other
extreme case is the indiscrete (or trivial) topology T = f;; S g.
Any subset of S containing a point p is a neighborhood of p. The complement of an open set is (by de nition) a closed set. A set which is open in a
23
topology
topology may be closed in another. It follows that ; and S are closed (and
open!) sets in all topologies.
The space (S , T ) is connected if ; and S are the only sets which are
simultaneously open and closed. In this case S cannot be decomposed into the union of
two disjoint open sets (this is di erent from path-connectedness). In the discrete topology
all open sets are also closed, so that unconnectedness is extreme.
Comment 2.1
2.8
Let f : A ! B be a function between two topological spaces A (the
domain) and B (the target). The inverse image of a subset X of B by f is the
set f < 1> (X ) = fa 2 A such that f (a) 2 X g. The function f is continuous
if the inverse images of all the open sets of the target space B are open sets
of the domain space A. It is necessary to specify the topology whenever one
speaks of a continuous function. A function de ned on a discrete space is
automatically continuous. On an indiscrete space, a function is hard put to
be continuous.
x
2.9
A topology is a metric topology when its open sets are the open balls
Br (p) = fq 2 S such that d(q; p) < rg of some distance function. The
simplest example of such a \ball-topology" is the discrete topology P (S ): it
can be obtained from the so-called discrete metric: d(p; q ) = 1 if p 6= q , and
d(p; q ) = 0 if p = q . In general, however, topologies are independent of any
distance function: the trivial topology cannot be given by any metric.
x
2.10
A caveat is in order here. When we say \metric" we mean a positivede nite distance function as above. Physicists use the word \metrics" for
some invertible bilinear forms which are not positive-de nite, and this practice is progressively infecting mathematicians. We shall follow this seemingly
inevitable trend, though it should be clear that only positive-de nite metrics
can de ne a topology. The fundamental bilinear form of relativistic Physics,
the Lorentz metric on Minkowski space-time, does not de ne true distances
between points.
x
We have introduced Euclidean spaces En in x2.2. These spaces, and
Euclidean half-spaces (or upper-spaces) En+ are, at least for Physics, the
most important of all topological spaces. This is so because Physics deals
x
2.11
24
continuity
mostly with manifolds, and a manifold (di erentiable or not) will be a space
which can be approximated by some En or En+ in some neighborhood of
each point (that is, \locally"). The half-space En+ has for point set Rn+ =
fp = (p1; p2 ; :::; pn) 2 Rn such that pn 0g. Its topology is that \induced"
by the ball-topology of En (the open sets are the intersections of Rn+ with
the balls of En ). This space is essential to the de nition of manifolds-withboundary.
2.12
A bijective function f : A ! B will be a homeomorphism if it is
continuous and has a continuous inverse. It will take open sets into open sets
and its inverse will do the same. Two spaces are homeomorphic when there
exists a homeomorphism between them. A homeomorphism is an equivalence relation: it establishes a complete equivalence between two topological
spaces, as it preserves all the purely topological properties. Under a homeomorphism, images and pre-images of open sets are open, and images and
pre-images of closed sets are closed. Two homeomorphic spaces are just the
same topological space. A straight line and one branch of a hyperbola are
the same topological space. The same is true of the circle and the ellipse.
A 2-dimensional sphere S 2 can be stretched in a continuous way to become
an ellipsoid or a tetrahedron. From a purely topological point of view, these
three surfaces are indistinguishable. There is no homeomorphism, on the
other hand, between S 2 and a torus T 2, which is a quite distinct topological
space.
Take again the Euclidean space En . Any isometry (distance{preserving
mapping) will be a homeomorphism, in particular any translation. Also
homothecies with reason 6= 0 are homeomorphisms. From these two properties it follows that each open ball of En is homeomorphic to the whole En .
Suppose a space S has some open set U which is homeomorphic to an open
set (a ball) in some En : there is a homeomorphic mapping : U ! ball,
f (p 2 U ) = x = (x1 ; x2; :::; xn). Such a local homeomorphism , with En as
target space, is called a coordinate mapping and the values xk are coordinates
of p.
x
x
2.13 S is locally Euclidean if, for every point p 2 S , there exists an open
set U to which p belongs, which is homeomorphic to either an open set in
25
homeo
morphism
coordinates
some Es or an open set in some Es+ . The number s is the dimension of S at
the point p.
2.14
We arrive in this way at one of the concepts announced at the beginning of this chapter: a (topological) manifold is a connected space on which
coordinates make sense.
x
A manifold is a topological space S which is
(i) locally Euclidean;
(ii) has the same dimension s at all points, which is then the
dimension of S , s = dim S .
Points whose neighborhoods are homeomorphic to open sets of Es+ and not
to open sets of Es constitute the boundary @S of S . Manifolds including
points of this kind are \manifolds{with{boundary".
The local-Euclidean character will allow the de nition of coordinates and
will have the role of a \complementarity principle": in the local limit, a
di erentiable manifold will look still more Euclidean than the topological
manifolds. Notice that we are indicating dimensions by m, n, s, etc, and
manifolds by the corresponding capitals: dim M = m; dim N = n, dim S =
s, etc.
2.15
Each point p on a manifold has a neighborhood U homeomorphic to
an open set in some En , and so to En itself. The corresponding homeomorphism
: U ! open set in En
x
will give local coordinates around p. The neighborhood U is called a coordinate neighborhood of p. The pair (U; ) is a chart, or local system of
coordinates (LSC) around p.
We must be more speci c. Take En itself: an open neighborhood V of a
point q 2 En is homeomorphic to another open set of En . Each homeomorphism u: V ! V 0 included in En de nes a system of coordinate functions
(what we usually call coordinate systems: Cartesian, polar, spherical, elliptic, stereographic, etc.). Take the composite homeomorphism x: S ! En ,
x(p) = (x1 ; x2; :::; xn) = (u1 Æ (p); u2 Æ (p); :::; un Æ (p)). The functions
26
manifold
xi = ui Æ : U ! E 1 will be the local coordinates around p. We shall use
the simpli ed notation (U, x) for the chart. Di erent systems of coordinate
functions require di erent number of charts to plot the space S . For E2 itself,
one Cartesian system is enough to chart the whole space: V = E2, u = the
identity mapping. The polar system, however, requires at least two charts.
For the sphere S 2 , stereographic coordinates require only two charts, while
the cartesian system requires four.
Suppose the polar system with only one chart: E2 ! R1+ (0; 2). Intuitively, close points (r; 0+ ) and (r; 2 ), for small, are represented by faraway points.
Technically, due to the necessity of using open sets, the whole half-line (r; 0) is absent, not
represented. Besides the chart above, it is necessary to use E2 ! R1+ ( ; + 2 ), with
arbitrary in the interval (0; 2 ).
Comment 2.2
Classical Physics needs coordinates to distinguish points. We see that
the method of coordinates can only work on locally Euclidean spaces.
Comment 2.3
2.16
As we have said, every time we write a derivative, a di erential, a
Laplacian we are assuming an additional underlying structure for the space
we are working on: it must be a di erentiable (or smooth) manifold. And
manifolds and smooth manifolds can be introduced by imposing progressively restrictive conditions on the decomposition which has led to topological spaces. Just as not every space accepts coordinates (that is, not not every
space is a manifold), there are spaces on which to di erentiate is impossible.
We arrive nally at the crucial notion by which knowledge on di erentiability
on Euclidean spaces is translated into knowledge on di erentiability on more
general spaces. We insist that knowledge of Analysis on Euclidean spaces is
taken for granted.
A given point p 2 S can in principle have many di erent coordinate neighT
borhoods and charts. Given any two charts (U; x) and (V; y ) with U V 6= ;,
T
to a given point p in their intersection, p 2 U V , will correspond coordinates x = x(p) and y = y (p). These coordinates will be related by a homeomorphism between open sets of En ,
x
y
Æ
x<
1>
: En ! En
which is a coordinate transformation, usually written y i = y i (x1; x2; :::; xn).
Its inverse is x Æ y < 1> , written xj = xj (y 1 ; y 2; :::; y n). Both the coordinate
27
coordinates
transformation and its inverse are functions between Euclidean spaces. If
both are C 1 (di erentiable to any order) as functions from En into En , the
two local systems of coordinates are said to be di erentially related. An atlas
S
on the manifold is a collection of charts f(Ua; ya )g such that a Ua = S .
If all the charts are di erentially related in their intersections, it will be a
di erentiable atlas. The chain rule
Æki =
@y i @xj
@xj @y k
says that both Jacobians are 6= 0.y
An extra chart (W; x), not belonging to a di erentiable atlas A, is said
to be admissible to A if, on the intersections of W with all the coordinateneighborhoods of A, all the coordinate transformations from the atlas LSC's
to (W; x) are C 1 . If we add to a di erentiable atlas all its admissible charts,
we get a complete atlas, or maximal atlas, or C 1 {structure. The extension
of a di erentiable atlas, obtained in this way, is unique (this is a theorem).
A topological manifold with a complete di erentiable atlas is
a di erentiable manifold.
di erentiable
manifold
2.17
A function f between two smooth manifolds is a di erentiable function (or smooth function) when, given the two atlases, there are coordinates
systems in which y Æ f Æ x< 1> is di erentiable as a function between
Euclidean spaces.
x
2.18
A curve on a space S is a function a : I ! S , a : t ! a(t), taking the
interval I = [0; 1] E1 into S . The variable t 2 I is the curve parameter.
If the function a is continuous, then a is a path. If the function a is also
di erentiable, we have a smooth curve.z When a(0) = a(1), a is a closed
x
This requirement of in nite di erentiability can be reduced to k-di erentiability (to
give a \C k {atlas").
yIf some atlas exists on S whose Jacobians are all positive, S is orientable. When 2{
dimensional, an orientable manifold has two faces. The Mobius strip and the Klein bottle
are non-orientable manifolds.
zThe trajectory in a brownian motion is continuous (thus, a path) but is not di erentiable (not smooth) at the turning points.
28
curves
curve, or a loop, which can be alternatively de ned as a function from the
circle S 1 into S . Some topological properties of a space can be grasped by
studying its possible paths.
This is the subject matter of homotopy theory. We shall need one concept
| contractibility | for which the notion of homotopy is an indispensable preliminary.
Let f; g : X ! Y be two continuous functions between the topological spaces X
and Y. They are homotopic to each other (f g) if there exists a continuous function
F : X I ! Y such that F (p; 0) = f (p) and F (p; 1) = g(p) for every p 2 X . The
function F (p; t) is a one-parameter family of continuous functions interpolating between
f and g, a homotopy between f and g. Homotopy is an equivalence relation between
continuous functions and establishes also a certain equivalence between spaces. Given any
space Z , let idZ : Z ! Z be the identity mapping on Z, idZ (p) = p for every p 2 Z . A
continuous function f : X ! Y is a homotopic equivalence between X and Y if there exists
a continuous function g : Y ! X such that g Æ f idX and f Æ g idY . The function
g is a kind of \homotopic inverse" to f . When such a homotopic equivalence exists, X
and Y are homotopic. Every homeomorphism is a homotopic equivalence but not every
homotopic equivalence is a homeomorphism.
Comment 2.4
A space X is contractible if it is homotopically equivalent to a point. More
precisely, there must be a continuous function h : X I ! X and a constant function
f : X ! X , f (p) = c (a xed point) for all p 2 X , such that h(p; 0) = p = idX (p) and
h(p; 1) = f (p) = c. Contractibility has important consequences in standard, 3-dimensional
vector analysis. For example, the statements that divergenceless uxes are rotational
(div v = 0 ) v = rot w) and irrotational uxes are potential (rot v = 0 ) v = grad )
are valid only on contractible spaces. These properties generalize to di erential forms (see
page 40).
Comment 2.5
2.19
We have seen that two spaces are equivalent from a purely topological point of view when related by a homeomorphism, a topology-preserving
transformation. A similar role is played, for spaces endowed with a di erentiable structure, by a di eomorphism: a di eomorphism is a di erentiable
homeomorphism whose inverse is also smooth. When some di eomorphism
exists between two smooth manifolds, they are said to be di eomorphic. In
this case, besides being topologically the same, they have equivalent di erentiable structures. They are the same di erentiable manifold.
x
2.20
Linear spaces (or vector spaces) are spaces allowing for addition and
rescaling of their members. This means that we know how to add two vectors
x
29
di eo
morphism
vector
space
so that the result remains in the same space, and also to multiply a vector by
some number to obtain another vector, also a member of the same space. In
the cases we shall be interested in, that number will be a complex number.
In that case, we have a vector space V over the eld C of complex numbers.
Every vector space V has a dual V , another linear space formed by all the
linear mappings taking V into C . If we indicate a vector 2 V by the \ket"
jv >, a member of the dual can be indicated by the \bra" < uj. The latter will
be a linear mapping taking, for example, jv > into a complex number, which
we indicate by < ujv >. Being linear means that a vector ajv > + bjw > will
be taken by < uj into the complex number a < ujv > + b < ujw >. Two
linear spaces with the same nite dimension (= maximal number of linearly
independent vectors) are isomorphic. If the dimension of V is nite, V and
V have the same dimension and are, consequently, isomorphic.
Comment 2.6
Every vector space is contractible. Many of the most remarkable proper-
ties of En come from its being, besides a topological space, a vector space. En itself and
any open ball of En are contractible. This means that any coordinate open set, which is
homeomorphic to some such ball, is also contractible.
A vector space V can have a norm, which is a distance function and
de nes consequently a certain topology called the \norm topology". In this case, V is a
metric space. For instance, a norm may come from an inner product, a mapping from
the Cartesian set product V V into C, V V ! C , (v; u) ! < v; u > with suitable
properties. The number kvk = (j < v; v > j)1=2 will be the norm of v 2 V induced by
the inner product. This is a special norm, as norms can be de ned independently of inner
products. When the norm comes from an inner space, we have a Hilbert space. When
not, a Banach space. When the operations (multiplication by a scalar and addition) keep
a certain coherence with the topology, we have a topological vector space.
Comment 2.7
Once in possession of the means to de ne coordinates, we can proceed to
transfer to manifolds all the (supposedly well{known) results of usual vector
and tensor analysis on Euclidean spaces. Because a manifold is equivalent
to an Euclidean space only locally, this will be possible only in a certain
neighborhood of each point. This is the basic di erence between Euclidean
spaces and general manifolds: properties which are \global" on the rst hold
only locally on the latter.
30
2.1.2 Vector and Tensor Fields
x 2.21 The best means to transfer the concepts of vectors and tensors from
Euclidean spaces to general di erentiable manifolds is through the mediation
of spaces of functions. We have talked on function spaces, such as Hilbert
spaces separable or not, and Banach spaces. It is possible to de ne many
distinct spaces of functions on a given manifold M , di ering from each other
by some characteristics imposed in their de nitions: square{integrability for
example, or di erent kinds of norms. By a suitable choice of conditions we
can actually arrive at a space of functions containing every information on
M . We shall not deal with such involved subjects. At least for the time
being, we shall need only spaces with poorly de ned structures, such as the
space of real functions on M , which we shall indicate by R(M ).
Of the many equivalent notions of a vector on En , the directional
derivative is the easiest to adapt to di erentiable manifolds. Consider the
set R(En ) of real functions on En . A vector V = (v 1; v 2; : : : ; v n) is a linear
operator on R(En ): take a point p 2 En and let f 2 R(En ) be di erentiable
in a neighborhood of p. The vector V will take f into the real number
x
2.22
V (f ) =
v1
@f
@x1
p
+ v2
@f
@x2
p
+ + vn
@f
:
@xn p
This is the directional derivative of f along the vector V at p. This action
of V on functions respects two conditions:
1. linearity: V (af + bg ) = aV (f ) + bV (g ), 8a; b 2 E1 and 8f; g 2 R(En );
2. Leibniz rule: V (f g ) = f V (g ) + g V (f ).
x
2.23 This conception of vector { an operator acting on functions { can
be de ned on a di erential manifold N as follows. First, introduce a curve
through a point p 2 N as a di erentiable curve a : ( 1; 1) ! N such that
a(0) = p (see page 29). It will be denoted by a(t), with t 2 ( 1; 1). When t
varies in this interval, a 1-dimensional continuum of points is obtained on N.
In a chart (U; x) around p, these points will have coordinates ai (t) = xi(t).
31
vectors
Consider now a function f 2 R(N ). The vector Vp tangent to the curve a(t)
at p is given by
i
d
@
dx
Vp (f ) =
(f Æ a)(t)
=
f:
dt
dt t=0 @xi
t=0
Vp is independent of f , which is arbitrary. It is an operator Vp : R(N ) ! E1 .
Now, any vector Vp , tangent at p to some curve on N , is a tangent vector
k
to N at p. In the particular chart used above, dxdt is the k-th component of
Vp . The components are chart-dependent, but Vp itself is not. From its very
de nition, Vp satis es the conditions (1) and (2) above. A tangent vector on
N at p is just that, a mapping Vp : R(N ) ! E1 which is linear and satis es
the Leibniz rule.
2.24 The vectors tangent to N
at p constitute a linear space, the tangent space Tp N to the manifold N at p. Given some coordinates x(p) =
(x1; x2; : : : ; xn ) around the point p, the operators f @[email protected] i g satisfy conditions (1)
and (2) above. More than that, they are linearly independent and consequently constitute a basis for the linear space: any vector can be written in
the form
@
Vp = Vpi i :
@x
The Vpi 's are the components of Vp in this basis. Notice that each coordinate
xj belongs to R(N ). The basis f @[email protected] i g is the natural, holonomic, or coordinate
basis associated to the coordinate system fxj g. Any other set of n vectors
feig which are linearly independent will provide a base for TpN . If there is
no coordinate system fy k g such that ek = @[email protected] k , the base feig is anholonomic
or non-coordinate.
x
2.25 TpN and En are
nite vector spaces of the same dimension and are
consequently isomorphic. The tangent space to En at some point will be
itself an En . Euclidean spaces are di eomorphic to their own tangent spaces,
and that explains in part their simplicity | in equations written on such
spaces, one can treat indices related to the space itself and to the tangent
spaces on the same footing. This cannot be done on general manifolds.
These tangent vectors are called simply vectors, or contravariant vectors.
The members of the dual cotangent space TpN , the linear mappings !p : TpN
! En , are covectors, or covariant vectors.
x
32
tangent
space
2.26 Given an arbitrary basis feig of TpN ,
f j g of TpN , its dual basis, with the property
there exists a unique basis
j (e ) = Æ j . Any ! 2 T N ,
i
p
i
p
i
i
is written !p = !p (ei ) . Applying Vp to the coordinates x , we nd Vpi
= Vp (xi ), so that Vp = Vp (xi ) @[email protected] i = (Vp )ei . The members of the basis
dual to the natural basis f @[email protected] i g are indicated by fdxi g, with dxj ( @[email protected] i ) =
Æij . This notation is justi ed in the usual cases, and extended to general
manifolds (when f is a function between general di erentiable manifolds, df
takes vectors into vectors). The notation leads also to the reinterpretation of
@f dxi , as a linear
the usual expression for the di erential of a function, df = @x
i
operator:
@f
df (Vp ) = i dxi (Vp ):
@x
In a natural basis,
@
!p = !p ( i )dxi :
@x
x 2.27 The same order of ideas can be applied to tensors in general: a tensors
tensor at a point p on a di erentiable manifold M is de ned as a tensor on
TpM . The usual procedure to de ne tensors { covariant and contravariant {
on Euclidean vector spaces can be applied also here. A covariant tensor of
order s, for example, is a multilinear mapping taking the Cartesian product
Tps M = TpM TpM TpM of Tp M by itself s-times into the set of real
numbers. A contravariant tensor of order r will be a multilinear mapping
taking the the Cartesian product Tpr M = TpM Tp M TpM of TpM
by itself r-times into E1 . A mixed tensor, s-times covariant and r-times
contravariant, will take the Cartesian product Tps M Tpr M multilinearly
into E1. Basis for these spaces are built as the direct product of basis for the
corresponding vector and covector spaces. The whole lore of tensor algebra is
in this way transmitted to a point on a manifold. For example, a symmetric
covariant tensor of order s applies to s vectors to give a real number, and is
indi erent to the exchange of any two arguments:
x
T (v1; v2; : : : ; vk ; : : : ; vj ; : : : ; vs ) = T (v1; v2; : : : ; vj ; : : : ; vk ; : : : ; vs ):
An antisymmetric covariant tensor of order s applies to s vectors to give a
real number, and change sign at each exchange of two arguments:
T (v1; v2; : : : ; vk ; : : : ; vj ; : : : ; vs ) =
33
T (v1; v2; : : : ; vj ; : : : ; vk ; : : : ; vs ):
2.28
Because they will be of special importance, let us say a little more on
such antisymmetric covariant tensors. At each xed order, they constitute
a vector space. But the tensor product ! of two antisymmetric tensors
! and of orders p and q is a (p + q )-tensor which is not antisymmetric,
so that the antisymmetric tensors do not constitute a subalgebra with the
tensor product.
x
2.29
The wedge product is introduced to recover a closed algebra. First
we de ne the alternation Alt(T) of a covariant tensor T, which is an antisymmetric tensor given by
x
Alt(T )(v1; v2; : : : ; vs) =
1X
(sign P )T (vp1 ; vp2 ; : : : ; vps );
s! (P )
the summation taking place on all the permutations P = (p1 ; p2 ; : : : ; ps ) of
the numbers (1,2,. . . , s) and (sign P) being the parity of P. Given two
antisymmetric tensors, ! of order p and of order q, their exterior product,
or wedge product, indicated by ! ^ , is the (p+q)-antisymmetric tensor
!^ =
(p + q )!
Alt(!
p! q !
):
With this operation, the set of antisymmetric tensors constitutes the exterior algebra, or Grassmann algebra, encompassing all the vector spaces of
antisymmetric tensors. The following properties come from the de nition:
(! + ) ^ = ! ^ + ^ ;
(2.1)
^ (! + ) = ^ ! + ^ ;
(2.2)
a(! ^ ) = (a! ) ^ = ! ^ (a ); 8a 2 R;
(2.3)
(! ^ ) ^ = ! ^ ( ^ );
(2.4)
! ^ = ( )@! @ ^ ! :
(2.5)
34
Grassmann
algebra
In the last property, @! and @ are the respective orders of ! and . If
f g is a basis for the covectors, the space of s-order antisymmetric tensors
has a basis
i
f i1 ^
i2
^ ^ is g; 1 i1; i2; : : : ; is dim TpM;
(2.6)
in which an antisymmetric covariant s-tensor will be written
!=
In a natural basis fdxj g,
!=
2.30
1
!
s! i1 i2 :::is
i1
^
i2
^ ^
is
:
1
!
dxi1 ^ dxi2 ^ ^ dxis :
s! i1 i2 :::is
Thus, a tensor at a point p 2 M is a tensor de ned on the tangent
space Tp M . One can choose a chart around p and use for Tp M and TpM the
natural bases f @[email protected] i g and fdxj g. A general tensor will be written
x
:::ir
T = Tji11ji22:::j
s
@
@xi1
@
@xi2
@[email protected]
dxj1
dxj2
dxjs :
In another chart, with natural bases f @[email protected] g and (dxj 0 ), the same tensor will
be written
0 0 :::i0
@
@
j10
j20
js0
r @
T = Tji101ji220 :::j
dx
dx
dx
0
0
i
i
s0 @xi01
@x r
@x 2
@xjs0
@xi1 @xi2
@xir @xj10 @xj20
=
@xi0r @xj1 @xj2 @xjs
@xi01 @xi02
@
@
@
dxj1 dxj2 dxjs ;
(2.7)
i
i
@x 1 @x 2
@xir
which gives the transformation of the components under changes of coordinates in the charts' intersection. We nd frequently tensors de ned as entities
@xjr0 for
whose components transform in this way, with one Lame coeÆcient @x
jr
each index. It should be understood that a tensor is always a tensor with respect to a given group. Just above, the group of coordinate transformations
was involved. General base transformations constitute another group.
0 0 :::i0
r
Tji101ji220 :::j
s0
35
2.31
Vectors and tensors have been de ned at a xed point p of a di er entiable manifold M . The natural basis we have used is actually f @[email protected] i pg. A
vector at p 2 M has been de ned as the tangent to a curve a(t) on M , with
a(0) = p. We can associate a vector to each point of the curve by allowing
the variation of the parameter t: Xa(t) (f ) = dtd (f Æ a)(t). Xa(t) is then the
tangent eld to a(t), and a(t) is the integral curve of X through p. In general,
this only makes sense locally, in a neighborhood of p. When X is tangent to
a curve globally, X is a complete eld.
x
2.32
Let us, for the sake of simplicity take a neighborhood U of p and
suppose a(t) 2 U , with coordinates (a1(t), a2(t), , am (t)). Then, Xa(t) =
dai @ , and dai is the component X i . In this sense, the eld whose integral
a(t)
dt @ai
dt
curve is a(t) is given by the \velocity" da
dt . Conversely, if a eld is given
by its components X k (x1(t), x2 (t), : : : , xm (t)) in some natural basis, its
integral curve x(t) is obtained by solving the system of di erential equations
k
X k = dxdt . Existence and uniqueness of solutions for such systems hold in
general only locally, as most elds exhibit singularities and are not complete.
Most manifolds accept no complete vector elds at all. Those which do are
called parallelizable. Toruses are parallelizable, but, of all the spheres S n ,
only S 1, S 3 and S 7 are parallelizable. S 2 is not.x
x
2.33
At a point p, Vp takes a function belonging to R(M ) into some real
number, Vp : R(M ) ! R. When we allow p to vary in a coordinate neighborhood, the image point will change as a function of p. By using successive
cordinate transformations and as long as singularities can be surounded, V
can be extended to M . Thus, a vector eld is a mapping V : R(M ) ! R(M ).
In this way we arrive at the formal de nition of a eld:
a vector eld V on a smooth manifold M is a linear mapping V : R(M ) !
R(M ) obeying the Leibniz rule:
x
X (f g ) = f X (g ) + g X (f ); 8f; g 2 R(M ):
We can say that a vector eld is a di erentiable choice of a member of TpM
at each p of M . An analogous reasoning can be applied to arrive at tensors
elds of any kind and order.
x This is the hedgehog theorem: you cannot comb a hedgehog so that all its prickles
stay at; there will be always at least one singular point, like the head crown.
36
vector
elds
x
2.34
Take now a eld X, given as X = X i @[email protected] i . As X (f ) 2 R(M ), another
eld as Y = Y i @[email protected] i can act on X(f). The result,
2
@X i @f
jXi @ f ;
+
Y
@xj @xi
@xj @xi
does not belong to the tangent space because of the last term, but the commutator
Y Xf = Y j
[X; Y ] := (XY
Y X) =
Xi
@Y j
@xi
Yi
@X j @
@xi @xj
does, and is another vector eld. The operation of commutation de nes a
linear algebra. It is also easy to check that
[X; X ] = 0;
(2.8)
[[X; Y ]; Z ] + [[Z; X ]; Y ] + [[Y; Z ]; X ] = 0;
(2.9)
the latter being the Jacobi identity. An algebra satisfying these two conditions is a Lie algebra. Thus, the vector elds on a manifold constitute, with
the operation of commutation, a Lie algebra.
2.1.3 Di erential Forms
x 2.35 Di erential forms{ are antisymmetric covariant tensor
elds on differentiable manifolds. They are of extreme interest because of their good
behavior under mappings. A smooth mapping between M and N take differential forms on N into di erential forms on M (yes, in that inverse order)
while preserving the operations of exterior product and exterior di erentiation (to be de ned below). In Physics they have acquired the status of
a new vector calculus: they allow to write most equations in an invariant
(coordinate- and frame-independent) way. The covector elds, or PfaÆan
{ On the subject, a beginner should start with H. Flanders, Di erential Forms, Academic Press, New York, l963; and then proceed with C. Westenholz, Di erential Forms in
Mathematical Physics, North-Holland, Amsterdam, l978; or W. L. Burke, Applied Di erential Geometry, Cambridge University Press, Cambridge, l985; or still with R. Aldrovandi
and J. G. Pereira, Geometrical Physics, World Scienti c, Singapore, l995.
37
Lie
algebra
forms, or still 1-forms, provide basis for higher-order forms, obtained by exterior product [see eq. (2.6)]. The exterior product, whose properties have
been given in eqs.(2.1)-(2.5), generalizes the vector product of E3 to spaces of
any dimension and thus, through their tangent spaces, to general manifolds.
2.36
The exterior product of two members of a basis f! i g is a 2-form,
typical member of a basis f! i ^ ! j g for the space of 2-forms. In this basis,
a 2-form F , for instance, will be written F = 12 Fij ! i ^ ! j . The basis for the
m-forms on an m-dimensional manifold has a unique member, ! 1 ^ ! 2 ! m .
The nonvanishing m-forms are called volume elements of M , or volume forms.
x
2.37
The name \di erential forms" is misleading: most of them are not
di erentials of anything. Perhaps the most elementary form in Physics is
the mechanical work, a PfaÆan form in E3 . In a natural basis, it is written
W = Fk dxk , with the components Fk representing the force. The total work
realized in taking a particle from a point a to point b along a line is
x
Wab [ ] =
Z
W=
Z
Fk dxk ;
and in general depends on the chosen line. It will be path-independent only
when the force comes from a potential U as a gradient, Fk = (grad U )k .
In this case W = dU , truly the di erential of a function, and Wab =
U (a) U (b). An integrability criterion is: Wab [ ] = 0 for any closed curve.
Work related to displacements in a non-potential force eld is a typical nondi erential 1-form. Another well-known example is heat exchange.
2.38
In a more geometric mood, the form appearing in the integrand of the
R
arc length ax ds is not the di erential of a function, as the integral obviously
depends on the trajectory from a to x, and is a multi-valued function of x.
The elementary length ds is a prototype form which is not a di erential,
despite its conventional appearance. A 1-form is exact if it is a gradient, like
! = dU . Being exact is not the same as being integrable. Exact forms are
integrable, but non-exact forms may also be integrable if they are of the form
fdU .
x
38
2.39
@f dxi = @f ^ dxi , which is a
The 0-form f has the di erential df = @x
i
@xi
1-form. The generalization of this di erential of a function to forms of any
order is the di erential operator d with the following properties:
x
1. when applied to a k-form, d gives a (k+1)-form;
2. d( + ) = d + d ;
3. d(
^
) = (d ) ^ + ( )@
4. d2 = dd
^ d(
0 for any form
), where @ is the order of ;
.
2.40
The invariant, basis-independent de nition of the di erential of a
k-form is given in terms of vector elds:
x
d (X0 ; X1 ; : : : ; Xk ) =
+
X
i<j
(
k
X
( )i Xi
i=0
i
+
) j ([X
h
i
(X0 ; X1 ; : : : ; Xi 1 ; X^ i ; Xi+1 : : : ; Xk )
i ; Xj ]; X0 ; X1 ; : : :
; X^ i ; : : : ; X^ j : : : ; Xk ):(2.10)
Wherever it appears, the notation X^ n means that Xn is absent. From this
de nition, or from the systematic use of the de ning conditions, we can
obtain the rst examples of derivatives:
if f is a function (0-form), df = @i f dxi (gradient) ;
if A = Ai dxi is a covector (1-form), then dA = 21 (@i Aj @j Ai) dxi ^ dxj
(rotational)
Comment 2.8
To grasp something about the meaning of
d2 0;
which is usually called the Poincare lemma, let us examine the simplest case, a 1-form in
a natural basis: = idxi. Its di erential is d = (d i) ^ dxi + i ^ d(dxi) = @@xji dxi ^ dxj
= 12 ( @@xji @@xij ) dxi ^ dxj . If is exact, = df (in components, i = @i f ) then
2
@f
@2f
d = d2f = 12
dxi ^ dxj
@xi @xj @xj @xi
and the property d2f 0 is just the symmetry of the mixed second derivatives of a
function. Along the same lines, if is not exact, we can consider
39
2
@2 i
@ i
2
j
k
i
1
d = j k dx ^ dx ^ dx = 2
@x @x
@xj @xk
@2 i
dxj ^ dxk ^ dxi = 0:
@xk @xj
Thus, the condition d2 0 comes from the equality of mixed second derivatives of the
functions i, related to integrability conditions.
x
2.41 A form
be written as
such that d = 0 is said to be closed. A form
= d for some is said to be exact.
which can
It is natural to ask whether every closed form is exact. The answer, given
by the inverse Poincare lemma, is: yes, but only locally. It is yes in Euclidean spaces,
and di erentiable manifolds are locally Euclidean. Every closed form is locally exact. The
precise meaning of \locally" is the following: if d = 0 at the point p 2 M , then there
exists a contractible (see below) neighborhood of p in which there exists a form (the
\local integral" of ) such that = d . But attention: if is another form of the same
order of and satisfying d = 0, then also = d( + ). There are in nite forms of which
an exact form is the di erential.
The inverse Poincare lemma gives an expression for the local integral of = d . In
order to state it, we have to introduce still another operation on forms. Given in a natural
basis the p-form
Comment 2.9
(x) = i1i2 i3 :::ip (x)dxi1 ^ dxi2 ^ dxi3 ^ ^ dxip
the transgression of
T
is the (p-1)-form
=
p
X
Z 1
j =1
0
( )j 1
dttp 1xij i1 i2 i3:::ip (tx)
dxi1 ^ dxi2 : : : ^ dxij 1 ^ dxij+1 ^ ^ dxip :
(2.11)
Notice that, in the x-dependence of , x is replaced by (tx) in the argument. As t ranges
from 0 to 1, the variables are taken from the origin up to x. This expression is frequently
referred to as the homotopy formula.
The operation T is meaningful only in a star-shaped region, as x is linked to the origin
by the straight line \tx", but can be generalized to a contractible region. Contractibility
has been de ned in Comment 2.5. Consider the interval I = [0; 1]. A space or domain X
is contractible if there exists a continuous function h : X I ! X and a constant function
f : X ! X , f (p) = c (a xed point) for all p 2 X , such that h(p; 0) = p = idX (p) and
h(p; 1) = f (p) = c. Intuitively, X can be continuously contracted to one of its points.
n
n
E is contractible (and, consequently, any coordinate neighborhood), but spheres S and
n
toruses T are not. The limitation to the result given below comes from this strictly local
40
property. Well, the lemma then says that, in a contractible region, any form
written in the form
= dT + T d :
can be
(2.12)
When
d =0;
(2.13)
= dT ;
(2.14)
so that is indeed exact and the integral looked for is just = T , always up to 's such
that d = 0. Of course, the formulae above hold globally on Euclidean spaces, which are
contractible. The condition for a closed form to be exact on the open set V is that V
be contractible (say, a coordinate neighborhood). On a smooth manifold, every point has
an Euclidean (consequently contractible) neighborhood | and the property holds at least
locally. The sphere S 2 requires at least two neighborhoods to be charted, and the lemma
holds only on each of them. The expression stating the closedness of , d = 0 becomes,
when written in components, a system of di erential equations whose integrability (i.e.,
the existence of a unique integral ) is granted locally. In vector analysis on E3 , this
includes the already mentioned fact that an irrotational ux (dv = rot v = 0) is potential
(v = grad U = dU ). If one tries to extend this from one of the S 2 neighborhoods, a
singularity inevitably turns up.
2.42 Let us
nally comment on the mappings between di erential manifolds and the announced good{behavior of forms. A C 1 function f : M ! N
between di erentiable manifolds M and N induces a mapping between the
tangent spaces:
x
! Tf (p)N:
If g is an arbitrary real function on N , g 2 R(N ), this mapping is de ned by
[f (Xp )](g ) = Xp (g Æ f )
(2.15)
for every Xp 2 TpM and all g 2 R(N ). When M = Em and N = En , f is the
f : Tp M
jacobian matrix. In the general case, f is a homomorphism (a mapping which
preserves the algebraic structure) of vector spaces, called the di erential of f .
It is also frequently written \df ". When f and g are di eomorphisms, then
(f Æ g ) X = f Æ g X . Still more important, a di eomorphism f preserves
the commutator:
f [X; Y ] = [fX; f Y ]:
41
(2.16)
Consider now an antisymmetric s-tensor wf (p) on the vector space Tf (p)N .
Then f determines a tensor on Tp M by
(f ! )p (v1 ; v2; : : : ; vs ) = !f (p)(f v1; f v2; : : : ; fvs ):
(2.17)
Thus, the mapping f induces a mapping f between the tensor spaces, working however in the inverse sense. f is called a pull-back and f , by extension,
push-forward. The pull-back has some wonderful properties:
f is linear;
(f Æ g ) = g Æ f .
f preserves the exterior product: f (! ^ ) = f ! ^ f ;
f preserves the exterior derivative: f (d! ) = d(f ! ).
Mappings between di erential manifolds preserve, consequently, the most
important aspects of di erential exterior algebra. The well-de ned behavior
when mapped between di erent manifolds makes of the di erential forms
the most interesting of all tensors. Notice that all these properties apply also
when f is simply some di erentiable transformation mapping M into itself.
42
pull
back
2.1.4 Metrics
x 2.43 The Euclidean space E3 consists of the set of triples R3 with the ball-
topology. The balls come from the Euclidean metric, a symmetric secondorder positive-de nite tensor g whose components are, in global Cartesian
coordinates, given by gij = Æij . Thus, E3 is R3 plus the Euclidean metric.
We use this metric to measure lengths in our everyday life, but it happens
frequently that another metric is simultaneously at work on the same R3.
Suppose, for example, that the space is permeated by a medium endowed
with a point-dependent isotropic refractive index n(p). Light rays will \feel"
the metric gij0 = n2 (p)Æij . To \feel" means that they will bend, acquire a
\curved" aspect. Fermat's principle says simply that light rays will become
geodesics of the new metric, the straightest possible curve if measurements
are made using gij0 instead of gij . As long as we proceed to measurements
using only light rays, distances { optical lengths { will be di erent from those
given by the Euclidean metric. Suppose further that the medium is some
compressible uid, with temperature gradients and all which is necessary to
render point-dependent the derivative
of the pressure with respect to the
@p
uid density at xed entropy, cs = @ . In that case, sound propagation
S
will be governed by still another metric, gij00 = c1s Æij . Nevertheless, in both
cases we use also the Euclidean metric to make measurements, and much
of geometric optics and acoustics comes from comparing the results in both
metrics involved. These words are only to call attention to the fact that there
is no such a thing like the metric of a space. It happens frequently that more
than one is important in a given situation.
2.44
Bilinear forms are covariant tensors of second order. The tensor
product of two linear forms w and z is de ned by (w z ) (X,Y) = w(X ) z (Y ).
The most fundamental bilinear form appearing in Physics is the Lorentz
metric on R4 (see the nal of this section).
Given a basis f! j g for the space of 1-forms, the products wi wj , with
i; j = 1; 2; : : : ; m, constitute a basis for the space of covariant 2-tensors, in
terms of which a bilinear form g is written g = gij wi wj . In a natural
basis, g = gij dxi dxj .
A metric on a smooth manifold is a bilinear form, denoted g (X; Y ), X Y
x
43
or < X; Y >, satisfying the following conditions:
1. it is indeed bilinear:
X (Y + Z ) = X Y + X Z
(X + Y ) Z = X Z + Y Z ;
2. it is symmetric:
X Y = Y X;
3. it is non-singular: if X Y = 0 for every eld Y, then X = 0.
