Relativitat General, o com la gravetat corba l`espai

Espacio, Tiempo y materia en
el Cosmos
J. Alberto Lobo
ICE/CISC-IEEC
Summary
Part I:
General
Relativity
Theory
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•
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•
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Concepts of motion: From Aristotle to Newton
Newton’s laws: Absolute Space and Time
Mach’s criticism and Mach’s Principle
Einstein’s Equivalence Principle
General Theory of Relativity:
– Curvature of space-time
– Field equations: Space, Time and Matter link together
Part II:
General
Relativistic
Cosmology
•
•
•
•
•
•
•
Newtonian Cosmology, Olbers’ paradox
General Relativistic Cosmology, an evolutionary paradigm
The Cosmological Principle
Friedmann equations
Big Bang Cosmology: parameters of the Universe
The Accelerating Universe
Outlook and future talks
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Cosmos
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Concepts of motion
Aristotle expressed in his treatise
“Physica” that a force is necessary
to keep bodies in motion, no matter
whether accelerated or not.
Galileo Galilei, some 1800 years on,
proved by experiment that uniform
motion does not require a force. And
that forces cause accelerations.
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Newton’s laws of Mechanics
Isaac Newton wrote a fundamental book in
1689: “Principia Mathematica Philosophiae
Naturalis”. There, he developed his theories
of motion and of Universal Gravitation.
Newton’s laws of mecahnics are explained in High School courses. We
recall them here now:
1.
2.
Under no forces, bodies move at constant speed, or stay at rest.
Forces cause accelerations, according to the equation:
F = ma
3.
Every force action has a reaction counterpart, equal and opposite.
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Let’s review the 1st law
Is it a consequence of the second? For, it may be argued,
if F = ma , and F = 0 ⇒ a = 0 and v = constant
Problem is: how can one possibly asceratain that F=0?
This is in fact a logically circular question:
F = 0 R v = constant
The first law actually defines the class of reference systems
in which the second law applies. These are called Inertial
reference systems. Newton inferred from here the existence
of Absolute Space (Scholium of his PM), and that it was the
cause of inertia.
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The bucket of water experiment
Rope wound
Rope unwinds
Water
at rest
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Water
rotates
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Inertia and Absolute Space
Newton derived important conclusions from his experiment:
Accelerated motion is an objective concept.
1. There is an absolute reference system which
enables the intrinsic definition of acceleration.
3. This is Absolute Space. It is therefore the causal
agent of inertia.
4. It cannot however be individuated.
5. “Fixed stars” are (perhaps) the best approximation...
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For the next 200 years...
D. Papin
J. Watt
U. Leverrier
P.S. Laplace
I. Kant
L. Boltzmann
Immense success of Newtonian Mechanics and its variety of
applications somewhat left in stand by those almost philosophical
discussions:
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•
•
•
Mechanical machines of many types: industry, etc.
Celestial Mechanics
Mechanical theory of Thermodynamics
...
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Ernst Mach’s criticism
What would happen if the rest of the
Universe should start revolving around
the bucket?
He conjectured the same would happen...
Mach went on to formulate what was to
become known as Mach’s Principle:
–The inertia of the bodies is determined
by the distribution of matter throughout the
Universe.
Unfortunately, Mach’s principle is not
any quantitative...
Ernst Mach
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General Relativity
The old riddle was solved by Einstein’s General Theory of Relativity,
towards the end of 1915.
1
8π G
Rµν − R gµν = − 4 Tµν
2
c
1950
Because that solution heavily relies on Gravitation –it actually constitutes
the new theory of gravitation–, it will be useful to review a few basic facts
about gravity...
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Bodies in free fall
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Port Aventura
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Simple theoretical analysis
Rope tension: T
Rope tension: T
F = Ma ≠ 0
⇓
a=g≠0
T +F = 0
⇓
a=0
Gravity force: F
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Gravity force: F
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A very contemporary example of Galileo’s experiment: The astronaut is in free fall
at fhe same rate as the spacecraft, so he feels weightlessly floating. Nevertheless,
gravity up there is not negligible: 8 m/sec2...
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“Hipotesis non fingo”
Let h = 5 m
v
h
L
L = vt
1 2
h = gt
2
⇒ L = 2hv 2 / g ⇒
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v = 700 m/s
L = 700 m
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v = 100 km/s
L = 100 km
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“Hipotesis non fingo”
But take a careful look at these numbers:
v = 100 km/s
L = 100 km
^^^^^^^^^
100 km
>100 km !!!
The ocean’s surface actually bends…
If the Earth were flat then L=100 km
But the Earth is not flat!! Hence L > 100 km !!!
