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i 11
GhiL t 1esis ~,.
JJ . u . l.. l .
• :.;,tll Oc lio JOI'
9 '.)
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G
CHAPTER 1
The Hoyle-Narli kar Theory
of Gravitation
1. I n t rod uction
Th e success of Maxwell's e qu a tion s ha s l ed to
electrod ynamics being n orma lly for mul a te d in te rms of fi elds
that hav e de g rees of f!eedom independent of the par tic l es i n
th em .
However, Gauss sugg ested t h at a n a c t ion-a t-a - di s t an ce
t heory in which the action trav e lled at a fi n it e v e loci ty
mi g ht b e p o s sible.
This idea was develo pe d by ~h eel er and
Fe ynman ( 4 , 2 ) who derived their t h eory from a n action-p ri ncip l e
that involved only direct int e r a ctions between p airs o f particl e s.
A
f e ature of this t h eory was tha t the 'p s eudo' -fie l ds
introduced ar e the half-ret a rded plus half- a dva nced f ie l ds
c l a cul a te d from ~he world-lin e s of the particl e s. However ,
W1ee ler a nd F eynman, and, in a different way , Hog a t th ( 3)
were ab le to s how that, provided certain c os molog ical
co nd itions were satis f ied, these fiel d s could comb ine t o
~ ive the observed field . Hoyle a n d Narlikar ( 4 ) extend e d the
tt1 eory to g eneral space - times a nd obtained simil a r t he or i es
for th e ir 'C' - field (5)and for the g ravit a tional f~ el d (G ).
It is with these t h eories t hat t h is c hapter is conc e r n e d .
. . . . •. . . •. . .I
It wi ll b e :3 hown tha t in an exp , . nding universe the
advanced f ields are infin ite, a nd t he ret a rded f ields f in i te .
~h i s is b e cause, unlike electric cha r g es, all nJas ses
hHV A
2. The Boundary Condition
Hoyle and Narlikar derive their theory from the
action :
where the integ r ation i s over the world-lines of part i cles
0.
I
b.. .. .
I n this expressi on
(
is a Gre en function .
that s atisfies the wave equation:
where
j
is t he determinant of
in the action
particles
G..)
A
3.j
. 3 inc e t he double s um
is s ymmetrical b etween all pairs of
b , on ly th&.t part of
s y rmne trical between c" and
b will
c;-(o... b)
that i s
cont ribute to the action
i.e. t he action c a n be written
A:
where
'.I'hus
~-.Ar ( C\,
(it
b)
=
must be the time-symmetric Green f un ction, and c a n
be written:
q~ = f
qre-t ,.. j qo..dv
where
the
Cic,._Jv
and
u,I.JJ~
a r e the r etarded c:.nd advanced Green f unct ion s .
re oui
ri n ~
that the action be stationa r vJ u n d er v ari a t i on s
-
o f the
3~
[£.Z.
~{Vl,(o.){X)/Vl(b)(x)](l<ik-
C\..
Hoyle a n d Narlikar obtain the field - equ a tion s :
>
±scK R)
-f b
.:
(b)J
(a.);r-
I
- 7: QLK
J
tv\ : r
fVL
>
i,,.me re
c on se q u e nc e of the particular choice of Green fu n ction, the
con traction of the field-equati ons i s satisfied identica l l :! .
~ her e a ~e thus only 9 equations for the 10 comp onents o f Q·
· '.
_J VJ
a nd the s y stem i s indetermina te.
Hoy l e and Narlikar t here fo re imp o se
f. m
[c..)
=
fYl 0
=const . ,
-
tl s the te nth equation. By then ma kin~ the ' smb oth-fluid '
.
.
L ~ 11A ( o.) {b) r-,
/\rt 2
appr oximation, tha t i s by putting L ~ 1 • v ··~ - - ,,~
,1;,
Q.:/b
-. -
(J)
they obt a in the Einstein f ield-e quati ons:
..!.... V\.. ~ {
~ f
O
R.
.. I<
-
.!_
:2.
R_
5"
V-
')
:
-
f ,. K
Ther e is an imp ort a nt d i ff erence, h owev er , b etwe e n thes e
field- e q u at ions in the d ir~ct-particle i n tera ction t ce ory
and in the u s ual g eneral theory of re l at ivit y .
In the
g eneral t he ory of r e lativity, any metric ~hat sat i s fies t he
the field-equations is admissibl e , but in the dire ct-particl a
interaction theory only t hose solutions of the f ield-equati on s
are admissible that satisfy t he additiona l requirement:
/VL,C) (
X.)