In a basis, g (Xi ; Xj ) = Xi Xj = gmn ! m (Xi ) ! n (Xj ), so that
gij = gji = g (Xi ; Xj ) = Xi Xj :
It is standard notation to write simply wi wj = w(i wj ) = 21 (wi wj + wj wi )
for the symmetric part of the bilinear basis, so that
g = gij wi wj
or, in a natural basis,
g = gij dxi dxj :
2.45 Given a
eld Y = Y i ei and a form z = zj wj in the dual basis,
z (Y ) = < z; Y > = zj Y j . A metric establishes a relation between vector
and covector elds: Y is said to be the contravariant image of a form z
if, for every X, g (X; Y ) = z (X ). Then gij Y j = zi. In this case, we write
simply zj = Yj . This is the usual role of the covariant metric, to lower
indices, taking a vector into the corresponding covector. If the mapping
Y ! z so de ned is onto, the metric is non-degenerate. This is equivalent
to saying that the matrix (gij ) is invertible. A contravariant metric g^ can
then be introduced whose components (denoted by g rs ) are the elements
of the matrix inverse to (gij ). If w and z are the covariant images of X
and Y, de ned in a way inverse to the image given above, then g^(w; z ) =
g (X; Y ). All this de nes on the spaces of vector and covector elds an internal
product (X; Y ) := (w; z ) := g (X; Y ) = g^(w; z ). Invertible metrics are called
semi-Riemannian. Although physicists usually call them just Riemannian,
x
44
mathematicians more frequently reserve this denomination to non-degenerate
positive-de nite metrics, with values in the positive real line R+ .
As the Lorentz metric is not positive de nite, it does not de ne balls and
is consequently unable to provide for a topology on Minkowski space-time
(whose topology is, by the way, unknown). A Riemannian manifold is a
smooth manifold on which a Riemannian metric is de ned. A theorem due
to Whitney states that it is always possible to de ne at least one Riemannian
metric on an arbitrary di erentiable manifold.
A positive de nite metric is presupposed in any measurement: lengths,
angles, volumes, etc. The length of a vector X is introduced as
kX k = (X; X )1=2:
nite when kX k = 0 does not imply X = 0. It is the case of
A metric is inde
Lorentz metric, which attributes zero length to vectors on the light cone.
The length of a curve : (a; b) ! M is then de ned as
L =
Z b
a
k ddt dtk:
Given two points p; q on a Riemannian manifold M , consider all the piecewise
di erentiable curves with (a) = p and (b) = q . The distance between
p and q is de ned as the in mum of the lengths of all such curves between
them:
d(p; q ) = inf
Z b
(t) a
k dC
dtk:
dt
(2.18)
In this way a metric tensor de nes a distance function on M .
2.46
The metrics referred to in the introduction of this section, concerned
with simpli ed models for the behavior of light rays and sound waves, are
both obtained by multiplying all the components of the Euclidean metric
by a given function. A transformation like gij ! gij0 = f (p) gij is called a
conformal transformation. In angle measurements, the metric appears in a
numerator and in a denominator and in consequence two metrics di ering
by a conformal transformation will give the same angles. Conformal transformations preserve the angles, or the cones.
x
45
To nd the angle made by two vector elds U and V at each point,
calculate jjU V jj2 = jjU jj2 + jjV jj2 - 2 U V = U U - V V - 2 jjU jjjjV jj cos UV , that is,
p
p
g (U V ) (U V ) = g U U + g V V - g U U g V V cos UV . Then,
g U U +p
g V V g (U V ) (U V )
p
UV = arccos ;
g U U g V V 0 = f (p) g .
which does not change if g is replaced by g
Comment 2.10
x
2.47 Geometry has had a very strong historical bond to metric. \Geome-
x
2.48 Minkowski spacetime is a 4-dimensional connected manifold on which
tries" have been synonymous of \kinds of metric manifolds". This comes from
the impression that we measure something (say, distance from the origin)
when attributing coordinates to a point. We do not. Only homeomorphisms
are needed in the attribution, and they have nothing to do with metrics.
We hope to have made it clear that a metric on a di erentiable manifold is
chosen at convenience.
a certain inde nite metric (the \Lorentz metric") is de ned. Points on spacetime are called \events". Being inde nite, the metric de nes only a pseudodistance function for any pair of points. Given two events x and y with Cartesian coordinates (x0; x1; x2 ; x3) and (y 0; y 1; y 2; y 3), their pseudo-distance will
be
s2 = x x = (x0
y 0)2
(x1
y 1)2
(x2
y 2)2
(x3
y 3)2 : (2.19)
This pseudo-distance is called the \interval" between x and y . Notice the
usual practice of attributing the rst place, with index zero, to the time{
related coordinate. The Lorentz metric does not de ne a topology, but establishes a partial ordering of the events: causality. A general spacetime will
be any di erentiable manifold S such that, at each point p, the tangent space
TpS is a Minkowski spacetime. This will induce on S another metric, with
the same set of signs (+,-,-,-) in the diagonalized form. A metric with that
set of signs is said to be \Lorentzian".
In General Relativity, Einstein's equations determine a Lorentzian metric
which will be \felt" by any particle or wave travelling in spacetime. They are
non-linear second-order di erential equations for the metric, with as source
an energy-momentum density of the other elds in presence. Thus, the metric
depends on the source and on the assumed boundary conditions.
46
Minkowski
spacetime
2.2 Pseudo-Riemannian Metric
2.49
Each spacetime is a 4{dimensional pseudo{Riemannian manifold. Its
main character is the fundamental form, or metric
x
g (x) = g dx dx :
(2.20)
This metric has signature 2. Being symmetric, the matrix g (x) = (g ) can
be diagonalized. Signature concerns the signs of the eigenvalues: it is the
numbers of eigenvalues with one sign minus the number of eigenvalues with
the opposite sign. It is important because it is an invariant under changes of
coordinates and vector bases. In the convention we shall adopt this means
that, at any selected point P , it is possible to choose coordinates fx g in
terms of which g takes the form
g (P ) =
+jg00 j
0
0
0
jg11 j
0
0
0
0
jg22 j
0
0 !
0
0
jg33 j
:
(2.21)
2.50
The rst example has been given above: it is the Lorentz metric of
Minkowski space, for which we shall use the notation
x
(x) = ab dxa dxb :
(2.22)
We are using indices ; ; ; : : : for Riemannian spacetime, and a; b; c; : : : for
Minkowski spacetime.
Minkowski space is the simplest, standard spacetime. Up to the signature, is an Euclidean space, and as such can be covered by a single, global
coordinate system. This system | the cartesian system | is the father of
all coordinate systems and just puts in the diagonal form
g (P ) =
+1
0
0
0
1
0
0
0
0
1
0
0 0
0
1
:
(2.23)
A metric is a real symmetric non{singular bilinear form. A bilinear form
takes two vectors into a real number: g(V; U ) = g V U 2 R. When symmetric, the
numbers g(V; U ) and g(U; V ) are the same. A metric de nes orthogonality. Two vectors
U and V are orthogonal to each other by g if g(V; U ) = 0. The metric components can be
disposed in a matrix (g ). It will have, consequently, ten components on a 4{dimensional
spacetime.
Comment 2.11
47
2.51
Given the metric, a vector eld V is timelike, spacelike or a null
vector, depending on the sign of g V V :
x
(
> 0 timelike
< 0 spacelike
=0
null
g V V (2.24)
V are the components of the vector V in the coordinate system fx g. Higher
indices indicate a contravariant vector. The metric can be used to lower
indices. From V , obtain a covariant vector, or covector, whose components
are V = g V . Einstein's summation convention is being used: repeated
higher{lower indices are summed over everytime they appear, without the
summation symbol.
2.52
As presented above, with lower indices, the metric is sometimes called
the contravariant metric. The elements of its inverse matrix are indicated by
g . Thus, with the Kronecker delta Æ (which is = 1 when = and zero
otherwise), we have
g g = g g = Æ :
x
The set fg g is then called the covariant metric, and can be used to raise
indices, or to get a contravariant vector from a covariant vector: V = g V .
The same holds for indices of general tensors. Frequently used notations are
g = jg j = det(g ). Of course, det(g ) = g 1 .
x
2.53
The norm ds of the in nitesimal displacement dx , whose square is
ds2 = g dx dx
(2.25)
is the in nitesimal interval, or simply interval. Given a curve with extreme
points a and b, the integral
L[ ] =
along
Z
ds =
Z b
a
ds
(2.26)
is a function(al) of , called its \length".
Again a name used by extension of the strictly Riemannian case, in
which this integrals is a true length. A strictly Riemannian metric determines a true
distance between two points. In the case above, the distance between a and b would be
the in mum of L[ ], all curves considered.
Comment 2.12
48
interval
The determinant of a metric matrix like (2.23) is always negative. In
cartesian coordinates, an integration over 4-space has the form
Comment 2.13
Z
V4
d4x:
In another coordinate system, a Jacobian turns up. We recall that what appears in
integration measures is the Jacobian up to the sign. Integration using a coordinate system
in which the in nitesimal length takes the form (2.25), and in which the metric has a
negative determinant, is given by the expression
Z
V4
p g d4x;
which holds in every case.
49
2.3 The Notion of Connection
Besides well{behaved entities like tensors (including metrics, vectors and
functions), a manifold contains other, not so well-behaved objects. The most
important are connections, essential to the notion of parallelism. We proceed
now to present the physicits' approach to connections.
2.54
What we understand by good behavior is: covariance under change
of coordinates. A scalar eld is invariant under change of coordinates. Take
the next case in complexity, a vector eld V . It will have components V in
a coordinate system fx g and components V in a coordinate system fy g.
The two sets are related by
x
@y V :
(2.27)
@x
This is the standard behavior. It de nes a vector by the group of coordinate
transformations. Tensors of any order reproduce it, index by index. The
metric, for example, has its components changed according to
V =
g =
@x @x
g :
@y @y (2.28)
Notice by the way that, contracting (2.27) with the gradient operator
V
@
@y ,
@
@y @
@
=
V
=V
:
@y
@x @y
@x
Thus, the expression
@
(2.29)
@x
is invariant under change of coordinates. The vector eld V , once conceived
as such a directional derivative, is an invariant concept. This is the notion of
vector eld used by the mathematicians: a directional derivative acting on
the functions de ned on the manifold.
Take now the derivative of (2.27):
V =V
@ @y @
@ @y V
=
V
+
V
@y @x @y @y @x
50
@y @x @
@x @ @y =
V
+
V
@x @y @x
@y @x @x
@x @ 2y @y @x @
=
V
+
V
@x @y @x
@y @x @x
@x @y @
@y @x @ 2y = V +
V
@y @x @x
@x @y @x @x
@
@x @y ) V = @y
@y @x
:
@
@x @ 2 y V + V
@x
@y @x @x
:
(2.30)
If alone, the rst term in the right{hand side would confer to the derivative a
good tensor status. The second term breaks that behavior: the derivative of
a vector is not a tensor. In other words, the derivative is not covariant. On
a manifold, it is impossible to tell, for example, whether a vector is constant
or not.
The solution is to change the very de nition of derivative by adding another structure. We add to each derivative an extra term involving a new
object :
@ D
= @ V+ V
V
)
V
@y Dy @y D
@
@
V )
V =
V +
V
@x
Dx
@x
and impose good behavior of the modi ed derivative:
(2.31)
D
D
= @x @y
V
V ;
Dy @y @x Dx
or
@ V +
@y V
@x @y = @y @x
@
V +
@x
V
:
(2.32)
We then compare with (2.30) and look for conditions on the object . These
conditions x the behavior of under coordinate transformations. must
transform according to
@x @y @x
=
@y @x @y 51
@x @ 2y + @y @x @x
;
covariant
derivative
or
@y @x
@x
@x @x @ 2y +
:
(2.33)
@x @y @y @y @y @x @x
This non{covariant behavior of the connection makes of (2.31) a well{
behaved, covariant derivative. We have used a vector eld to nd how
should behave but, once is known, covariant derivatives can be de ned on
general tensors.
There are actually in nite objects satisfying conditions (2.33), that is,
there are in nite connections. Take another one, 0 . It is immediate that
is a tensor under coordinate transformations.
the di erence 0 The covariant derivative of a function (tensor of zero degree) is the usual
derivative, which in the case is automatically covariant. Take a third order
mixed tensor T . Its covariant derivative will be given by
=
D T = @ T +
T T T :
(2.34)
The rules to calculate, involving terms with contractions for each original
index, are fairly illustrated in this example. Notice the signs: positive for
upper indices, negative for lower indices. The metric tensor, in particular,
will have the covariant derivative
D g = @ g
g
g :
(2.35)
When the covariant derivative of T is zero on a domain, T is \self{parallel"
on the domain, or parallel{transported. An intuitive view of this notion will
be given soon (see below, Fig.2.1, page 54). It exactly translates to curved
space the idea of a straight line as a curve with maintains its direction along
all its length.
If the metric is parallel{transported, the equation above gives the metricity condition
@ g =
g
+
g
=
+
=2
where the symbol with lowered index is de ned by
compact notation for the symmetrized part
()
=
1
2
f
52
+
g;
() ;
= g
(2.36)
and the
(2.37)
parallel
transport
has been introduced. The analogous notation for the antisymmetrized part
[]
=
1
2
f
g
(2.38)
is also very useful.
Another convenient notational device: it it usual to indicate the common
derivative by a comma. We shall adopt also one of the two current notations
for the covariant derivative, the semi-colon. The metricity condition, for
example, will have the expressions
g; = g;
2
()
= 0:
(2.39)
The bar notation for the covariant derivative, gj = g; , is also found in
the literature.
2.4 The Levi{Civita Connection
There are, actually, in nite connections on a manifold, in nite objects behaving according to (2.33). And, given a metric, there are in nite connections
satisfying the metricity condition. One of them, however, is special. It is
given by
Æ
= 12 g [email protected] g + @ g
@ g g :
(2.40)
It is the single connection satisfying metricity and which is symmetric in
the last two indices. This symmetry has a deep meaning. The torsion of a
connection of components is a tensor T of components
T =
=
2
[] :
(2.41)
Connection (2.40) is called the Levi{Civita connection. Its components are
just the Christo el symbols we have met in x1.15. It has, as said, a special
relationship to the metric and is the only metric{preserving connection with
zero torsion. Standard General Relativity works only with such connections.
We can now give a clear image of parallel-transport. If the connection is
related to a de nite-positive metric | so that angles can be measured | a
parallel-transported vector keeps the angle with the curve the same all along
(Fig.2.1).
53
X
θ
X
V
θ
V
θ
X
V
γ
Figure 2.1: If the connection is related to a de nite-positive metric, a paralleltransported vector keeps the angle with the curve all along.
Notice that, with the notations introduced above, a general connection
will have components of the form
=
()
2 T :
(2.42)
Connections with zero torsion, = () are usually called symmetric
connections.
We have said that a curve on a manifold M is a mapping from the real
line on M . The mapping will be continuous, di erentiable, etc, depending
on the point of interest. We shall be mostly interested in curves with xed
endpoints. A curve with initial endpoint p0 and nal endpoint p1 is better
de ned as a mapping from the closed interval I = [0; 1] into M :
: [0; 1] ! M ; u ! (u) ;
(0) = p0 ; (1) = p1 :
(2.43)
The variable u 2 [0; 1] is the curve parameter. A loop, or closed curve, with
p0 = p1 , can be alternatively de ned as a mapping of the circle S 1 into M . We
shall be primarily concerned with di erentiable, or smooth curves, de ned
as those for which the above mapping is a di erentiable function. A smooth
curve has always a tangent vector at each point. Suppose the tangent vectors
at all the points is time-like (see Eq. (2.24)). In that case the curve itself is
said to be time-like. In the same way, the curve is said to be space-like or
light-like when the tangent vectors are all respectively space-like and null.
We can now consider derivatives along a curve , whose points have coordinates x (u). The usual derivative along is de ned as
d
dx @
=
du du @x
54
curves
again
The covariant derivative along { also called absolute derivative | is de ned
by
D dx
=
D ;
(2.44)
Du du
and apply to tensors just in the same way as covariant derivatives. Thus, for
example,
DV
dV
dx
=
+
V
:
(2.45)
Du
du
du
Let us go back to the invariant expression of a vector eld, Eq.(2.29).
It is an operator acting on the functions de ned on the manifold. If the
manifold is a di erentiable manifold, and fx g is a coordinate system, the
@x = Æ . We say then
set of operators f @[email protected] g is linearly independent, and @x
@
that the set of derivative operators f @x g constitute a base for vector elds.
Such a base is called natural, or coordinate base, as it is closely related to the
coordinate system fx g. Any vector eld can be expressed as in Eq.(2.29),
@y @ .
with components V . Under a change of coordinate system, @[email protected] = @x
@y
@
The base member e = @x is in this way expressed in terms of another base,
the set f @[email protected] g. The latter constitutes another coordinate base, naturally
related to the coordinate system fy g. The components V transform in the
converse way, so that V is, as already said, an invariant object. Vector bases
on 4{dimensional spacetime are usually called tetrads (also vierbeine, and
four{legs).
Tetrad elds are actually much more general { they need not be related
to any coordinate system. Put h = @@xy , so that e = @[email protected] = h @[email protected] . Of
course, the members of the base fe g commute with each other, [e ; e ] = 0.
This is also the suÆcient the condition for a base to be natural in some
coordinate system: if [e ; e ] = 0, then there exists a coordinate system fx g
such that e = @[email protected] for = 0; 1; 2; 3. These things are simpler in the so{called
dual formalism, which uses di erential forms.
As said in x2.20, to every real vector space V corresponds another vector
space V , which is its dual. This V is de ned as the set of linear mappings
from V into the real line R. Given a base fe g of V , there exists always
a base fe g of V which is its dual, by which we mean that e (e ) = Æ .
The base dual to f @[email protected] g is the set of di erential forms fdx g, each dx being
understood as a linear mapping satisfying dx ( @@ ) = Æ .
Take the di erential of a function f : df = @[email protected] dx . Applying the above
@f . An arbitrary di erential 1{form (also
rule, it follows that df ( @[email protected] ) = @x
55
dual
space
again
called a covector eld) is written ! = ! dx in base fdx g. The above dual
base will have members e = dx = h dy , with h = @x
@y . As it always
2
happen that d 0, the condition for a covector base to be naturally related
to a coordinate system is that de = 0. Each dx transforms according to
dx = @x
@y dy , and the above expression for a covector ! is invariant. We
shall later examine general tetrads in detail.
56
2.5 Curvature Tensor
2.55
A connection de nes covariant derivatives of general tensorial objects. It goes actually a little beyond tensors. A connection de nes a
covariant derivative of itself. This gives, rather surprisingly, a tensor, the
Riemann curvature tensor of the connection:
x
R = @
@
+
:
(2.46)
It is important to notice the position of the indices in this de nition. Authors di er in that point, and these di erences can lead to di erences in the
signs (for example, in the scalar curvature de ned below). We are using
all along notations consistent with the di erential forms. There is a clear
antisymmetry in the last two indices,
R =
R :
2.56
Notice that what exists is the curvature of a connection. Many connections are de ned on a given space, each one with its curvature. It is
common language to speak of \the curvature of space", but this only makes
sense if a certain connection is assumed to be included in the very de nition
of that space.
x
2.57
The above formulas hold on spaces of any dimension. The meaning
of the curvature tensor can be understood from the diagram of Figure 2.2.
x
First, build an in nitesimal parallelogram formed by pieces of geodesics,
indicated by dx, dx , dx0 and dx0 .
Take a vector eld X with components X at a corner point P . Paralleltransport X along dx. At its extremity, X will have the value X X dx
Parallel-transport that value along dx0 . It will lead to a value which
we denote X 0 .
Back at the starting point, parallel-transport X now rst along dx
and then along dx0 . This will lead to a eld value which we denote
X 00 .
57
In a at case, X 00 = X 0 . On a curved case, the di erence between
them is non-vanishing, and given by
ÆX = X 00
X 0 =
R X dx dx :
(2.47)
This is the in nitesimal case, in the limit of vanishing parallelogram
and with the value of R at the left-lower corner.
µ
µ
ν
−Γ νλ
µ
dx λ
µ
δ
dx' ν
µ
dx' λ
dx λ
µ
dx ν
Figure 2.2: The meaning of the Riemann tensor.
x
2.58 Other tensors can be obtained from the Riemann curvature tensor
by contraction. The most important is the Ricci tensor
R = R = @
@
+
:
(2.48)
This tensor is symmetric in the case of the Levi-Civita connection:
Æ
Æ
R = R :
(2.49)
In this case, which has a special relation to the metric, the contraction with
it gives the scalar curvature
Æ
Æ
R = g R :
58
(2.50)
2.6 Bianchi Identities
x
2.59
Take the de nition (2.46) in the case of the Levi-Civita connection,
Æ
R = @
Æ
@
Æ
+
Æ Æ
Æ Æ
:
(2.51)
The metric can be used to lower the index . Calculation shows that
Æ
Æ
R = g R = @
Æ
Æ
@
Æ
+
Æ
Æ
Æ
Æ
;
(2.52)
Æ
where = g . The curvature of a Levi-Civita connection has some
special symmetries in the indices, which can be obtained from the detailed
expression in terms of the metric:
Æ
Æ
Æ
Æ
Æ
Æ
R = R = R = R ;
R = R :
(2.53)
(2.54)
In consequence of these symmetries, the Ricci tensor (2.48) is essentially the
only contracted second-order tensor obtained from the Riemann tensor, and
the scalar curvature (2.50) is essentially the only scalar.
x
2.60 A detailed calculation gives the simplest way to exhibit curvature.
Consider a vector eld U with components U and take twice the covariant derivative, getting U ; ; . Reverse then the order to obtain U ; ; and
compare. The result is
U
; ;
U
; ;
Æ
=
R
U :
(2.55)
Curvature turns up in the commutator of two covariant derivatives.
x
2.61 Detailed calculations lead also to some identities. Of of them is
Æ
Æ
Æ
R + R + R = 0 :
(2.56)
Another is
Æ
Æ
Æ
R; + R; + R; = 0
59
(2.57)
(notice, in both cases, the cyclic rotation of the three last indices). The last
expression is called the Bianchi identity. As the metric has zero covariant
derivative, it can be inserted in this identity to contract indices in a convenient way. Contracting with g , it comes out
Æ
Æ
Æ
R; + R ; = 0:
R;
Further contraction with g yields
Æ
Æ
which is the same as
or
Æ
R ;
R;
Æ
R ; = 0;
Æ
2R ; = 0
R ;
hÆ
R 1 2 Æ
Æi
R
;
= 0:
(2.58)
This expression is the \contracted Bianchi identity". The tensor thus \covariantly conserved" will have an important role. Its totally covariant form,
Æ
1
2
G = R
Æ
g R ;
(2.59)
is called the Einstein tensor. Its contraction with the metric gives the scalar
curvature (up to a sign).
g G =
x
Æ
R:
(2.60)
2.62 When the Ricci tensor is related to the metric tensor by
Æ
R = g ;
(2.61)
where is a constant, It is usual to say that we have an Einstein space. In
Æ
Æ
that case, R = 4 and G = g . Spaces in which R is a constant are
said to be spaces of constant curvature. This is the standard language. We
insist that there is no such a thing as the curvature of space. Curvature is a
characteristic of a connection, and many connections are de ned on a given
space.
Some formulas above hold only in a four-dimensional space. We shall in
the following give some lower-dimensional examples. It should be kept in
mind that, in a d-dimensional space, g g = d. When d = 2 for example,
g G 0. On a two-dimensional Einstein space, G 0.
60
2.6.1 Examples
Calculations of the above objects will be necessary to write the dynamical
equations of General Relativity. Such calculations are rather tiresome. In
order to get a feeling, let us examine some low-dimensional examples. We
shall look at the 2-dimensional sphere and at two 2-dimensional hyperboloids.
In e ect, two surfaces of revolution can be got from the hyperbola shown in
Figure 2.3: one by rotating around the vertical axis, the other by rotating
around the horizontal axis (Figure 2.4). The rst will have two separate
sheets, the other only one. They are obtained from each other by exchanging
the squared vertical coordinate (below, z 2 ) by the squared coordinates of the
resulting surface (below, x2 + y 2). The spacetimes of de Sitter are higherdimensional versions of these hyperbolic surfaces.
Figure 2.3: Hyperbola. Two distinct surfaces of revolution can be obtained,
by rotating either around the vertical or the horizontal axis.
ned as the set of points ofpE3 satisfying x2 +
y 2 + z 2 = a2 in cartesian coordinates. This means z = a2 x2 y 2 , and
consequently
x
2.63 The sphere S 2 is de
dz =
dx x
a2 x2
p
y2
61
dy y
:
a2 x2 y 2
p
H
(1,1)
H2
H
Figure 2.4: The two hyperbolic surfaces. The second is the complement of
the rst, rotated of 90o .
The interval dl2 = dx2 + dy 2 + dz 2 becomes then
dy 2 x2 + 2 dx dy x y
a2 x2 y 2
It is convenient to change to spherical coordinates
dl2 =
a2 dx2 + a2 dy 2
dx2 y 2
x = a sin cos ; y = a sin sin ; z = a cos :
The interval becomes
dl2 = a2 d 2 + sin2 d2 :
The corresponding metric is given by
g = (g ) =
a2
0
0 a2 sin2 !
;
Æ
with obvious inverse. The only non-vanishing Christo el symbols are 122 =
Æ
Æ
- sin cos and 212 = 2 21 = cot . The only non-vanishing components of
62
Æ
the Riemann tensor are (up to symmetries in the indices) R1 212 = sin2 and
Æ
Æ
Æ
R2 112 = 1. The Ricci tensor has R11 = 1 and R22 = sin2 , so that it can
be represented by the matrix
Æ
(R ) =
1 0
0 sin2 !
=
1
a2
(g ) :
Consequently, the sphere is an Einstein space, with = a12 . Finally, the
scalar curvature is
Æ
2
R = a2 :
Not surprisingly, a sphere is a space of constant curvature. As previously said,
the Einstein tensor is of scarce interest for two-dimensional spaces. Though
we shall not examine the geodesics, we note that their equations are
 = sin cos _ 2 ;  =
2 cot __ ;
and that constant and are obvious particular solutions. Actually, the
solutions are the great arcs.
2.64 The two-sheeted hyperboloid H 2 is de ned as the locus of those
points of E3 satisfying x2 + y 2 z 2 = a2 in cartesian coordinates. The line
x
element has the form
dx2 + dy 2
dz 2 =
a2 dx2 + a2 dy 2
dy 2 x2 + 2 dx dy x y
a2 x2 y 2
dx2 y 2
Changing to coordinates
x = a sinh cos ; y = a sinh sin ; z = a cosh the interval becomes
dl2 = a2 d2 + sinh2 d2 :
The metric will be
g = (g ) =
a2
0
2
0 a sinh2 63
!
:
:
Æ
Æ
The only non-vanishing Christo el symbols are 1 22 = - sinh cosh and 2 12
Æ
= 221 = coth . The only non-vanishing components of the Riemann tensor
Æ
Æ
are (up to symmetries in the indices) R1 212 = sinh2 and R2112 = 1. The
Ricci tensor constitutes the matrix
Æ
(R ) =
0
1
0
sinh2 !
:
1
a2 .
We see that H 2 is an Einstein space, with =
Æ
The scalar curvature is
2
:
a2
H 2 is a space of constant, but negative, curvature. It is similar to the sphere,
but with a imaginary angle. An old name for it is \pseudo-sphere".
R=
2.65 The one-sheeted hyperboloid H (1;1) is de ned as the locus of those
points of E3 satisfying x2 + y 2 z 2 = a2 in cartesian coordinates. The line
x
element has the form
dz 2
dx2
a2 dx2 + a2 dy 2
dy 2 =
dy 2 x2 + 2 dx dy x y dx2 y 2
:
a2 x2 y 2
Changing to coordinates
x = a cosh cos ; y = a cosh sin ; z = a sinh the interval becomes
dl2 = a2
d 2 + sinh2 d2 :
g = (g ) =
a2
0
2
0 a cosh2 The metric will be
!
:
Æ
Æ
The only non-vanishing Christo el symbols are 1 22 = sinh cosh and 2 12
Æ
= 2 21 = tanh . The only non-vanishing components of the Riemann tensor
Æ
Æ
are (up to symmetries in the indices) R1 212 = cosh2 and R2 112 = 1. The
Ricci tensor constitutes the matrix
Æ
(R ) =
0
1
64
0
cosh2 !
:
We see that H (1;1) is an Einstein space, with = a12 . The scalar curvature is
Æ
2
R = a2 :
H (1;1) is a space of constant positive curvature, like the sphere.
x
2.66 The rotating disk We shall now consider the rotating disk of x1.13
from two points of view: in (2 + 1) dimensions (two for the rotation plane,
one for time); and only the 3-dimensional space section. Curiously, the two
cases will have quite di erent results: the (2; 1) case gives zero curvature,
while the 3-dimensional space is curved.
Take rst the (2; 1) case, with coordinates (ct; R; ). The metric will be
0
g = (g ) = B
@
1
!2 R2
c2
0
!R2
c
0
0
!R2
c
1
0
R2
1
C
A
:
Æ
2
Æ
Æ
The only non-vanishing Christo el symbols are 211 = !c2R , 2 13 = 2 31
Æ2
Æ3
Æ3
! , and Æ 3 = Æ 3 = 1 . All the com,
=
R
,
=
=
= !R
33
12
21
23
32
c
cR
R
ponents of the Riemman tensor vanish, so that the 3-dimensional spacetime
is at.
Take now the 3-dimensional space, with coordinates (x; y; z ). The metric
will be
0
1
1 + fy 2
fxy 0
g = (g ) = B
fxy 1 + fx2 0 C
@
A ;
0
0
1
Æ
2 2
with f = c2 !2!(xc2 +y2 )2 . All the Christo el symbols of type 3 ij are zero, but
the other have some rather lengthy expressions. The same is true of the Ricci
tensor. We shall only quote the scalar curvature, which is (with r2 = x2 + y 2 )
R=
2 c4 r 4 ! 6
2 c8 ! 2 (3 + 2 r2 ! 2 ) + c6 (4 r2 ! 4 2 r4 ! 6 )
:
(c2 r2 ! 2 )2 (r2 ! 2 c2 (1 + r2 ! 2 ))2
This means that the space is curved. A simpler expression turns up for the
particular values ! = 1, c = 1:
6
R=
:
1 r2
65
We see that there is a singularity when r ! !c. Exactly the same results
come out if we consider the pure 2-dimensional case, the plane rotating disk.
This corresponds to dropping the last column and row of the metric above.
x
2.67
In all these examples,
1. a set point S is rst de ned by a constraint on the points of an ambient
space E ;
2. the metric is de ned by the restriction, on the subset, of the metric of
the ambient space;
3. such a metric is said to be induced by the imbedding of S in E .
66
Chapter 3
Dynamics
3.1 Geodesics
3.1
Curves de ned on a manifold provide tests for many of its properties.
In Physics, they do still more: any observer will be ultimately represented
by some special curve on spacetime. We have obtained the geodesic equation
before. We had then used implicitly the assumption that ds 6= 0. The approach which follows though more involved, has the advantage of including
the null geodesics, for which ds = 0.
Our aim is to obtain certain priviledged curves as extremals of certain
functionals, or functions de ned on the space of curves. Such functional
involve integrals along each curve, like
x
F[ ] =
Z
f:
A simple stratagem allows to manipulate such functionals as if they were
functions. To do it, it is necessary to give a label to each curve, so that
varying the curve becomes a simple change of label. This can be done by
considering, beyond the initial family of possible, curves, another family of
\transversal" curves.
x
3.2 We shall be interested in families of curves, like the curves
in Figure 3.1.
, and
We have indicated by a0 and a1 the initial and the nal
See J. L. Synge, Relativity: The general theory, North{Holland, Amsterdam, 1960.
67
congru
ences
c1
b1
c0
v
a1
b0
γ
a0
β
u
α
σ
u1
ρ
u0
Figure 3.1: The family of curves , , is parametrized by u. The crossed
family of curves , is parametrized by v . The second family is a \variation"
of the rst.
endpoints of the piece of which will be of interest, and analogously for the
other curves. The curve parameter u is indicated as \going along" them,
with u0 and u1 the initial and nal endpoints. We say that the curves form a
congruence. The variation leading from to and to can be represented
by another parameter v . It is useful to consider v as the parameter related to
another set of curves, such that each curve of the second congruence intersects
every curve of the rst, and vice versa. We have drawn only and , which
go through the endpoints of the rst family of curves. The points on all the
curves constitute a double continuum, a two{parameter domain which we
shall indicate by (u; v ). It will have coordinates x (u; v ) = (u; v ). For a
xed value of v , v (u) = (u; v = constant) will describe a curve of the rst
family. For a xed value of u, u (v ) = (u= constant; v ) will describe a curve
of the second family. The second family is, by the way, called a \variation"
of any member of the rst family. There will be vectors along both families,
such as
dx
dx
U =
and V =
:
du
dv
68
If the connection is symmetric, the crossed absolute derivatives coincide:
DV
DU
=
:
(3.1)
Du
Dv
Let us rst x v at some value, thereby choosing a xed curve of the rst
family and consider, along that curve, the function(al)
I [v] = 12 (u1
u0 )
R u1
u0 g
dx dx
du du
du = 21 (u1
R u1
u0
u0 )
g
U U du :
(3.2)
On the two-parameter domain, variations of that curve become simple derivatives with respect to v . By conveniently adding antisymmetric and symmetric
pieces, we have the series of steps
d
I [v] = (u1
dv
= (u1
= (u1
u0 )
Z u1
u0 )
Z u1
u0
u0
D
g U
U du = (u1
Dv
d g U V
du
u0 ) g U V
u1
u0
du (u1
du (u1
u0)
u0 )
u0 )
Z u1
Z u1
u0
g
u0
Z u1
V
u0
g
g U
V
DV
du
Du
D
U du
Du
D
U du
Du
(3.3)
We are interested in curves with the same endpoints, and shall now collapse
the parameter v . In Figure 3.1, a0 = b0 = c0 and a1 = b1 = c1 . This
corresponds to putting V = 0 at the endpoints. The rst contribution in
the right{hand side above vanishes and
d
I [v] = (u1 u0)
dv
Z u1
u0
g
V
D
U du :
Du
(3.4)
We now call geodesics the curves with xed endpoints for which I is stationary. As in the last integrand V is arbitrary except at the endpoints, such
curves must satisfy
Æ
D
dU
U =
+
Du
du
U U =0;
(3.5)
geodesic
equation
which is the same as
Æ
d2 x
+
du2
dx dx
= 0:
du du
69
(3.6)
This is the geodesic equation we have met before. It says that the vector
U = dxdu has vanishing absolute derivative along the curve. This means
that it is parallel-transported along the curve. The only direction that can
be attributed to a curve is, at each point, that of its \velocity" U . This
leads to a much better name for such a curve: self-parallel curve. It keeps
the same direction all along. Geodesics (solutions of the geodesic equation)
play on curved spaces the role the straight lines have on at spaces.
Comment 3.1
Why is the name \self-parallel" better ? There are at least two reasons:
1. the word \geodesic" has a very strong metric conotation; its original meaning was
that of a shortest length curve; but length, real length, only is de ned by a positivede nite metric, not our case;
2. the concept has actually no relation to metric at all; such curves can be de ned for
any connection; connection is a concept quite independent of metric, and de nes
parallelism; for general connections, only \self-parallel" makes sense.
x
3.3
The geodesic equation has the rst integral
g
U U = C:
(3.7)
This is to say that, along the curve,
d g
du
Comment 3.2
x
3.4
U U
= 0:
Prove it for the connection given by (2.40).
Comparison with the interval expression (2.25) shows that
ds = C 1=2du:
Eq.(3.6) is invariant under parameter changes of type
u ! u0 = au + b :
(3.8)
The constant C can, consequently, be rescaled to have only the values 1 and
0. This means that, unless C = 0, the interval parameter s can be used as
the curve parameter. That parameter is the proper time, and we recall that
70
U = dxds is the the four{velocity. The choice of C = 1 allows contact to be
made with Special Relativity, as (3.7) becomes
U 2 = U U = U U = g U U = 1:
(3.9)
The case C = 0 includes trajectories of particles with vanishing masses, in
special light{rays. As long as we keep in mind this exception, we can rewrite
the geodesic equation in the forms
D
dU
U =
+
Ds
ds
U U =
d2 x
+
ds2
dx dx
= 0:
ds ds
(3.10)
Once things have been interpreted in this way, we say that the velocity
is covariantly derived along . This is supported by Eq.(2.44), which now
shows the absolute derivative as the covariant derivative projected, at each
point, on the velocity.
3.5
In the case of massive particles, we can use the freedom allowed by
(3.8) to choose another parameter. Introduce s by
x
u0)1=2s=L :
u u0 = (u1
Then, comparison with(2.26) shows that I (v ) = 21 L2 . This leads to the
variational principle
ÆL = Æ
Z
ds = 0 ;
(3.11)
which we have found before.
x
3.6 If we look back at what has been done in x1.15, we see that we have
there got (3.10) from (3.11). A massive particle, without additional structure
(for instance, supposing that the e ect of its spin is negligible, or zero) will
follow the geodesic equation. That is actually the standard approach. It is
enough to replace, in all the discussion, the expression \in the presence of a
metric" by the expression \in the presence of a gravitational eld". It holds,
wee see now, for massive particles, for which ds 6= 0.
71
x
3.7
x
3.8 The expression
Eqs.(3.6) and (3.10) are second{order ordinary di erential equations.
Existence and unicity theorems of the theory of di erential equations state
i
that, given starting coordinates xi and \velocities" dx
du at a point P , there
will be a curve (u), with < u < for some > 0, which goes through P
( (P ) = 0) and which is unique.
dU
+
U U
(3.12)
ds
is the covariant acceleration, the only to have a meaning in an arbitrary
coordinate system. The existence of the above rst integral is equivalent to
A =
U A=g
U A =0:
(3.13)
As in Special Relativity the acceleration is, at each point of , orthogonal
to the velocity. As the velocity is parallel (or tangent) to at each point,
we can say that the acceleration is orthogonal to . A curve is self{parallel
when its acceleration vanishes.
3.9
We have above supposed a Levi{Civita connection. All the terminology comes from the rst historical case, the Levi{Civita connection of a
strictly Riemannian metric. We have said that any vector which is parallel{
transported along a curve keeps the same angle with respect to the velocity
all along it. The concept of self{parallel curve keeps its meaning for a general linear connection. A self{parallel curve is in that case de ned as a curve
satisfying (3.10).
x
3.10
Let us go back to the rst integral given in Eq.(3.7). It has a very
deep meaning. As the momentum of a particle of mass m is P = mc U ,
we rewrite it as
x
m2c2 g U U = g P P = m2c2 C :
(3.14)
mc g U = @ S:
(3.15)
Write now
72
This means dS = mc g U dx . This gradient form, applied to the tangent
vector U = U @[email protected] , gives dS (U ) = mc C: That reveals the meaning of S for
the present case of a massive particle: with the choice C = 1 announced in x
3.4, it is the action written in terms of the sole coordinates along the curve,
as it appears in Hamilton-Jacobi theory:
P = @ S :
(3.16)
The geodesic curve is, at each point, orthogonal to the surface S = constant passing through that point. In e ect, dS = mc g U dx is (up to
the sign) just the action we have used before, but here along the curve. The
expression
g @ S @ S = m2 c2
(3.17)
is the relativistic Hamilton-Jacobi equation for the free particle. It is possible
to recover the geodesic equation from it. Let us see how.