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“Hipotesis non fingo”
Is it thinkable to shoot the bullet fast enough that it eventually hits
the pirate from the back ???
Like this:
mmm...
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“Hipotesis non fingo”
Perhaps the Moon is, after all, an example of that...:
Somebody must have shot it from where it revolves around the Earth...
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“Hipotesis non fingo”
Newton’s Law of Universal Gravitation
v=0
v small
v large
GMm
F= 2
r
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Let’s pause for a moment...
...and write down the two main results seen so far:
1.
The definition of Inertial Reference System is circular,
therefore Newton’s laws –and Special Relativity laws,
too— suffer from a severe logical inconsistency.
2.
Gravitational fields impinge the same acceleration on
all bodies, regardless of their mass and/or composition.
Inertial Reference Systems may not have a prvileged status,
any Reference System should be equally appropriate to describe
Physical laws. General, rather than Special Relativity must be the
ultimate concept in Physics.
It is not possible to tell whether a given Reference System is
accelerated by a gravitational field or by some other agent.
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Equivalence Principle
In 1915, Einstein came up with a far reaching conjecture which
settled the basis for a new theory of gravity, and disposed of the
epistemological inconsistencies of Newtonian Mechanics.
The Equivalence Principle
The laws of Physics are those of Special Relativity when
expressed in a Reference System which is falling freely in
a gravitational field.
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Equivalence Principle
-- The EP disposes of the servitudes of needing Inertial Systems.
General Relativity is soundly established instead.
-- Inertial and gravitational masses must be equal.
-- Absolute space is a superfluous concept.
-- Freely falling particles (generally) follow curved trajectories.
These are straight lines if no gravitational fields are present,
and shortest distance curves (or geodesics) if they are.
-- Space-time is therefore not absolute, it has geometric
properties determined by the distribution of gravitating
mass and energy.
-- The old tenet of Euclidean Geometry thus gives way to the
rules of Riemannian Geometry.
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Field equations and geodesics
The EP sets the general framework. But a very important question
is still pending:
How does matter distribution actually determine geometry?
The answer is provided by Einstein’s field equations:
1
8π G
Rµν − R g µν = − 4 Tµν
2
c
which is a complicated set of non-linear partial differential equations
for the field unknowns gµν.
Once the gµν’s are known, the trajectories of test particles and light
rays are determined by the geodesic equation:
x µ + Γ µρσ x ρ xσ = 0
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Bending of light rays
A truly remarkable prediction of GR is the bending of light rays:
just like material particles, light rays (photons) follow non-straight
line trajectories as they come close to gravitating objects.
Take for example a ray which grazes the Sun surface:
4GM :
Einstein’s formula: δφ = 1.76 arc - seconds
2
Rc
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Eddington’s 1919 observations
Night
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Eddington’s 1919 observations
Day
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Solar
Eclipse
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Gravitational lenses
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End of Part I
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Summary
Part I:
General
Relativity
Theory
•
•
•
•
•
Concepts of motion: From Aristotle to Newton
Newton’s laws: Absolute Space and Time
Mach’s criticism and Mach’s Principle
Einstein’s Equivalence Principle
General Theory of Relativity:
– Curvature of space-time
– Field equations: Space, Time and Matter linked together
Part II:
General
Relativistic
Cosmology
•
•
•
•
•
•
•
Newtonian Cosmology, Olbers’ paradox
General Relativistic Cosmology, an evolutionary paradigm
The Cosmological Principle
Friedmann equations
Big Bang Cosmology: parameters of the Universe
The Accelerating Universe
Outlook and future talks
A Lobo, 14-ii-2007
Espacio, Tiempo y Materia en el
Cosmos
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Cosmology: Preamble
Cosmology is concerned with the entire Universe –a big problem
indeed, and a very old one, too...
The scientific approach to Cosmology is however much younger,
about 300 years only.
Newtonian Cosmology faces a number of insurmountable problems:
• Space must be infinite: what would be there beyond its limits
otherwise?
• Likewise, time must be infinite: the Universe is thus eternal.
• As a consequence, the gravitational potential is also infinite.
This creates a problem of stability versus infinitely long life.
• Olbers’s paradox: the sky is dark at night...
General Relativistic Cosmology fixes these issues because it
attributes dynamic properties to space and time.
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Contemporary Cosmology
Modern Cosmology –so called Standard, or Big Bang model– is
based upon a few major observational facts:
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•
•
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The Universe is very highly isotropic –at large scales.
The Universe is expanding at the present epoch.
The Universe is filled with thermal radiation of ~2.7o Kelvin.
Abundance of light elements: H—73%, He—26%.