£ ) C/:,:, °'" ) cl a
-::.
i
i
£ Jr C ;; V.
0..
This requirement is highly r e strictive;
( -:r:.) °') clo--
it will b e sh own
tha t it i s not satisfied for the cosmolog ical solutions of
the ~instein field-equations, a nd it appears tha t it c t nnot
b e satisfied for any mod.els o.f the univer;:;e tho.t either
contain an infinite amount of matte r o~ undergo infinite
exp ansion.
The difficulty is simila r to that oc curring in
Newtonian t ~c ory when it is recognized that the universe
mi g ht be infinite.
The Newtonian p ot ential
.r.Jhere
1
l
i s the de nsity.
t
obeys the equ ation:
In an infinite stat ic univ e r se,
f
the source a l ways h a s th e same Gi g n.
would be infini t e, s i nce
The d if fi cul ty was r es ol-
ved wh en it was re a lized t h ~~ t the u n i v ers8 was exp and. ins , .s ince
in an ex-oancling universe the retarded s olution o f the ab ove
equation i s finite by a sort o f ' red-shift' e ffect. The
advanced solution will be infinite by a ' b lu e - shift' eff e ct.
fhis i s unimp ortant in Newtoni an the6ry, s i n ce one i s fr ee
t o choo se the soluti on of the equation and so ma y i gn ore the
infinite advanced solution and tak e simply the finite
retarded solution.
Bi milarly in the direct-particle interaction t h eory the
lh, - field satisfies the equati on :
Om ~
[R.tvt -
N
whe re /\) i :3 the density of world-line s of particles.
.As i n
the Newtonian c a s e , one ma y expect that the effect of the
exp a~sion of the univers e will be to ma k e th e retard ed sol u ti on
fini te and the advanced solution infinite.
However, one i s
now not free to choose the finite retarde d solution, f or th e
equation i s derived from a d irect-pa rticle interacti on action pr inciple symmetric between pairs of particles, an d one mu st
c h oo se fo r
s olutions.
fV\...
half the s1ilm of th e retarded and adva nced
Vi e would expect t u.is t o be infinit e , and this i s
shown to be so in the n ext section.
The Robert son-Walk er cosmolog ica l metrics have the
form
S inc e t aey are conformal l y f l at , on e c an c h oos e c o ordi n ates
in wh ich
t h ey become
ds2
e:i. , f
5l l. [' cl. vc 2 - d f 1 .,. Fvld
: J1~ VJ dx°'- cLx h
I o.b
.i.
· 2
cl
r1(\. 0
i s the fl a t-sp a ce metric tensor and
whe re
- -~ · RU:)
;rr,
·t
+r K <'f ·r-
-
r) iJ L t Ji <'t - r )2JJ
I
T
For example, for the Einstein-de Sitter univer s e
k •
o, R. (t) _,,
..fl ,
R • (-~
r
-=-
I
( vz
AA i r o
YT
2.
-=
·7 J t
(0
L. /; ,( oO )_
-<- 't <
o0 ),
J_ )
3
For the steady-state ( d e Sitt e r ) universe
K
=-
0/
Rtt) ;
·.·..t
e,
(7)
( -<:D~l:LoO )
--
Jl_
R
·-
:.
(
I
- d::J -<.
.:.-!'-
c
o)
L...
C
r- --
(<f
f
a
·-
Te 7" )
c;* (~) h)
'l'he Green fu nction
O
·- L-
c; -~(o.) h)
ob ey s t be equation
Z Rq-lk (~, b J
-T
e1'i" }x
From t his it f ollows t hat
; " - J-dx ·o:··-
b
b
"~= "-,
t :1en
J)_ .L ( °' b )
S' ) . =
d ~~ 7 ~
~
If we let
J L
(>'
.)
.'l 'hi s is s imply th e flat- space Gr r-> en function e qu ation , ancl.
he nce
*.
G ( l 0j
1 ,
vl J.
1/
.-
f) ~ Jl- l1.J /. J_(~ 'fA-r
S-,r
.
The
1
/V\.. .~ fi e l
(11,
. .- !lCt
d i s g iven by
U, )
~
- Jl(t1)f
r ·~ - 'f,
Jl ( "C.'1.) (
f(; "'N / - 5 Jx:
~')
'< =
f]
,..
l ( fVL.t(!,I:.
j
r
/J,t o.r) V:
)
For universes without cre at ion ( e.g. the Einst e i n-de Bitter
3n. .