Consider both sides of the expression
d
d
[g U ] = U = U @ U :
du
du
The left-hand side (LHS) is
d
d
d
(g U ) = g U + U g
du
du
du = g
d U + U U @ g :
du
The last piece is symmetric in the indices and , so that we can write
d
d
(g U ) = g U + 12 U U (@g + @ g ) :
du
du
Now to the right-hand side:
LHS =
RHS = U @U = U @ @ S = U @ @ S = U @ U =
U @ U ;
using @ [g U U ] = @ [constant] = 0 in the last step. Notice that here we
have the only contribution of Hamilton-Jacobi theory: we have used the fact
(3.16) that the momentum is a derivative to justify the exchange @ U =
@ U . Now, @ [g U U ] = 0 is also
0 = (@ g ) U U + 2 g U @ U ) U @ U =
73
1 (@ g ) U U 2 ;
Hamilton
Jacobi
equation
so that
RHS =
U @ U = 21 (@ g ) U U :
Now, LHS = RHS gives
g
d 1
U + 2 (@ g + @ g
du
@ g ) U U = 0 ;
or
d 1 U + 2 g [@ g + @ g @ g ] U U = 0 ;
du
the announced result. All this can be repeated step by step, but starting
from
g @ S @ S = 0 :
eikonal
equation
(3.18)
This equation has a di erent meaning: it is the eikonal equation, with S now
in the role of the eikonal. The geodesic, in this case, is the light-ray equation.
The Hamilton-Jacobi and the eikonal equations allow thus a uni ed view
of particle trajectories and light rays. The geodesic equation comes
from the Hamilton-Jacobi equation, as the equation of motion of a
massive particle;
from the eikonal equation, as the trajectory of a light ray.
3.11 As we have said, curves are of fundamental importance.
They not
only allow testing many properties of a given space. In spacetime, every
(ideal) observer is ultimately a time{like curve.
x
The nub of the equivalence principle is the concept of observer:
An observer is a timelike curve on spacetime, a world{line.
Such a curve represents a point-like object in 3-space, evolving in the timelike 4-th \direction". An object extended in 3-space would be necessarily
represented by a bunch of world{lines, one for each one of its points. This
mesh of curves will be necessary if, for example, the observer wishes to do
some experiment. For the time being, let us take the simplifying assumption
above, and consider only one world{line. This is an ideal, point-like observer.
If free from external forces, this line will be a geodesic.
74
observer
And here comes the crucial point. Given a geodesic going through a
point P ( (0) = P ), there is always a very special system of coordinates
(Riemannian normal coordinates) in a neighborhood U of P in which the
components of the Levi-Civita connection vanish at P . The geodesic is, in
this system, a straight line: y a = ca s. This means that, as long as traverses
U , the observer will not feel gravitation: the geodesic equation reduces to the
2 a
a
forceless equation duds = ddsy2 = 0. This is an inertial observer in the absence
of external forces. If = 0, covariant derivatives reduce to usual derivatives.
If external forces are present, they will have the same expressions they have
in Special Relativity. Thus, the inertial observer will see the force equation
dua = F a of Special Relativity (see Section 3.7).
ds
3.12 There is actually more.
Given any curve , it is possible to nd a
local frame in which the components of the Levi-Civita connection vanish
along . That observer would not feel the presence of gravitation.
x
3.13 How
point{like is a real observer ? We are used to say that an
observer can always know whether he/she is accelerated or not, by making
experiments with accelerometers and gyroscopes. The point is that all such
apparatuses are extended objects. We shall see later that a gravitational
eld is actually represented by curvature and that two geodesics are enough
to denounce its presence (x 3.46).
x
3.14
As repeatedly said, the principle of equivalence is a heuristic guiding
precept. It states that, as long as the dimensions involved in the de nition of
an observer are negligible, an observer can choose his/hers coordinates so that
everything (s)he experiences is described by the laws of Special Relativity.
x
75
3.2 The Minimal Coupling Prescription
3.15
The equivalence principle has been used up to now in a one-way trip,
from General Relativity to Special Relativity. Some frame exists in which the
connection vanishes, so that covariant derivatives reduce to simple derivatives. This can be used in the opposite sense. Given a special-relativistic
expression, how to get its version in the presence of a gravitational eld ?
The answer is now very simple: replace common derivatives on any tensorial
object by covariant derivatives. Symbolically, this is represented by a rule,
x
@ ) @ +
:
(3.19)
This comma ) semi-colon rule is the minimal coupling prescription. To this
must be added the already discussed passage from at to curved metric,
ab ) g :
(3.20)
Once acccepted, these two rules allow to translate special-relativistic laws,
or equations, into expressions which hold in the presence of a gravitational
eld.
3.16 Conservation of energy
is one of the most important laws of
Physics. Its special-relativistic version states that the energy-momentum
tensor has vanishing divergence:
x
@ T = T ; = 0:
(3.21)
In the presence of a gravitational eld, this becomes
@ T +
T
+
T
= 0;
(3.22)
being understood that every other derivative and any metric factor appearing in T are equally replaced according to (3.19) and (3.20). Notice that,
because of the metricity condition (2.39), the metric can be inserted in or
extracted from covariant derivatives at will. Equation (3.22) is usually represented in the comma-notation, as
T ; = 0 :
76
(3.23)
rules
An exercise: A dust cloud (or incoherent uid) is a uid formed by
massive particles ignoring each other. It is a gas without pressure | only
energy is present. Special Relativity gives its energy{momentum the form
x
3.17
T = U U ;
(3.24)
where is the energy density. The 4{vector U is a eld representing the
velocities of the uid stream{lines.
Comment 3.3
The following results are immediate:
T U = U ; T U U = ; g T = :
The covariant divergence of T must vanish:
D T = U D ( U ) + U D U = U D ( U ) + D U = 0:
Ds
(3.25)
Contract this expression with U : as U U = 1,
D U = 0:
Ds
As the metric can be inserted into or extracted from the covatiant derivative
without any modi cation (because it is preserved by the Levi-Civita conD U = 0. We get in consequence a continuity
nection), it follows that U Ds
equation, or energy ux \conservation":
D (U ) + U
D (U ) = 0:
Taken into (3.25), this leads to the geodesic equation for the stream{lines:
D U =0:
Ds
x 3.18 The covariant derivative D of a scalar eld is just the usual
derivative, D = @ . But the derivative @ is by itself a vector. The
Laplacian operator, or (its usual name in 4 dimensions) D'Alembertian, is
D @ = @ @ + @ . The contracted form of the Levi-Civita
connection has a special expression in terms of the metric. From Eq. (2.40),
= 21 g [email protected] g + @ g
@ g g = 12 g @ g = 12 tr[g 1 @ g];
77
dust
cloud
where the matrix g = (g ) and its inverse have been introduced. This is
= 21 tr[@ ln g] = 21 @ tr[ln g]:
For any matrix M, tr[ln M] = ln det M, so that
= 12 @ ln[det g] = @ ln[ g ]1=2 = p1 g @
p
g;
(3.26)
where g = j det gj. The D'Alembertian becomes
1
= @ @ + p
or
1
=p
g
g
[@
p
@ [
p
g ]@ ;
Laplace
Beltrami
operator
g @ ] :
(3.27)
Laplaceans on curved spaces are known since long, and are called LaplaceBeltrami operators.
Another example: the electromagnetic
notice that, due to (3.18) and using (3.26),
x
3.19
@ A ) A ; =
p1 g @ [p
eld.
To begin with, we
g A ] :
(3.28)
This is, by the way, the general form of the covariant divergence of a fourvector. The eld strength F is antisymmetric and, due to the symmetry of
the Christofell symbols, remain the same:
F = @ A
@ A + [
Æ
Æ
]A
= @ A
@ A :
In consequence, only the metric changes in the Lagrangean Lem =
= 14 ac bd Fab Fcd of Special Relativity: it becomes
Lem =
1 g F F :
4
1 F F ab
4 ab
(3.29)
Maxwell's equations, which in Special Relativity are written
@F + @ F + @ F = F; + F; + F; = 0
78
(3.30)
covariant
divergence
@ F = F ; = j ;
(3.31)
F ; + F; + F; = 0
(3.32)
F ; = j :
(3.33)
take the form
The latter deserves to be seen in some detail. Let us rst examine the
derivative
Æ
Æ
F ; = @ F + F + F :
Æ
The last term vanishes, again because is symmetric. If another connection were at work, a coupling with torsion would turn up: the last term
would be ( 12 T F ). The rst two terms give Maxwell's equations in
the form
p
1
F ; = p @ [ g F ] = j :
(3.34)
g
The energy-momentum tensor of the electromagnetic eld, which is
Tab = cd Fac Fbd + 14 ab ec fd Fef Fcd
in Special Relativity, becomes
T = g F F + 14 g g g FF :
Comment 3.4
Notice
(3.35)
g T = 0:
3.20 A
uid of pressure p and energy density has an energy-momentum
tensor generalizing (3.24):
x
T ab = (p + ) U a U b
p ab :
energy
momentum
This changes in a subtle way. Its form is quite analogous,
T = (p + ) U U but the 4-velocities are U =
dx ,
ds
pg ;
with ds the modi ed interval.
79
(3.36)
Comment 3.5
With respect to the dust cloud, only the trace changes:
T U = U ; T U U = ; T = g T = 3 p :
If the gas is ultrarelativistic, the equation of state is = 3 p and, consequently, T = 0, as
for the electromagnetic eld.
3.21
We have seen in x 3.17 that the streamlines in a dust cloud are
geodesics. The energy-momentum density (3.36) di ers from that of dust by
the presence of pressure. We can repeat the procedure of that paragraph in
order to examine its e ect. Here, T ; = 0 leads to
x
D
DU (p + ) + (p + )
+ (p + )U U ;
; =
Ds
Ds
Contraction with U leads now to
T U
(U ); =
@ p = 0:
p U ; ;
so that the energy ux is no more conserved. Taking this expression back
into that of T ; = 0 implies
DU Dp
= @ p U = (g U U )@ p :
Ds
Ds
More will be said on this force equation in x 3.50.
(p + )
x
3.22 Suppose that T is symmetric, as in the examples above. De
quantity
It is immediate that
and
L = x T x T :
L ; = 0
U L = (xU 80
x U ):
(3.37)
ne the
3.3 Einstein's Field Equations
3.23
The Einstein tensor (2.59) is a purely geometrical second-order tensor
which has vanishing covariant derivative. It is actually possible to prove that
it is the only one. The energy-momentum tensor is a physical object with the
same property. The next stroke of genius comes here. Einstein was convinced
that some physical chacteristic of the sources of a gravitational eld should
engender the deformation in spacetime, that is, in its geometry. He looked
for a dynamical equation which gave, in the non-relativistic, classical limit,
the newtonian theory. This means that he had to generalize the Poisson
equation
x
V = 4G
(3.38)
within riemannian geometry. The G has second derivatives of the metric,
and the energy-momentum tensor contains, as one of its components, the
energy density.
He took then the bold step of equating them to each other, obtaining
what we know nowadays to be the simplest possible generalization of the
Poisson equation in a riemannian context:
R
1
2
g R =
8G
c4
T :
(3.39)
This is the Einstein equation, which xes the dynamics of a gravitational
eld. The constant in the right-hand side was at rst unknown, but he xed
it when he obtained, in the due limit, the Poisson equation of the newtonian
theory (as will be seen in x3.31 below).
A text on gravitation has always a place for the value of G. At present
time, the best experimental value for the gravitational constant is
G = 6:67390 10 11m3 =kg=sec2 :
Comment 3.6
The uncertainty is 0:0014%. From that Earth's and Sun's masses can be obtained. The
values are M = 5:97223(0:00008) 1024kg and M = 1:98843(0:00003) 1030kg. The
apparatus used (by Jens H. Gundlach et al, University of Washington, 2000) is a moderm
version of the Cavendish torsion balance.
x
3.24 Contracting (3.39) with g , we
R=
8G
c4
81
nd
T;
(3.40)
eld
equation
where T = g T . This result can be inserted back into the Einstein equation, to give it the form
R =
8G T
c4
1 g T:
2 (3.41)
3.25 Consider
the sourceless case, in which T = 0. It follows from
the above equation that R = 0 and, therefore, that R = 0. Notice that
this does not imply R = 0. The Riemann tensor can be nonvanishing
even in the absence of source. Einstein's equations are non-linear and, in
consequence, the gravitational eld can engender itself. Absence of gravitation is signalled by R = 0, which means a at spacetime. This case |
Minkowski spacetime | is a particular solution of the sourceless equations.
Beautiful examples of solutions without any source at all are the de Sitter
spaces (see subsection 4.3.3 below).
x
x
3.26 It is usual
x
3.27 In reality, the Einstein tensor (2.59) is not the most general parallel-
to introduce test particles, and test elds, to probe a
gravitational eld, as we have repeatedly done when discussing geodesics.
They are meant to be just that, test objects. They are supposed to have no
in uence on the gravitational eld, which they see as a background. They
do not contribute to the source.
transported purely geometrical second-order tensor which has vanishing covariant derivative. The metric has the same property. Consequently, it is in
principle possible to add a term g to G , with a constant. Equation
(3.39) becomes
R
( 12 R + )g =
8G
c4
T :
(3.42)
From the point of view of covariantly preserved objects, this equation is as
valid as (3.39). In his rst trial to apply his theory to cosmology, Einstein
looked for a static solution. He found it, but it was unstable. He then
added the term g to make it stable, and gave to the name cosmological
constant. Later, when evidence for an expanding universe became clear, he
called this \the biggest blunder in his life", and dropped the term. This
is the same as putting = 0. It was not a blunder: recent cosmological
82
evidence claims for 6= 0. Equation (3.42) is the Einstein's equation with a
cosmological term. With this extra term, Eq,(3.41) becomes
R =
8G T
c4
83
1 g T
2 g :
(3.43)
3.4 Action of the Gravitational Field
3.28
Einstein's equations can be derived from an action functional, the
Hilbert-Einstein action
x
S [g ] =
Z
p
g R d4 x :
(3.44)
It is convenient to separate the metric as soon as possible, as in R = g R .
Variations in the integration measure, which is metric-dependent, are conp g. Taking variations with respect to the metric,
centrated in the Jacobian
ÆS [g ]
=
Æg Z
p
Æ g g R d4 x +
Æg Z
p
Æg g R d4 x +
Æg
Z
p
g g ÆR 4
d x:
Æg The rst term is
p
Æ g
Æg
=
1
p
2
Æ( g)
1 Æ (exp[tr ln g])
1
Ætr ln g
= p
= p
exp[tr ln g]
g Æg
2 g
Æg
2 g
Æg Æ ln g
1
Æg
= p ( g ) tr = p ( g ) tr g 1 2 g
Æg
2 g
Æg
p gg :
1
Æg
1
Æg p
= p ( g ) g (
g
)
g
=
= 12
2 g
Æg
2 g
Æg
The last term can be shown to produce a total divergence and can be dropped.
The rst two terms give
1
ÆS [g ] p =
g R
Æg 1
2 g
R :
This is the left-hand side of (3.39). In the absence of sources, Einstein's
equation reduces to
p
x
g R
1
2
g R = 0:
(3.45)
3.29 A few comments:
1. variations have been taken with respect to the metric, which is the
fundamental eld
84
Hilbert
action
2. it is perhaps strange that the variation of R , which encapsulates the
gravitational eld-strength, gives no contribution
3. If a source is present, with Lagragian density L and action
Ssource =
Z
p
g L d4 x;
(3.46)
its contribution to the eld equation, that is, its modi ed energymomentum tensor density, will be given by
p
Z
p
Æ
g T = Ssource =
Æg
ÆL
g d4 x +
=
Æg
Consequently,
Z
L
p
Z
Æ p
g
Æg L
p g ÆL
Æ g 4
d
x
=
Æg Æg ÆL 1
+ g L:
Æg 2 T =
d4 x
1g
2 L
:
(3.47)
4. Einstein's equation (3.42) with a cosmological term comes, in and analogous way, from the action
S [g ] =
x
Z
p
g (R + 2) d4 x:
(3.48)
3.30 We have used in x1.15 a dimensional argument to write the action of
a test particle and of its coupling with an electromagnetic potential. Nave
dimensional analysis can be very useful in eld theories. Not only do they
help keeping trace of factors in calculations, but also lead to deep questions in
the problem of quantization. Actually, nave dimensional considerations are
enough to exhibit a fundamental di erence between gravitation and all the
other basic interactions. Indeed, all the coupling constants are dimensionless
quantities (think of the electric charge e), except that of gravitation. This
p gR, whose dimension di ers from
is related to the Lagrangean density
those of the other theories. Let us be nave for a while, and consider usual
mechanical dimensions in terms of mass (M), length (L) and time (T). The
dimension of a velocity will be represented as [v ] = [c] = [LT 1 ]; that of a
force as [F ] = [MLT 2 ]; an energy will have [E ] = [F L] = [ML2T 2 ]; a
85
dimensions
pressure, [p] = [F L 2] = [EL 3 ]; an action, [S ] = [~] = [ET ] = ML2 T 1 .
Metric is dimensionless, so that [ds] = [dx] = [L]. The fact that a quantity
is dimensionless will be represented as in [e] = [g ] = [0].
Field theory makes use of natural units (seemingly forbidden by international law), in which ~ = 1 and c = 1. In that scheme, which greatly simplify
discussions, actions and velocities are dimensionless. In consequence, [L] =
[T ] = [M 1 ] and only one mechanical dimension remains. Usually, the mass
M is taken as the standard reference. A new set turns out. As examples,
[F ] = [2], [E ] = [M ], [ds] = [M 1 ]. We say then that force has \numeric"
dimension zero, energy has dimension one, length has dimension minus one.
quantity
usual dimension natural dimension numeric
mass
M
M
+1
length
L
M 1
-1
1
time
T
M
-1
1
0
velocity
LT
M
0
2
acceleration
LT
M
+1
2
2
force
MLT
M
+2
Newton's G
M 1 L3 T 2
M 2
-2
2
2
1
energy
ML T
M
+1
action
ML2 T 1
M0
0
1
2
2
pressure
ML T
M
-2
g
M 0 L0 T 0
M0
0
1
ds
L
M
-1
0
0
0
0
charge e
M LT
M
0
A
ML2 T 2
M
+1
2
2
F , E, B
MLT
M
+2
T
ML 1 T 2
M 2
-2
Rp
4
0
0
0
0
gF F d x
M LT
M
0
L 1
M
+1
2
2
R , R
L
M
+2
Rp
4
2
2
gRdx
L
M
-2
86
Fields representing elementary particles have, in general, natural dimenRp
sion + 1. Notice that
g R d4 x has not the dimension of an action.
Actually, in order to give coherent results, it must be multiplied by a con3
stant of dimension MT 1 , actually 16cG .
3.5 Non-Relativistic Limit
x
3.31 A massive particle follows the geodesic equation (3.10),
dU
d2 x
dx dx
+
U U = 2 +
= 0;
ds
ds
ds ds
which comes from the rst term in action (1.7),
S=
mc
Z
ds :
(3.49)
(3.50)
To compare with the non-relativistic case, we recall that the motion of a
particle in a gravitational eld is in that case described by the Lagrangian
mc2 +
L=
This means the action
S=
Z
mv 2
2
mV :
v2 V
+
dt :
2c c
mc c
(3.51)
Comparison with (3.50) shows that necessarily
v2 V
+
2c c
ds = c
dt
which, neglecting all the smaller terms, gives
ds2
2V
= 1+ 2
c
c2 dt2
dx2 :
(3.52)
non
retativistic
metric
We see that, in the non-relativistic limit,
g11 = g22 = g33 = 1 ; g00 = 1 +
87
2V
:
c2
(3.53)
According to Special Relativity, the energy-momentum density tensor (3.36)
reduces to the sole component
T00 = c2 ;
(3.54)
where is the mass density. The trace has the same value, T = c2 . If we
use Eq.(3.41), we have
R =
8G Æ0Æ 0
c2
1 g :
2 In particular,
4G
:
(3.55)
c2
The other cases are Ri6=j = 0 and Rii = 4GV=c4 = in our approximation.
We can then proceed to a careful calculation of R00 as given by (2.48), using
(3.53) and (2.40). It turns out that the terms in
are of at least second
0
order in v=c. The derivatives with respect to x = ct are of lower order if
compared with the derivatives with respect to the space coordinates xk , due
k
to the presence of the factors 1=c. What remains is R00 = @@xk00 . It is also
1 @V
00
found that k 00 12 g kj @g
@xj . But this is = c2 @xk . Thus,
R00 =
1 @ @V
1
= 2 V :
2
k
k
c @x @x
c
Comparison with (3.55) leads then to
R00 =
V = 4G :
(3.56)
Poisson
recovered
(3.57)
This shows how Einstein's equation (3.39) reduces to the Poisson equation
(3.38) in the non-relativistic limit. By the way, the above result con rms the
value of the constant introduced in (3.39).
3.32
Notice that the non-relativistic limit corresponds to a weak gravitational eld. A strong eld would accelerate the particles so that soon the
small-velocity approximation would fail.
x
3.33
If the cosmological constant is nonvanishing, then we should use
(3.43) to obtain R , instead of Eq.(3.41) as above. An extra term g00
x
88
appears in R00. The Poisson equation becomes V = 4G c2. To
examine the e ect of this term in the non-relativistic limit, we can separate
the potential in two pieces, V = V1 + V2 , with V1 = 4G . Then, V2 =
c2 has for solution V2 = c2r2 =6, leading to a force F (r) = c2 r=3.
As > 0 by present-day evidence, this harmonic oscillator-type force is
repulsive. This is a universal e ect: the cosmological constant term produces
repulsion between any two bodies.
3.34
Let us now examine what happens to the geodesic equation. First of
all, the four-velocity U has the components ( ; v=c) in Special Relativity,
p
with = 1= 1 v2=c2 . In the non-relativistic limit,
x
1 + 21
(1 + 21
U
v2
c2
;
v2 ;
c2
v=c) :
i
1 d
; 2
c dt
Concerning the interval ds, all 3-space distances jdxj are negligible if compared with cdt, so that ds2 c2 dt2. Consequently,
dU
ds
d
d(ct)
h
1 v2
2 c2
v
:
Now, in order to use the geodesic equation, we have to calculate the components of the connection given in Eq.(2.40),
Æ
= 21 g [email protected] g + @ g
@ g g :
To begin with, we notice that g 11 = g 22 = g 33 = 1 and, to the order we
are considering, g 00 = 1 2V=c. Given the metric (3.53), only the derivatives
with respect to the space variables of g00 are 6= 0. We arrive at the general
expression
Æ
k 0 0 Æ 0Æ 0 Æ k @ V :
1 k 0
= c2 (Æ Æ + Æ Æ ) Æ
k
The only non-vanishing components are
Æk
00
=
Æ0
0k
=
Æ0
The geodesic equations are:
89
k0
=
1
c2
@k V :
1. for the time-like component,
Æ0 k 0
dU 0 Æ 0 d h 1 v2 i
0=
+ U U k0 U U
2 +2
2
c
ds
d(ct)
h
i
dtd 12 vc32 + 2 c13 @k V vk :
Both terms are of order 1=c3 , equally negligible in our approximation.
2. the space-like components are more informative:
dU k Æ k 1 d k Æ k 0 0 1 d k 1
0=
+ U U 2
v + 00U U = c2 v + c2 @k V ;
ds
c dt
dt
which is the force equation
d k
v = @kV :
(3.58)
dt
x 3.35 In the non-relativistic limit, only g00 remains, as well as R00 . Einstein's equation greatly enlarge the the scope of the problem. What they
achieve is better stateid in the words of Mashhoon (reference of the footnote
in page 5):
\the newtonian potential V is generalized to the ten components of the metric; the acceleration of gravity is replaced by
2V
the Christo el connection; and the tidal matrix @[email protected]@x
j is replaced
by the curvature tensor; the trace of the tidal matrix is related
to the local density of matter ; the Riemann tensor is related to
the energy{momentum tensor"
3.6 About Time, and Space
3.6.1 Time Recovered
x 3.36 A gravitational eld is said to be constant when a reference frame
exists in which all the components g are independent of the \time coordinate" x0 . This coordinate, by the way, is usually referred to as \coordinate
time", or \world time". The non-relativistic limit given above is an example
of constant gravitational eld, as the potential V in (3.53) is supposed to be
time-independent.
90
force
equation
3.37
Let us now examine the relation between proper time and the \coordinate time" x0 = ct. Consider two close events at the same point in space.
As d~x = 0, the interval between them will be ds = cd , where is time as
seen at the point. This means that ds2 = c2 d 2 = g00 (dx0 )2 , or that
x
1p
p
g00 dx0 = g00 dt :
(3.59)
c
As long as the coordinates are de ned, the time lapse between two events at
the same point in space will be given by
d =
1
=
c
Z
pg
00
dx0 :
(3.60)
This is the proper time at the point. In a constant gravitational eld,
1p
g x0 :
(3.61)
c 00
Once a coordinate system is established, it is the coordinate time which is
seen from \abroad". Nevertheless, a test particle plunged in the eld will see
its proper time.
=
x
3.38 Consider some periodic phenomenon taking place in a constant grav-
itational eld. Its period will be given by the formula above and, as such,
will be di erent when measured in coordinate time or with a \proper" clock.
Its proper frequency will have, for the same reason, the values
!
!= p0 :
(3.62)
g00
In the non-relativistic limit, (3.53) will lead to
! = !0 1
x
V
c2
:
(3.63)
3.39 Consider a light ray goint from a point \1" to a point \2" in a weak
constant gravitational eld. If the values of the potential are V1 and V2 at
points \1" and \2", its proper frequency will be !1 = !0 1 Vc21 at point
\1" and !2 = !0 1 Vc22 at point \2". It will consequently change while
91
moving from one point to the other. As the potential is a negative function,
V = jV j, this change will be given by
! = !2
!1 = !0
jV2j jV1j c2
:
(3.64)
If the eld is stronger in \1" than in \2", jV1 j > jV2 j and !2 < !1 . The
frequency, if in the visible region, will become redder: this is the phenomenon
of gravitational red-shift.
3.40
This e ect provides one of the three \classical" tests of General Relativity. The others will be seen later: they are the precession of the planets
perihelia (x 4.12) and the light ray deviation (x 4.13).
x
3.6.2 Space
x 3.41 In Special Relativity, it is enough to put dx0 = 0 in the interval ds to
get the in nitesimal space distance dl. The relationship between the proper
time and the coordinate time is the same at every point. This is no more
the case in General Relativity. The standard procedure to obtain the space
interval runs as follows (see Figure 3.2). Consider two close points in space,
P = (x ) and Q = (x + dx ). Suppose Q sends a light signal to P , at which
there is a mirror which sends it back through the same space path. With
time measured at Q, the distance between Q and P will be dl = c=2. The
interval, which vanishes for a light signal, is given by
ds2 = gij dxi dxj + 2 g0j dx0 dxj + g00 dx0dx0 = 0 :
(3.65)
Solving this second-order polynomial for dx0 , we nd the two values
q
1
j
g0j dx
(g0i g0j gij g00 )dxi dxj ;
=
g00
q
1
0
j
dx(2) =
g0j dx + (g0i g0j gij g00 )dxi dxj :
g00
The interval in coordinate time from the emission to the reception of the
signal at Q will be
2 q
dx0(2) dx0(1) =
(g0i g0j gij g00 )dxi dxj :
g00
dx0(1)
92
red
shift
The corresponding proper time is obtained by (3.59):
d =
2
p
c g00
q
(g0i g0j
gij g00)dxi dxj :
The space interval dl = c=2 is then
dl =
s
g0i g0j
g00
gij dxi dxj ;
or
dl2 =
i dxj
ij dx
; with
ij
gij +
=
g0i g0j
:
g00
(3.66)
This is the space metric. Curiously enough, the inverse is simpler: it so
happens that
ij
=
g ij :
This can be seen by contracting g ij with
(3.67)
jk .
x0 dx02
x0 + ∆ x0
x0
x0 dx01
P
Q
Figure 3.2: Q sends a light sign towards a mirror at P , which sends it back.
3.42
Finite distances in space have no meaning in the general case, in
R
which the metric is time-dependent. If we integrate dl and take the in mum
(as explained in x 2.45), the result will depend on the world-lines. Only
constant gravitational elds allow nite space distances to be de ned.
x
93
3.43
When are two events simultaneous ? In the case of Figure 3.2, the
instant x0 of P should be simultaneous to the instant of Q which is just in
the middle between the emission and the reception of the signal, which is
x
x0 + x0 = x0 + 21 [dx0(2) + dx0(1) ] :
Thus, the di erence between the coordinate times of two simultaneous but
spatially distinct events is
g0i i
dx :
x0 =
g00
Time ows di erently in di erent points of space. This relation allows one to
synchonize the clocks in a small region. Actually, that can be progressively
done along any open line. But not along a closed line: if we go along a closed
curve, we arrive back at the starting point with a non-vanishing x0. This
is not a property of the eld, but of the general frame. For any given eld, it
is possible to choose a system of coordinates in which the three components
g0i vanish. In that system, it is possible to synchonize the clocks.
x
3.44 Suppose that a reference system exists in which
synchronous
system
1. the synchronization condition holds:
g0i = 0
(3.68)
g00 = 1:
(3.69)
2. and further
Then, the coordinate time x0 = ct in that system will represent proper time
in all points covered by the system. That system is called a synchronous
system. The interval will, in such a system, have the form
ds2 = c2dt2
i j
ij dx dx
(3.70)
and
ij
=
94
gij :
(3.71)
Direct calculation shows that 00 = 0 in such a coordinate system. This
has an important consequence: the four-velocity u = dxds , tangent to the
world-line xi = 0 (i = 1, 2, 3)) has components (u0 = 1; ui = 0) and satis es
the geodesic equation. Such geodesics are orthogonal to the surfaces ct =
constant. In this way it is possible to conceive spacetime as a direct product
of space (the above surfaces) and time. This system is not unique: any
transformation preserving the time coordinate, or simple changing its origin,
will give another synchronous system.
3.7 Equivalence, Once Again
Æ
3.45 Consider a Levi-Civita connection de ned on a manifold M and
a point P 2 M . Without loss of generality, we can choose around P a
coordinate system fx g such that x (P ) = 0. Such a system will cover a
x
neighbourhood N of P (its coordinate neighbourhood), and will provide dual
holonomic bases f @[email protected] ; dx g, with dx ( @[email protected] ) = Æ, for vector elds and 1-forms
(covector elds) on N . Any other base fea; ea such that ea (eb) = Æba g, will
be given by the components of its members in terms of such initial holonomic
bases, as fea = ea (x) @[email protected] ; ea = ea (x)dx g. Take another coordinate system
fyag around P , with a coordinate neighborhood N 00 such that P 2 N \
N 00 6= ;. This second coordinate system will de ne another holonomic base
@
a = dy a = @ya dx g. It will be enough for our purposes
fea = @[email protected] = @x
;
e
a
@y @x
@x
to consider inside the intersection N \ N 00 a non-empty sub{domain N 0 , small
enough to ensure that only terms up to rst order in the x 's can be retained
in the calculations.
Æ
Let us indicate by (x) the components of in the rst holonomic
Æ
base, and by a b (x) the components of referred to base f @[email protected] ; dy a g. These
components will be related by
Æ
(x) =
@y b @x @ @y c
@x Æ a
(
x
)
+
;
@y a b @x @y c @x @x
(3.72)
@y a Æ @x @y a @ @x
(
x
)
+
:
@x
@y b @x @x @y b
(3.73)
or
Æa
b (x) =
95
Æ
Let us indicate by = (P ) the value of the connection components
at the point P in the rst holonomic system. On a small enough domain N 0
the connection components will be approximated by
Æ
(x)
=
+ x [@
Æ
]P
to rst order in the coordinates x . Choose now the second system of coordinates
y a = Æa x + 12 Æa
x x :
Then,
@y a
a + Æ a x ; @x = Æ =
Æ
a
@x
@y a
Taken into (3.73), these expressions lead to
Æa
b (x)
h
= ÆaÆb @
Æ
(P )
(3.74)
Æa
c
Æc y
i
:
x :
(3.75)
We see that at the point P , which means fx = 0g, the connection compoÆ
nents in base [email protected] vanish: ab (P ) = 0. The curvature tensor at P , however,
is not zero:
Æ
R (P ) = @
Æ
(P )
@
Æ
(P ) +
:
It is thus possible to make the connection to vanish at each point by a suitable choice of coordinate system. The equation of force, or the geodesic
equation, acquire the expressions they have in Special Relativity. The curvature, nevertheless, as a real tensor, cannot be made to vanish by a choice
of coordinates.
Furthermore, we have from Eq.(3.74)
d a a y = Æ U + ( ) U x ;
ds
d2 a a d d a
y = Æ
U + ( ) U U + Æ ( ) x
U :
ds2
ds
ds
Then, at P ,
dy a
= Æa U (P ) ;
ds
96
d2 y a
= ÆaA :
2
ds
Suppose a self{parallel curve goes through P in N 0 with velocity U . Then,
a
2
at P , dyds = Æa U (P ) = constant = U a (P ) and dsd 2 y a = 0. As by Eq.(3.74)
y a(P ) = 0, this gives for the geodesic
y a = U a (P ) s
in some neighborhood around P . The geodesic is, in this system, a straight
line.
3.8 More About Curves
3.8.1 Geodesic Deviation
x 3.46 Curvature can be revealed by the study of two nearby geodesics. Let
us take again Eq. (3.1), rewritten in the form
U V
;
=V U
(3.76)
;
The deviation between two neighboring geodesics in Figure 3.1 is measured by the vector parameter = V ÆV , with ÆV a constant. It gives
the di erence between two points (such as a1 and b1 in that Figure) with
the same value of the parameter u. Or, if we prefer, relates two geodesics
corresponding to V and V + ÆV .
Let us examine now the second-order derivative
D2 V
= (U V ; ); U :
Du2
Use of Eq.(3.76) allows to write
D2 V
= (V U ; ); U = V ; U U ; + V U
Du2
=U ; V U ; +V U ; ; U ;
; ;
U
where once again use has been made of Eq.(3.76). Now, Eq.(2.55) leads to
D2 V
= (U ; V U
Du2
;
+V U U
97
; ;
)
Æ
V U R
U
geodesic
system
= V (U
;
Æ
U ); + R
U U V :
The rst term on the right-hand side vanishes by the geodesic equation. We
have thus, for the parameter ,
Æ
D2 = R U U :
(3.77)
2
Du
This the geodesic deviation equation.
As a heuristic, qualitative guide: test particles tend to close to each other
in regions of positive curvature and to part from each other in regions of
negative curvature.
3.8.2 General Observers
x 3.47 Let us go back to the notion of observer introduced in x 3.11.
An
ideal observer | a timelike curve | will only feel the connection, and that
can be made to vanish along a piece of that curve. Nevertheless, a real
observer will have at least two points, each one following its own timelike
curve. It will consequently feel curvature | that is, the gravitational eld.
3.48 Let us go back to curves, with a mind to observers. Given a curve
,
it is convenient to attach a vector basis at each one of its points. The best
bases are those which are (pseudo-)orthogonal. This is to say that, if the
members have components ha , then
x
g ha hb = ab :
(3.78)
Consider a set of 4 vectors e0; e1; e2; e3 at a point P on , satisfying the
following conditions:
De0
De3
De1
De2
= be3;
= ce1 + be0;
= de2 ce3;
= d e1 : (3.79)
Ds
Ds
Ds
Ds
These are called the Frenet{Serret conditions. The choice of non-vanishing
parameters is such as to allow the 4{vectors to be orthogonal. Actually, these
four vectors are furthermore required to be orthonormal at each point of :
e0 e0 = 1; e1 e1 = e2 e2 = e3 e3 = 1 :
98
(3.80)
We shall always consider timelike curves, with velocity U = e0. In the Frenet{
Serret language, e1 ; e2; e3 will be the rst, second and third normals to at
P . The parameters b; c; d are real numbers, called the rst, second and third
curvatures of at P .
By what we have said above, be3 = A, the acceleration, which is orthogonal to the velocity. The absolute value of the rst curvature of at P is,
thus, the acceleration modulus: jbj = jAj.
For a geodesic, b = c = d = 0. The case b = constant, c = d = 0
corresponds to an hyperbola of constant curvature and the case b = constant,
c =constant, d = 0, to a helix.
3.49
Of course, most curves are not geodesics. A geodesic represents an
observer in the absence of any external force. An observer may be settled on
a linearly accelerated rocket, or turning around Earth, or still going through
a mad spiral trajectory. An orthogonal tetrad de ned as above remains an
orthogonal tetrad under parallel transport, which also preserves the components of a vector in that tetrad. There is, however, a problem with parallel
transport: if we take, as above, e0 as the velocity U at a point P , e0 will not
be the velocity at other points of , unless is a geodesic.
There are other kinds of \transport" which preserve orthogonal tetrads
and components. There is one, in special, which corrects the mentioned
problem with the velocity:
The Fermi-Walker derivative of a vector V is de ned by
DF W V DV DV =
b V (e0 e3 e0 e3 ) =
V (U A U A ): (3.81)
Ds
Ds
Ds
A vector V is said to be Fermi{Walker{transported along a curve of
velocity eld U if its the Fermi-Walker derivative vanishes,
DF W V DV =
V (U A U A ) = 0 :
(3.82)
Ds
Ds
W U = 0, and also that DF W = D if the
We see that, applied to U , DFDs
Ds
Ds
X = 0
curve is a geodesic. Take two vector elds X and Y such that DFW
Ds
Y = 0. Then, it follows that the component of X along Y is
and DFW
Ds
preserved along :
D(X Y ) d(X Y )
=
= 0:
Ds
ds
x
99
Fermi
Walker
transport
In particular, if e0 = U at P , then e0 will remain = U along the curve if it is
Fermi-Walker transported.
3.8.3 Transversality
x 3.50 Given a curve whose
tangent velocity is U , it is interesting to
introduce a \transversal metric" by
h = g
U U :
(3.83)
Transversality is evident: h U = 0. In particular, h A = A . A projector
is an operator P satisfying P 2 = P . Matrices (h ) = (g h ) = (Æ
U U ) are projectors: h h = h . They satisfy h h = h g h =
h g (g U U ) = h . Notice that g h = h h = h = 3. The
energy{momentum tensor (3.36) of a general uid can be rewritten as
T = (p + )U U
p g = U U
p h :
(3.84)
The transversal metric extracts the pressure:
T h =
3 p:
The Einstein equations with this energy{momentum tensor give, by contraction,
4G
( + 3p) :
(3.85)
c4
We have seen in x 3.21 that the streamlines of a general uid | unlike those of
a dust cloud | are not geodesics. The equation of force (3.37) has, actually,
the form
DU (p + )
= h @ p :
(3.86)
Ds
The pressure gradient, as usual, engenders a force. In the relativistic case the
force is always transversal to the curve. Here, it is the transversal gradient
that turns up. The equation above governs the streamlines of a general uid.
R U U =
100
uid
stream
lines
3.8.4 Fundamental Observers
x 3.51 On a pseudo{Riemannian spacetime, there exists always a family of
world{lines which is preferred. They represent the motion of certain preferred
observers, the fundamental observers and the curves themselves are called the
fundamental world{lines. Proper time coincides with the line parameter, so
that he 4-velocities along these lines are U = dxds and, consequently U 2
= U U = 1. The time derivative of a tensor T :::::: is dsd T :::::: =
d :::
dx
:::
::: ds = (T
::: ); U . This is not covariant. The covariant time
dx T
derivative, absolute derivative, is
D :::
T ::: = (T :::::: ); u :
Ds
For example, the acceleration is
D U = U ; U :
A =
Ds
Using the Christo el connection, it is easily seen that U A = 0. This property is analogous to that found in Minkowski space, but here only has an
invariant sense if acceleration is covariantly de ned, as above.