The Standard Model rests on two fundamental hypotheses:
• General Relativity theory
• Cosmological Principle.
It also needs many other sources of scientific knowlwdge, such as High
Energy Physics, Chemistry, Thermodynamics, Astrophysics, Statistics, etc.
Cosmology is definitely a very pluri-disciplinar activity.
This talk addresses only the “GR connection”.
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The Cosmological Principle
The Cosmological Principle is the assumption that:
1.
2.
The Universe is large scale isotropic.
Us, humans, are standard observers of the Universe
as regards its large scale properties.
Hence the Universe is:
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⎧isotropic
⎪
⎨ and
⎪⎩homogeneous
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Friedmann-Robertson-Walker
The CP has an immediate mathematical consequence as to which is
the large scale geometry of the Universe:
⎡ dr 2
2
2
2
2 ⎤
+ r (dθ + sin θ dϕ ) ⎥
ds = c dt − a (t ) ⎢
2
⎣1 − kr
⎦
2
2
2
2
(FRW)
where a(t) is the scale factor of the Universe, and k is a trichotomic
constant:
a (t ) > 0
Universe expand
a (t ) < 0
Universe
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s
shrinks
⎧+1 closed, spherical
⎪
k = ⎨ 0 open, flat
⎪⎩−1 open, hyperbolic
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The expanding Universe
Whether the Universe actually
expands or shrinks, i.e., whether
a (t ) > 0 or a (t ) < 0
is to be determined by
observation.
It appears that, at the current
epoch, the Universe is in an
expansive phase.
Time
advances
in this
direction
FRW geometry
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This is determined by the fact that light
from remote galaxies arrives red-shifted
to us.
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Cosmological red-shift
Cosmological red-shift
Here
Far away
Even farther
λreception − λemission
z=
,
λemission
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measure of redshift
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Cosmological red-shift
This was fisrst discovered in 1929 by Edwin Hubble, who interpreted
redshift as an indication that galaxies move away from us.
Hubble also inferred that the expansion speed
is proportional to distance:
v = H 0d ,
Hubble’s law
where H0 is known as Hubble’s constant:
H 0 75 km / sec⋅ Mpc
In the light of FRW geometry, redshift is an evolutionary effect:
z=
a (treception )
a (temission )
−1
Redshift is therefore a measure of age and size of the evolving Universe
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The geometry of space
k = +1
k=0
k = −1
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Friedmann equations
The evolution of the Universe is described by Einstein’s equations:
1
8π G
Rµν − R g µν + Λg µν = − 4 Tµν
c
2
where Λ is the so called Cosmological Constant.
Tµν includes both ponderable matter and radiation in the Universe.
Two magnitudes characterise their contribution: density and pressure.
Expanding Einstein’s equations one finds Friedmann’s equations. In
the present time, the Universe is matter-dominated, i.e., pressure is
negligible. The following obtains:
a 2 8πGρ k Λ
,
H ≡ 2 =
+ 2+
a
3
a
3
2
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p ρc 2
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Friedmann equation
39
Λ=0 Cosmological evolution
ρ0 < ρc
ρ0 = ρc
ρ0 > ρc
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Λ=0 Cosmological evolution
There is a critical density, which determines whether the Universe
is spherical, hyperbolic or flat:
⎧ k = +1 if ρ 0 > ρ c spherical
3H 02 ⎪
ρc =
, ⎨ k = 0 if ρ 0 = ρ c flat
8π G ⎪ k = −1 if ρ < ρ
hyperbolic
0
c
⎩
But we do not know which of the three is true, as measuring ρ0 is
a very difficult task. Indeed, under
forms of matter...
ρ0
go both visible and dark
To complicate things further, the Universe was recently seen to
be under accelerated expansion!!
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Parameters of the Universe
It is customary to define dimensionless parameters which characterise
the evolution of the Universe in terms of the critical density. They are:
ΩM =
8πGρ0
3H 02
matter
ΩΛ =
Λ
3H 02
dark energy, or Λ
Ωk =
k
H 02
Curvature
The Friedmann equation is then rewritten as:
3
2
H 2 ( z ) = ⎡Ω M (1 + z ) + Ω k (1 + z ) + Ω Λ ⎤ H 02
⎣
⎦
and gives rise to complicated casuistics, as we see next...
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Possible states of the Universe
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Composition of the Universe
We live in a Universe which is:
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•
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•
•
•
•
Spatially flat
Therefore of infinite volume
In accelerated expansion
Dominated by dark energy
With only 4% of visible matter
13 billion years old
Will expand forever
This model is certainly not free of
problems... To discuss them is the
objective of this series of talks.
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LISA and the GW Universe
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End of Presentation
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