~
n,. ~
const. For
universe ),
N
,i ·-
J
fV.: .
u n ive rs es ·•.Ji th creation (ste adv s t ate )
V
~
I
( '1; )
o.. ) \/,
:;
fi I
n-l (f 1}..NJ'l3 ( 't~)
I
'
~
c onst .
(L,::.
6:
7.cLl"
'tT T
·,T~
w~ere the integ ration is over the f u t ure lig~b cone .
Th i s
will n orma lly be i nf inite in a n expan d i ng univ e r se , e ~g . i n
the ~ i n stein-de 0 i t ter universe.
n. ( 'C A -
'f ) cl 1 ~
I
::::.
I n the s~eady-state universe
.. ( ·,/.\I
LI
)-i ·f
o -
rL ( ~
-
)3 ('2.i - 'l, ) cl '1'.,_ .
i ~
L.,
"
By cQntrast, on the other hand, we h a ve
fvl..J-e.t
(
'-1:,)
=
; · · f N123 ''-" ,, ,--
J2 - ('t.)
'l.. '
4 rr r
wher e the integ ration is ove r t he past lig ht con e.
CL
r
Thi s wi ll
normally b e finite , e.g. in the ~inste in-de ,:i i t t e r un_i.. v ers e
whi l e i n the steady-state univ er se
~ (i
l-
L. I
)-'f ·r:; n.(~' )3('2.
l
·-CO
i
~ (L
·-
I
Thu s it c o..n ·be s e e n th;-;.t the solution
2
~
i
Iv\.. ::.
const . o f t b ~
equation
i s n ot, in a cosmolog ical me tric, the half-adva nc ed plu s
half-retarded solution since this would b e i nf i n it e.
In fact ,
i n the c ase of the Einstein-de Gitte r a nd steady- state metri cs ,
it is the pure ret a rded solu t ion.
4.
The ' C' - Fi e l d
Hoy le and Narlikar derive t heir d ire ct-p a rticl e
inte raction theory of the ' C' -field from t he a ct ion
where the suff ixes
.As
~:. {G\. b)
1
"'
G
b
Q
.
r e fer t o d i ff e r e nt i a tion of
I
on th e world-l ines of
C\
1
6
r e1j 9ective1 ~, .
i s a Gr ee n fun c ti on ob eying the equation
DG(X,X') - s~cx)X')
~-ie define
11
.. I\
D..nd the matter-c urre nt
J '"'( ~)
Then
C (t_) .
oc
c
.J
by
_is a
4(
j , b) cL b '".
Sf:. (
X,
J) J
'< ( 'j )J 1~ / -
3 dX
\
- J- '" J 1,
We thu s see ·that t he sourc es of t he
1
C 1 -fi e l d are the pl a c es
whe re ma tter is creat ed or destroyad .
i1.s in tl10 c ase of the
'n '-field , t he Green function
must be time-symmetric, tbat is "'-
/\.
.
q'Ca-, b) . ± Gtet(°', h) ,_ i ( ,.),. c°', b
)
Hoyle a nd Narlikar cl a im th8t if the action of th e
1
C' -fielc:i. i s i n cluded a lon,~ vlit h the a ctio n of the
1
/'I'\.,,
' -fie l d ,
a unive rs e Hill be obt ained t i-1:_;. t 2-.pproxim a t es t o the 3tead:y -
st a te univ erse on a l a r g e s cale a lthoug h t here may ·be loc al
i rre ~ularities.
In this univers e , the v a lue of C will be
f{ n i Ge and its gradient time-like and of unit magn i t ude.
Given this universe, we may check it for consiste nc y by
cl a culatini th e a dvan ~ed and retarded 'C'-fields and f i nd ing
if their sum is finite.
',,Je
sha ll not do t his d irec tl y but
will show that the advanc ed field is infini te while the re tarded field is finite.
Consj_der a region in space-time bounded by a thr ee d imen s ional space-like hypersurface ~
and t i1e pas t lig ht cone
Qf
z.
at the present time ,
of some p oint
P
to the future
5)
By Gaus s 's theorem
we /-s d:x". (
)i.·t-P
J.i cl S
d n,
Let the advanced field produced by sources within
__) C'
r' '
L
, and h0nc e
~
will b e zero on z...
and
'1:hen
c'
drt-
( J h;K
Jv
./- q
J
cL x.
4 -
JD.
- r(
But
j
is the rate of creation of matter=
f'L- (
const. ) i n
)K
the s tead y-state unive r se, a n d hence
2-Q_' cl
Jn,,
AS
the p oint
?
s -: ·n/V .
is taken further into the futur e , the v olume
of the :reJion V tends to infinity.
hy p.ersurf ace
effects.