At each point P , under a condition given below, a fundamental observer
has a 3-dimensional space which it can consider to be \hers/his own": its
rest{space. Such a space is tangent to the pseudo{Riemannian spacetime
and, as time runs along the fundamental world{line, orthogonal to that line
at P (orthogonal to a line means orthogonal to its tangent vector, here U ).
At each point of a world{line, that 3-space is determined by the projectors
h .
It is convenient to introduce the notations
U(; ) = 12 (U; + U ; ) ; U[; ] =
1
2
(U;
U ; )
for the symmetric and antisymmetric parts of U; . There are a few important
notions to be introduced:
the vorticity tensor
! = h h U[;] = U[; ] + U[ A ] ;
(3.87)
it satis es ! = ![ ] = - ! and ! U = 0; it is frequently indicated
by its magnitude ! 2 = 12 ! ! 0.
101
the expansion tensor
= h h U(;) = U(; )
U( A ) ;
(3.88)
its transversal trace is called the volume expansion; it is = h =
= U ; , the covariant divergence of the velocity eld; it measures
the spread of nearby lines, thereby recovering the original meaning of
the word divergence; in the Friedmann model, turns up as related to
the Hubble expansion function by = 3H (t).
= 31 h = ( ) is the symmetric trace{free shear tensor;
it satis es u = 0 and = 0 and its magnitude is de ned as
2 = 21 0. Notice = 2 2 + 31 2 .
Decomposing the covariant derivative of the 4-velocity into its symmetric
and antisymmetric parts, U; = U(;) + U[;] , and using the de nitions
(3.87) and (3.88), we nd
U; = ! + + A U ;
(3.89)
U; = ! + + 13 h + A U :
(3.90)
or
x
3.52 With the above characterizations of the energy density and the pres-
sure, the Einstein equations reduce to the Landau{Raychaudhury equation.
Let us go back to Eq.(2.55) and take its contracted version
U
U
; ;
Æ
R U :
=
; ;
Contracting now with U ,
U U
; ;
U U
; ;
(3.91)
Æ
R U U :
=
Æ
D
U ; (U U ; ); + U ; U ; + R U U = 0 ;
Ds
Æ
d
)
U ; A ; +U ; U ; +R U U =0 :
ds
To obtain U ; U ; , we notice that
)
=
1 U
;
2
U
;
+U
102
;
U
;
AA
;
Landau
Raychaudhury
equation
! ! =
1 U
;
2
U
;
U
;
U
;
A A
:
It follows that
U
;
U
;
= 1
! ! = 2 2 2 ! 2 + 2 :
3
The equation acquires the aspect
Æ
1
d
+ 2 ( 2 ! 2 ) + 2 A ; + R U U = 0 :
(3.92)
ds
3
Up to this point, only de nitions have been used. Einstein's equations lead,
however, to Eq.(3.85), which allows us to put the above expression into the
form
d
4G
= 13 2 2 2 ! 2 + A;
( + 3p) + :
(3.93)
ds
c4
The promised condition comes from a detailed examination which shows
that, actually, only when ! = 0 there exists a family of 3-spaces everywhere
orthogonal to U . In that case, there is a well{de ned time which is the same
over each 3-space.
From the above equation, we can see the e ect of each quantity on expansion: as we proceed along a fundamental world{line, expansion
decreases (an indication of attraction) with higher values of
expansion itself
shear
energy content
increases (an indication of repulsion) with higher values of
vorticity
second-acceleration
cosmological constant .
103
3.9 An Aside: Hamilton-Jacobi
3.53
The action principle we have been using [say, as in Eq.(1.5), or (3.51)]
has a \teleological" character which brings forth a causal problem. When we
look for the curve which minimizes the functional
x
S[ ] =
Z t1
t0
Ldt =
Z
PQ
Ldt ;
we seem to suppose that the behavior of a particle, starting from a point P
at instant t0, is somehow determined by its future, which forcibly consists
in being at a xed point Q at instant t1. Another notion of action exists
which avoids this diÆculty. Instead of as a functional S is conceived, in that
version, as a function
S (q1(t); q2(t); :::; qn(t); t) =
Z t
t0
dt0 L [q1(t0); q2 (t0); :::; qn(t0 ); q_1(t0); q_2(t0); :::; q_n(t0)]
(3.94)
of the nal time t and the values of the generalized coordinates at that instant
for the real trajectory. The particle, by satisfying the Lagrange equation at
each point of its path, automatically minimizes S . In e ect, taking the
variation
ÆS =
Z t
@L i @L i
@L i @ @L i
0
0
dt
Æq + i Æ q_ = dt
Æq + 0
Æq
@q i
@ q_
@q i
@t @ q_i
t0
t0
Z t
Z
@ @L
Æq i
@t0 @ q_i
t
@L i t
@L
@ @L
=
Æq + dt0
Æq i :
i
i
0
i
@ q_
@q
@t @ q_
t0
t0
The second term vanishes by Lagrange's equation. In the rst term, Æq i(t0 ) =
0, so that
@L
ÆS = i Æq i = pi Æq i ;
(3.95)
@ q_
which entails
@S
pi = i :
(3.96)
@q
104
We have been forgetting the time dependence of S . The integral (3.94) says
that L = dS
dt . On the other hand,
dS @S @S i @S
=
+ i q_ =
+ pi q_i:
dt @t @q
@t
Consequently,
@S
= L pi q_i =
@t
Thus, the total di erential of S will be
dS = pi dq i
Variation of the action integral
Z
S=
H:
(3.97)
Hdt :
pi dq i
Hdt
(3.98)
(3.99)
leads indeed to Hamilton's equations:
ÆS =
Z Æpi
dq i + p
i
dÆq i
@H i
Æq dt
@q i
Z @H
Æp dt
@pi i
dq i
dÆq i @H i @H
=
Æpi + pi
Æq
Æp dt
dt
dt
@q i
@pi i
i
Z dpi @H
dq @H
i
+
Æq dt:
=
Æpi
dt @pi
dt @q i
An integration by parts was performed to arrive at the last espression which,
to produce ÆS = 0 for arbitrary Æq i and Æpi , enforces
dq i @H dpi
=
;
=
dt @pi
dt
x
@H
:
@q i
(3.100)
3.54 Hamilton's equations are invariant under canonical transformations
leading to new variables Qi = Qi(q i ; pj ; t), Pi = Pi (q i ; pj ; t), H 0 (Qi ; Pj ; t).
This means that if
Z
Æ pi dq i Hdt = 0 ;
then also
Æ
Z
Pi dQi
105
H 0 dt = 0
must hold. In consequence, the two integrals must di er by the total di erential of an arbitrary function:
pi dq i
Hdt = Pi dQi
H 0 dt + dF :
F is the generating function of the canonical transformation. It is such that
dF = pi dq i
Pi dQi + (H 0
H )dt :
Therefore,
pi =
@F
; Pi =
@q i
@F
@F
; H0 = H +
:
i
@Q
@t
(3.101)
In the formulas above, the generating function appears as a function of the
old and the new generalized coordinates, F = F (q; Q; t). We obtain another
generating function f = f (q; P; t), with the new momenta instead of the new
generalized coordinates, by a Legendre transformation: f = F + Qi Pi , for
which
df = dF + dQi Pi + Qi dPi = pi dq i + Qi dPi + (H 0
H )dt :
In this case,
@f
@f
@f
; Qi =
; H0 = H +
:
(3.102)
i
@q
@Pi
@t
Other generating functions, related to other choices of arguments, can of
P
course be chosen. For instance, the function g = i q iQi generates a simple
interchange of the initial coordinates and momenta.
pi =
x
3.55 Let us go back to Eq. (3.97),
@S
+ H (q 1; q 2; :::; q n; p1 ; p2 ; :::; pn; t) = 0 :
(3.103)
@t
By Eq. (3.96), the momenta are the gradients of the action function S .
Substituting their expressions in the above formula, we nd a rst order
partial di erential equation for S ,
@S
@S @S
@S
+ H q 1; q 2; :::; q n; 1 ; 2 ; : : : ; n ; t = 0 :
@t
@q @q
@q
106
(3.104)
This is the Hamilton-Jacobi equation. The general solution of such an equation depends on an arbitrary function. The solution which is important for
Mechanics is not the general solution, but the so-called complete solution
(from which, by the way, the general solution can be recovered). That solution contains one arbitrary constant for each independent variable, (n+1) in
the case above. Notice that only derivatives of S appear in the equation. One
of the constants (C below) turns up, consequently, isolated. The complete
solution has the form
S = f q 1; q 2; :::; q n; a1; a2; : : : ; an ; t + C:
(3.105)
We have indicated the arbitrary constants by a1 ; a2; : : : ; an and C .
The connection to the mechanical problem is made as follows. Consider
a canonical transformation with generating function f , taking the original
variables (q 1; q 2; :::; q n; p1 ; p2 ; :::; pn) into (Q1; Q2; :::; Qn; a1; a2; :::; an). This is
a transformation of the type summarized in Eq. (3.102), with a1; a2; :::; an as
the new momenta. From those equations and (3.105),
pi =
@S
@S
@S
; Qi =
; H0 = H +
=0:
i
@q
@Pi
@t
(3.106)
To get the vanishing of the last expression use has been made of Eq. (3.104).
Hamilton equations havee then the solutions Qn = constant, ak = constant.
@S it is possible to obtain back
From the equations Qi = @a
i
q k = q k Q1; Q2 ; :::; Qn; a1; a2; : : : ; an ; t ;
that is, the old coordinates written in terms of 2n constants and the time.
This is the solution of the equation of motion. Summing up, the procedure
runs as follows:
given the Hamiltonian, one looks for the complete solution (3.105) of
the Hamilton-Jacobi equation (3.104);
once the solution S is obtained, one derives with respect to the constants ak and equate the results to the new constants Qk ; the equations
@S are algebraic;
Qi = @a
i
107
that set of algebraic equations are then solved to give the coordinates
q k (t);
the momenta are then found by using pi = @[email protected] .
x 3.56 For conservative systems, H is time-independent.
pends on time in the form
The action de-
S (q; t) = S (q; 0) E t:
It follows that
@S (q; 0)
q 1; q 2; :::; q n;
;
@q 1
@S (q; 0)
@S (q; 0)
H
; :::;
=E ;
(3.107)
2
@q
@q n
which is the time-independent Hamilton-Jacobi equation.
The same happens whenever some integral of motion is known from the
start. Each constant of motion is introduced as one of the constants. For
instance, central potentials, for which the angular momentum J is a constant,
will lead to a form
S (r; t) = S (r; 0) Et + J:
These are particular cases, in which time or an angle are cyclic variables.
In e ect, suppose some coordinate q(c) is cyclic. This means that q(c) does
not appear explicitly in the Hamiltonian nor, consequently in the HamiltonJacobi equation. The corresponding momentum is therefore constant, p(c)
= @[email protected](c) = a(c). It is an integral of motion, and S = S (remaining variables)
+a(c)q(c).
3.57
Of course, a suitable choice of coordinate system is essential to isolate
a cyclic variable. For a particle in a central potential, spherical coordinates
are the obvious choice. In the Hamiltonian
p2
1 2 p2
+ U (r) ;
H=
p + +
2m r r2 r2 sin2 the variables and , besides t, are absent. We shall use the knowledge that
the angular momentum J = mr2 _ is a constant, and start from the simpler
planar Hamiltonian
m h 2 2 _ 2i
p2
J2
H=
r_ + r + U (r) = r +
+ U (r) :
2
2m 2mr2
x
108
The time-independent Hamilton-Jacobi equation is then
@Sr
@r
2
J2
= 2m(E
r2
+
Thus,
S=
Now,
@S
@E
E t+J +
@S
@J
r
Z
dr 2m(E
J2
:
r2
U (r))
= C gives
t=
And
U (r)):
Z
mdr
2m[E U (r)]
q
C:
J2
r2
= C 0 gives
= C0 +
Z
r2
J dr
2m[E U (r)]
q
J2
r2
:
The constants can be chosen = 0, xing simply the origins of time and angle.
The rst equation,
t=
Z
m dr
2m[E U (r)]
q
J2
r2
;
(3.108)
gives implicitly r(t). The second,
=
Z
r2
J dr
2m[E U (r)]
q
J2
r2
;
(3.109)
gives the trajectory.
We see that what is actually at work is an e ective potential Ueff (r) =
J 2 , including the angular momentum term. The values of r for
U (r) + 2mr
2
which Ueff (r) = E represent \turning points". If the function r(t) is at rst
decreasing, it becomes increasing at that value, and vice versa.
x
3.58 Classical planetary motion There are two general kinds of motion:
limited (bound motion) and unlimited (scattering). We shall be concerned
here only with the rst case, in which r(t) has a nite range rmin r(t) 109
rmax. In one turn, that is, in the time the variable takes to vary from rmin
to rmax and back to rmin , the angle undergoes a change
= 2
Z rmax
rmin
J
r2
dr
2m[E U (r)]
q
J2
r2
:
(3.110)
The trajectory will be closed if = 2m=n. A theorem (Bertrand's) says
that this can happen only for two potentials, the Kepler potential U (r) =
K=r and the harmonic oscillator potential U (r) = Kr2 . We shall here limit
ourselves to the rst case, which describes the keplerian motion of planets
around the Sun. For U (r) = K=r, Eq.(3.109) can be integrated to give
= arccos
8
<
:
J
r
mK
J
1
q
9
=
2mE + mJ2 K2 2 ;
:
(3.111)
This trajectory corresponds to a closed ellipse. In e ect,
q introduce the \el2
J
EJ 2 . Equation
lipse parameter" p = mK and the \eccentricity" e = 1 + 2mK
2
(3.111) can then be put into the form
p
r=
;
(3.112)
1 + e cos which is the equation for the ellipse. The above choice of the integration
constants corresponds to = 0 at r = rmin , which is the orbit perihelium.
Suppose we add another potential (for example, U 0 = K 0=r3 , with K 0 small.
The orbit, as said above, will be no more closed. Staring from = 0 at
r = rmin , the orbit will reach the value r = rmin at 6= 0 in the rst
turn, and so on at each turn. The perihelium will change at each turn.
This efect (called the perihelium precession) could come, for example, from
a non-spherical form of the Sun. A turning gas sphere can be expected to be
oblate. Observations of the Sun tend to imply that its oblateness is negligible,
in any case insuÆcient to answer for the observed precession of Mercury's
perihelium. We shall see later (x4.12) that General Relativity predicts a value
in good agreement with observations.
2
recall that the equation of the ellipse in cartesian coordinates is xa2 +
pa2 We
= 1, e = a b2 and p = b2=a. For a circle, a = b and e = 0.
Comment 3.7
y2
b2
110
3.59
We have already seen the main interest of the Hamilton-Jacobi formalism: in the relativistic case, the Hamilton-Jacobi equation (3.17) for a
free particle coincides, for vanishing mass, with the eikonal equation (3.18).
The formalism allows a uni ed treatment of test particles and light rays.
x
111
Chapter 4
Solutions
Einstein's equations are a nightmare for the searcher of solutions: a system of ten coupled non-linear partial di erential equations. Its a tribute to
human ingenuity that many (almost thirty to present time) solutions have
been found. The non-linear character can be interpreted as saying that the
gravitational eld is able to engender itself. The equations have non-trivial
solutions even in the absence of sources. Actually, most known solutions are
of this kind. We shall only examine a few examples, divided into two categories: \small scale solutions", which are of \local" interest, idealized models
for stars (which gives an idea of what we mean by \small") and objects alike;
and \large scale solutions", of cosmological interest.
4.1 Transformations
In tackling the big task of solving so diÆcult a problem, it is not surprising
that the hunters have always supposed a high degree of symmetry. Let us
begin with a short comment on symmetries of spacetimes.
4.1 Let us look for the condition for a vector
eld to generate a symmetry
of the metric. Consider an in nitesimal transformation
x
x0 = x + " (x) ; " (x) << 1 ;
(4.1)
arbitrary as longs as " is an arbitrary, though small, function of x. To the
A treatment of the subject is given in the chapter 13 of S. Weinberg, Gravitation and
Cosmology, J. Wiley, New York, 1972.
112
rst order in ",
@x0
+ @" :
=
Æ
@x
@x
Under such a tranformation, the metric components will change according
to
@x0 @x0
@"
@"
0
0
g (x ) = g (x) = g (x) Æ + Æ + @x @x
@x
@x
@"
@"
g (x) + g (x) @x
+ g (x) @x :
On the other hand, always keeping only the rst-order terms,
g 0 (x0) = g 0 (x + ") g 0 (x) + " (x)@ g 0 (x) g 0 (x) + " (x)@g (x) :
Equating both expressions,
@"
(x) @" ;
+
g
@x
@x
from which we obtain the variation of the metric components at a xed point,
(x) @g :
(x) = g 0 (x) g (x) = g (x) @" + g (x) @"
Æg
"
@x
@x
@x
(4.2)
g 0 (x) + " (x)@g (x) = g (x) + g (x)
We now calculate the covariant derivative of " , conveniently separating
the pieces comming from the Christo el symbol:
"; = @ "
1 2 " @ g
+ 21 " (@ g @ g ) :
We see then that
"; + " ; = @ " + @ "
" @ g :
(4.3)
Therefore,
(x) = "; + " ; :
Æg
(4.4)
This gives the change in the functional form of g under the transformation
(x) = 0,
generated by the eld " = " @ . The condition for a symmetry is Æg
that is,
"; + " ; = 0 :
113
(4.5)
Killing
equation
This is the Killing equation. Fields satisfying it are called Killing elds. They
generate transformations preserving the metric, which are called isometries,
or motions. Applied to the Lorentz metric, the ten generators of the Poincare
group are found.y
x
4.2 We shall here quote three theorems:
the rst says that the maximal number of isometries in a d-dimensional
space is d(d + 1)=2.z Consequently, a given spacetime has at most 10
isometries.
the second theorem says that this maximal number can only be attained
if the scalar curvature R is a constant. There are only three kinds of
spacetimes with 10 isometries: Minkowski spacetime, for which R = 0,
and the two families of de Sitter spacetimes, one with R > 0 and the
other with R < 0.
a third theorem states that the isometries of a given metric constitute a
group (group of isometries, or group of motions). The Poincare group
is the group of motions of Minkowski space.
4.3
The converse procedure may be useful in the search of solutions: impose a certain symmetry from the start, and nd the metrics satisfying
Eq.(4.3) for the case,
x
" @ g = @ " + @ " :
(4.6)
It should be said, however, that the Killing equation is still more useful in
the study of the simmetries of a metric given a priori.
4.4
The above procedure is a very particular case of a general and powerful method. How does a transformation acts on a manifold ? We are used
to translations and rotations in Euclidean space. The same transformations,
plus boosts and time translations, are at work on Minkowski space: they
preserve the Lorentz metric. For these we use generators like, for example,
x
y See for example W.R. Davis & G.H. Katzin, Am. J. Phys. 30 (1962) 750.
z L. P. Eisenhart, Riemannian Geometry, Princeton University Press, 1949.
114
L = x @ x @ , which is a vector eld. This can be extended to general
manifolds: given a group of transformations, each generator is represented
on the manifold by a vector eld X . This vector eld presides on the innitesimal transformations undergone by every tensor eld by the so called
Lie derivative, an operation denoted LX . The calculation above must be
repeated for each type of tensor T : take a transformation like (4.1), compare
T 0(x0) obtained from the tensor behavior with T 0(x0 ) obtained as a Taylor
series, etc. The general result is
ab:::r
(LX T )ab:::r
ef:::s = X (Tef:::s )
ib:::r
(@i X a )Tef:::s
ai:::r
(@i X b )Tef:::s
Lie
derivative
ab:::i
::: (@iX r )Tef:::s
ab:::r + (@ X i )T ab:::r + ::: + (@ X i )T ab:::r :
+(@e X i )Tif:::s
f
s
ei:::s
ef:::i
(4.7)
The requirement of invariance is LX T = 0. For T a vector eld, LX T is
just the commutator:
LX V = [X; V ] :
The vector V is invariant with respect to the transformations engendered by
X if it commutes with X .
Equation (4.2) is just the Lie derivative of g with respect to the eld "
= " @ :
@"
@"
@g L" g (x) = g 0 (x) g (x) = g (x) + g (x) " (x) :
@x
@x
@x
(4.8)
We shall not go into the subject in general.x Let us only state a property
which holds when the tensor T is a di erential form. For that we need a
preliminary notion.
x
4.5 Given a vector
(p-1)-form iX
eld X, the interior product of a p-form by X is that
, which, for any set of elds fX 1 ; X 2; : : : ; X p 1 g, satis es
iX (X 1 ; X 2 ; : : : ; X p 1 ) = (X; X 1 ; X 2 ; : : : ; X p 1 ):
If
(4.9)
is a 1-form, it gives simply its action on X :
iX = < ; X > = (X ):
x A very detailed account can be found in B. Schutz,Geometrical Methods of Mathematical Physics, Cambridge University Press, Cambridge, 1985.
115
interior
product
The interior product of X by a 2-form is that 1-form satisfying iX =
(X; Y ) for any eld Y . For a form of general degree, it is enough to know
that, for a basis element,
iX
1
^ 2 ^ :::^
p =
p
X
j =1
( )j
1 1
^ 2 ^ : : : [iX
j]
^ :::^
p:
4.6 The promised result is as follows:
if ! is a di erential form, then its
Lie derivative has a simple expression in terms of the exterior derivative and
the interior product:
LX ! = d[iX ! ] + iX [d! ] :
x
Notice that LX preserves the tensor character: it takes an r-covariant, scontravariant tensor into another tensor of the same type.
4.2 Small Scale Solutions
Life is much simpler when a system of coordinates can be chosen so that
invariance means just independence of some of the coordinates. In that case,
Eq.(4.8) reduces to the last term and the intuitive property holds: the metric
components are independent of those variables.
4.2.1 The Schwarzschild Solution
x 4.7 Suppose we look for a solution of the Einstein equations which has
spherical symmetry in the space section. This would correspond to central
potentials in Classical Mechanics. It is better, in that case, to use spherical
coordinates (x0; x1; x2 ; x3) = (ct; r; ; ). This is one of the most studied of
all solutions, and there is a standard notation for it. The interval is written
in the form
ds2 = e c2dt2
r2 (d2 + sin2 d2 ) edr2 :
The contravariant metric is consequently
g=
0 e
B 0
(g ) = B
B
@ 0
0
0
e
0
0
116
0
0
r2
0
0
0
0
2
r sin2 (4.10)
1
C
C
C
A
(4.11)
and its covariant counterpart,
0
g
B
e
= (g ) = B
B
1
@
0
e
0
0
0
0
0
0
0
r
0
0
0
0
2
r sin 2 2
1
C
C
C
A
:
(4.12)
We have now to build Einstein's equations. The rst step is to calculate
the components of the Levi-Civita connection, given by Eq.(2.40). Those
which are non-vanishing are:
0
00
1
00
=
1 d
2 cdt
= 12 e
1
2
3
0
;
d
dr
22
10
1
;
=
2
13
=
3
01
1 d
2 dr
01
=
=
1
10
=
0
;
1 d
2 cdt
11
= 21 e
;
1
11
;
1
33
= re sin2 ;
21
= r1 ;
2
33
=
31
= 1r ;
3
23
=
= re
12
0
=
=
d
cdt
;
1 d
2 dr
;
sin cos ;
3
32
= cot :
(4.13)
As the second step, we must calculate the Ricci tensor of Eq.(2.48) and
the Einstein tensor (2.59). We list those which are non-vanishing:
G0 0 =
1
r2
G2
2
e
1
r2
1 d r dr
=
G3
1
2
e
e
3
=
+ 12
; G1 0 =
h
d2 dr2
d2 c2 dt2
1
r
e
d
cdt
; G1 1 =
2 1 d
+ 21 ( d
dr ) + r ( dr
+
1
2
d )2
( cdt
1 d d
2 cdt cdt
d )
dr
:
1
r2
1 d
r dr
i
1 ( d d )
2 dr dr
e
+ r12
(4.14)
Each G should now be imposed to be equal to 8cG
4 T . The source could
be, for example, an electromagnetic eld, in which case Eq.(3.35) would be
used. Or the uid inside a star, with the source given by Eq.(3.36) supplemented by an equation of state. Notice that the symmetry requirements
made above would be satis ed not only by a static star, but also by a radially
117
pulsating one. We shall here consider the so-called \external", or \vacuum"
solution for this case. We shall put T = 0, which is the case outside the
star. The four di erential equations following from G = 0 in (4.14) reduce
in that case to only three (see Comment 4.1 below):
G0 0 = 0
)
e
1
r2
G1 1 = 0
)
e
1
r2
+ 1r
)
d
dt
G1 0 = 0
1 d = 1
r dr
r2
;
d = 1
dr
r2
;
(4.15)
=0:
A rst result from Eqs.(4.15) is that is time-independent. Taking the
di erence between the rst two equations shows that
d
d
=
:
dr
dr
Substituting this back in those equations lead to
(4.16)
d
:
(4.17)
dr
Equation (4.16) says that + is independent of r, and is consequently
a function of time alone: + = f (t). In the interval (4.10), it is always
possible to rede ne the time by an arbitrary transformation t = (t0), which
corresponds to adding an arbitrary function of t to . The choice of a new
R
time coordinate t0 = 0t e f (t)=2dt corresponds to changing ! 0 = + f (t).
This means that it is always possible to choose the time coordinate so as to
have + = 0.
e = 1 + r
Time-independence of entails the vanishing of the last line in (4.14).
Using (4.16) in (4.17) and taking the derivative implies that also the one-but-last line
vanishes. This shows that the equation G22 = G33 = 0 is indeed redundant.
Comment 4.1
Integration of the only remaining equation, which is (4.17) rewritten with
= ,
d
e =1+r
;
dr
leads then to
RS
e = e = 1
;
r
118
where RS is a constant. Far from the source, when r ! 1, we have e =
e ! 1, so that the metric reduces to that of Minkowski space. Large values
of r means a weak gravitational eld. To x the constant RS , it is enough
to impose that, at those values of r the solution reduce to the newtonian
approximation, g00 = 1 + 2V=c2 , with V = GM=r and M the mass of
source body. It follows that
RS =
2GM
:
c2
(4.18)
The interval (4.10) is therefore
ds2
= 1
= 1
2GM 2 2
c dt
c2 r
RS 2 2
c dt
r
r2 (d2 + sin2 d2 )
r2 (d2 + sin2 d2 )
1
dr2
(4.19)
2GM
c2 r
1
dr2
RS
r
:
(4.20)
This is the solution found by K. Schwarzschild in 1916, soon after Eintein
had presented his nal version of General Relativity. It describes the eld
caused, outside it, by a symmetrically spherical source. We see that there is
a singularity in the metric components at the value r = RS . The parameter
RS , given in Eq.(4.18), is called the Schwarzschild radius. Its value for a
body with the mass of the Sun would be RS 3 km. For a body with
Earth's mass, RS 0:9 cm. For such objects, of course, there exists to real
Schwarzschild radius. It would be well inside their matter distribution, where
T 6= 0 and the solution is not valid.
x
4.8 The above solution has been obtained in the absence of the cosmolog-
ical constant. Its presence would change it to{
ds2
= 1
2GM
c2 r
2 2 2
r c dt
3
r2 (d2 + sin2 d2 )
1
dr2
2GM
c2 r
r2 :
3
(4.21)
{ See the x96 of R.C. Tolman, Relativity, Thermodynamics and Cosmology, Dover, New
York, 1987.
119
If we compare with Eq.(3.53), we nd the potential
MG
r
V =
c2 r2
:
6
(4.22)
Eq.(3.58) would then lead to
d
v=
dt
We recognize Newton's law in the
cosmological term has the aspect
produces a repulsive force.
MG 1 2
(4.23)
+ 3 c r :
r2
rst term of the righ-hand side. The extra,
of a harmonic oscillator but, for > 0,
4.9
The eld, just as in the newtonian case, depends only on the mass
M . At a large distance of any limited source, the eld will forget details
on its form and tend to have a spherical symmetry. The interval above is
approximately given, at larges distances, by
RS 2 2 2 ds2 c2 dt2 dr2 r2 (d2 + sin2 d2 )
dr + c dt :
(4.24)
r
The last term is a correction to the Lorentz metric and the above interval
should be the asymptotic limit, for large values of r, of any eld created by
any source of limited size. We see that the Schwarzschild coordinate system
used in (4.20) is \asymptotically Galilean": Schwarzschild's spacetime tends
to Minkowski spacetime when r ! 1.
As g0j = 0 in Eq.(3.66), the 3-dimensional space sector induced by (4.20)
will have the interval
dr2
+ r2 (d2 + sin2 d2 ) ;
(4.25)
d 2 =
R
S
1
r
x
to be compared with the Euclidean interval
d 2 = dr2 + r2 (d2 + sin2 d2 ) :
(4.26)
At xed and , that is, radially, the distance between two points P and
Q standing outside the Schwarzschild radius will be
Z Q
P
dr
q
1
RS
r
120
> rQ
rP :
(4.27)
On the other hand, the proper time will be
r
p
d = g
00
dt =
RS
dt < dt :
r
1
(4.28)
We see that d = dt when r ! 1. And we see also that, at nite distances
from the source, time \marches slower" than time at in nity. This di erence
between proper time and coordinate time arrives at an extreme case near the
Schwarzschild radius.
x
4.10
We can make some checking on the results found. Given the metric
0
B
B
B
B
@
1
0
0
0
RS
r
0
0
0
r2
0
1
1 RrS
0
0
0
0
0
2
r sin2 1
C
C
C
C
A
;
we can proceed to the laborious computation of the Christo eln and Riemann
components. We nd, for example,
R1 212 =
RS
; R1 313 =
r2 (r RS )
RS
; R1 414 =
2r
RS sin2 2r
(R 2 r) sin2 R sin2 RS
; R2 424 = S
; R3 434 = cos2 + S
:
2r
2r
r
All components of the Ricci tensor vanish, as they should for an exterior,
T = 0 solution. The scalar curvature, of course, vanishes also. This is a
good illustration of the statement made in x 3.25, by which T = 0 implies
R = 0 but not necessarily R = 0. It is also a good example of a
fundamental point of General Relativity: it is the non-vanishing of R that indicates the presence of a gravitational eld.
R2 323 =
4.11
We can, furthermore, examine the space section. With coordinates
(r; ; ), the metric is
x
0
1
0
0
0
r2
0
2
0 r sin2 R
B 1 rS
B
@
121
0
1
C
C
A
:
The Christo eln form the matrices
0
(
1
ij )
=B
@
RS
2r(RS r)
0
2
(
ij )
=B
@
3
ij )
1
r
0
(Rij ) = B
@
0
0
1
r
0
0
r) sin2 (RS
0
0
sin cos 0
0 0
=B
@
The Ricci tensor is given by
0
r
1
r
0
0
(
RS
0
0
0
1
0
r
0 cot cot 0
RS
r2 (RS r)
0
0
0
RS
2r
0
1
C
A
1
C
A
1
C
A
0
0
RS sin2 2r
:
1
C
A
:
Thus, the Ricci tensor of the space sector is non-trivial. The scalar curvature,
however, is zero.
x
4.12 Perihelium precession The Hamilton-Jacobi equation (3.17) and
the eikonal equation (3.18) provide, as we have seen, a uni ed approach to
the trajectories of massive particles and light rays. Let us rst examine the
motion of a particle of mass m in the above gravitational eld. As angular
momentum is conserved, it will be a plane motion, with constant . For
reasons of simplicity, we shall choose the value = =2. With the metric
(4.19), the Hamilton-Jacobi equation g (@ S )(@ S ) = m2c2 acquires the
form
RS
@S 2 1 @S 2
RS 1 @S 2
1
m2 c2 = 0 :
1
r
@ct
r
@r
r2 @
(4.29)
The solution is looked for by the Hamilton-Jacobi method described in Section 3.9. With some constant energy E and constant angular momentum J ,
we write
S=
Et + J + Sr (r):
122
(4.30)
This, once inserted in (4.29), gives
Sr =
Z
(
E2
dr 2 1
c
RS
r
2
J2
m2c2 + 2
c
1
RS
r
1 )1=2
: (4.31)
@S = constant,
By the method, r = r(t) is obtained from the equation @E
from which comes
Z
dr
E
q
(4.32)
ct = 2
:
2
2 mc
R
R
J
E
S
S
1 + m2 c2 r2 1
1
r
mc2
r
The trajectory is found from
=
Z
@S
@J
J
q
r2 E 2
c2
= constant, which gives
dr
m2 c2 + Jr22 1
RS r
:
(4.33)
This leads to an elliptic integral. We are putting the additive integration
constants, which merely x the origins of the coordinates ct and , equal to
zero.
We should compare the above results with their non-relativistic counterparts given in Eqs.(3.108), (3.109). However, in order to calculate the small
corrections given by the theory to the trajectories of the planets turning
around the Sun, it is wiser to make approximations in (4.31) before taking
@S . We shall suppose radial distances very large with respect
the derivative @J
to the Schwarzschild radius: RS << r. We also change the integration varip
able to r0 = r2 rRS (and drop the pirmes afterwards). Writing E 0 for
the non-relativistic energy, we nd
Sr =
Z
E02
1
0
dr 2 + 2mE + 2m2 MG + 4E 0 MRS
c
r
1
J2
2
r
1=2
3 m2c2 R2 S
2
(4.34)
The term in 1=r2 will produce a secular displacement of the orbit perihelium.
The remaining terms cause changes in the relationships between the fourmomentum of the particle and the newtonian ellipse. We shall be interested
r
only in the perihelium precession. The trajectory is determined by + @S
@J
123
= constant. The variation of Sr in one revolution is, in the approximation
considered,
3m2c2 R2S @ Sr
Sr = Sr(0)
:
4J
@J
Sr(0) is the closed ellipse case. The variation of the angle in one revolution
will be
@ Sr
=
:
@J
Taking into account that
@ Sr(0)
= (0) = 2 ;
@J
we nd
6G2 m2M 2
3m2c2 R2S
=
2
+
:
2J 2
c2 J 2
The last piece gives the precession Æ. It is usual to express it in terms of the
ellipse parameters. If the great axis is a and the eccentricity is e, we have
= 2 +
J2
= a(1 e2) :
GMm2
The perihelium precession is then
Æ =
6GM
:
a(1 e2)c2
For the Earth, this is a very small variation: in seconds of arc, 3:800 per
century. For Mercury, it is 43:000 per century. This is in good agreement with
measurements.
x
4.13 Light-ray deviation The eikonal equation (3.18) is just the Hamil-
ton-Jacobi equation (3.17) with m = 0. The trajectory will still be given
by Eq.(4.33). The interpretation is, of course, quite another. S is now the
eikonal, the energy E must be replaced by the light frequency !0 , and it is
convenient to introduce a new constant, the impact parameter = Jc=!0 .
Thus,
=
Z
r2
q
dr
1
2
124
1
r2
1
RS r
:
(4.35)
This gives r = = cos | a straight line passing at a distance of the
coordinate origin | in the non-relativistic case RS = 0. The procedure to
analyse the small corrections due to RS 6= 0 is analogous to that used for
m 6= 0. We go back to Eq. (4.31),
Z
n
1 o1=2
!0
2
2
2
2
Sr (r) =
dr r (r RS )
r
rRS
:
c
With the same transformations used previously, this becomes
Z
p
!0
dr 1 2 r 2 + 2RS r 1 :
Sr (r) =
c
Expanding in powers of RS r 1 ,
Z
dr
RS !0
RS =0 + RS !0 arccosh r :
R
=0
S
p
=
S
Sr Sr
+
r
c
c
r2 2
(4.36)
(4.37)
(4.38)
The deviation undergone by a ray coming from a large distance R down to
a distance and then again to the same distance R will be
R !
r
Sr = SrRS =0 + 2 S 0 arccosh :
(4.39)
c
To get the variation in the angle , it is enough to take the derivative with
respect to J = !0 =c:
@ SrRS =0
R R
(4.40)
+ 2 p S2 2 :
@J
R The term corresponding to the straight line has = . Taking the asymptotic limit R ! 1,
R
(4.41)
= + 2 S :
This gives a deviation towards the centre of an angle
R
4GM
Æ = 2 S =
:
(4.42)
c2
For a light ray grazing the Sun, this gives Æ = 1:7500 . This e ect has been
observed by a team under the leadership of Eddington during the 1919 solar
eclipse, at Sobral. It has been considered the rst positive experimental test
of General Relativity.
=
@ Sr
=
@J
125
x
4.14 The event horizon We can compare the energy of the particle as
seen by an observer using the Schwarzschild coordinate system with the its
energy as seen in the proper frame. The rst is, as previously seen, E = @S
@t ,
p
@S
@
@S
@S
2
and the second is E0 = @ = mc . But @t = @t @ , so that E = g00 mc2,
or
r
E=
mc2
RS
:
r
1
(4.43)
Let us examine the case of a particle falling in purely radial motion ( = 0,
= 0, ) J = 0) towards the center r = q
0. If it starts from a point r0 at
2
the instant t0, its energy will be E = mc 1 Rr0S . At a moment t and a
distance r, Eq.(4.32) gives
c(t t0 ) =
r
Z r0
RS
r0
1
dr0
RS r0
1
r
q
RS
r0
RS
r0
:
(4.44)
This coordinate time diverges when r ! RS . Seen from an external observer,
the particle will take an in nite time to arrive at the Schwarzschild radius.
On the other hand, we can calculate the proper interval of time for the same
thing to happen: it will be
c(
From (4.44),
0 ) =
dt
=
dr
Z r0
r
ds =
r
) g00
r
s
dr0
RS
r0 1
1
c2
Z r0
dt
dr
2
g00 c2
0 ) =
r
2
+ g11 :
dr
RS
r
Thus,
c(
dt
dr0
q
RS RS
r
r
+ g11 =
Z r0
q
1
dr0
RS
r0
RS
r0
RS
r0
:
RS
r0
:
(4.45)
This is a convergent integral, giving a nite value when r ! RS . The
particle, looking at things from its own proper frame and measuring time in
126
its own clock, arrives at the Schwarzschild radius in a nite interval of time.
It will even arrive at the center r = 0 in nite time.
Suppose now a gas of particles, each one in the conditions above. They
will all fall towards the center. Each will do it in a nite interval of time in
its own frame. The gas will eventually colapse. A quite distinct picture will
be seen from the coordinate, asymptotically at frame. Seen from a distant
observer, the particles will never actually traverse the Schwarzschild radius.
All that happens inside the Schwarzschild sphere lies \beyond the in nity of
time". The sphere is a horizon, technically called an event horizon.
We have seen (in x 3.37 and ensuing paragraphs) how coordinate time
and proper time are related. Here we have an extreme di erence, given by
Eq.(4.28). Consider a light source near the Schwarzschild radius. Seen from
a distant observer, it will be strongly red-shifted. It will actually be more
and more red-shifted as the source is closer and closer to the radius. The
red-shift will tend to in nity at the radius itself: the external observer cannot
receive any signal from the sphere surface.
x
4.15 In consequence of all that has been said, a massive object can even-
tually fall inside its own Schwarzschild sphere. In a real star, such a gravitational colapse is kept at bay by the centrifugal e ects of pressure caused
by the energy production through nuclear fusion. A massive enough star
can, however, collapse when its energy sources are exhausted. Once inside
the radius, no emission will be able to scape and reach an external observer.
Particles and radiation will go on falling down to the sphere, but nothing will
get out. Such a collapsed object, such a black hole, will only be observable by
indirect means, as the emission produced by those external particles which
are falling down the gravitational eld.
4.16
It should be said, however, that the singularity in the components
of the metric does not imply that the metric itself, an invariant tensor, be
singular. A real singularity must be independent of coordinates and should
manifest itself in the invariants obtained from the metric. For example, the
determinant is an invariant. It is g = r4 sin2 . This shows that the point
r = 0 is a real singularity, in which the metric is no more invertible. The
Schwarzschild sphere, however, is not. It is, as seen, an event horizon, but
x
127
black
hole
not a singularity. This seems to have been rst noticed by Lema^tre in 1938,
and can be veri ed by transforming to other coordinate systems. Take for
instance the family of transformations
Z
Z
f (r)dr
dr
0
0
ct = ct ;
r
=
ct
+
;
(4.46)
R
R
1 rS
(1 rS )f (r)
involving an arbitrary function f (r) and which lead to
ds2 =
1
1
RS
r
0
0
(c2 dt 2
f (r)2
f 2 dr 2) r2 (d2 + sin2 d2 ) :
The Schwarzschild singularity
will disappear for an f such that f ( RrS ) = 1.
q
The better choice is f (r) = RrS , which gives a synchomous system. Notice
that there are two possible choices of the signs in (4.46). One leads to an
expanding reference system, the other to a contracting frame. In e ect, the
upper signs in (4.46) give
Z
Z
Z
r
2r3=2
= 1=2 :
RS RS
This shows that the system contracts in the old system:
r0
ct0 =
f (r)2 )
RS )f (r) =
r
(1
dr
(1
r=
R1S=3
3 0
(r
2
dr
=
f (r)
2=3
ct0 )
:
dr
(4.47)
As r 0, r0 ct0, the equality corresponding to the center real singularity.
The singularity would correspond to the value
3 0
(r ct0) = RS :
2
The interval takes the form given by Lema^tre
ds2
=
0
c2dt 2
h
dr0 2
3
0
2RS (r
i2=3
ct0)
3 0
(r
2
4=3
ct0)
R2S=3(d2 + sin2 d2 ) :
(4.48)
The Schwarzschild singularity does not turn up. To get a feeling, we can use
Eq.(4.47) to rewrite the interval in mixed coordinates:
RS 0 2 2 2
0
ds2 = c2dt 2
dr r (d + sin2 d2 ) :
(4.49)
r
128
Lema^tre
line
element
I
r=
t'
0
r=
RS
nt
ta
ns
r=
co
r'
Figure 4.1: In Lema^tre coordinates, a particle can go through the
Schwarzschild radius in a nite amount of time. The cones become narrower
as r0 decreases.
By Eq.(4.47), to each value r = constant in the old coordinates corresponds a straight line r0 = a + ct0 in the new system (see Figure 4.1). These
lines indicate, consequently, immobile particles in the old reference frame.
Lema^tre's system is synchonous (x 3.44). Geodesics are represented by
vertical lines, as the dashed line in the diagram. A free particle can fall
through the Schwarzschild radius and attain the center at r = 0.
Light cones have an interesting behavior. Consider a radial (d = 0; d =
0) light ray. Equation (4.49) gives, for ds = 0, the two (future and past)
cones, one for each sign in
r
dt0
RS
c 0 =
:
(4.50)
dr
r
We see that the cone solid angle becomes smaller for smaller values of r.
Consider again the lines r = constant, indicating immobility in the original
dt0 = 1, and there are two
system of coordinates. Their inclination is c dr
0
possibilities:
for r > RS ,
dt < 1: a line r = constant through the vertex lies
then c dr
0
inside the cone; immobility is consequently causally possible;
0
129
for r < RS ,
dt > 1: a line r = constant through the vertex lies
then c dr
0
outside the cone; immobility is causally impossible; any particle will
fall to the center.
0
The cones de ned in (4.50) have one more curious behavior: they will deform
themselves progressively as they approach the center. The derivative becomes
in nite at r = 0. The lines representing the cones in the Figure will meet
the line r = 0 as vertical lines.
If we choose the lower signs in (4.46), the line element will be (4.48), but
with t0 ! t0 :
ds2
0
= c2 dt 2
h
dr0 2
i2=3
3
0
0
2RS (r + ct )
4=3
3 0
(r + ct0)
2
R2S=3(d2 + sin2 d2 ) :
(4.51)
An analogous discussion can be done concerning the behavior of cones and
the issue of immobility. Immobility is still forbidden inside the radius. The
di erence is that, instead of falling fatally towards the center, a particle will
inexorably draw away from it. Contrary to the previous case, the system
expands in the old system:
r
x
= R1S=3
2=3
3 0
(r + ct0)
2
:
(4.52)
4.17 A reference frame is complete when the world line of every particle
either go to in nity or stop at a true singularity. In this sense, neither of the
above coordinate systems is complete. The Schwarzschild coordinates do not
apply to the interior of the sphere. An outside particle in the contracting
Lema^tre system can only fall down towards the centre: initial conditions
in the opposite sense are not allowed. Just the contrary happens in the
expanding Lema^tre systems. Both leave some piece of space unattainable.
That a complete system of coordinates does exist was rst shown by Kruskal
and Fronsdal. We shall here only mention a few aspects of this question.k
In such kind of system no singularity at all appears in the Schwarzschild
radius. The coordinates are given as implicit functions of those used above.
130
Lema^tre
line
element
II
In a form given by Novikov, the metric is looked for in a form generalizing
the mixed-coordinate interval (4.49). New time and radial variables ; R are
de ned so as to put the line element in the form
ds2
= c2 d 2
R2
1 + S2 dr2
R
r2 (; R)(d2 + sin2 d2 )
Novikov
line
element
(4.53)
and supposing a dust gas as source. The coordinates ; R are given implicitly
and in parametric form by
R
R2
r = S 1 + S2 (1 cos );
2
R
R2
R
= S 1 + S2
2
R
3=2
(
+ sin ):
(4.54)
(4.55)
The parameter take values in the interval 2; 0. When it runs from 2 to
0 the time variable increases monotonically, while r increases from zero up
to a maximum value
r = RS
R2
1 + S2
R
(4.56)
and then decreases back to zero. The Kruskal diagram of Figure 4.2 sumarizes the whole thing. Coordinates ; R are complete, so that all situations
are described. The Schwarzschild coordinates describe only situations external to the line r = RS . Contracting Lema^tre coordinates cover the shaded
area, expanding coordinates cover the domain which appears shaded after
specular vertical re ection. The small arrows indicate the forcible sense particles follow inside the Schwarzschild sphere: contracting in the upper side,
expanding in the lower one.
We have seen in x 3.17 that dust particles follow geodesics. The system is
synchronous (x 3.44), so that such geodesics are the vertical lines R =constant
in the diagram. An example is shown as a dashed line. Starting at = 0
k Details are given in an elegant form in L.D. Landau & E.M. Lifshitz, Theorie des
Champs, 4th french edition, x 103.
131
Kruskal
diagram
3
r=R
r=
0
S
τ
2
R
1
0
r = RS
r=
Figure 4.2: Kruskal diagram.
(point 1 in the Figure), a particle attains the center r = 0 in a nite time
R
R2
= S 1 + S2
2
R
3=2
:
(4.57)
Starting with outward initial conditions at the center, a situation corresponding to the lower part of the diagram, a particle will follow a trajectory like
the dashed line. It will cross the Schwarzschild radius in the outward direction, attain a maximal coordinate distance given by (4.56) at = 0 (point
1 again), and then fall back, crossing the Schwarzschild radius in an inward
progression at point 2 and reaching the center at point 3.
132
4.3 Large Scale Solutions
4.18
Two of the four fundamental interactions of Nature | the weak and
the strong | are of very short range. Electromagnetism has a long range
but, as opposite charges attract each other and tend thereby to neutralize, the eld is, so to speak, "compensated" at large distances. Gravitation
remains as the only uncompensated eld and, at large scales, dominates.
This is why Cosmology is deeply involved with it. We shall now examine
two of the main solutions of Einstein's equations which are of cosmological import. The Friedmann solution provides the background for the socalled Standard Cosmological Model, which | despite some diÆculties |
gives a good description of the large-scale Universe during most of its evolution. It represents a spacetime where time, besides being separated from
space, is position{independent, and space is homogeneous and isotropic at
each point. The Universe \begins" with a high-density singularity (the \bigbang") and evolves through two main periods, one radiation-dominated and
another matter-dominated, which lasts up to present time. There are difculties both at the beginning and at present time. The rst problem is
mainly related to causality, and could be solved by a dominant cosmological
term. The second comes from the recent observation that a cosmological
term is, even at present time, dominant. It is consequently of interest to
examine models in which the cosmological term gives the only contribution.
These are the de Sitter universes, which have an additional theoretical advantage: all calculation are easily done. A more detailed account of both the
Friedmann and the de Sitter solutions, mainly concerned with cosmological
aspects, can be found in the companion notes Physical Cosmology.
x
4.3.1 The Friedmann Solutions
x 4.19 We have seen under which conditions the second term in the interval
ds2 = g00 dx0dx0 + gij dxi dxj represents the 3-dimensional space. We shall
suppose g00 to be space{independent. In that case, the coordinate x0 can be
chosen so that the time piece is simply c2 dt2, where t will be the coordinate
time. It will be a \universal time", the same at every point of space.
The \Universe", that is, the space part, is supposed to respect the Cos133
Standard
model
mological Principle, or Copernican Principle: it is homogeneous as a whole.
Homogeneous means looking the same at each point. Once this is accepted,
imposing isotropy around one point (for instance, that point where we are) is
enough to imply isotropy around every point. This means in particular that
space has the same curvature around each point. There are only 3 kinds of
3{dimensional spaces with constant curvature:
the sphere S 3, a closed space with constant positive curvature;
the open euclidean space E3 of zero curvature (that is, at).
the open hyperbolic space S 2;1, or (a pseudo{sphere, or sphere with
imaginary radius), whose curvature is negative; and
These three types of space are put together with the help of a parameter
k : k = +1 for S 3 , k = 1 for S 2;1 and k = 0 for E3. The 3{dimensional line
element is then, in convenient coordinates,
dl2 =
dr2
+ r2 d2 + r2 sin2 d2 :
1 kr2
(4.58)
The last two terms are simply the line element on a 2-dimensional sphere S 2
of radius r | a clear manifestation of isotropy. Notice that these symmetries
refer to space alone. The \radius" can be time-dependent.
4.20
The energy content is given by the energy-momentum of an ideal
uid, which in the Standard Model is supposed to be homogeneous and
isotropic:
x
T = (p + c2) U U
p g :
(4.59)
Here U is the four-velocity and = =c2 is the mass equivalent of the energy
density. The pressure p and the energy density are those of the matter (visible
plus \dark") and radiation. When p = 0, the uid reduces to \dust".
4.21
We can now put together all we have said. Instead of a time{
dependent radius, it is more convenient to use xed coordinates as above,
x
134
and introduce an overall scale parameter a(t) for 3{space, so as to have the
spacetime line element in the form
ds2 = c2 dt2
a2 (t)dl2:
(4.60)
Thus, with the high degree of symmetry imposed, the metric is entirely xed
by the sole function a(t). The spacetime line element will be
ds2
=
c2 dt2
a2 (t)
dr2
+ r2 d2 + r2 sin2 d2 :
1 kr2
Friedmann
Robertson
Walker
interval
(4.61)
This is the Friedmann{Robertson{Walker line element.
4.22
The above line element is a pure consequence of symmetry considerations. We have now to impose the dynamical equations. Recent data,
as said, point to a non-vanishing cosmological constant. It is, consequently,
wiser to use the Einstein equations in the form (3.42). The extreme simplicity of the model is re ected in the fact that those 10 partial di erential
equations reduce to 2 ordinary di erential equations (in the variable t) for
the scale parameter.
In e ect, once (4.59) is used, the eld equations (3.42) reduce to the two
Friedmann equations for a(t):
x
a_ 2
4G
c2 2
+
a
= 2
3
3
c2
a =
3
4G
3p
+ 2
3
c
kc2 ;
(4.62)
a(t) :
(4.63)
It will be convenient to absorb the length dimensionality in a(t), so that the
variable r can be seen as dimensionless. The cosmological constant has the
dimension (length) 2 . The second equation determines the concavity of the
function a(t). This has a very important qualitative consequence when = 0. In that case, for normal sources with > 0 and p 0, a is forcibly
negative for all t and the general aspect of a(t) is that of Figure 4.3. It will
consequently vanish for some time tinitial. Distances and volumes vanish at
that time and densities become in nite. This moment tinitial is taken as the
135
Friedmann
equations
Big
Bang
beginning, the \Big Bang" itself. It is usual to take tinitial as the origin of
the time coordinate: tinitial = 0. If > 0, there is a competition between
the two terms. It may even happen that the scale parameter be 6= 0 for all
nite values of t.
a
7
6.5
6
5.5
5
4.5
1.2 1.4 1.6 1.8
2
t
2.2 2.4
Figure 4.3: Concavity of a(t) for = 0.
Combining both Friedmann equations, we nd
d
=
dt
3
a_ p
+ 2 ;
a
c
(4.64)
which is equivalent to
d 3
(a ) + 3 p a2 = 0:
(4.65)
da
This equation re ects the energy{momentum conservation: it can be alternatively obtained from T ; = 0.
4.23
It is convenient to introduce two new functions, in terms of which
the equations assume simpler forms. The rst is the Hubble function
x
H (t) =
a_ (t) d
= ln a(t) ;
a(t) dt
(4.66)
whose present-day value is the Hubble constant
H0 = 100 h km s 1 Mpc 1 = 3:24 10
136
18
hs
1
:
Hubble
function
&
constant
The parameter h, of the order of unity, encapsulates the uncertainty in
present-day measurements, which is large (0:45 h 1). The second is
the deceleration function
aa
a
1
a
q (t) =
=
=
:
(4.67)
2
2
a_
a_ H (t)
H (t) a
Equivalent expressions are
H_ (t) =
H 2 (t) (1 + q (t)) ;
d 1
= 1 + q (t) :
dt H (t)
(4.68)
Uncertainty is very large for the deceleration parameter, which is the present{
day value q0 = q (t0). Data seem consistent with q0 0. Notice that we are
using what has become a standard notation, the index \0" for present-day
values: H0 for the Hubble constant, t0 for present time, etc.
The Hubble constant and the deceleration parameter are basically integration constants, and should be xed by initial conditions. As previously
said, the present{day values are used.
x
4.24 There are a few other basic numerical parameters, internal to the
theory. The main one is the critical mass density, a simple function of the
Hubble constant:
3H02
= 1:878 10 26 h2 kg m 3 :
(4.69)
8G
We shall see later why this density is critical: in the = 0 case, the Universe
is nite or in nite if (the mass equivalent of) its energy content is respectively
larger or lesser than crit . Still another function can be de ned, the baryon
density function
crit =
b (t) =
(t) 8G(t)
=
:
crit
3H02
Now, the = 0 Universe is nite (closed) or in nite (open) if the present
value b0 = b (t0) (the \baryon density parameter") is respectively > 1 or
1. Observational data give
0:0052
b0
h2
137
0:026
(4.70)
deceleration
parameter
for the matter contained in visible objects. There is a strong evidence for the
existence of invisible matter, usually called \dark matter". This is indicated
by an analogous parameter m , which is believed to be at most 0:3. In
any case matter, visible or dark, seems insuÆcient to close the Universe.
4.25 Some important special cases deserve to be mentioned:
Einstein static model Historically the rst model of modern Cosmology:
x
a(t) = 1=2 is constant, k = +1, = 4G=c2 , p = 0; everything is
constant in time; a rst proposal, with = 0, was shown to be unstable
by Eddington; Einstein added a 6= 0 to stabilize it; thus, this model
showed that no static solution existed without a 6= 0;
Milne model a(t) = t, k =
1, + p = 0; this last \equation of state" is
equivalent to a cosmological constant and, consequently, to an empty
Universe;
de Sitter Universe a(t) = eHt, k = 0, H a constant; empty, steady{state,
constant curvature; shall be examined in detail later on;
Einstein{de Sitter model a(t) = C t2=3, k = 0, = 0; when only dust is
considered, C is a constant; this is the simplest non-empty expanding
Universe; = 0 = 1, age 2=(3H0 ); see more in Section 4.35;
= 0 models with ordinary matter and radiation, the Universe expands
inde nitely for k = 1 and k = 0; if k = 1, it expands up to a
maximum radius (for us, in the future) and then contracts to the \Big
Crunch".
x
4.26 The at Universe Let us, as an exercise, examine in some detail
the particular case k = 0. The Friedmann{Robertson{Walker line element is
simply
ds2 = c2 dt2 a2 (t)dl2 ;
138
(4.71)
where dl2 is the Euclidean 3-space interval. In this case, calculations are
much simpler in cartesian coordinates. The metric and its inverse are
0
(g ) =
B
B
B
@
0
(g )
1
0
0
0
0
a2(t)
0
0
1
0
0
0
B
=B
B
@
0
0
a2(t)
0
0
0
0
a2(t)
0
0
0
0
0
0
a 2(t)
0
0
Æ
a 2 (t)
0
1
C
C
C
A
;
1
C
C
C:
A
a 2(t)
In the Christo el symbols
, the only nonvanishing derivatives are those
0
with respect to x . Consequently, only Christo el symbols with at least one
Æ
index equal to 0 will be nonvanishing. For example, k ij = 0. Actually, the
only Christo els 6= 0 are:
Æ
Æ
k = Æ k 1 a_ ;
0 = Æ 1 a a_ :
0j
ij
ij
j c a
c
The nonvanishing components of the Ricci tensor are
3 a
H 2 (t)
R00 =
=
3
q (t) ;
c2 a
c2
Æ
Æ
Rij = ij2 [aa + 2a_ 2] = ij2 a2 H 2 (t)[2 q (t)] :
c
c
In consequence, the scalar curvature is
"
R = g 00 R00 + g ij Rij =
a
a_
6 2 +
ca
ca
2 #
=6
H 2 (t)
[q (t) 1] :
c2
The nonvanishing components of the Einstein tensor G =R
1
2
Rg are
a_ 2
Æij 2
; Gij =
[_a + 2aa] :
G00 = 3
ca
c2
Let us consider the sourceless case with cosmological constant. The Einstein
equations are then
G00
a_
g00 = 3
ca
2
= 0 ; Gij
139
gij =
Æij 2
[a_ + 2aa a2] = 0 :
c2
Subtracting 3 times one equation from the other, we arrive at the equivalent
set
c2 2
c2
a_ 2
a = 0 ; a
a=0:
(4.72)
3
3
These equations are (4.62) and (4.63) for the case under consideration. They
are the same as
c2
c2
H 2 (t) =
; H 2 (t) q (t) =
;
(4.73)
3
3
or to
c2
; q (t) = 1 :
(4.74)
H 2 (t) =
3
Both parameter{functions are actually constants, so that H0 =
= 1. Of the two solutions, a(t) = a0 eH0 (t t0 ) , only
a(t) = a0
eH0(t t0 )
q
= a0 e
c2 (t t0 )
3
q
c2
3
and q0
(4.75)
would be consistent with expansion. Expansion is a fact well established
by observation. This is enough to x the sign, and the model implies an
everlasting exponential expansion. Notice that the scalar curvature is R =
4, as is always the case in the absence of sources.
x
4.27 The red{shift The red-shift z is de
ned by
a(t0)
:
(4.76)
a(t)
This formula represented the rst great success of the model. For a light
ray ds = 0, so that, from (4.60), dl = acdt
(t) . Suppose we observe light coming
from a distant galaxy at xed and . If it is emitted at a point with radial
coordinate r1 , the coordinate distance down to us will be
1+z =
d(r1 ) =
But this is also
Z r1
0
p
1
dr
kr2
=
d(r1 ) =
8
>
<
>
:
arcsin r1 (k = 1)
r1
(k = 0)
arcsinh r1 (k = 1) :
Z t0
t1
140
cdt
;
a(t)
where t1 and t0 are respectively the emission and the reception times. Consider a wave maximum which departs at t1 and arrives here at t0; and the
next maximum with depart at t1 + Æt1 (so that Æt1 is the wave period at
emission) and arrival at t0 + Æt0 (so that Æt0 i s the wave period at reception).
Then, for this second maximum,
d(r1 ) =
Z t0
t1
cdt
=
a(t)
It follows that
Z t0 +Æt0
t1 +Æt1
cdt
=
a(t)
Z t1 +Æt1
Z t0
t1
cdt
a(t)
Z t1 +Æt1
t1
cdt
+
a(t)
Z t0 +Æt0
t0
cdt
:
a(t)
Z
t0 +Æt0 cdt
cdt
=
:
a(t)
a(t)
t0
t1
As the wave periods Æt1 and Æt0 are small in comparison with the time scales
involved,
Æt1
Æt
= 0 :
a(t1) a(t0)
In terms of the frequency, we obtain the law
1 a(t0)
=
;
0 a(t1)
(4.77)
(t) a(t) = constant:
(4.78)
or
The red{shift is de ned as z = 011 = 01 1, from which follows (4.76).
Hubble's discovery, in 1929, of a consistent red{shift of light coming from distant galaxies is one of the greatest landmarks in Cosmology. This has, since
then, been systematically and extensively con rmed for larger and larger
numbers of objects. The astrophysicist analyses the spectrum from some
distant object, recognizes the lines emitted by some atom (say, Calcium) and
compares with those found in laboratory. She/he nds a systematic shift of
the lines, given by (4.76) or (4.78).
x
4.28 Energy density and red-shift Given an equation of state in the
form p = p(), equation (4.65) can be integrated to give
1 R d
a
1 + z = 0 = e 3 0 +p() :
a(t)
141
(4.79)
For example, for a pure radiation content the equation of state is p = 13 , so
that
z = 0 (1 + z )4 :
The energy density of dust matter, with p = 0, will behave according to
z = 0 (1 + z )3 :
Recall that Eq.(4.65) is a mere consequence of energy conservation. These
results are independent of the parameters k and . The relationship between
z and the Hubble function will be found later [see Eq.(4.86)].
x
4.29 Hubble's law If the distance between two objects is l(t0) today, it
was
a(t)
a(t0)
at some t < t0. This distance will change with time according to
l(t) = l(t0)
a(t) a_ (t)
l_(t) = v (t) = l(t0)
= H (t) l(t):
a(t0) a(t)
At present time, this recession velocity is given by Hubble's law
v (t0) = H0 l0 :
(4.80)
The velocity of recession v between two objects is proportional to their distance. The law holds for objects rather close to us, for which t t0.
x
4.30 Age of the Universe From (4.66) in the form dt = a1
calculate the total time from tinitial = 0,
Z t0
0
dt =
Z a0
0
da
=
a H [t(a)]
Z 1
1
da ,
H
we can
d(1 + z )
:
(1 + z )H (1 + z )
This \age of the Universe" must be larger than the ages of its somehow
structured constituents. It would be a disaster for the model if Earth, for
example, were found to be older than the Universe. The formula shows a
strong dependence on the behaviour of H (t). We shall see later that a dust{
lled Universe, for example, gives an age = 2=(3H0 ). For h = 1, this means
142
6:5 billion years. What we can say is that di erent contents and parameter
values give an age in the interval 5 | 20 billion years.
There are independent methods to determine the ages of stars, of certain
star clusters and of the Earth. There are, for the time being, large uncertainties in these numbers. Despite some alarms turning up from time to time,
the numbers seem consistent with the model.
x
as
4.31
In terms of H (t), the Friedmann equations (4.62, 4.63) can be written
H2
a
H_ =
a
which is the same as
kc2 c2
+
;
a2
3
4G
=2
3
H2 =
H 2 (t)q (t) =
4 G +
4 G
3
+3
(4.81)
p kc2
+ 2 ;
c2
a
p
c2
(4.82)
c2
;
3
(4.83)
a_ (t) = H (t) a(t) ;
(4.84)
d
p
= 3H (t) + 2 :
(4.85)
dt
c
Taking time derivatives of Eq.(4.79) and comparing with the last expression,
H (t)(1 + z ), which integrates to
one arrives at dz
dt =
1+z =e
Rt
t0 H (t)dt
:
(4.86)
The Friedmann equations acquire their most convenient form in terms of
dimensionless variables. It is enough to divide them by H02 and introduce
still two functions:
c2
kc2
and
(
t
)
=
:
(4.87)
=
k
3H02
a2H02
The rst Friedmann equation becomes
H2
=
2
H0 crit
kc2
c2
+
a2 H02 3H02
143
b (t) +
k (t) +
;
(4.88)
which gives on present-day values the constraint 1 =
b0 + k0
+
0
crit
-
kc2
a20 H02
+
= 1:
, or
(4.89)
In terms of H (t), equation (4.82) becomes
H_
3 + p=c2
kc2
=
+
:
H02
2 crit
a2H02
Let us choose for time and length the units
H0 1 = 3:086 1017 h 1 sec = 0:9796 1010 h
c2
3H02
(4.90)
1
years ;
c
= 9:258 1025 h 1 m :
(4.91)
H0
These units are natural in Cosmology. The rst is of the same order of
magnitude of the Universe age, in whatever model; the second, of its \causal
size" (distance which light would travel during the age). The equations
acquire the simpler forms
k
H2 =
+ = b (t) + k (t) + (4.92)
2
crit a (t) 3
3 2 3
p
1 k
H +
:
(4.93)
2
2
2
crit c
2 a2 (t)
Notice that the density is absent in the last expression. In fact, it is hidden
in the rst term of the right{hand side.
H_ =
4.32
The pressure p and the mass density in (4.59), (4.62) and (4.63) are
supposed (in the standard model) to be only those of matter and radiation
nowadays present in the Universe. This replaces the \gas" obtained from the
smearing of the energy content of large domains.
x
4.33
The values = 0, k = 0 lead to very simple solutions and are helpful
in providing a qualitative idea of the general picture. They will provide a
reference case. We shall exhibit later the exact analytic general solution for
H (z ) and an implicit solution for a(t). Nevertheless, in order to get a rmer
grip on the relevant contributions and the role of each term, let us shortly
review the thermal history of the Universe.
x
144
reference
case
4.3.2 Thermal History
x 4.34 The present{day content of the Universe consists of matter (visible or
not) and radiation, the last constituting the cosmic microwave background.
The energy density of the latter is very small, much smaller than that of
visible matter alone. Nevertheless, it comes from the equations of state that
radiation energy increases faster than matter energy with the temperature.
Thus, though matter dominates the energy content of the Universe at present
time, this dominance ceases at a \turning point" time in the past (equation
(4.115)). At that point radiation takes over. At about the same time, hydrogen | the most common form of matter | ionizes. The photons of
the background radiation establish contact with the electrons and the whole
system is thermalized. Above that point, there exists a single temperature.
And, above the turning point, the dominating photons increase progressively
in number while their concentration grows by contraction. The opportunity
for interactions between them becomes larger and larger. When they approach the mass of an electrons, pair creation sets up as a stable process.
Radiation is now more than a gas of photons: it contains more and more
electrons and positrons. Concomitantly, nucleosynthesis stops. As we insist
in going up the temperature ladder, the photons, which are more and more
energetic, break the composite nuclei. The nucleosynthesis period is the most
remote time from which we have reasonably sure information nowadays.
x
4.35 Matter{dominated era All during this period kT << 1 GeV , so
that protons are non-relativistic. Matter pressure is supposed to be given
by the ideal non{degenerate gas expression pb = nb kTb . It appears in Friedmann's equations only in the combination
n
b + pb =c2 = nb [m + kTb=c2 ] = 2b [mc2 + kTb ] :
c
In the prevailing non-relativistic regime, pb is quite negligible. Dust pervades
the Universe. Putting p = 0, the equations (4.92) and (4.93) become very
simple for the reference case = 0, k = 0:
H_ =
3 2
H =
2
145
3 b
:
2 crit
The general solution is
1
3
= t+C :
H (t) 2
x
(4.94)
4.36 The dust Universe Let us here make an exercise, studying what is
called \the dust Universe". It is an unrealistic model which supposes matter
domination all the time. We x at the \beginning" Ht=0 = 1, then the
integration constant C vanishes and the exact solution is
2
H (t) = :
3t
This means that
da 2 dt
=
:
a 3 t
The expressions relating the Hubble function, the expansion parameter, the
red-shift, and the density follow immediately (we reinsert H0 for convenience):
H2
= (1 + z )3 :
2
H0
a(t)
t
=
a(t0)
t0
t
1+z = 0
t
H2
b
= 2
crit H0
2=3
b0
2=3
(4.95)
;
2
=
3H0 t
= (1 + z )3
2=3
b0
;
:
(4.96)
(4.97)
The age of the Universe can be got from (4.96), by putting z = 0. One obtains
t0 = 2=(3H0 ), which is 6:5 109 years, a rather small number. Always
in the reference case, this dust{dominated Universe is just the Einstein{de
Sitter Universe.
There are, anyhow, many serious aws in this exercise-model: it supposes
that matter dominates down to t 0. This is far from being the case:
radiation dominates at the early stages. Furthermore, the protons cannot, of
course, be non-relativistic at the high temperatures of the "beginning".
146
4.37 Matter{domination: k = 0, = 0
Let us go back to the general
solution (4.94) and x the integration constant by the present value
x
1
1
3
=
= t0 + C :
H (t0) H0 2
The solution, now more realistic, will be
H (t) =
Integrations of
da
a
1+
H0
3 H (t
2 0
t0)
:
(4.98)
= H (t)dt leads then to
1+z =
a0
1
:
=
3
a(t) [1 + 2 H0 (t t0 )]2=3
(4.99)
Equation (4.97) keeps holding. The middle equality, because it is simply
the Friedmann equation (4.88) for the reference case = 0, k = 0. The last
equality, because it comes anyhow from energy conservation, as has been seen
in x 4.28. Using that equality, and the value (4.69) of the critical density, we
nd
b = 1:878 10
26
(1 + z )3
b0
h2 [kg m 3 ]
(4.100)
Dividing by the proton mass, the number density is
nb = 11:2 (1 + z )3
b0
h2 [m 3 ]
(4.101)
Actually, b0 = 1 in the reference case we are considering. This gives a few
nucleons per cubic meter at present time. The age of the Universe is basically
the same: we look for the time t corresponding to z ! 1, and nd t0 t
= 2=(3H0 ). Finally, for time ranges small in comparison to H0 1 , we nd a
linear relation between red-shif and time:
z H0 (t0 t) :
(4.102)
This expression is valid for nearby objects. As to (4.98) and (4.99), they hold
from the turning point down to present times (provided, we insist, k = 0 and
= 0).
147
4.38 Matter{domination: k = 0, 6= 0
Recent evidence for k = 0
and a nonvanishing cosmological constant at present time gives to this case
the prominent role.
Let us insert (4.86) into (4.97), to get
x
R
3 tt0 H (t)dt
= 0 e
and then insert this expression into (4.81):
H2
4G
=2
0 e
3
This gives, by the way,
H02
R
3 tt0 H (t)dt
+
4G
c2
=2
0 +
:
3
3
Taking the time derivative, it comes
dH 3 c2
=
dt 2 3
Integration leads to the expression
r
H (t) =
q
c2
3
q
c2
3
c2
3
c2
:
3
+ H0
+ H0
H2
q
q
3 3c2 (t t0 )
e
q
:
c2
e3 3 (t t0 ) +
q
c2
3
H0
c2
3
H0
:
(4.103)
This expression for H (t) gives H = H0 when t ! t0 and tends to (4.98) when
! 0. To have it in terms of more accessible parameters, we may use the
rst de nition of Eq.(4.87) in the form c2=3 = H02 to rewrite it as
H (t) = H0
p
p
p
p (t t )
0
+ 1 e3H0
p
+ 1 e3H0
(t t0 )
+
p
p
1
:
1
(4.104)
To get the relation with z , we notice that (4.81), which is
H2
gives
H2
4G
c2
=2
0 (1 + z )3 +
= H02 b (1 + z )3 +
3
3
c2
4G
=2
0 (1 + z )3 ) H02
3
3
148
;
c2
4G
=2
0 ;
3
3
which together imply
c2
3
c2
3
H2
H02
Alternatively,
= (1 + z )3 :
b+ (1 + z )3
=1;
H 2 H02 = 2
:
H0 (1
)
(4.105)
It remains to use (4.103) to obtain
1+z = 4
c2 1=3
3
q
c2
3
+ H0
e
q
c2 (t t0 )
3
q
c2
e3 3 (t t0) +
q
c2
3
H0
2=3
;
(4.106)
which is the same as
1 + z = (4
)
1=3
p
eH0
+ 1 e3H0
p (t t )
0
p (t t )
0 +
p
2=3
1
:
(4.107)
The scale parameter a(t)=a0 is just the inverse.
x
4.39 Radiation{dominated era For photons the temperature behaves
as a frequency. This comes from the equations for the black{body radiation,
which give an average energy per photon < h > / kT . In e ect, the
energy density is e / (kT )4, whereas the number density is n / (kT )3.
As the red{shift gives < h > = < h0 > (1 + z ), kT must have that same
behaviour,
T = T 0 (1 + z ):
For example, hydrogen recombination takes place at T = T = Tb 3000o K .
This, with the present{day value T 0 = 2:7 measured for the background
radiation, means (1 + z ) = 1:1 103 . Using the equation of state for the
blackbody radiation,
48
e = 3 3 (4) (kT )4 ;
h c
149
the mass-equivalent density = e c2 is given by
= 2:1 10 7 T 4 h 2 = 1:1 10 5 (1 + z )4 h 2 :
(4.108)
crit
The number appearing as a factor will turn up frequently. We shall introduce
a notation for it:
N = 1:1 10
5
h 2:
(4.109)
Consider again in the reference case k = 0, = 0. Because e = 3 p and = ec2 , Eqs.(4.92, 4.93) become
H_ = 2 H 2 = 2
:
crit
Automatically, in units (4.91),
H 2 = N (1 + z )4 :
(4.110)
Solving the di erential equation is only necessary to x the relation between
the time parameter and the red{shift. The solution is
1
H (t) = ;
2t
which implies t0 = 1=(2H0 ) and
p
a(t)
t
1
;
=p
=
2
NH0 t :
(4.111)
t0
a0
N (1 + z )2
Notice that these equations are not expected to meet present{day values.
We are solving the problem of a Universe dominated by radiation, which
only happens at very early times. Thus, these formulae are expected to hold
much before the turning point given below. It is useful to have at hand the
numerical expressions (in seconds):
q
t=
6:8 109 a(t)
4:65 1019
p ; a = 1:47 10
;
1
+
z
=
(1 + z )2
t
0
10
p
t : (4.112)
Before recombination (that is, for higher z 's) there is thermal equilibrium
between matter and radiation, because electrons are free and the mean free
path of the photons is very small. An estimate of the energy per photon at
a certain z can be obtained from kT 0 = 2:3 10 10 MeV , which leads to
kT = kT 0(1 + z ) = 2:32 10
150
10 (1 + z ) MeV:
(4.113)
For example, an energy of 4MeV corresponds to z 2 1010 .
In Kelvin degrees, the last equation above is
T = 2:7 (1 + z ) :
(4.114)
Let us make a simple order{of{magnitude estimate concerning recombination. If we take for the hydrogen recombination temperature T 3000o K ,
we nd z 1100 for the recombination time. Equation (4.101) gives then
nbR 1:5 1010 h2
b0
[cm 6]
for the number density of protons at that time. Suppose now the medium to
be neutral as a whole. After recombination, each proton has one electron to
neutralize it. Before recombination, the number of free electrons is equal to
that of protons.
x
4.40 The turning point The scale parameter, and consequently the red{
shift, behave quite di erently in a matter{dominated Universe (4.96) and in
a radiation{dominated one (4.111). Radiation becomes dominant at high
z 's. The turning point, or change of regime, takes place when ' b , or
[comparing (4.97) and (4.108)]
1 + z ' 9:1 104
This corresponds to t ' 1:8 10
8
2
b0 =H0
b0
h2 :
= 5:5 109
(4.115)
2
b0
sec.
4.41
Let us now go back to the general equations (4.92) and (4.93). Thermalization makes the whole system dependent of a single temperature. Much
of the discussion can be made in terms of energies, by using (4.113) and noticing that, if T is in o K , then kT is given in MeV 's as
x
kT (MeV ) = 8:617 10
4.42
11
T (o K ) :
It is frequently more convenient to use the variable z . Recall that,
from (4.76) and (4.66),
dz
(1 + z )H (z ) =
:
dt
Extracting b0 from (4.89) and using (4.81),
k
k
H 2 (z ) = N (1 + z )4 2 (1 + z )2 + + (1 + z )3 (1 N + 2
) :
a0
a0
(4.116)
x
151
This holds, it is good to keep in mind, if radiation provides all the pressure.
In the reference (k = 0, = 0) cases,
H 2 = N (1 + z )4 + (1 N )(1 + z )3 :
Using 1 + z = a0=a, we have
da q 4
a = Na0 ka2 +
dt
whose solution is given by
t = t0 +
Z a(t)
a(t0)
q
Na40
ky 2 +
a4 + a30(1 N + ka0 2
y dy
y 4 + y 1 N + ka0 2
(4.117)
)
a;
3
a0
: (4.118)
4.3.3 de Sitter Solutions
x 4.43 de Sitter spacetimes are hyperbolic
spaces of constant curvature.
They are solutions of vacuum Einstein's equation with a cosmological term.
There are two di erent kinds of them: one with positive scalar Ricci curvature, and another one with negative scalar Ricci curvature. As the calculations are remarkably simple, we shall give a fairly detailed account.
We shall denote by R the de Sitter pseudo-radius, by ( ; ; =
0; 1; 2; 3) the Lorentz metric of the Minkowski spacetime, and A (A; B; : : : =
0; : : : ; 4) will be the Cartesian coordinates of the pseudo-Euclidean 5{spaces.
There are two types of spacetime named after de Sitter:
1.
de Sitter spacetime dS (4; 1): hyperbolic 4-surface whose inclusion
in the pseudo{Euclidean space E4;1 satisfy
AB A B = 4 2 = R2 :
(4.119)
It is a one-sheeted hyperboloid (a 4-dimensional version of the surface
seen in x 2.65) with topology R1 S 3 , and | within our conventions | negative scalar curvature. Its group of motions is the pseudo{
orthogonal group SO(4; 1)
2.
anti{de Sitter spacetime dS (3; 2):
hyperbolic 4-surface whose inclusion in the pseudo{Euclidean space E3;2 satisfy
AB A B = + 4 2 = R2 :
152
(4.120)
It is a two-sheeted hyperboloid (a 4-dimensional version of the space
seen in x 2.64) with topology S 1 R3 , and positive scalar curvature.
Its group of motions is SO(3; 2)
With the notation 44 = s, both de Sitter spacetimes can be put together in
AB A B = + s 4 2 = sR2 ;
(4.121)
where we have the following relation between s and the de Sitter spaces:
s = 1 for dS (4; 1)
s = +1 for dS (3; 2).
4.44 The metric Let us
nd now the line element of the de Sitter spaces.
The most convenient coordinates are the stereographic conformal. The passage from the Euclidean A to the stereographic conformal coordinates x
( ; ; = 0; 1; 2; 3) is done by the transformation:
x
= x ; 4 = R(1 2 );
(4.122)
(a sign in the last expression would have no consequence for what follows)
with (x) a function of x which we shall determine. Two expressions
preparatory to the calculation of the line element can be immediately obtained by taking di erentials:
d = x d + dx ;
(4.123)
and
d 4 2 = 4R2 d 2 :
as
(4.124)
Let us introduce 2 = x x and rewrite the de ning relation (4.121)
2 2 + s
4 2 = sR2 :
Equating ( 4 )2 got from (4.122) and (4.125), we nd
1
=
2 :
1 + s 4R2
153
(4.125)
(4.126)
Notice that from this expression it follows that
d =
2
s
2
2 d = s
2 x dx ;
2R
4R2
from which another preparatory result is obtained:
2 x dx =
s
4R2
2
(4.127)
d :
(4.128)
Now, the de Sitter line element is
d2 = AB d A d B = d d + s d 4
2
;
or, by using (4.123),
d2 = (x d + dx ) x d + dx + s d 4
2
:
Expanding and using (4.124),
d2 = x x d
2 + 2
dx dx + s 4R2 d
2
x dx d +
2
:
Now, using (4.128),
d2
=
2
dx dx +
2
s
4R2
d
2
+ s 4R2 d
2
;
and then (4.126),
d2 =
2
dx dx +
s 4R2 d
2
+ s 4R2 d
2
;
so that nally
d2 = g dx dx ;
where the metric g
(4.129)
is
g =
2
=
h
1
2
1 + s 4R2
i2
:
The de Sitter spaces are, therefore, conformally at (see
conformal factor given by 2 (x).
154
(4.130)
x 2.46), with the
4.45
The Christo el symbol corresponding to a conformally at metric
g with conformal factor 2 (x) has the form
x
Æ
=
Æ Æ + Æ Æ @ ln
(x) :
(4.131)
Taking derivatives in (4.126), we nd for the de Sitter spaces
Æ
s 2R2 Æ + Æ =
Æ x :
(4.132)
The Riemann tensor components can be found by taking the following
Æ
steps. First, take the derivative of the de Sitter connection
:
Æ
@
Æ
Æ Æ +
s 2R2 Æ + Æ =
Æ
@ ln
sx Æ
R
2 2
s 2R2 Æ + Æ Æ + x4Rx4 Æ + Æ s 2R2 Æ + Æ Æ + 4xR4x 2 Æ g + Æ g
Æ + 4R14 2 Æ x x + Æ x x
2R2 Æ + Æ =
=
s
=
s 2R2 Æ + Æ =
Æ
g Æ
g x x :
Indicating by [ ] the antisymmetrization (without any factor) of the included indices, we get
@
Æ
@
Æ
[ Æ] + 4R14
s 2R2 Æ[ ]
=
x Æ[ x]
2
x g [ x]
s R2 Æ[ ] + 4R14 2 x Æ[ x] x g [ x] :
This is the contribution of the derivative terms.
The product terms are
=
Æ
Æ
Æ
2 = 4R4
Æ
Æ
Æ + Æ Æ
Æ
= 4R14
2
Æ s
+ Æ s
Æs x xs ;
2 2 g
x Æ[ x] + x x[ g] +
A provisional expression for the curvature is, therefore,
Æ
R
= @
Æ
@
Æ
155
+
Æ
Æ
Æ
Æ
[ Æ]
:
s R2 Æ[ ] + 4R14 2 x Æ[ x] x g [ x]
+ 4R14 2 x Æ[x] + x x[g] + 2 2 g [ Æ] :
=
The rst two terms in the last line just cancel the last two in the line above
them. Therefore,
Æ
R
s R2 Æ[ ] + 4R14
2 2 g
=
=
s R2 [ Æ] + 4R1 4 2 2 [ Æ]
=
s R2 + 4R1 4 2 2 [ Æ]
2
[ Æ]
= s R2 4Rs 2 2 1 [Æ]
Using (4.126), we nd that the bracketed term is =
Æ
R
=
s Æ
2
s R2 [ Æ] =
g R2
Æ g
. Therefore, we get
:
(4.133)
The Ricci tensor will be
Æ
R = 3R2s g
(4.134)
and the scalar curvature,
Æ
R=
12 s
R2
:
(4.135)
The de Sitter spacetimes are spaces of constant curvature. We can now
make contact with the cosmological term. From the expressions above, we
nd that
Æ
Æ
3s
1
(4.136)
R 2 g R + R2 g = 0:
Thus, the de Sitter spaces are solutions of the sourceless Einstein's equations
with a cosmological constant
=
3s
R
2
=
Æ
R =4:
(4.137)
Notice the relationships to the de Sitter and the anti-de Sitter spaces:
s = 1 for the de Sitter space dS (4; 1) ! > 0
s = +1 for the anti-de Sitter space dS (3; 2) ! < 0 .
156
4.46
We have said (x 3.46) that positive scalar curvature tends to make
curves to close to each other, and just the contrary for negative curvature.
The relative sign in (4.137) shows that the cosmological constant has the
opposite e ect: > 0 leads to diverging curves, < 0 to converging ones.
This actually depends on the initial conditions. Let us look at the geodetic
deviation equation,
x
Æ
D2 X
=R
2
Du
U
U X :
(4.138)
Using Eq.(4.133),
D2 X
Ds2
=
s
R2 Æ g Æ g
U U X =
s
R2 [Æ
U U ] X =
s
R2 h X :
(4.139)
We have recognized the transversal projector h (see x 3.50). By the
geodesic equation, the component of X along U will have vanishing contributions tothe left-hand side. Consequently, only the transversal part X?
will appear in the equation, which is now
D 2 X? s
D2 X? RÆ
D 2 X? X
=
X
=
(4.140)
+
+
R2 ?
12 ?
3 X? = 0 :
Ds2
Ds2
Ds2
Negative leads to oscillatory solutions. Positive can lead both to contracting and expanding congruences. If two lines are initially separating,
they will separate inde nitely more and more.
x
4.47 In ationary solution de Sitter spacetimes are homogeneous and
isotropic. Equations (4.62-4.64) hold consequently for de Sitter cosmology,
with vanishing and p:
a_ 2
c2 2
=
a
3
a =
kc2 ;
c2
a(t) :
3
157
(4.141)
(4.142)
The solutions are given by
a+
p
or
a2
h
a=
3k= = a0 +
a0 +
p
h
a20
2 a0 +
3k=
p
q
i2
3k= ec
a20
p (t
3
p (t t )
0 + 3k=
3
i
p
c (t t0 )
e2 c
a20 3k= e
3
t0 )
;
:
(4.143)
(4.144)
With k = 0, we have (4.72), whose solution is
p (t
a(t) = a(0) ec
3
t0 )
:
(4.145)
The sign in the exponent has been chosen to provide for expansion. A solution of the type (4.145) is called \in ationary". More precisely, it is called
\in ationary of exponential type", to di erentiate it from other explosive
solutions. Notice
p (t0
1 + z = ec
3
t)
:
(4.146)
There are two supposedly equivalent simple scenarios for exponential ination:
1. no sources, and cosmological constant. Eq.(4.145) shows then that for
in ation necessarily > 0 if k = 0.
2. no cosmological constant, source with special equation of state
p=
(4.147)
Eq.(4.64) shows then that and p are constant. It turns out that R < 0.
It is as if there was an e ective cosmological constant = T=4 = . Or
we may say that a cosmological constant is equivalent to a source uid
with an e ective pressure (4.147). This would also agree with (4.135).
This brings the problem back to a pure de Sitter case.
As > 0 is essential for the de Sitter in ation, we must have, from
(4.137), the sign s = 1. Therefore, only the de Sitter space dS (4; 1)
158
can lead to in ation. The expression of the solution is suggestive: (4.145)
becomes
a(t) = a(0) ec(t
t0 )=R:
(4.148)
The cosmological constant is, in this
case, difectly related to the Hubp
ble constant: H (t) H0 = c=R = c =3. When t = 0, the expansion
parameter has a minimum value
amin = a(0) e
H0 t0
6= 0:
(4.149)
There is no initial singularity. The corresponding red-shift is not in nite.
We have been using carefully two coordinate systems. The most convenient system for cosmological considerations is the so{called comoving system, in which the Friedmann equations, in particular, have been written. In
that system the scale parameter appears in its utmost simplicity. The stereographic coordinates are of interest for de Sitter spaces. We could perform
a transformation between the two systems, but that is not really necessary:
we have only taken scalar parameters from one system into the other. The
only exception, Eq. (4.136), is a tensor which vanishes in a system and,
consequently, vanishes also in the other. We shall say a few more words on
coordinate systems in x 4.49.
x
4.48 Physical meaning of As already seen, the Friedmann equations
in the presence of a cosmological constant term are
2
a_
a
8G
=
3
3
k
(4.150)
a 4G
=
( + 3p) :
(4.151)
a 3
3
Let us then consider the following simple model. Take a small space region
bounded by a sphere of radius \r". The distance between the centre and
any point on its surface is determined by the physical distance a(t)r0, which
satis es the equation
d2 (ar0) = ar0
dt2
3
4G
( + 3p) ar0 :
3
159
(4.152)
or
d2 (ar0) = ar0
dt2
3
GM
(ar0)2
F + FN ;
(4.153)
where
4
( + 3p)(ar0 )3
3
is the total mass inside the sphere.
The second term of (4.153) is the ordinary Newtonian force
M=
(4.154)
GM
:
(ar0)2
FN =
The rst term, caused by the cosmological constant , is also a gravitational
force, but with the rather unusual form
ar :
3 0
This eld of force is global and homogeneous. The intensity of the force
depends on the distance between the interacting particles as (ar0). The
further away are the particles, the larger is the force. This is a more elaborate
approach to the non-relativistic considerations given in x 4.8.
F =
4.49 The expression (4.130) for the metric is very di
erent from the original one. De Sitter has found it in another coordinate system, in the form
x
ds2
= 1
2 2 2
r c dt
3
r2 (d2 + sin2 d2 )
1
dr2
r2
3
:
(4.155)
This expression is just the Schwarzschild solution in the presence of a term, Eq.(4.21), when the source mass tends to zero. There are many other
coordinates and metric expressions of interest for di erent aims. One of
them exhibits clearly the in ationary property above discussed:
ds2 = c2dt2
ect=R dx2 + dy 2 + dz 2 :
(4.156)
A few, included that given below, are given by R.C. Tolman, Relativity, Thermodynamics and Cosmology, Dover, New York, 1987, x 142.
160
Chapter 5
Tetrad Fields
5.1 Tetrads
For each source, set of symmetries and boundary conditions, there will be a
di erent solution of Einstein's equations. Each solution will be a spacetime.
It will be interesting to go back and review the general characteristics of
spacetimes. While in the process, we shall revisit some previously given
notions and reintroduce them in a more formal language.
5.1
A spacetime S is a four{dimensional di erentiable manifold whose
tangent space (x 2.24) at each point is a Minkowski space. Loosely speaking,
we implant a Lorentz metric ab on each tangent space. Bundle language
is more speci c: it considers the tangent bundle T S on spacetime, an 8dimensional space which is locally the direct product of S and a typical
bre representing the tangent space. For a spacetime, the typical ber is
the Minkowski space M . The ber is \typical" because it is an \ideal" (in
the platonic sense) Minkowski space. The relationship between the typical
Minkowski ber and the spaces tangent to spacetime is established by tetrad
elds (see Figure 5.1). A tetrad eld will determine a copy of M on each
tangent space. M is considered not only as a at pseudo-Riemannian space,
but as a vector space as well. We are going to use letters of the latin alphabet,
a; b; c; : : : = 0; 1; 2; 3, to label components on M , and those of the greek
alphabeth, ; ; ; : : : = 0; 1; 2; 3 for spacetime components. The rst will be
called \Minkowski indices", the latter \Riemann indices".
x
161
5.2
We shall need an initial vector basis in M . We take the simplest one,
the standard \canonical" basis
x
K0 = (1; 0; 0; 0)
K2 = (0; 0; 1; 0)
K1 = (0; 1; 0; 0)
K3 = (0; 0; 0; 1) :
Each Ka is given by the entries (Ka )b = Æab . Each tetrad eld h will be a
mapping
h:M
! T S;
h(Ka ) = ha :
The four vectors ha will constitute a vector basis on S . Actually, this is a
h (tetrad)
Tp S
xa K a
hj
Kj
xa h a
X=
hk
p
Kk
<- 1 >
h
ideal
Minkowski
space
<- 1 >
ce
spa
ink
nt M
e
tang
U(α}
x
x
ki
ows
S = spacetime
Figure 5.1: The role of a tetrad eld.
local a air: given a point p 2 S , and around it an euclidean open set U ,
the ha will constitute a vector basis not only for the tangent space to S at
p, denoted TpS , but also for all the Tq S with q 2 U . The extension from p
162
to U is warranted by the di erentiable structure. The dual forms hb , such
that hb (ha ) = Æ ba , will constitute a vector basis on the cotangent space at
p, denoted TpS . The dual base fha g can be equally extended to all the
points of U . For example, each coordinate system fxg on U will de ne a
\natural" vector basis [email protected] = @[email protected] g, with its concomitant covector dual
basis, fdx g. This is a very particular and convenient tetrad eld, frequently
called a \trivial" tetrad, given by ea = e(Ka ) = Æa @ . It is usual to x a
coordinate system around each point p from the start, and in this sense this
basis is indeed \natural".
Another tetrad eld, as the above generic fha g and its dual fhb g, can be
written in terms of [email protected] g as
ha = ha @ and hb = hb dx ;
(5.1)
hb ha = Æ b a and ha ha = Æ :
(5.2)
with
The components ha (x) have one label in Minkowski space and one in spacetime, constituting a matrix with the inverse given by ha (x). It is usual to
designate the tetrad by the sets fha (x)g or fha g, but their meaning should
be clear: they represent the components of a general tetrad eld in a natural
basis previously chosen.
x
5.3 The tetrad Minkowski labels are vector indices, changing under Lor-
entz transformations according to
ha = a b hb ;
(5.3)
ha (x) = a b (x) hb (x) :
(5.4)
0
0
or, in terms of components,
0
0
For each tangent Minkowski space, a0 b is constant. It will, however,
depend on the point of spacetime, which we indicate by its coordinates x
= fx g. This is better seen if we contract the last expression above with
163
hc (x), to obtain the Lorentz transformation in terms of the initial and nal
tetrad basis:
a b (x) = ha (x) hb (x) :
0
0
(5.5)
Equation (5.4) says that each tetrad component behaves, on each Minkowski
ber, as a Lorentz vector. For each xed Riemann index , ha transforms
according to the vector representation of the Lorentz group. A Lorentz (actually, co-)vector on a Minkowski space has components transforming, under
a Lorentz transformation with parameters cd , as
a = a b (x) b = exp
0
0
i
2
cd J a0 b
cd
b
:
(5.6)
Here, each Jcd is a 4 4 matrix representing one of the Lorentz group generators:
[Jcd ]a b = i cb Æ a d
0
db Æ a c :
0
0
(5.7)
This means also that, for each xed Riemann index , the ha 's constitute a
Lorentz basis (or frame) for M .
5.4 A tetrad
eld converts tensors on M into tensors on spacetime, transliterating one index at a time. A general Lorentz tensor T , transforming
according to
x
T a b c ::: = a a b b c c : : : T abc::: ;
0 0 0
0
0
0
will satisfy automatically T a b c ::: = ha hb hc : : : T ::: , which shows how
tetrad elds can \mediate" Lorentz transformations. As an example of that
transmutation, a tetrad will produce a eld on spacetime out of a vector in
Minkowski space by
0 0 0
0
0
0
(x) = ha (x) a(x) :
(5.8)
As the tetrad, in its Minkowski label, transforms under Lorentz transformation as a vector should do, (x) is Lorentz{invariant. Another case of
\tensor transliteration" is
(x) = ha0 (x) a b (x) hb (x) = Æ 0
164
(5.9)
[using (5.5)]. Thus, there is no Lorentz transformation on spacetime itself.
The Minkowski indices | also called \tetrad indices" | are lowered by
the Lorentz metric ab :
ha = ab hb :
An important consequence is that the Lorentz metric ab is transmuted into
the Riemannian metric
g = ab ha hb :
(5.10)
Of course, also g (as any component of a spacetime tensor) is Lorentz{
invariant. Here, given ab , the tetrad eld de nes the metric g . Di erent tetrad elds transmute the same ab into di erent spacetime pseudoRiemannian metrics.
5.5 The members of a general tetrad eld fhag, as vector
a Lie algebra (see x 2.34) with a commutation table
x
[ha; hb ] = cc ab hc :
elds, will satisfy
(5.11)
The structure coeÆcients cc ab measure its anholonomicity | they are sometimes called \anoholonomicity coeÆcients" | and are given by
cc ab = [ha (hb ) hb (ha )] hc :
(5.12)
If fhag is holonomous, cc ab = 0, then ha = dxa for some coordinate system
fxag, and
dxa = a b dxb :
Expression (5.10) would, in that holonomic case, give just the Lorentz metric
written in another system of coordinates. This is the usual choice when we
are interested only in Minkowski space transformations, because then a0 b
= @xa0 [email protected] In this case the tetrad components can be identi ed with the
Lame coeÆcients of coordinate transformations, and the metric g will be
the Lorentz metric written in a general coordinate system.
We have up to now left quite inde nite the choice of the tetrad eld
itself. In fact its choice depends on the physics under consideration. Trivial
tetrads are relevant when only coordinate transformations are considered.
A non{trivial tetrad reveals the presence of a gravitational eld, and is the
fundamental tool in the description of such a eld.
0
0
165
5.2 Linear Connections
Let us examine, in a purely descroptive way, the transformation properties
of a linear connection. A linear connection is a 1-form with values in the
linear algebra, that is, the Lie algebra of the linear group GL(4; R) of all
invertible real 4 4 matrices. This means a matrix of 1-forms. A Lorentz
connection is a 1-form with values in the Lie algebra of the Lorentz group,
which is a subgroup of GL(4; R). All connections of interest to gravitation
(the Levi-Civita in particular) are ultimately linear connections.
5.2.1 Linear Transformations
x 5.6 A linear transformation of N variables xr is an invertible transformation of the type
xr = M r s xs :
0
0
(5.13)
In this case (M r0 s ) is an invertible matrix. Linear transformations form
groups, which include all the groups of matrices. The set of invertible N N
matrices with real entries constitutes a group, called the real linear group
GL(N; R).
x
5.7 Consider the set of N N matrices. This set is, among other things, a
vector space. The simplest of such matrices will be those whose entries
are all zero except for that of the -th row and -th column, which is 1:
( )Æ = Æ Æ Æ :
(5.14)
An arbitrary N N matrix K can be written K = K .
The 's have one great quality: they are linearly independent (none
can be written as linear combinations of the other). Thus, the set f g
constitutes a basis (the \canonical basis") for the vector space of the N N
matrices. An advantage of this basis is that the components of a matrix K
as a vector written in basis f g are the very matrix elements: (K ) =
K .
x
5.8 Consider now the product of matrices: it takes each pair (A; B ) of ma-
trices into another matrix AB . In our notation, matrix product is performed
166
coupling lower{right indices to higher{left indices, as in
Æ
= Æ
=Æ
Æ
;
(5.15)
where in ( )Æ , is the column index.
The structure of vector space (x 2.20) includes an addition operation and
its inverse, subtraction. Once the product is given, another operation can be
de ned by the commutator [A; B ] = AB BA, the subtraction of two products. The Lie algebra of the N N real matrices with the operation de ned
by the commutator is called the real N -linear algebra, denoted gl(N; R). The
set f g is called the canonical base for gl(N; R). A Lie algebra is summarized by its commutation table. For gl(N; R), the commutation relations
are
; = f(
)( )
(Æ ) Æ
:
(5.16)
The constants appearing in the right-hand side are the structure coeÆcients,
whose values in the present case are
f(
)( )
(Æ )
= Æ Æ ÆÆ Æ Æ ÆÆ :
(5.17)
The choice of index positions may seem a bit awkward, but will be convenient
for use in General Relativity. There, linear connections
and Riemann
curvatures R will play fundamental roles. These notations are quite well{
established. Now, a linear connection is actually a matrix of 1-forms =
, with the components
being usual 1-forms, just
dx . The
rst two indices refer to the linear algebra, the last to the covector character
of . A Riemann curvature is a matrix of 2-forms, R = R , where each
R is an usual 2-form R = 12 R dx ^ dx . In both cases, we talk of
\algebra{valued forms". In consequence, the notation we use here for the
matrices seem the best possible choice: in order to have a good notation for
the components one must sacri ce somewhat the notation for the base.
The invertible N N matrices constitute, as said above, the real linear
group GL(N; R). Each member of this group can be obtained as the exponential of some K 2 gl(N; R). GL(N; R) is thus a Lie group, of which
gl(N; R) is the Lie algebra. The generators of the Lie algebra are also called,
by extension, generators of the Lie group.
167
5.2.2 Orthogonal Transformations
x 5.9 A group of continuous transformations preserving an invertible real
symmetric bilinear form (see x 2.44) on a vector space is an orthogonal
group (or, if the form is not positive{de nite, a pseudo{orthogonal group).
A symmetric bilinear form is a mapping taking two vectors into a real
number: (u; v) = u v , with = . It is represented by a symmetric
matrix, which can always be diagonalized. Consequently, it is usually presented in its simplest, diagonal form in terms of some coordinates: (x; x) =
x x . Thus, the usual orthogonal group in E3 is the set SO(3) of rotations
preserving (x; x) = x2 + y 2 + z 2 ; the Lorentz group is the pseudo{orthogonal
group preserving the Lorentz metric of Minkowski spacetime, (x; x) = c2 t2
- x2 - y 2 - z 2 . These groups are usually indicated by SO( ) = SO(r; s), with
(r; s) xed by the signs in the diagonalized form of . The group of rotations
in n-dimensional Euclidean space will be SO(n), the Lorentz group will be
SO(3; 1), etc.
5.10
Given the transformation x 0 = 0 x , to say that \ is preserved"
is to say that the distance calculated in the primed frame and the distance
calculated in the unprimed frame are the same. Take the squared distance in
0
0
0
0
the primed frame, 0 0 x x , and replace x and x by their transformation
expressions. We must have
x
0 0
0
0
x x =
0 0
0
0
x x =
x x ;
8x :
(5.18)
This is the group{de ning property, a condition on the 0 's. When is
an Euclidean metric, the matrices are orthogonal | that is, their columns
0
are vectors orthogonal to each other. If is the Lorentz metric, the 's
belong to the Lorentz group. We see that it is necessary that
=
0 0
0
0
=
0
0 0
0
:
The matrix form of this condition is, for each group element ,
T = ;
where T is the transpose of .
168
(5.19)
5.11
There is a corresponding condition on the members of the group Lie
algebra. For each member A of the algebra, there will exist a group member such that = eA . Taking = I +A+ 21 A2+: : : and T = I +AT + 12 (AT )2 +: : :
in the above condition and comparing order by order, we nd that A must
satisfy
x
AT =
1
A
(5.20)
and will consequently have vanishing trace: tr A = tr AT = - tr (
= - tr ( 1 A) = - tr A ) tr A = 0.
1
A )
5.12 If is de
ned on an N -dimensional space, the Lie algebras so( ) of
the orthogonal or pseudo{orthogonal groups will be subalgebras of gl(N; R).
Given an algebra so( ), both basis and entry indices can be lowered and
raised with the help of . We de ne new matrices by lowering labels
with : ( )Æ = Æ Æ . Their commutation relations become
x
[ ; Æ ] = Æ
The generators of so( ) will then be J
relations
[J ; J Æ ] = Æ J + J
Æ :
=
Æ
ÆJ
(5.21)
- , with commutation
JÆ:
(5.22)
These are the general commutation relations for the generators of the orthogonal or pseudo{orthogonal group related to . We shall meet many cases in
what follows. Given , the algebra is xed up to conventions. The usual
group of rotations in the 3-dimensional Euclidean space is the special orthogonal group, denoted by SO(3). Being \special" means connected to the
identity, that is, represented by 3 3 matrices of determinant = +1.
The group O(N ) is formed by the orthogonal N N real matrices. SO(N )
is formed by all the matrices of O(N ) which have determinant = +1. In
particular, the group O(3) is formed by the orthogonal 3 3 real matrices.
SO(3) is formed by all the matrices of O(3) which have determinant = +1.
The Lorentz group, as already said, is SO(3; 1). Its generators have just the
algebra (5.22), with the Lorentz metric.
169
5.2.3 Connections, Revisited
x 5.13 Suppose we are given the connection by components
a
the rst
two indices being \algebraic" and the last a Riemann index. This supposes a
basis in the linear algebra and a basis of vector elds on the manifold. Taking
the canonical basis fab g for the algebra, and a holonomic vector basis fdx g
on the manifold, for example, the connection is given in invariant form by
= 21 a b
a
b dx
:
b ,
(5.23)
For reasons which will become clear later, the set of components f a b g will
be called spin connection.
Connections have been introduced in in Section 2.3 through their behavior
under coordinate tranformations, that is, under change of holonomic tetrads.
We proceed now to a series of steps extending that presentation to general,
holonomic or not, tetrads. First, we (i) change from Minkowski indices to
Riemann indices by
a
b
!
= h a
a
b
b h + ha @ ha :
(5.24)
This generalizes Eq.(2.33). Then, we (ii) change again through a Lorentz{
transformed tetrad, ! a0 b0 = ha0 h b0 + ha0 @ h b0 , which means
that
a0
b0 = a a
a
0
b
1 b
b0
+ a c @ 0
1 c
b0
:
(5.25)
This gives the e ect on a b of a Lorentz transformation a0 a = ha0 h a . In
0
0
the notation adopted, a a changes V a into V a ; we can write simply c b0 for
the inverse ( 1 )c b0 , understanding that
V c = c b0 V b = 0
Now, start instead with
Minkowski indices by
!
a
b
,
1 c
b0 V
b0
:
and (iii) change from Riemann indices to
= ha h b
+ ha @ hb ;
and (iv) go back to modi ed Riemann indices by
0
0 0
= h a
0
a
b
b h 0
170
+ h c @ hc 0 :
0
(5.26)
Consequently,
0
0 0
= h a ha 0
0 a
h b + h a h @ h b hb 0 + h c @ hc 0 ;
0
or
0
= B 0
0 0
0
B 0
+ B @ B 0 :
0
(5.27)
This is the e ect of a change of basis given by B 0 = h0 a ha .
x
5.14
Vector elds transform according to (5.6):
e (x) = he (x) (x) = e b hb (x) (x) = e b b (x) :
0
0
0
0
(5.28)
What happens to their derivatives? Clearly, they transform in another way:
@
0
e
= @
h
0
e
i
b
b + e b @ b :
0
As the name indicates, the covariant derivative of a given object is a
derivative modi ed in such a way as to keep, under transformations, just the
same behavior of the object. Here, it will have to obey
D0 e = e b D b :
0
0
(5.29)
5.15
The way physicists introduce a connection is as a \compensating
eld", an object a b with a very special behavior whose action on the eld,
once added to the usual derivative, produces a covariant result. In the present
case we look for a connection such that
x
@e +
0
e0
d0
d0 = e b @ b +
0
d
d b
:
A direct calculation shows then that the required behavior is just (5.25),
which can be written also as
a0
b0 = a d Æ dc @ +
0
5.16
d
c
c
b0
:
(5.30)
As a rule, all indexed objects are tensor components and transform
accordingly. A connection, written as a b , is an exception: it is tensorial
in the last index, but not in the rst two, which change in the peculiar
x
171
way shown above. Any connection transforming in this way will lead to a
covariant derivative of the form
ra = @a +
a
b
b = he ee a +
a
be b
:
(5.31)
Applied in particular to ha , it gives
rha = @ha +
a
b
b h ;
b
:
applied to its inverse ha ,
rha = @ha
a hb
It is easily checked that a b = 2i cd (Jcd )a b , with Jcd given in (5.7). It has
its values in the Lie algebra of the Lorentz group. The spin connetion a b
is, for this reason, said to be a Lorentz connection. We can de ne the matrix
i a
b
a
= 2 b Ja whose entries are b . Then,
ra = @a + 2i
cd
a b
(Jcd ) b a
]
= [@ +
:
We nd also that
= ha a
b
b h + hb @ hb (5.32)
is Lorentz invariant, that is,
h
ha0 Æ a b0 @ +
0
a0
i
b0 hb = ha [Æ ab @ +
0
a
b
b ] h :
Notice that the components a b and refer to di erent spaces. Equation
(5.32) describes how changes when the algebra indices are changed into
Riemann indices. It should not be confused with (5.25), which relates the
connection components in two Lorentz{related frames.
Equation (5.32) is frequently written as the vanishing of a \total covariant
derivative" of the tetrad:
Comment 5.1
@ ha a
a
c
h + c h = 0 :
172
5.17
As the two rst indices in a b are not \tensorial", the behavior of
is very special. On the other hand, the indices in Jab are tensorial. The
consequence is that the contraction = 2i ab Jab is not Lorentz invariant.
Actuallly,
x
a0 b0
Ja 0 b0
= a d @ db Ja0 b0 +
0
0
ab
Jab
:
Again, decomposing in terms of the tetrad, we nd
h
a0 b0
0
+ hb @
0
ha
i
Ja0 b0 =
ab
b
a + h @ h Jab
:
Thus, what is really invariant is
=
1 ab + hb @ ha J
ab
2
:
(5.33)
5.18
The \archaic" approach to connections is more intuitive and very
suggestive. It is worth recalling, as it complements the above one. It starts
with the assumption that, under an in nitesimal displacement dx , a eld
su ers a change which is proportional to its own value and to dx . The
proportionality coeÆcient is an \aÆne coeÆcient" (an oldish name !),
so that
x
Æ (x) =
(x) (x) dx
:
If we introduce the entries of the matrix Jab , we verify that this is the same
as
Æ (x) =
1
2
ab
(x) (Jab ) (x) dx
:
Consequently, the variation in the functional form of will be
(x) = Æ (x) @ (x) dx =
Æ
r(x) dx ;
which de nes the covariant derivative
r (x) = @(x) +
5.19 When r = 0, we say that the
(x) :
eld is \parallel transported". In
(x) = 0, or 0(x) = (x). In parallel transport,
that case, we have Æ
x
173
the functional form of the eld does not change. To learn more on the
meaning of parallel-transport and its expression as r = 0, let us look at
the functional variation of the vector eld along a curve of tangent vector
(x) applied to the eld U = d=ds:
(velocity) U . It will be the 1-form Æ
D
:
ds
The purely-functional variation along the curve will vanish if r = 0. This
is what is meant when we say that is parallel-transported along a curve :
the eld is transported in such a way that it su ers no change in its functional
form. The change coming from the argument,
(x)[U ] = [@ +
Æ
] dx [U ] = U r =
d dx =
@ = U @ ;
ds
ds
is exactly compensated by the term U .
When = U itself, we have the acceleration DU =ds. The condition
of no variation of the velocity- eld along the curve will lead to the geodesic
equation
DU = U [@ U +
ds
which is an equation for .
U ] =
dU +
ds
U
U
=0;
5.2.4 Back to Equivalence
x 5.20 We have seen in Section 3.7 how a convenient choice of coordinates
leads to the vanishing of the Levi-Civita connection at a point. Nevertheless,
we had in x 3.11 introduced an observer as a timelike curve, and quali ed
that notion in the ensuing paragraphs. A timelike curve is, actually, an
ideal observer, which is point-like in any local space section. Real observers
are extended in space and can always detect gravitation by comparing the
neighboring curves followed by hers/his parts (x 3.46). Ideal observers can
be arbitrary curves (subsection 3.8.2) and, eventually, can have well-de ned
space-sections all along (subsection 3.8.4).
We shall now see how a tetrad fHa g can be chosen so that, seen from the
frame it represents, the connection can be made to vanish all along a curve.
174
Looking from that frame, an ideal observer will not feel the gravitational
eld.
5.21
Take a di erentiable curve which is an integral curve of a eld U ,
with
= dxds = d ds(s) . The condition for the connection to vanish along ,
a ( (s)) = 0, will be
b
x
U
U @ Ha ( (s)) +
(
(s)) U Ha ( (s)) = 0;
that is,
d
H ( (s)) + ( (s)) U Ha ( (s)) = 0:
(5.34)
ds a
This is simply the requirement that the tetrad (each member Ha of it) be
parallel-transported along . Given a curve and a linear connecion, any
vector eld can be parallel-transported along . The procedure is then very
simple: take, in the way previously discussed, a point P on the curve and
nd the corresponding trivial tetrad fHa (P )g; then, parallel-transport it
along the curve.
For the dual base fH a g, the above formula reads
x
5.22
d a
H ( (s))
ds Take (5.34) in the form
H b (x)
and contract it with U :
(
(s)) U H a ( (s)) = 0:
d H (x) =
ds b
(x)U
(5.35)
d (x)U U :
H (x) =
(5.36)
ds b
Seen from the tetrad, the tangent eld will be U b = H b U , and the above
formula is
d
U b Hb (x) + (x)U U = 0;
(5.37)
ds
or
d
d
D Hb (x) U b = U (x) + (x)U U =
U (x):
(5.38)
ds
ds
Ds
U H b (x)
175
This shows that, if is a self-parallel curve, then
d a
U = 0:
ds
(5.39)
This is the equation for a geodesic, as seen from the frame fHa = Ha D U (x) = F and
If an external force is present, then m Ds
@
@x
g.
d a
U = F a:
(5.40)
ds
This means that, looking from that tetrad, the observer will see the laws of
Physics as given by Special Relativity.
m
5.23
Consider now a Levi-Civita geodesic. In that case there exists a
preferred tetrad ha , which is not parallel-transported along the curve. Its
deviation from parallelism is measured by the spin connection. Indeed, from
(5.32),
x
d h +
ds b
U hb = ha a
b
U :
dc
Uc :
(5.41)
This is the same as
Æ
d b
h = hb ds U
hd b
Æ
(5.42)
We call U the velocity as seen from the frame fha g. Equation (5.32) is
actually a representation of the Equivalence Principle. Let us write it into
still another form,
D
d
hb = hb +
Ds
ds
d
ds
hb =
a
b
d
ds
ha :
(5.43)
This holds for any curve with tangent vector ( dsd ). It means that the frame
fhag can be parallel-transported along no curve. The spin connection forbids
it, and gives the rate of change with respect to parallel transport.
x
5.24 One of the versions of the Principle | the etymological one | says
that a gravitational eld is equivalent to an accellerated frame. Which frame
? We see now the answer: the frame equivalent to the eld represented by
the metric g = abha hb is just the anholonomous frame fha g. Another
176
piece of the Principle says that it is possible to choose a frame in which the
connection vanishes. Let us see now how to change from the \equivalent"
frame fha g to the free-falling frame fHa g. Contracting (5.41) with H a ,
d h + U H a hb = H a hc c b U ;
ds b
which is the same as
d
d a
a
a
a
H hb
hb
H
U H = H hc
ds
ds H a
c
b
U :
The second term in the left-hand side vanishes by Eq.(5.35), so that we
remain with
d
H a hb H a hc c b U = 0 :
(5.44)
ds
What appears here is a point-dependent Lorentz transformation relating the
metric tetrad ha to the frame Ha in which vanishes:
H a = a b hb ; H a = ab hb :
(5.45)
a b = H a hb :
(5.46)
This means
This relation holds on the common domain of de nition of both tetrad elds.
Taking this into (5.44), we arrive at a relationship which holds on the interÆ
section of that domain with a geodesics of the Levi-Civita connection :
Æ
Æ
d a
b ac c bd U d = 0 :
(5.47)
ds
This equation gives the change, along the metric geodesic, of the Lorentz
transformation taking the metric tetrad ha into the frame Ha . The vector
formed by each row of the Lorentz matrix is parallel-transported along the
line. Contracting with the inverse Lorentz transformation
( 1 )a b = ha Hb ;
(5.48)
the expression above gives
Æa
bd
Æ
d
d
U d = ( 1 )a c ds c b = ( 1 ds )ab :
177
(5.49)
This is, in the language of di erential forms,
Æa d d
d
1
a
= ( d) b
:
(5.50)
bd h
ds
ds
Thus on the points of the curve, the connection has the form of a gauge
Lorentz vacuum:
Æa
= ( 1 d)a b
(5.51)
It is important to stress that this is only true along a curve | a onedimensional domain | so that curvature is not a ected. Curvature, the real
gravitational eld, only manisfests itself on two-dimensional domains. Seen
from the frame ha, the geodesic equation has the form
d Æ a Æa Æ b Æc
+ bc U U = 0;
(5.52)
ds U
which is the same as
Æ
d Æa
d
+ ( 1
)a b U b = 0:
U
ds
ds
This expression, once multiplied on the left by , gives
Æ
Æ
d Æa d
d
c a
+ (c a ) U a = (c a U a ) = 0:
(5.53)
U
ds
ds
ds
This is Eq. (5.39) for the present case. Summing up: at each point of the
curve/observer there is a Lorentz transformation taking the accelerated frame
fhag, equivalent to the gravitational eld, into the inertial frame fHag, in
which the force equation acquires the form it would have in Special Relativity.
These considerations can be enlarged to general Lorentz tensors. Take,
for instance, a second order tensor:
b
Æ
T ab = a c b d T cd :
Taking dds of this expression leads to
d Æ ab Æ a Æ cb Æ b Æ ac
d cd
+ c T + c T = ( 1 )a c ( 1 )b d
T :
T
ds
ds
Æ
The covariant derivative according to the connection , which is seen from
the frame ha , is the Lorentz transform of the simple derivative as seen from
the inertial frame Ha , in which vanishes.
Summing up again: at each point of the curve/observer there is a Lorentz
transformation taking the accelerated frame fha g, equivalent to the gravitational eld, into the free falling frame fHa g, in which all tensorial (that is,
covariant) equations acquire the form they would have in Special Relativity.
This is the content of the Equivalence Principle.
178
5.2.5 Two Gates into Gravitation
x 5.25 Starting with a given nontrivial tetrad
eld fha g, two di erent but
equivalent ways of describing gravitation are possible. In the rst, according
to (5.10), the nontrivial tetrad eld is used to de ne a Riemannian metric g ,
from which we can contruct the Levi{Civita connection and the corresponding curvature tensor. As the starting point was a nontrivial tetrad eld, we
can say that such a tetrad is able to induce a metric structure in spacetime,
which is the structure underlying the General Relativity description of the
gravitational eld. We have seen that fhag is not parallel-transported by the
Levi-Civita connection.
On the other hand, a nontrivial tetrad eld can be used to de ne a very
special linear connection, called Weitzenbock connection, with respect to
which the tetrad fha g is parallel. For this reason, this kind of structure has
received the name of teleparallelism, or absolute parallelism, and is the stageset of the so called teleparallel description of gravitation. The important
point to be kept from these considerations is that a nontrivial tetrad eld is
able to induce in spacetime both a teleparallel and a Riemannian structure.
In what follows we will explore these structures in more detail.
x
5.26
It is
Let us consider now the covariant derivative of the metric tensor g .
rg = @g
g
g
;
or, by using (5.24)
rg = ha hb (
ab
+
ba )
:
(5.54)
Therefore, the metricity condition
rg = 0
(5.55)
will only hold when the connection is either purely antisymmetric [(pseudo)
orthogonal],
ab
=
179
ba
;
or when it vanishes identically:
ab
=0:
The Levi{Civita connection falls into the rst case and, as we are going
to see, the Weitzenbock connection into the second case. This means that
both connections preserve the metric.
180
Chapter 6
Gravitational Interaction of the
Fundamental Fields
6.1 Minimal Coupling Prescription
The interaction of a general eld with gravitation can be obtained through
the application of the so called minimal coupling prescription, according to
which the Minkowski metric must be replaced by the riemannian metric
ab ! g = ab ha hb ;
(6.1)
and all ordinary derivatives must be replaced by Fock-Ivanenko covariant
derivatives [1],
@
! D
= @
i ab
! S
2 ab
(6.2)
where ! ab is a connection assuming values in the Lie algebra of the Lorentz
group, usually called spin connection, and Sab is a Lorentz generator written in a representation appropriate to the eld . This \double" coupling
prescription is a characteristic property of the gravitational interaction as
for all other interactions of Nature only the derivative replacement (6.2) is
necessary.
Let us explore deeper this point. The metric replacement (6.1) is a consequence of the local invariance of the lagrangian under translations of the
tangent{space coordinates [2]. This part of the prescription, therefore, is
related to the coupling of the eld energy{momentum to gravitation, and is
universal in the sense that it is the same for all elds. On the other hand,
181
the derivative replacement (6.2) is a consequence of the invariance of the
lagrangian under local Lorentz transformations of the tangent{space coordinates. This part of the prescription, therefore, is related to the coupling of
the eld spin to gravitation, and is not universal because it depends on the
spin contents of the eld.
Another important point of the above coupling prescription is that the
metric change (6.1) is appropriate only for integer{spin elds, whose lagrangian is quadratic in the eld derivative, and consequently a metric tensor
is always present to contract these two derivatives. For half-integer spin elds,
however, the lagrangian is linear in the eld derivative, and consequently no
metric will be present to be changed. In this case, the metric change must
be replaced by the equivalent rule in terms of the tetrad eld,
ea
! ha ;
(6.3)
where ea is the trivial tetrad
@xa
;
(6.4)
=
@x
and ha is a nontrivial tetrad representing a true gravitational eld.
Besides being more fundamental than the metric change (6.1), the tetrad
change (6.3) allows the introduction of a full coupling prescription which encompasses both the metric | or equivalently the tetrad | and the derivative
changes. It is given by
ea
@a ! Da ha
D = ha
@
i ab
! S
;
2 ab
(6.5)
This coupling prescription is general in the sense that it holds for both integer
and half{integer spin elds. For the case of integer spin elds, it yields
automatically the metric replacement (6.1). For the case of half{integer spin
elds, it yields automatically the tetrad replacement (6.3).
6.2 General Relativity Spin Connection
As is well known, a tetrad eld can be used to transform Lorentz into spacetime indices, and vice{versa. For example, a Lorentz vector eld V a is related
to the corresponding spacetime vector V through
V a = ha V :
182
(6.6)
It is important to notice that this applies to tensors only. Connections, for
example, acquire an extra vacuum term under such change [3]:
! a b = ha ! hb + ha @ hb :
(6.7)
On the other hand, because they are used in the construction of covariant
derivatives, connections (or potentials, in physical parlance) are the most
important personages in the description of an interaction. Concerning the
speci c case of the general relativity description of gravitation, the spin conÆ
nection, denoted here by ! ab = Aa b , is given by [4]
Æ
Aab = ha Æ
hb
+ ha
@ hb
ha rÆ hb :
(6.8)
Æ
We see in this way that the spin connection Aa b is nothing but the Levi{
Civita connection
Æ
= 21 g [@ g + @ g
@g ]
(6.9)
rewritten in the tetrad basis. Therefore, the full coupling prescription of
general relativity is
Æ
Æ
@a ! D a ha D (6.10)
with
i Æ ab
S
(6.11)
2 A ab
the general relativity Fock{Ivanenko [1] covariant derivative operator.
Now, comes an important point. The covariant derivative (6.11) applied
to a general Lorentz tensor eld reduces to the usual Levi-Civita covariant
derivative of the corresponding spacetime tensor. For example, take again a
vector eld V a for which the appropriate Lorentz generator is [5]
Æ
D = @
(Sab )c d = i (Æ c a bd
Æ cb ad ) :
(6.12)
It is then an easy task to verify that [6]
Æ
Æ
DV a = ha rV :
(6.13)
On the other hand, no Levi{Civita covariant derivative can be de ned for
half-integer spin elds [7]. For these elds, the only possible form of the
183
covariant derivative is that given in terms of the spin connection. For a
Dirac spinor , for example, the covariant derivative is
Æ
D
i Æ ab
A S ;
2 ab
= @
(6.14)
where
1
i
Sab = ab = [ a ; b ]
(6.15)
2
4
is the Lorentz spin-1/2 generator, with a the Dirac matrices. Therefore, we
may say that the covariant derivative (6.11), which take into account the spin
contents of the elds as de ned in the tangent space, is more fundamental
than the Levi{Civita covariant derivative in the sense that it is able to describe the gravitational coupling of both tensor and spinor elds. For tensor
elds it reduces to the Levi{Civita covariant derivative, but for spinor elds
it remains as a Fock{Ivanenko derivative.
6.3 Application to the Fundamental Fields
6.3.1 Scalar Field
Let us consider rst a scalar eld in a Minkowski spacetime, whose lagrangian is
L = 12
ab
@a @b 2 2 ;
with
=
mc
~
(6.16)
:
(6.17)
The corresponding eld equation is the so called Klein{Gordon equation:
@a @ a + 2 = 0 :
(6.18)
In order to get the coupling of the scalar eld with gravitation, we use
the full coupling prescription
Æ
@a ! D a ha
Æ
D = ha
184
@
i Æ ab
S
:
2 A ab
(6.19)
For a scalar eld, however,
Sab = 0 ;
(6.20)
and the coupling prescription in this case becomes
! [email protected] :
@a
(6.21)
Applying this prescription to the lagrangian (6.16), we get
L =
p
2
Then, by using the identity
p
@
g=
g
p
g @
@ g
g @ g 2 2 :
p
g
Æ
2
it is easy to see that the corresponding eld equation is
Æ
Æ
;
+ 2 = 0 ;
where
Æ
(6.22)
= r @ p1 g @ p
(6.23)
(6.24)
g g @
(6.25)
Æ
is the Laplace{Beltrami operator, with r the Levi{Civita covariant derivative. We notice in passing that it is completely equivalent to apply the
minimal coupling prescription to the lagrangian or to the eld equations.
Furthermore, we notice that, in a locally inertial coordinate system, the rst
derivative of the metric tensor vanishes, the Levi{Civita connection vanishes
as well, and the Laplace{Beltrami becomes the free{ eld d'Alambertian operator. This is the usual version of the (weak) equivalence principle.
6.3.2 Dirac Spinor Field
In Minkowski spacetime, the spinor eld lagrangian is
ic~ a
@a
@a a
mc2 :
2
The corresponding eld equation is the Dirac equation
L
=
i~ a @a
mc = 0 :
185
(6.26)
(6.27)
In the context of general relativity, the coupling of a Dirac spinor with
gravitation is obtained through the application of the full coupling prescription
Æ
Æ
i Æ ab
@a ! D a ha D = ha @
;
(6.28)
A S
2 ab
where now Sab stands for the spin-1/2 generators of the Lorentz group, given
by
ab i
= [ a; b] :
(6.29)
2
4
The spin connection, according to Eq.(6.8), is written in terms of the tetrad
eld as
Sab =
Æ
Aa b = ha
Æ
hb
+ ha
@ hb
:
(6.30)
Applying the above coupling prescription to the free lagrangian (6.26), we
get
L =p
where
= ea ic~ gc
2
a
Æ D
Æ
D
m c2
;
(6.31)
is the local Dirac matrix, which satisfy
f
; g = 2abhahb = 2g :
(6.32)
The corresponding Dirac equation can be obtained through the use of the
Euler-Lagrange equation
@L
@
Æ
D
@L
Æ =0:
@ (D )
(6.33)
The result is the Dirac equation in a Riemann spacetime
i~
Æ
D
mc = 0 :
(6.34)
6.3.3 Electromagnetic Field
In Minkowski spacetime, the electromagnetic eld is described by the lagrangian density
Lem = 14 FabF ab ;
(6.35)
186
where
Fab = @aAb
@b Aa
(6.36)
is the Maxwell eld strength. The corresponding eld equation is
@aF ab = 0 ;
(6.37)
which along with the Bianchi identity
@a Fbc + @c Fab + @bFca = 0 ;
(6.38)
constitute Maxwell's equations. In the Lorentz gauge @aAa = 0, the eld
equation (6.37) acquires the form
@c @ c Aa = 0 :
(6.39)
In the framework of general relativity, the form of Maxwell's equations can
be obtained through the application of the full minimal coupling prescription
(6.5), which amounts to replace
@a ! ha
Æ
D = ha
@
i Æ ab
S
;
2 A ab
(6.40)
Æb c ad )
(6.41)
with
(Sab )c d = i (Æa c bd
the vector representation of the Lorentz generators. For the speci c case of
the electromagnetic vector eld Aa, the Fock-Ivanenko derivative acquires
the form
Æ
Æ
DAa = @Aa + Aab Ab :
(6.42)
It is important to remark once more that the Fock{Ivanenko derivative
is concerned only to the local Lorentz indices. In other words, it ignores the
spacetime tensor character of the elds. For example, the Fock{Ivanenko
derivative of the tetrad eld is
Æ
Æ
D ha = @ha + Aab hb :
(6.43)
Substituting
Æ
Æ
Aa b = ha r hb ;
187
(6.44)
we get
Æ
Dha =
Æ
a
h :
(6.45)
As a consequence, the total covariant derivative of the tetrad ha , that is, a
covariant derivative which takes into account both indices of ha , vanishes
identically:
Æ
@ ha + Aa b hb Æ
a
h =0:
(6.46)
Now, any Lorentz vector eld Aa can be transformed into a spacetime
vector eld A through
A = ha Aa ;
(6.47)
where A transforms as a vector under a general spacetime coordinate transformation. Substituting into equation (6.42), and making use of (6.45), we
get
Æ
Æ
DAa = ha rA :
(6.48)
We see in this way that the Fock{Ivanenko derivative of a Lorentz vector eld
Ac reduces to the usual Levi{Civita covariant derivative of general relativity.
This means that, for a vector eld, the minimal coupling prescription (6.40)
can be restated as
Æ
@a Ac ! ha hc r A :
(6.49)
Therefore, in the presence of gravitation, the electromagnetic eld lagrangian
acquires the form
1p
g F F ;
4
Lem =
where
Æ
Æ
r A @A
F = r A
(6.50)
@ A ;
(6.51)
the connection terms canceling due to the symmetry of the Levi{Civita connection in the last two indices. The corresponding eld equation is
Æ
rF = 0 ;
188
(6.52)
Æ
or equivalently, assuming the covariant Lorentz gauge r A = 0,
Æ Æ
r rA
Æ
R A = 0 :
(6.53)
Analogously, the Bianchi identity (6.38) can be shown to assume the form
@ F + @ F + @ F = 0 :
(6.54)
We notice in passing that the presence of gravitation does not spoil the U(1)
gauge invariance of Maxwell theory. Furthermore, like in the case of the
scalar eld, it results the same to apply the coupling prescription in the
lagrangian or in the eld equations.
189
Chapter 7
General Relativity with Matter
Fields
7.1 Global Noether Theorem
Let us start by brie y reviewing the results of the global | or rst |
Noether's theorem [8]. As is well known, the global Noether theorem is
concerned with the invariance of the action functional under global transformations. For each of such invariances, Noether's theorem determines a
conservation law. In the speci c case of the invariance under a global translation of the spacetime coordinates, the corresponding Noether conserved
current is the canonical energy{momentum tensor
a b =
@L
@
@@a b
Æ ab L ;
(7.1)
with L the lagrangian of the eld .
On the other hand, in the case of the invariance under a global rotation of the spacetime coordinates | that is, under a Lorentz transformation
| the corresponding Noether conserved current is the canonical angular{
momentum tensor
J abc = Mabc + S abc ;
(7.2)
where
Mabc = xb ac
190
xc a b ;
(7.3)
is the orbital angular{momentum, and
@L
S abc = i @@
a
Sbc
(7.4)
is the spin angular{momentum, with Sbc the generators of Lorentz transformations written in the representation appropriate for the eld . Notice that
Mabc is the same for all elds, whereas S abc depends on the spin contents of
the eld .
Notice that the canonical energy{momentum tensor ab is not symmetric
in general. However, using the Belinfante procedure [9] it is possible to de ne
a symmetric energy{momentum tensor for the spinor eld,
ab = ab
1 cab
@' ;
2 c
(7.5)
where
'cab = 'acb = S cab + S abc
S bca :
(7.6)
It can be easily veri ed that
@c 'cab = ab
ba ;
(7.7)
which together with (7.5) show that ab is in fact symmetric.
7.2 Energy{Momentum as Source of Curvature
An old and controversial problem of gravitation is the conservation of energy{
momentum density for both gravitational and matter elds. Concerning the
energy{momentum tensor of matter elds, it becomes problematic mainly
when spinor elds are present [10]. In order to explore deeper these problems, we are going to study the de nition as well as the conservation law
of the gravitational energy{momentum density of a general matter eld. By
gravitational energy{momentum tensor we mean the source of gravitation,
that is, the tensor appearing in the right hand{side of the gravitational eld
equations. For the speci c case of a spinor eld, this energy{momentum
tensor is sometimes believed to acquire a genuine non{symmetric part. As
the left hand{side of the gravitational eld equations are always symmetric, this would call for a generalization of general relativity. However, the
191
gravitational energy{momentum tensor is actually always symmetric, even
for a spinor eld, which shows the consistency and completeness of general
relativity.
Let us consider the lagrangian
L = LG + L
;
(7.8)
where
Æ
c4 p
gR
(7.9)
16G
is the Einstein{Hilbert lagrangian of general relativity, and L is the lagrangian of a general matter eld . The functional variation of L in relation
to the metric tensor g yields the eld equation
LG =
Æ
R
where
Æ 4G
1
g R= 4 T ;
2
c
ÆL
T = p2 g Æg
(7.10)
(7.11)
is the gravitational energy{momentum tensor of the eld . The contravariant components of the gravitational energy{momentum tensor is
ÆL
T = p2 g Æg
:
(7.12)
@L
@ @g (7.13)
In these expressions,
@L
ÆL
= Æg
@g
@
is the Lagrange functional derivative. In general relativity, therefore, energy
and momentum are the source of gravitation, or equivalently, are the source
of curvature. As the metric tensor is symmetric, the gravitational energy{
momentum tensor obtained from either expression (7.11) or (7.12) is always
symmetric. These expressions yield the energy{momentum tensor not only
in the case of the presence of a gravitational eld, but also in the absence.
In the absence of a gravitational eld, a transition to curvilinear coordinates
must be done before the calculation of T . Of course, the metric tensor
in this case will not represent a true gravitational eld, but only e ects of
coordinates.
192
7.3 Energy{Momentum Conservation
Let us obtain now the conservation law of the gravitational energy{momentum tensor of a general source eld . Denoting by L the lagrangian of the
eld , the corresponding action integral is written in the form
Z
1
S=
L d4x :
(7.14)
c
As a spacetime scalar, it does not change under a general transformation of
coordinates. Of course, under a coordinate transformation, the eld will
change by an amount Æ . Due to the equation of motion satis ed by this
eld, the coeÆcient of Æ vanishes, and for this reason we are not going to
take these variations into account. For our purposes, it will be enough to
consider only the variations in the metric tensor g . Accordingly, by using
Gauss theorem, and by considering that Æg = 0 at the integration limits,
the variation of the action integral (7.14) can be written in the form [11]
Z
Z
1 ÆL
1 ÆL
4
ÆS =
Æg d x =
Æg d4 x ;
(7.15)
c Æg
c Æg
with Æ L =Æg the Lagrange functional derivative (7.13). But, we have already seen in section 7.2 that
p g
ÆL
=
T ;
(7.16)
Æg 2 where T is the gravitational energy{momentum tensor of the eld . Therefore, we have
Z
Z
p
1
1
4
Æg p g d4 x :
T
gd x=
T
(7.17)
ÆS =
Æg
2c
2c
On the other hand, under a spacetime general coordinate transformation
x ! x0 = x + ;
(7.18)
with small quantities, the components of the metric tensor change according to
0 (x ) g (x ) = g @ Æg g
g @ @g ;
(7.19)
where only terms linear in the transformation parameter have been kept.
Substituting into (7.17), we get
Z
1
g @ g @ @ g p g d4 x :
ÆS =
T
(7.20)
2c
193
Integrating by parts the second and third terms, neglecting integrals over
hypersurfaces, and making use of the symmetry of T , we get
1
ÆS =
c
Z or equivalently
1
ÆS =
c
p
@ ( g T )
Z h
p g T d4x ;
1
@g
2
p gT )
@ (
Then, by using the identity
@
p
we get
1
ÆS =
c
g=
Z
p
Æ
g
Æ
T p
Æ
r T p
i
g d4 x :
(7.21)
(7.22)
;
(7.23)
g d4 x :
(7.24)
Therefore, from both the invariance condition ÆS = 0 and the arbitrariness
of , it follows that
Æ
r T = 0 :
(7.25)
It is important to remark that this is not a true conservation law in the
sense that it does not lead to a charge conserved in time. Instead, it is
an identity satis ed by the gravitational energy{momentum tensor, usually
called Noether identity [8]. The sum of the energy{momentum of the gravitational eld t and the energy{momentum of the matter eld T is a truly
conserved quantity. In fact, this quatity satis es
@
p
g (t + T ) = 0 ;
(7.26)
which, by using Gauss theorem, yields the true conservation law
dq
=0;
dt
with
q =
Z
t0 + T 0 the conserved charge.
194
(7.27)
p
g d3 x
(7.28)
7.4 Examples
7.4.1 Scalar Field
Let us take the lagrangian of a scalar eld in a Minkowski spacetime,
L =
1 ab
@a @ b 2
2 2 ;
(7.29)
with given by (6.17). From Noether's theorem we nd that the corresponding canonical energy{momentum and spin tensors are given respectively by
a b = @ a @b Æ ab L ;
(7.30)
S abc = 0 :
(7.31)
and
As a consequence of the vanishing of the spin tensor, the canonical energy{
momentum tensor of the scalar eld is symmetric, and is conserved in the
ordinary sense:
@a a b = 0 :
(7.32)
In the presence of gravitation, the scalar eld lagrangian is given by
L =
p
2
g
g @
@ By using the identity
p
Æ
p
2 2 :
(7.33)
g
g Æg ;
2 the dynamical energy{momentum tensor is found to be
g=
p gT = p
g @ @ g L :
(7.34)
Like in the free case, it is symmetric, and conserved in the covariant sense:
Æ
rT = 0 :
195
(7.35)
7.4.2 Dirac Spinor Field
The Dirac spinor lagrangian in Minkowski spacetime is
ic~ a
L =
@a
@a a
mc2 :
(7.36)
2
From the rst Noether's theorem one nds that the corresponding canonical
energy{momentum and spin tensors are given respectively by
a b =
and
S abc =
ic~ a
@b
2
@b a c~ a
Sbc + Sbc
2
;
(7.37)
a ;
(7.38)
with
bc i
= [ b; c ] :
(7.39)
2
4
It should be noticed that, in contrast to the scalar eld case, the canonical
energy{momentum tensor for the Dirac spinor is not symmetric. As already
discussed, however, we can use the Belinfante procedure [9] to construct a
symmetric energy{momentum tensor for the spinor eld, which is given by
1 cab
ab = ab
@' ;
(7.40)
2 c
with
Sbc =
'cab = 'acb = S cab + S abc
S bca :
(7.41)
In the presence of gravitation, the spinor eld lagrangian is
L =p
i~ Æ
gc
D
2
Æ
D mc :
(7.42)
The dynamical energy{momentum tensor of the spinor eld, according to the
de nition (7.11), is found to be
1Æ
' ;
2 D T = ~
where
ic~ ~ =
2
D
196
D (7.43)
(7.44)
is the canonical energy{momentum tensor modi ed by the presence of gravitation, and ' is still given by (7.6), but now written in terms of the spin
tensor modi ed by the presence of gravitation:
S bc = c4~
a ha bc + bc aha :
(7.45)
Equation (7.43) is a generalization of the Belinfante procedure for the
presence of gravitation. In fact, through a tedious but straightforward calculation we can show that
D = g ~ g ~ ;
(7.46)
from which we see that the dynamical energy{momentum tensor T of the
Dirac eld, that is, the Euler{Lagrange functional derivative of the spinor
lagrangian (7.42) with respect to the metric, is always symmetric.
7.4.3 Electromagnetic Field
In Minkowski spacetime, the electromagnetic eld is described by the lagrangian density
1
F F ab :
(7.47)
4 ab
The corresponding canonical energy{momentum and spin tensors are given
respectively by
Lem =
a b = [email protected] Ac F ac + Æ ab Fcd F cd ;
(7.48)
and
S abc = F abAc
F ac Ab :
(7.49)
As in the spinor case, the canonical energy{momentum tensor a b of the
electromagnetic eld is not symmetric. By using the Belinfante procedure,
however, it is possible to de ne the symmetric energy{momentum tensor,
ab = ab
1 cab
@' ;
2 c
(7.50)
where
'cab = 'acb = S cab + S abc
197
S bca :
(7.51)
As can be easily veri ed,
'cab = 2F ca Ab :
(7.52)
Consequently,
ab
=4
F ac F b
1 ab
cd
c + Fcd F
4
(7.53)
is in fact symmetric, and conserved in the ordinary sense:
@a a b = 0 :
(7.54)
In the presence of gravitation, the electromagnetic eld lagrangian is
1p
g F F ;
4
and the dynamical energy{momentum tensor
Lem =
(7.55)
T = p2 g ÆÆgLem
is found to be
T =
F
F
(7.56)
1
g F F :
4 (7.57)
It is symmetric and covariantly conserved:
Æ
r T = 0 :
198
(7.58)
Part II
Teleparallel Gravity
199
Chapter 8
Teleparallel Equivalent of
General Relativity
8.1 Preliminary Considerations
This part of the text is an attempt to describe the present state-of-art of
Teleparallel Gravity, and like any text on a not yet completely established
subject, it is strongly biased by the author's point of view. For example, it
is assumed that, at least up to now, there exists no compelling experimental
evidence of the necessity of including torsion, besides curvature, in the description of the gravitational interaction. Consequently, the basic guideline
of our description will be to look for a gauge theory which results completely
equivalent to general relativity. This theory, given by a gauge theory for
the translation group, is what we call Teleparallel Gravity, or Teleparallel
Equivalent of General Relativity. As such, it is not an alternative theory
to general relativity, but simply a di erent description of the same theory.
Of course, there exists in the literature a lot of di erent alternative theories
for the gravitational interaction, which are much more general than general
relativity. These models, however, will not be considered here.
The rst attempt to unify gravitation and electromagnetism was made by
Weyl in 1918 [12]. Despite not successful, this beautiful proposal introduced
for the rst time the notions of gauge transformations and gauge invariance,
and can be considered as the seed that has grown to what is known today as
Gauge Theories. On the other hand, the notion of absolute parallelism (or
teleparallelism) was introduced by Einstein in the late twenties as another
attempt to unify gravitation and electromagnetism. This attempt of uni cation did not succeed either, but like Weyl's work, it introduced concepts
200
which remains important up to the present day. It is interesting to notice
that Einstein did not agree with Weyl's theory: Although your idea is so
beautiful, I have to declare frankly that, in my opinion, it is impossible that
the theory corresponds to nature. Weyl did not agree with Einstein's theory
either: I prefer not to believe in distant parallelism for a number of reasons.
First, my mathematical intuition objects to accepting such an arti cial geometry; I nd it diÆcult to understand the force that would keep the local tetrads
at di erent points and in rotated positions in a rigid relationship. This difference of opinions led to an intense exchange of letters between Einstein
and Weyl, which involved also other physicists, like Pauli for example, who
wrote to Weyl ... let me emphasize that side of matter concerning which I
am in full agreement with you: your incorporation of spinor theory into gravitational theory. I am as dissatis ed as you are with distant parallelism and
your proposal to let the tetrads rotate independently at di erent space-points
is a true solution. A very good account of the history underlying the birth
of both gauge theories and teleparallelism can be found in Ref. [13].
Concerning the notion of teleparallelism, no new advances were made
in the following three decades. Then, works by Mller [14], Pellegrini &
Plebanski [15], Hayashi & Nakano [16], and Hayashi & Shirafuji [17] produced
a revival of those ideas, which since then have received considerable attention,
mainly in the context of gauge theories for the translation and Poincare the
groups. Other approaches, based on the gauge cation of the Lorentz group
have also been proposed mainly by Utiyama [18] and Kibble [19]. An almost
exhaustive list of publications on the gauge approach to gravitation up to
1995 can be found in [20].
8.2 General Concepts
According to the point of view of the teleparallel equivalent of general relativity, curvature and torsion are able to provide each one equivalent descriptions
of the gravitational interaction. Conceptual di erences, however, show up.
According to general relativity, curvature is used to geometrize spacetime,
and in this way successfully describe the gravitational interaction. Teleparallelism, on the other hand, attributes gravitation to torsion, but in this case
torsion accounts for gravitation not by geometrizing the interaction, but by
acting as a force. This means that, in the teleparallel equivalent of general relativity, there are no geodesics, but force equations quite analogous
to the Lorentz force equation of electrodynamics [2]. Gravitational interaction, thus, can be described alternatively in terms of curvature, as is usually
done in general relativity, or in terms of torsion, in which case we have the
201
so called teleparallel gravity. Whether gravitation requires a curved or a
torsioned spacetime, therefore, turns out to be a matter of convention.
The important point of teleparallel gravity is that it corresponds to a
gauge theory for the translation group [2]. Due to the peculiar character
of translations, any gauge theory including these transformations will di er
from the usual internal gauge models in many ways, the most signi cant being
the presence of a tetrad eld. On the other hand, a tetrad eld can naturally
be used to de ne a linear Weitzenbock connection, which is a connection
presenting torsion, but no curvature. A tetrad eld can also naturally be used
to de ne a riemannian metric, in terms of which a Levi{Civita connection
can be constructed. As is well known, it is a connection presenting curvature,
but no torsion. It is important to keep in mind that torsion and curvature
are properties of a connection [3], and many di erent connections can be
de ned on the same space [21]. Therefore one can say that the presence of
a nontrivial tetrad eld in a gauge theory induces both a teleparallel and a
riemannian structures in spacetime. The rst is related to the Weitzenbock,
and the second to the Levi{Civita connection.
The question then arises: What is the reason for gravitation to present
two equivalent descriptions? The answer is related to the quite peculiar
property of gravitation, the so called Universality. Let us explore this point
in more details. Like any other interaction of Nature, gravitation presents a
description in terms of a gauge theory. In fact, teleparallel gravity is known
to be a gauge theory for the translation group, with torsion playing the role of
force. On the other hand, universality of gravitation means that all particles
of Nature feel gravity the same. In other words, bodies with di erent masses
will feel a di erent gravitational force in such a way that all these bodies will
acquire the same acceleration. As a consequence of this property, it turns
out possible to describe gravitation not as a force, but as a deformation of
spacetime. More precisely, according to this view, a gravitational eld is
supposed to produce a curvature in spacetime, the gravitational interaction
being achieved in this case by letting particles to follow the geodesics of
spacetime. This is the approach used by the general relativity description
of gravitation. It is important to notice that only an interaction presenting
the Universality property can be described by a geometrization of spacetime.
Incidentally, for historical reasons, the metric description provided by general
relativity was discovered before the gauge description provided by teleparallel
gravity.
202
8.3 Gauge Transformations
As usual for gauge theories, spacetime is assumed to be the base{space of
teleparallel gravity. We use the greek alphabet (, , , = 1; 2; 3; 4)
to denote indices related to this space. Its coordinates, therefore, will be
denoted by x , and its metric by g . At each point of spacetime, there is a
tangent space attached to it, given by a Minkowski space, which is the ber of
the corresponding tangent{bundle. We use the latin alphabet (a, b, c, =
1; 2; 3; 4) to denote indices related to this space. Its coordinates, therefore,
will be denoted by xa , and its metric by ab . As gauge transformations take
place in this space, these will also be the algebra indices of the gauge model.
The holonomic derivatives in these two spaces can be identi ed by
@ = (@ xa ) @a ; @a = (@a x ) @ ;
(8.1)
where @ xa is a trivial holonomic tetrad, with @a x its inverse.
A gauge transformation is de ned as a local translation of the ber coordinates,
x0a = xa + a(x ) ;
(8.2)
with a (x ) | a for short | the corresponding parameters. It can be written
in the form x0 = Ux, with U an element of the translation group. For an
in nitesimal translation,
U = 1 + ÆaPa ;
(8.3)
with Æa representing the in nitesimal parameters, and Pa = @a standing for
the generators of in nitesimal translations, which satisfy
[Pa ; Pb ] = 0 :
(8.4)
In terms of these generators, the in nitesimal version of transformation (8.2)
becomes
Æxa = Æb Pb xa :
(8.5)
Let us study now the coupling of a general matter eld to gravitation.
As we have already seen, the gravitational coupling prescription includes
two parts: the coupling of the eld energy{momentum to gravity, and the
coupling of the eld spin to gravity. To start with, we are going to consider
here only the rst part, that is, the coupling of the eld energy{momentum
203
density to gravity. Accordingly, let us consider a scalar eld , which under
a gauge transformation changes according to
0(x ) = U (x ) :
(8.6)
The corresponding in nitesimal transformation, therefore, is
Æ = ÆaPa ;
(8.7)
with Æ standing for the functional change at the same x , which is the
relevant transformation for gauge theories. It is important to remark that, as
a derivative operator, the translation generators are able to act on any source
eld through their arguments. Therefore, every source eld in Nature will
respond to their action, and consequently will couple to the gauge potentials.
Consequently, provided the coupling of the spin of the eld to gravitation is
neglected, every source eld in Nature will feel gravitation the same way. This
is the origin of the universality property of the gravitational interaction.
In order to de ne the gauge covariant derivative of (x ), we must rst
introduce the gauge potentials of the model, which will be denoted by
B = B a Pa :
(8.8)
Using the general de nition of gauge covariant derivatives,
Æ
;
(8.9)
Æa
where the velocity of light c was introduced for dimensional reasons, the
covariant derivative of (x ) turns out to be
D = @ + c
2Ba
D = @ + c
2B a
Pa :
(8.10)
By using Eq.(8.1), this covariant derivative can be rewritten in the form
D = ha @a ;
(8.11)
ha = @ xa + c 2 B a D xa
(8.12)
where
is a tetrad eld. Notice that the gravitational eld appears as the nontrivial
part of the tetrad [19]. Making use of Eqs.(8.5) and (8.14), it is easy to see
that, as in fact it should be, the tetrad is gauge invariant:
h0a = ha :
204
From the covariance requirement for D , we can get the gauge transformation of the potentials:
B0 = UB U
1
+ [email protected] U
1
:
(8.13)
The corresponding in nitesimal transformation is
B 0a = B a
c2 @ Æa :
(8.14)
The eld strength F a is de ned as the covariant derivative of the gauge
potential B a , and analogously to the U (1) electromagnetic gauge theory, it
reads
F a = @ B a
@ B a :
(8.15)
Notice that, as expected for an abelian theory, F a is invariant under a
gauge transformation:
F 0a = F a :
Consequently, its ordinary and gauge covariant derivatives coincide.
The expression for the gauge covariant derivative operator of source elds,
D = ha @a ;
(8.16)
is actually the de nition of a non{holonomous basis. In the absence of gravitation, ha becomes trivial, and it reduces to the coordinate basis @ appearing in Eq.(8.1). This new basis, induced by the presence of the gravitational
eld, satisfy the commutation relation
[D ; D ] = c
2
F a Pa :
(8.17)
Therefore, as usual in gauge theories, the commutator of covariant derivative
operators yields the eld strength F a . There is a di erence, though: as we
are going to see, in contrast to the usual gauge theories, the eld strength
here will be directly related to the spacetime geometry.
8.4 Spacetime Geometric Structures
The presence of a nontrivial tetrad eld allows to de ne the linear Weitzenbock connection [3]
= ha @ ha ;
205
(8.18)
which is a connection presenting torsion, but no curvature. As a consequence
of this de nition, the eld strength F a can be written in the form
F a = c2 ha T ;
(8.19)
T =
(8.20)
where
is the torsion of the Weitzenbock connection. Moreover, the commutation
relation (8.17) acquires the form
[D ; D ] = T D ;
(8.21)
which shows that torsion plays also the role of the non{holonomy of the gauge
covariant derivative.
The presence of a Weitzenbock connection allows the introduction of a
spacetime covariant derivative which, acting for example on a spacetime covariant vector V , reads
r V @ V
V
:
(8.22)
From this de nition, one can easily see that, as a consequence of Eq.(8.18),
r ha = @ ha
ha 0:
(8.23)
This is the condition of absolute parallelism, which implies that the spacetime
underlying a translational gauge theory is naturally endowed with a teleparallel structure. In other words, a Weitzenbock four{dimensional manifold [22]
is present when considering a gauge theory for the translation group [17].
Besides the teleparallel structure, the presence of a nontrivial tetrad induces also a riemannian structure in spacetime. As the tetrad satis es
ha ha = Æ ; ha hb = Æ ab ;
(8.24)
if tangent-space indices are raised and lowered with the lorentzian metric
ab, spacetime tensor indices will necessarily be raised and lowered with the
riemannian metric
g = ab ha hb :
With this metric, a linear connection can be introduced,
Æ
1 = g [@ g + @ g @ g ] ;
2
206
(8.25)
(8.26)
which is the Levi{Civita connection of general relativity. As is well known,
it is a connection presenting curvature, but no torsion. Its curvature
Æ
Æ
R = @
Æ
+
Æ
( $ )
(8.27)
is directly related to the presence of the gravitational eld. The Levi{Civita
connection allows the introduction of another spacetime covariant derivative
which, acting for example on a spacetime covariant vector V , reads:
Æ
Æ
r V = @ V
As can be easily veri ed, both connections
V :
and
(8.28)
Æ
preserve the metric:
Æ
r g = r g = 0 :
Æ
Substituting now g into
,
=
we obtain the relation
Æ
+ K ;
(8.29)
where
1 T + T T (8.30)
2
is the contorsion tensor. Notice that the curvature of the Weitzenbock connection vanishes identically:
K =
R = @
Substituting
+
( $ ) 0 :
(8.31)
as given by Eq.(8.29), we get
Æ
R = R + Q 0;
(8.32)
where
Q = D K D K + K K K K (8.33)
is a tensor written in terms of the Weitzenbock connection only. Here, D is
the teleparallel covariant derivative, which is nothing but the Levi-Civita covariant derivative of general relativity rephrased in terms of the Weitzenbock
connection. In other words, it is the covariant derivative (8.28) with the
Levi{Civita connection replaced by
Æ
=
207
K :
(8.34)
Acting on a spacetime vector V , therefore, its explicit form is [6]
D V @ V + (
K ) V :
(8.35)
Notice that this covariant derivative, despite equivalent to the Levi{Civita
covariant derivative, depends on the Weitzenbock connection only. For this
reason, it is called the teleparallel equivalent of the Levi{Civita covariant
derivative.
From these considerations, we see that the presence of a nontrivial tetrad
eld induces in spacetime both a teleparallel and a riemannian structures.
These two structures are closely related to each other, from where an interÆ
esting interpretation can be obtained: the contribution R coming from
the Levi{Civita connection compensates exactly the contribution Q coming from the Weitzenbock connection, yielding an identically zero total curvature tensor R . This is a constraint satis ed by the Levi{Civita and
Weitzenbock connections, and is the fulcrum of the equivalence between the
riemannian and the teleparallel descriptions of gravitation.
8.5 Lagrangian and Field Equations
As usual in gauge theories, the dynamics of the gauge elds can be obtained
from a lagrangian quadratic in the eld strength,
LG = 16hG 14 F a F b g Nab ;
(8.36)
where G is the gravitational constant, and h = det(ha ) is the jacobian of the
transformation (8.16). Due to the presence of the tetrad eld, algebra and
spacetime indices can be changed into each other, and consequently appear
mixed up in the lagrangian. This means that
Nab = ab g ab hc hc
must include all cyclic permutations of a; b; c. A simple calculation shows
that
Nab = ab hc hc + 2 ha hb 4 ha hb :
(8.37)
Substituting in (8.36), the gauge eld lagrangian turns out to be
LG = 16hG F a F b g 41 hd hd ab + 21 ha hb
208
ha hb :
(8.38)
Then, by using Eq.(8.19), it becomes
hc4 1 LG = 16G 4 T T + 21 T T T
T :
(8.39)
In order to obtain the vacuum eld equations, it is convenient to rewrite the
lagrangian (8.39) in the form [23]
4
LG = 16hcG S T ;
(8.40)
where
1 1 (T + T T )
g T
g T :
(8.41)
4
2
By performing variations in relation to the gauge eld B a , we obtain the
teleparallel version of the gravitational eld equations,
S =
@ (hSa )
where Sa 4G (hj a ) = 0 ;
c4
(8.42)
haS . Analogously to the Yang-Mills theories,
@L
c4 h hj G =
h S T
h L
a
@ha 4G
a
a
G
(8.43)
stands for the gauge current, which in this case represents the energy and
momentum of the gravitational eld [24].
Now, by using Eq.(8.18) to express @ ha , the eld equations (8.42-8.43)
can be rewritten in a purely spacetime form,
@ (hS )
4G (ht ) = 0 ;
c4
(8.44)
where now
c4 h S Æ LG
(8.45)
4G stands for the canonical energy-momentum pseudotensor of the gravitational
eld. We remark that, despite not explicitly apparent, as a consequence
of the local Lorentz invariance [25] of the gauge Lagrangian LG , the eld
equation (8.44) is symmetric in ().
ht =
209
8.6 Equivalence with General Relativity
Let us take again the gravitational gauge lagrangian, as given by Eq.(8.40),
4
LG = 16hcG S T ;
(8.46)
with S given by Eq.(8.41). This lagrangian depends on the Weitzenbock
connection only. By using the relation (8.29), however, it can be rewritten
in terms of the Levi{Civita connection only. Up to a total divergence, the
result is the Hilbert{Einstein Lagrangian of general relativity
c4 p Æ
gR ;
(8.47)
16G
where the identi
g has been made. We see in this way that
the lagrangian of a gauge theory for the translation group is equivalent to
the Hilbert{Einstein Lagrangian of general relativity.
As a consequence of the equivalence between the lagrangians, the corresponding eld equations must also be equivalent. In fact, take the gauge eld
equation written in the form
4G @ (hS )
(ht ) = 0 ;
(8.48)
c4
with (ht ) given by Eq.(8.45). This equation depends on the Weitzenbock
connection only. Again, by using the relation (8.29), it can be rewritten in
terms of the Levi{Civita connection only. As expected, due to the equivalence
between the corresponding lagrangians, it is the same as Einstein's equation:
Æ
Æ
1
(8.49)
R 2 Æ R = 0 :
From the point of view of the dynamics of the gravitational eld, therefore,
a gauge theory for the translation group is completely equivalent to general
relativity. As we are going to see in Chapter 10, this equivalence remains
true in the presence of matter elds.
L=
p
cation h =
8.7 Energy{Momentum Density of Gravitation
The de nition of an energy-momentum density for the gravitational eld
is one of the oldest and most controversial problems of gravitation. As a
210
true eld, it would be natural to expect that gravity should have its own
local energy{momentum density. However, it is usually asserted that such a
density can not be locally de ned because of the equivalence principle [26].
As a consequence, any attempt to identify an energy-momentum density for
the gravitational eld leads to complexes that are not true tensors. The rst
of such attempt was made by Einstein who proposed an expression for the
energy-momentum density of the gravitational eld which was nothing but
the canonical expression obtained from Noether's theorem [27]. Indeed, this
quantity is a pseudotensor, an object that depends on the coordinate system.
Several other attempts have been made, leading to di erent expressions for
the energy{momentum pseudotensor for the gravitational eld [25, 28, 11].
Despite the existence of some controversial points related to the formulation of the equivalence principle [29], it seems true that, in the context of general relativity, no tensorial expression for the gravitational energy-momentum
density can exist. However, as our results show, in the gauge context, the
existence of an expression for the gravitational energy-momentum density
which is a true spacetime and gauge tensor turns out to be possible. Accordingly, the absence of such expression should be attributed to the general
relativity description of gravitation, which seems to be not the appropriate
framework to deal with this problem [30].
In spite of some skepticism [26], there has been along the years a continuous interest in this problem [31]. In particular, there is the so called quasilocal
approach which is highly clarifying [32]. According to this approach, for each
gravitational energy-momentum pseudotensor, there is an associated superpotential which is a hamiltonian boundary term. The energy{momentum dened by such a pseudotensor does not really depend on the local value of the
reference frame, but only on the value of the reference frame on the boundary
of a region | then its quasilocal character. As the relevant boundary conditions are physically acceptable, this approach validates the pseudotensor
approach to the gravitational energy{momentum problem.
Due to the fundamental character of the geometric structure underlying
gauge theories, the concept of currents, and in particular the concepts of
energy and momentum, are much more transparent when considered from
the gauge point of view [33]. Accordingly, we consider gravity as described
by the so called teleparallel equivalent of general relativity. As we have
already seem, the gravitational eld equations are
@ (hSa )
4G (hj a ) = 0;
c4
211
(8.50)
where, analogously to the Yang-Mills theories [5],
@ LG c4 h =
h S T @ha 4G a hj a ha LG
(8.51)
stands for the gravitational gauge current. The term (hSa ) is called superpotential in the sense that its derivative yields the gauge current (hj a ).
Because of the anti-symmetry of Sa in the last two indices, (hj a ) is conserved as a consequence of the eld equation:
@ (hj a ) = 0:
(8.52)
Making use of the identity
@h h
= h(
K ) ;
(8.53)
K ) ja = 0 ;
(8.54)
this conservation law can be rewritten as
D j a @ j a + (
where D is the teleparallel version of the covariant derivative [6], de ned by
Eq.(8.35). Now, as can be easily checked, j a transforms covariantly under
a general spacetime coordinate transformation, and is invariant under local
(gauge) translation of the tangent-space coordinates. This means that ja is
a true spacetime and gauge tensor.
Let us now proceed further and nd out the relation between the above
gauge approach and general relativity. By using Eq.(8.18) to express @ha ,
the eld equation (8.50) can be rewritten in a purely spacetime form,
@ (hS )
4G (ht ) = 0 ;
c4
(8.55)
where now
c4
h S Æ LG
(8.56)
4G
stands for the teleparallel version of the canonical energy-momentum pseudotensor of the gravitational eld.
It is important to remark that the canonical energy-momentum pseudotensor t is not simply the gauge current j a with the algebraic index
\a" changed to the spacetime index \". It incorporates also an extra term
coming from the derivative term of Eq. (8.50):
ht =
t
=
ha
j
c4
a+
4G
212
S
:
(8.57)
We see thus clearly the origin of the connection-term which transforms the
gauge current j a into the energy-momentum pseudotensor t . Through
the same mechanism, it is possible to appropriately exchange further terms
between the derivative and the current terms of the eld equation (8.55),
giving rise to di erent de nitions for the energy-momentum pseudotensor,
each one connected to a di erent superpotential (hS ). Like the gauge
current (hj a ), the pseudotensor (ht ) is conserved as a consequence of the
eld equation:
@ (ht ) = 0 :
(8.58)
However, in contrast to what occurs with ja , due to the pseudotensor character of t , this conservation law can not be rewritten with a covariant
derivative.
Because of its simplicity and transparency, the teleparallel approach to
gravitation seems to be much more appropriate than general relativity to deal
with the energy problem of the gravitational eld. In fact, Mller already
noticed a long time ago that a satisfactory solution to the problem of the
energy distribution in a gravitational eld could be obtained in the framework of a tetrad theory. In our notation, his expression for the gravitational
energy-momentum density is [34]
ht =
@L
@ ha
@@ha Æ L ;
(8.59)
which is nothing but the usual canonical energy{momentum density yielded
by Noether's theorem. Using for L the gauge Lagrangian (8.46), it is an easy
task to verify that Mller's expression coincides exactly with the teleparallel
energy{momentum density appearing in the eld equation (8.55-8.56). Since
ja is a true spacetime tensor, whereas t is not, we can say that the gauge
current ja is an improved version of the Mller's energy-momentum density
t . Mathematically, they can be obtained from each other by Eq. (8.57).
According to the quasilocal approach, to any energy-momentum pseudotensor there is an associated superpotential which is a hamiltonian boundary term [32]. On the other hand, the teleparallel eld equations explicitly
exhibit both the superpotential and the gravitational energy{momentum complex. We see then that, in fact, by appropriately exchanging terms between
the superpotential and the current terms of the eld equation (8.55), it is possible to obtain di erent gravitational energy{momentum pseudotensors with
their associated superpotentials. In this context, the above results can be
rephrased according to the following scheme. First, notice that the left-hand
side of the eld equation (8.55) as a whole is a true tensor, though each one
213
of its two terms is not. Then if we extract the spurious part from the rst
term | so that it becomes a true spacetime and gauge tensor | and add
this part to the second term | the energy-momentum density | it becomes
also a true spacetime and gauge tensor. We thus arrive at the gauge-type
eld equation (8.50), with (hSa ) as the superpotential, whose corresponding
expression for the conserved energy{momentum density for the gravitational
eld, given by ja , which is a true spacetime and gauge tensor.
8.8 Bianchi Identities
Analogously to the Maxwell theory, the rst Bianchi identity of the gauge
theory for the translation group is given by
@ F a + @ F a + @ F a = 0 :
(8.60)
By using Eqs.(8.19) and (8.33), it can be rewritten in a purely spacetime
form:
Q + Q + Q = 0 :
(8.61)
Then, by making use of relation (8.32), it is easy to verify that it coincides
with the rst Bianchi identity of general relativity:
Æ
Æ
Æ
R + R + R = 0 :
(8.62)
It is important to remark at this point that, in contrast to the usual internal
gauge theories, teleparallel gravity does not present the duality symmetry.
In other words, the eld equation (8.55) does not coincide with the Bianchi
identity written for the dual F~ a = (1=2) F a of F a [3].
As is well known, general relativity has also a second Bianchi identity,
Æ Æ
Æ Æ
Æ Æ
r R + r R + rR = 0 ;
whose contracted form is
Æ hÆ
R
r
i
1 Æ Æ
2 R
= 0:
(8.63)
(8.64)
The corresponding second Bianchi identity of teleparallel gravity is
D Q + D Q + D Q = 0 :
214
(8.65)
It is equivalent to (8.63), as can be seen by using relations (8.29) and (8.32).
Then, through a tedious but straightforward calculation, the contracted form
of the identity (8.65) is found to be
D @ (hS
)
4G (ht ) = 0 :
c4
(8.66)
This identity means simply that the teleparallel covariant derivative of the
sourceless eld equation (8.55) vanishes identically.
215
Chapter 9
Equation of Motion of Spinless
Particles
9.1 Gravitational Lorentz Force
We consider now the motion of a particle of mass m in a gravitational eld
described by teleparallel gravity. As far as teleparallel gravity corresponds
to a gauge theory for the translation group, we can rely on an analogy with
the electromagnetic case to obtain the interaction of the particle with the
gravitational eld. Thus, analogously to what occurs in electrodynamics [11],
we assume the interaction of the particle with the gravitational eld to be
described by the action
c
2
Z b
a
B a pa dx ;
(9.1)
with the integration taken along the world line of the particle. In this expression, pa is the Noether conserved charge of a free particle under the
transformation of the gauge group [35, 36]. In other words, pa is the four{
momentum
pa = m c ua
of the particle, where
dxa Dxa
a
u =
=
(9.2)
d
ds
is the tangent{space four{velocity, with
d = (ab dxa dxb )1=2
216
(9.3)
the Minkowski invariant interval, and
ds = (g dx dx )1=2
(9.4)
the spacetime invariant interval. In the de nition (9.2), Dxa is the non{
holonomous basis
Dxa = ha dx :
(9.5)
We recall that the tangent{space and the spacetime four{velocities are related
by
ua = ha u ;
(9.6)
where
dx
:
(9.7)
ds
The complete action describing a particle of mass m in a gravitational
eld, therefore, is given by
u =
S=
Z bh
a
i
m a
B ua dx ;
c
m c d
(9.8)
where the rst term represents the action of a free particle, and the second
the interaction action. Notice that, in writing this action, we have already
assumed the equality between inertial and gravitational masses, as stated by
the (weak) equivalence principle [25]. The corresponding equation of motion
can be obtained through a functional variation of the action (9.8). This is
done by considering a spacetime variation
x ! x + Æx ;
(9.9)
and assuming that the action is invariant under such transformation: ÆS = 0.
From this condition, we get
Z b
a
mc
ha
dua
d
m a
F u u
c a
m a B u @ ua Æx ds = 0 : (9.10)
c By using the identity
c 2 B a = ha 217
@ xa ;
the last term of the Eq.(9.10) can be rewritten in the form
mc
Z b
a
ua @ ua
1
d
Æx ds :
ds
Then, as ua ua = 1, we see that
ua @ ua = ua @ ua 0 ;
and consequently the contribution of this term vanishes. From the arbitrariness of Æx, we get then the following equation of motion [2]
dua
= c 2 F a ua u :
(9.11)
ds
This is the gravitational analog of the electromagnetic Lorentz force. Its
solution determines the trajectory of the particle under the in uence of a
gravitational eld. It is interesting to notice that, while in the electromagnetic case the particle four{acceleration is proportional to the relation e=m,
with e its electric charge, in the gravitational case the mass disappears from
the equation of motion. This is, of course, a consequence of the assumed
equivalence between the gravitational and inertial masses.
As we are going to see, there are two di erent ways of interpreting the
force equation (9.11). In fact, it can be rewritten alternatively in terms
of magnitudes related to the teleparallel or to the riemannian structures of
spacetime, giving rise respectively to the teleparallel and the metric description of the particle motion.
ha 9.2 Torsion: Force Equation
We start with the rst alternative, which corresponds to transform algebra
into spacetime indices in such a way to get the force equation (9.11) written
in terms of the Weitzenbock connection only. By using Eq.(8.19), as well as
the relation
du
du
(9.12)
ha a = a u u ;
ds
ds
with a the spacetime four{acceleration of the particle, it reduces to
du
ds
u u = T u u :
218
(9.13)
The left{hand side of this equation is the Weitzenbock covariant derivative of
u along the world{line of the particle. The presence of the torsion tensor on
its right hand{side means essentially that torsion plays the role of an external
force in teleparallel gravity. According to this description, therefore, the only
e ect of the gravitational eld is to induce a torsion in spacetime, which will
then be the responsible for determining the trajectory of the particle.
It is worthy noticing that, by using the de nition (8.20) of torsion, the
force equation (9.13) can be rewritten in the form
du
(9.14)
u u = 0 :
ds
Notice however that, as is not symmetric in the last two indices, the
left hand{side is not the covariant derivative of the four{velocity along the
trajectory of the particle, and consequently this equation is not a geodesic
equation. This means that spinless particles do not follow geodesics in the
induced Weitzenbock spacetime.
9.3 Curvature: Geodesic Equation
Let us transform again algebra into spacetime indices, but now in such a way
to get the force equation (9.11) written in terms of the Levi{Civita connection
only. Following the same steps used earlier, we get
du
(9.15)
u u = T u u :
ds
Then, by taking into account the symmetry of u u under the exchange
( $ ), we can rewrite it in the form
du
(9.16)
u u = K u u :
ds
As K is skew{symmetric in the rst two indices, Eq.(9.16) becomes
du
( K ) u u = 0 :
(9.17)
ds
Then, by using Eq.(8.29), we get nally
du Æ
(9.18)
u u = 0 :
ds
This is precisely the geodesic equation of general relativity, which means that
the trajectories followed by spinless particles are geodesics of the induced
Riemann spacetime. According to this description, therefore, the only e ect
of the gravitational eld is to induce a curvature in spacetime, which will
then be the responsible for determining the trajectory of the particle.
219
9.4 Weak Equivalence Principle
We discuss now the (weak) equivalence principle in both teleparallel gravity and general relativity. Let us consider rst the teleparallel equation of
motion,
du
(9.19)
u u = 0 :
ds
In a locally inertial coordinate system, the rst derivative of the metric tensor
vanishes: @ g = 0. Since g = ha ha , we have
@ (ha ha ) = ha @ ha + ha @ ha = 0 ;
(9.20)
which means that, in this coordinate system, the Weitzenbock connection
becomes skew{symmetric in the rst two indices:
=
:
(9.21)
As u u is symmetric under ( $ ), the second term of Eq.(9.19) vanishes,
and we get the equation of motion of a free particle. This is the teleparallel
version of the (weak) equivalence principle.
Let us consider now the geodesic equation of general relativity:
du Æ
(9.22)
u u = 0 :
ds
In a locally inertial coordinate system, the rst derivative of the metric tensor
vanishes, the Levi{Civita connection vanishes as well, and consequently the
geodesic equation (9.22) becomes the equation of motion of a free particle.
This is the usual general relativity version of the (weak) equivalence principle.
9.5 Equivalence of the Actions
There is a further interesting point related to the functional action yielding
the gravitational Lorentz force. The action integral (9.8) can be written in
the form
S = mc
Z b
d
a
ds
+c
2
Ba
By introducing the constraint
ha ua u = 1 ;
220
ua
u
ds :
(9.23)
it becomes
S = mc
Z b
d
a
ds
ha
ua u + c
2
Ba
ua
u
ds ;
(9.24)
or equivalently
S = mc
Z b
d
a
ds
ha
+c
2
Ba
ua u ds :
(9.25)
Now, from (9.6) we get the identity
@xa d a
= h ;
@x ds which reduces the action functional to the form
S = mc
or
S=
Z b
a
ha ua u ds ;
Z b
a
m c ds :
(9.26)
(9.27)
(9.28)
This is the general relativity version of the action. Notice that in this case,
the gravitational interaction is included in the line element ds. This is in
agreement with general relativity, according to which the gravitational interaction is described by a geometrization of spacetime. On the other hand, in
the teleparallel functional action (9.8), the gravitational interaction is not included in the line element, but is taken into account through a direct coupling
of the gravitational eld B a with the particle four{momentum pa . This is
in agreement with the gauge approach to gravitation of teleparallel gravity.
221
Chapter 10
Teleparallel Interaction of the
Fundamental Fields
10.1 Teleparallel Spin Connection
In order to obtain the teleparallel version of the minimal coupling prescription, it is necessary to nd rst the correct teleparallel spin connection. Inspired in the de nition (6.8), which gives the general relativity spin connection, it is usual to de ne the teleparallel spin connection in the form
Aa b = ha hb
+ ha @ hb ha r hb ;
(10.1)
where is the Weitzenbock connection, and r the corresponding covariant derivative. However, as a consequence of the absolute parallelism
condition (8.23), we see that Aab = 0. This does not mean, however, that
in teleparallel gravity the spin connection vanishes. It only means that we
have made a wrong guess.
Let us then adopt a di erent procedure to look for the teleparallel spin
connection. Our basic guideline will be to nd a coupling prescription which
results equivalent to the coupling prescription of general relativity. This can
be achieved by taking Eq.(8.29) and rewriting it in the tetrad basis. By using
the transformation property of connections, we get
Æ
Aa b = K a b + 0 ;
(10.2)
K a b = ha K hb ;
(10.3)
where
222
and where we have already used the fact that
ha hb
+ ha
@ hb
=0 :
Notice that the zero connection appearing in Eq.(10.4) is crucial in the sense
that it is the responsible for making the right hand-side a true connection.
Therefore, based on these considerations we can say that the teleparallel spin
connection is given by minus the contorsion tensor plus a zero-connection [37]:
Aab = K a b + 0 :
(10.4)
Æ
It is important to remark that, di erently from Aa b which depends on the
Levi-Civita connection only [see Eq.(6.8)], Aab depends on the Weitzenbock
connection only.
Like any connection assuming values in the Lie algebra of the Lorentz
group, Aa b is anti-symmetric in the rst two indices. Furthermore, under a
local Lorentz transformation, it changes according to [37]
ÆAab =
Dab ;
(10.5)
where
D ab = @ab + Aac cb
Ac b a c
(10.6)
is the covariant derivative with Aab as the connection. Equation (10.5) is the
standard gauge potential transformation of non-abelian gauge theories [5].
We see in this way that, in fact, Aa b plays the role of a Lorentz connection in
teleparallel gravity. In other words, Aa b is the spin connection of teleparallel
gravity. Accordingly, the teleparallel Fock-Ivanenko derivative operator is to
be written in the form
D = @ + 2i K ab Sab :
(10.7)
Notice that (10.6) is a particular case of this covariant derivative, obtained
by taking Sa b as the spin-2 representation of the spin generators [5].
10.2 Teleparallel Coupling Prescription
As we have already discussed, the interaction of any physical system with
gravitation can be obtained through the application of the so called minimal
223
coupling prescription, according to which the Minkowski metric must be
replaced by the riemannian metric
ab ! g = ab ha hb ;
(10.8)
and all ordinary derivatives must be replaced by Fock-Ivanenko covariant
derivatives,
i
@ ! D = @ + K ab Sab :
(10.9)
2
According to the discussion presented in section 6.1, the metric replacement
(10.8) is related to the local invariance of the lagrangian under translations of
the tangent{space coordinates [2]. This part of the prescription, therefore, is
related to the coupling of the eld energy{momentum to gravitation, and is
universal in the sense that it is the same for all elds. On the other hand, the
derivative replacement (10.9) is related to the invariance of the lagrangian
under Lorentz transformations of the tangent{space coordinates. This part
of the prescription, therefore, is related to the coupling of the eld spin to
gravitation, and is not universal because it depends on the spin contents of
the eld.
As is well know, the tetrad eld is considered to be more fundamental than
the metric tensor in the sense that it determines the metric unambiguously.
Accordingly, a more general coupling prescription, valid for both tensor and
spinor elds, can be obtained by replacing the metric change (10.8) by an
equivalent rule in terms of the tetrad:
ea
! ha ;
(10.10)
where ea is the trivial tetrad
@xa
;
(10.11)
@x
and ha is a nontrivial tetrad representing a true gravitational eld. Therefore, a full coupling prescription which encompasses both the metric and the
derivative replacements can be de ned as
ea =
@a ! Da ha
D = ha
i
@ + K ab Sab ;
2
(10.12)
As already discussed in chapter 6, this coupling prescription is general in the
sense that it holds for both integer and half{integer spin elds.
224
The covariant Fock{Ivanenko derivative operator (10.7) presents all necessary properties to be considered as yielding the fundamental coupling prescription in teleparallel gravity. For example, it transforms covariantly under
local Lorentz transformations:
D0 = U D U
1
:
Another important property is that the teleparallel coupling prescription dened by the covariant derivative (10.7) turns out to be completely equivalent
to the usual minimal coupling prescription of general relativity. Actually, it is
just what is needed so that it turns out to be the minimal coupling prescription of general relativity rephrased in terms of magnitudes of the teleparallel
structure. Like in general relativity, the covariant derivative (10.7) is the
only one available for spinor elds in teleparallel gravity. For tensor elds,
on the other hand, there is also a spacetime covariant derivative which acts
in the corresponding spacetime tensors. As an example, let us consider again
a Lorentz vector eld V a. By using the generator (6.12), it is an easy task
to verify that
D V a = ha D V ;
(10.13)
where
D V = @ V + (
K ) V (10.14)
is the teleparallel version of the Levi-Civita covariant derivative, which is
nothing but the Levi-Civita covariant derivative rephrased in terms of the
Æ
Weitzenbock connection [6]. As the Levi{Civita covariant derivative r is
know to satisfy
Æ
Æ
D V a = ha r V ;
(10.15)
Æ
with D the Fock{Ivanenko derivative of general relativity, we can see again
why D is called the teleparallel equivalent of the Levi{Civita covariant
derivative. We mention in passing that, by taking the equation of motion of
a free particle,
u @ u = 0 ;
and changing @ by D , we get the gravitational analog of the gravitational
Lorentz force equation [2], as can be seen by comparing with Eq.(9.16).
225
Due to the fact that the Weitzenbock connection vanishes when written
in the tetrad basis, it is usually asserted that, for spinor elds, the covariant
derivative
D = @ 2i Aab Sab
(10.16)
coincides with the ordinary derivative: D = @ [17]. However, there are
some problems associated to this coupling prescription. First, it is not compatible with the coupling prescription of general relativity, which is somehow
at variance with the equivalence between the corresponding gravitational lagrangians. Second, it results di erent to apply this coupling prescription
in the lagrangian or in the eld equation, which is a rather strange result.
Summing up, we could say that there is no compelling arguments supporting
the choice of Aab = 0 as the spin connection of teleparallel gravity.
Several other arguments favor the conclusion that the teleparallel coupling
prescription is that given by (10.7), that is, by
D = @ + 2i K ab Sab :
(10.17)
First, D is covariant under local Lorentz transformations. Second, in contrast to the coupling prescription (10.16), it results completely equivalent
to apply the minimal coupling prescription in the lagrangian or in the eld
equation. And third, it yields results which are self-consistent, and agree
with those of general relativity. For example, it is well know that the total
covariant derivative of the tetrad eld vanishes [7],
@ hb +
Æ
hb
Æ
Aa b ha = 0 ;
(10.18)
which is actually the same as (6.8). The teleparallel version of this expression
Æ
Æ
can be obtained by changing and Aab by their teleparallel counterparts.
This means to write
@ hb + (
K ) hb + K a b ha = 0 ;
(10.19)
where we have dropped the zero connection for simplicity of notation. However, the contorsion terms cancel out, yielding the absolute parallelism condition
@ hb +
hb
=0;
(10.20)
which is the fundamental equation of teleparallel gravity. This shows the
consistency of identifying the teleparallel spin connection as minus the contorsion tensor plus the zero connection. Finally, as the coupling prescription
226
(10.17) is covariant under local Lorentz transformation and is equivalent to
the minimal coupling prescription of general relativity, teleparallel gravity
with this coupling prescription turns out to be completely equivalent to general relativity, even in the presence of spinor elds.
10.3 Application to the Fundamental Fields
10.3.1 Scalar Field
Let us consider rst a free scalar eld in a Minkowski spacetime. Its
lagrangian is
L = 12
ab
@a @b 2 2 ;
with
mc
(10.21)
:
(10.22)
@ L
=0;
@ (@c )
(10.23)
=
~
By using the Euler-Lagrange equation
@ L
@
@c
the corresponding eld equation is found to be
@[email protected] a + 2 = 0 ;
(10.24)
which is the so called Klein{Gordon equation.
To obtain the coupling of a scalar eld with gravitation in teleparallel
gravity, we apply the full teleparallel coupling prescription
@a ! Da ha
D = ha
i
@ + K ab Sab ;
2
(10.25)
to the free lagrangian (10.21). The result is
L = h2
ab
Da Db 2 2 ;
L = h2
g
D D 2 2 ;
(10.26)
or equivalently
227
(10.27)
where we have used Eq.(8.25). In the speci c case of a scalar eld, for which
Sab = 0, we have that
D = @ :
(10.28)
Using the identity
@ h = h ha @ ha h
;
(10.29)
it is easy to show that the corresponding eld equation is
+ 2 = 0 ;
(10.30)
where
= h 1 @ (h @ ) @ @ +
@
(10.31)
is the teleparallel version of the Laplace{Beltrami operator. Because is
not symmetric in the last two indices, the last term of the above expression is
not the Weitzenbock covariant divergence of @ . From Eq.(8.20), however,
we can write
=
+ T ;
(10.32)
and the expression for may be rewritten in the form
= (r + T ) @ ;
(10.33)
where
r = @ +
(10.34)
is now the correct expression for the Weitzenbock covariant divergence of
@ . Making use of the identity
T = K ;
(10.35)
easily obtained from the de nition of the contorsion tensor, the teleparallel
version of the scalar eld equation of motion turns out to be
D @ + 2 = 0 ;
with D the teleparallel covariant derivative (8.35).
228
(10.36)
Finally, it is interesting to notice that, in terms of the Weitzenbock covariant derivative, the teleparallel Klein{Gordon (10.36) can be rewritten
as
r @ + 2 = K @ :
(10.37)
In this form, it shows clearly that in teleparallel gravity a scalar eld, through
its derivative @ , couples to, and therefore feels torsion. Moreover, it reveals
that torsion plays a role similar to an external force [2], quite analogous to
the role played by the electromagnetic eld in the Lorentz force equation. We
notice furthermore that it results completely equivalent to use the minimal
coupling prescription either in the lagrangian or in the eld equation.
10.3.2 Dirac Spinor Field
In Minkowski spacetime, the spinor eld lagrangian is
ic~ a
@a
@a a
2
Substituting in the Euler-Lagrange equation
L
=
@L
@
@c
mc2 :
(10.38)
@L
=0;
@ (@c )
(10.39)
mc = 0 :
(10.40)
we obtain the free Dirac equation
i~ a @a
Let us turn now to teleparallel gravity. The coupled Dirac spinor lagrangian can be obtained by applying the full teleparallel coupling prescription
@a ! Da = ha D ;
(10.41)
to the free lagrangian (10.38). The result is
where
L
ic~ =h
2
=
a h ,
a
D
D m c2
;
(10.42)
and
D
i
= @ + K ab Sab ;
2
229
(10.43)
is the teleparallel Fock-Ivanenko derivative, with
i
Sab ab = [ a ; b ]
(10.44)
2
4
the spin-1/2 representation of the Lorentz generators. Substituting L in the
Euler{Lagrange equation
@L
@L
D
=0;
(10.45)
@
@ D and using the identity D (h ) = 0, we obtain the Dirac equation in the
Weitzenbock spacetime:
i~
D
mc = 0 :
(10.46)
A point that deserves to be mentioned is that it results completely equivalent to apply the minimal coupling prescription to the lagrangian or to the
eld equation. Another point of interest is to notice that the Dirac spinor
couples to torsion both through the energy{momentum and through the spin
of the Dirac eld.
10.3.3 Electromagnetic Field
In Minkowski spacetime, the electromagnetic eld is described by the lagrangian density
Lem = 41 FabF ab ;
(10.47)
where
Fab = @aAb
@b Aa
(10.48)
is the Maxwell eld-strength. The corresponding eld equation is
@aF ab = 0 ;
(10.49)
which along with the Bianchi identity
@a Fbc + @c Fab + @bFca = 0 ;
(10.50)
constitutes Maxwell's equations. In the Lorentz gauge @a Aa = 0, the eld
equation (10.49) acquires the form
@c @ c Aa = 0 :
230
(10.51)
Let us now obtain Maxwell's theory in teleparallel gravity. This is done
by applying the full coupling prescription
@a ! Da ha
D = ha
i
@ + K ab Sab ;
2
(10.52)
K c d Ad :
(10.53)
with D the teleparallel Fock{Ivanenko derivative operator. In the speci c
case of the electromagnetic vector eld, Sab is the vector representation (6.12),
and we get
D Ac = @Ac
To obtain the corresponding covariant derivative of the spacetime vector eld
A , we substitute Ad = hd A in the right-hand side. The result is
DAc = hcD A ;
(10.54)
where
D A = @ A + (
K ) A
(10.55)
is the teleparallel covariant derivative. This means that, for the speci c case
of a vector eld, the teleparallel version of the minimal coupling prescription
(10.52) can alternatively be stated as
@a Ab ! ha hb D A :
(10.56)
In terms of the teleparallel structure, therefore, the free lagrangian (10.47)
becomes
(10.57)
Lem = h4 F F ;
where now
F = D A
D A :
(10.58)
Using the explicit form of D , given by (8.35), and the de nitions of torsion
and contorsion tensors, it is easy to verify that
F = @ A
@ A :
(10.59)
We notice in passing that this tensor is invariant under the U(1) electromagnetic gauge transformations. The corresponding eld equation is
D F = 0 ;
231
(10.60)
which yields the rst pair of Maxwell's equation in teleparallel gravity. Equivalently, assuming the teleparallel Lorentz gauge D A = 0, and using the
commutation relation
[D ; D ] A = Q A ;
(10.61)
where Q = Q , with Q given by equation (8.33), we obtain
D D A + Q A = 0 :
(10.62)
On the other hand, by using the same coupling prescription in the Bianchi
identity (10.50), the teleparallel version of the second pair of Maxwell's equation becomes
@ F + @ F + @ F = 0 :
(10.63)
It is important to notice that, in the context of the teleparallel equivalent
of general relativity, the electromagnetic eld is able to couple to torsion, and
that this coupling does not violate the U(1) gauge invariance of Maxwell's
theory. Furthermore, using the relation (8.29), it is easy to verify that the
teleparallel version of Maxwell's equations, which are equations written in
terms of the Weitzenbock connection only, are completely equivalent to the
usual Maxwell's equations in a riemannian background, which are equations
written in terms of the Levi{Civita connection only.
232
Chapter 11
Teleparallel Gravity with
Matter Fields
11.1 Energy-Momentum as Source of Torsion
In the description of Teleparallel Gravity in Chapter 8, only the sourceless
eld equations were studied. Now, we are going to obtain the same eld
equations, but in the presence of matter elds. To get the complete eld
equations, we have to consider the lagrangian
4h
L = 16c G
S T + LM ;
(11.1)
with LM the lagrangian of a general matter eld. Variation in relation to the
gauge potential B a yields the following eld equations
4G 4G
@ (hS )
(ht ) = 4 (hT ) ;
(11.2)
4
c
c
where
ÆL
ÆL
hT = c2 ha Ma ha aM
(11.3)
ÆB Æh is the dynamical energy{momentum tensor of the matter eld, with
@ LM
Æ LM @ LM
= a @
(11.4)
a
Æh @h @@ ha the Euler{Lagrange functional derivative. We see in this way that, in teleparallel gravity, energy and momentum play the role of the source of gravitational eld. In other words, in the same way energy and momentum are the
233
source of curvature in general relativity, they will be the source of torsion in
teleparallel gravity.
11.2 Energy{Momentum Conservation
We are going to obtain now the conservation law of the dynamical energy{
momentum tensor of a general source eld, whose action integral is written
in the form
Z
Z
1
1
4
S=
LM d x = c LM h d4x ;
(11.5)
c
where h = det(ha ). As a spacetime scalar, it does not change under a general
transformation of coordinates. Of course, under a coordinate transformation,
the matter eld will change by an amount Æ . Due to the equation of
motion satis ed by this eld, however, the coeÆcient of Æ vanishes, and for
this reason we are not going to take these variations into account. For our
purposes, it will be enough to consider only the variations in the tetrad eld
ha . Therefore, we have
1
ÆS =
c
Z @ LM
@ha
@ LM
@
) Æha d4 x; :
@ (@ ha (11.6)
But,
@ LM
@ha
@
@ LM
@ (@ha )
ÆÆhLM = hT a ;
a
(11.7)
is the dynamical energy{momentum tensor of the matter eld. Therefore, ÆS
takes the form
Z
Z
1
1
a
4
T Æha h d x = c T a Æha h d4 x :
(11.8)
ÆS =
c
On the other hand, under a spacetime general coordinate transformation
x ! x0 = x + ;
(11.9)
with a small quantity, the components of the tetrad eld will change
according to
Æha h0a (x ) ha (x ) = ha @
234
@ ha :
(11.10)
But, from the absolute parallelism condition, we have
@ ha =
ha
:
(11.11)
+ T ;
(11.12)
+ T ) :
(11.13)
+ T ] h d4 x :
(11.14)
Substituting in (11.10), and using that
=
we get
Æha = ha (@ +
Expression (11.8) then becomes
1
ÆS =
c
Z
T [@ +
Now, as a consequence of the local Lorentz invariance of the action (11.5),
the energy{momentum tensor T is symmetric [25]. Due to this symmetry,
and to the property
T + T = K = K ;
(11.15)
the above expression can be rewritten in the form
Z
1
ÆS =
T [@ + ( K ) ] h d4x :
(11.16)
c
Integrating by parts the rst term, and neglecting the surface term, this
equation becomes
Z
1
ÆS =
c
Using the relation
@ (hT ) (
@(hT ) = @ T + (
we obtain nally
1
ÆS =
c
Z
K )hT d4 x :
K )T h ;
D T h d4 x ;
(11.17)
(11.18)
(11.19)
with D the teleparallel covariant derivative. Due to the arbitrariness of the
parameter , the condition ÆS = 0 implies
D T = 0 :
235
(11.20)
Therefore, in teleparallel gravity, it is not the Weitzenbock covariant derivative r, but the teleparallel covariant derivative D that yields the correct
conservation law for the energy{momentum tensor of matter elds. It should
be remarked that this conservation law is equivalent to the corresponding
conservation law of general relativity,
Æ
rT @T +
Æ
Æ
T T = 0 ;
(11.21)
as can be easily veri ed by using relation (8.29). Furthermore, it is important
to note that the conservation law (11.20) agrees with the second Bianchi
identity [see (8.66)],
4G D @ (hS )
(ht ) = 0 :
(11.22)
c4
In fact, as D h = 0, this identity, together with the eld equation (11.2),
lead to the conservation law (11.20) for the energy{momentum tensor of the
matter elds.
It is important to remark once more that (11.20) is not a true conservation
law in the sense that it does not lead to a charge conserved in time. Instead,
it is an identity satis ed by the dynamical energy{momentum tensor, usually
called Noether identity [8]. As already discussed in section 7.3, the sum of
the energy{momentum density of the gravitational eld t and the energy{
momentum density of the matter eld T satis es a true conservation law:
@ [h (t + T )] = 0 :
By using Gauss theorem, we get
dq
=0;
dt
which is a conservation law for the charge
q =
Z
t0 + T 0 h d3 x :
(11.23)
(11.24)
(11.25)
11.3 Examples
11.3.1 Scalar Field
Before passing to the teleparallel case, we brie y review the basic properties
of a scalar eld in a Minkowski spacetime. The lagrangian is
L = 12 ab @a @b 22 ;
(11.26)
236
with given by (6.17). From Noether's theorem we nd that the corresponding canonical energy{momentum and spin tensors are given respectively by
a b = @ a @b Æ ab L ;
(11.27)
S abc = 0 :
(11.28)
and
As a consequence of the vanishing of the spin tensor, the canonical energy{
momentum tensor of the scalar eld is symmetric, and is conserved in the
ordinary sense:
@a a b = 0 :
(11.29)
Let us now turn to teleparallel gravity, in which case the lagrangian of
the scalar eld is
L = h2 h c hc @ @ 22 :
(11.30)
The dynamical energy{momentum tensor of the scalar eld, according to the
de nition,
ÆL
hT = ha (11.31)
Æha
is found to be
h T = h @ @ Æ L :
(11.32)
Like the canonical energy{momentum tensor, it is symmetric as a consequence of the vanishing of the spin tensor.
11.3.2 Dirac Spinor Field
Let us start by brie y reviewing the basic properties of a Dirac spinor in a
Minkowski spacetime. The free lagrangian is
L = ic2~ [email protected] @a a
mc2 ;
(11.33)
with a the Dirac matrix. From Noether's theorem one nds that the corresponding canonical energy{momentum and spin tensors are given respectively by
ic~ a
@b
@b a ;
(11.34)
a b =
2
237
and
S abc =
c~ a
Sbc + Sbc
2
a ;
(11.35)
with
bc i
= [ b; c ] :
(11.36)
2
4
We notice in passing that a b is not symmetric. However, using the Belinfante
procedure [9] one can de ne a symmetric energy{momentum tensor for the
spinor eld,
1 cab
ab = ab
@' ;
(11.37)
2 c
where
Sbc =
'cab = 'acb = S cab + S abc
S bca :
(11.38)
Let us now turn to teleparallel gravity. The coupled lagrangian of the
spinor eld is
L
ic~ a =h
ha D
2
D ha a
where
mc2
;
(11.39)
i
= @ + K b c b c ;
(11.40)
4
is the teleparallel Fock{Ivanenko covariant derivative.
The dynamical energy{momentum tensor of the spinor eld, according to
the de nition (11.3), is found to be
(11.41)
T a = hahb ~b 12 hbhc D'cba ;
where now
ic~ b
b ~b =
D
D
(11.42)
2
is the canonical energy{momentum tensor modi ed by the presence of gravitation, and 'cba is still given by (11.38), but now written in terms of the spin
tensor modi ed by the presence of gravitation:
(11.43)
S bc = c4~ ahabc + bc aha :
D
238
Multiplying (11.41) by ha yields
T = ~
1
D ' ;
2 (11.44)
where
D ' = hahb hc D 'cba ;
with D the teleparallel covariant derivative de ned in (8.35). From the
conservation of the energy{momentum tensor of matter elds we have
1
D D ' = 0 ;
2
D T D ~
(11.45)
so that
1
1
1
(11.46)
D ~ = D D ' = [D ; D ]' = Q S ;
2
4
4
with Q given by (8.33). We notice that, although the symmetric energy{
momentum tensor is conserved, the canonical energy{momentum tensor is
not [38]. Its covariant derivative gives the product of the curvature{like
tensor Q with the spin tensor, which is a kind of teleparallel version of
the Papapetrou coupling [39].
Equation (11.44) is a generalization of the Belinfante procedure for the
presence of gravitation. In fact, through a tedious but straightforward calculation we can show that
D = g ~ g ~ ;
(11.47)
from which we see that the dynamical energy{momentum tensor T of the
Dirac eld, that is, the Euler{Lagrange functional derivative of the spinor
lagrangian (11.39) with respect to the tetrad eld, is always symmetric. We
mention in passing that the same result could be obtained from the local
Lorentz invariance of the spinor lagrangian [25].
It is interesting to remark that, according to the teleparallel coupling
prescription (11.40), the spinor eld lagrangian (11.39) can be rewritten in
the form
L
ic~ =h
@
2
@ 1 bc K S bc
2
mc2
:
(11.48)
We see in this way that, as is usually claimed, the spin of the Dirac eld
couples to the contorsion tensor. It should be noticed that the rst term of
239
Eq.(11.48) does not represent a free lagrangian, but a lagrangian from which
only the spin coupling to gravity has been removed. Furthermore, we see
that the spin tensor, analogously to the energy{momentum tensor, has the
simple de nition [40]
1 ÆL
:
h ÆK bc S bc =
(11.49)
11.3.3 Electromagnetic Field
In Minkowski spacetime, the electromagnetic eld is described by the lagrangian density
1
FabF ab :
(11.50)
4
The corresponding canonical energy{momentum and spin tensors are given
respectively by
Lem =
a b = 4 @b Ac F ac + Æ a b Fcd F cd ;
(11.51)
and
S abc = F abAc
F ac Ab :
(11.52)
As in the spinor case, the canonical energy{momentum tensor a b of the
electromagnetic eld is not symmetric. By using the Belinfante procedure,
however, it is possible to de ne the symmetric energy{momentum tensor,
1 cab
@' ;
2 c
ab = ab
(11.53)
where
'cab = 'acb = S cab + S abc
S bca :
(11.54)
In fact, as can be easily veri ed,
'cab = 2F ca Ab ;
(11.55)
and consequently
ab
=4
F ac F b
1 ab
cd
c + Fcd F
4
240
(11.56)
is a tensor symmetric, and conserved in the ordinary sense:
@a a b = 0 :
(11.57)
Let us now turn to teleparallel gravity. The electromagnetic eld lagrangian is
Lem =
h
F F ;
4 (11.58)
with
F = @ A
@ A :
(11.59)
I this case, the corresponding dynamical energy{momentum tensor, according
to the de nition,
hT = ha is found to be
T
=4
F F +
Æ Lem
Æha (11.60)
1 Æ F F :
4
(11.61)
We see from this expression that the dynamical energy{momentum tensor for
the electromagnetic eld formally coincides with the Belinfante tensor, and
is consequently symmetric.
241
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