J)
However, the
tends to a finite limit owing to horizon
'l 1h ere f or e the g r adie nt
dC
d n.,.
I
mu st be infinite.
A similar c al cul at ion s hows t he gradient of the r et a r d e d
fie l d to be finite.
Thei r s ums cannot therefore ~ i~e t he
fie l d of unit g radient required by t he Hoy le-Narlikar lheory .
It is worth n oting that this result was obt a ined
without assump tions of a smooth d i s tribution of matter o r
o f conformal.~ fla tness.
5.
Con clus ion
It i s one of the weaknes se s of the Einst e in the ory of
relativity tha t a lthough it furnishes field equ at ions i t d oes
not p rovid e bounda ry cond itions for t hem.
Thu s i t d o e s n ot
Give a unique model f or the universe but al lows a who le series
of models .
Clearly a t heory th&t provided boundary conii ti on s
and t i.1.u s re-.stricted t h e possible solutions wou l d be ve r y
attra ctive .
The Hoyle-Narlikar t he ory d o es j u st tha t (t he
J..
't
1.. /'v\ )
re q uir ement that /r1, == .i. fY\- e.- t
;l
o...d-./.
is
equivalent to a b~und ary conditi on).
Unfortuna t ely , a s we
have see n above, this cond ition exclud e s tho s e ffi Od els t hat
seem ~ o correspond to the a ctual univers e , n amely t he
Robertso;1 - vJ hlker models.
The calculations g iven a bove h a ve considered thA u n i vcrs0
as being f ill e d \•t i t h a nnif orm d i strlbution of ma tter.
'_;_' h is
i s l eg itima t e if we are able t o make the ' smooth-f l uid '
approximat ion to obt a in the ~ i n s tein e qu at ions .
Al ternative l y
i f t his app rox ima tion i s invalid, it cann ot b e s aid that the
.t ne ory y ields the Eins t ein equations.
I t mig ht possibly be tha t
local irre3 ul ar i ties could make
fi n ite, but this has cert a i n l j not be en dem on strated
and seems unlike ly i n v iew of the f a ct t h ~t , in the Hoy leNarlikar dire ct-particl e i nte raction theory of t h e ir ' C ' - f ield ,
which i s derived from a very similar a ction-princ i ple , i t can
be shown
without assuming a smooth distribution tha t the
advanced 'C' field \•1 ill be infinite in an expand inr., u niverse
with creation.
The re a son tha t it is possible to formul ate a dire c t p a rticle interaction theory of el ec trodynamic s t hat does n ot
enc ount e r this difficulty ~f having t he advs nced s olu t ion
i n finite is that in ele c trodyn amics there are eo.u a1 numbe r s
of sources of p ositive and n egative sie;n.
rrhei r .f:i.P-lds can
c a ncel each other out and the total field can b e zero apart
from local irreg ulariti es .
Thi s sug~ est th a t .::, p ossible
1·Jay
to save the Hoyle-Narlikar theory would be to allow ma ss es o f
both po s itive and neg a tive s i g n.
The action wou l d be
are gravitational charg es a n al oz ou s to
el e c tric char·_-;es.
Particles of positive
'r" ' -f ield a nd particles of negative
q_
i
in a posit i ve
in a ne 6 ativ c
I
r/\ '
field wou ld have the normal g ravit at ional prope r ti ~s, th8t is,
tne:1 would have positive gravitational and inertia l masses.
A particle of n e Bative
sti ll f ol lo v1 a g eod esic.
a parti cle o f p ositive
i
in a p o s i tive
'M
' -f i e l d wGu ld
'r here for e it wo l1ld be &t trac t ed bz/
q_
It s own gravi t ~t i onal effect
howe ver woul d be to repel a l l other p a rti cles.
Thu s it wou ld
ha v8 t he propert ies of the negativ e mas s described by Bondi(~)
tha t is, n egative g r a vit at ional ma ss and n egat ive inertial
mass .
S i n ce t he re does n ot seem to be any ma t t er having
t o.e se pr op e rties in our re g ion of space ( where M .-:!J:-
ca ns t .
there mu s t cle a rly be separa tion on a very larc e sc a le.
It
would n ot be p os s ibl e to ide n tify particl es o f ne gative
t
>0
wi th ant ima t t er, sinc e it is known that antimatter has positiv e
in e rt ial mass .
Hwever, the introdu ction o f ne g ative masse s
wou ld p rob ab l y raise more difficu lt i es than i t ~ ul d s olve.
)
REFERENCES
12
1. J . A .1/11eeler and
R. P . Feynman
Rev. Mod.Phys .
2. J . A. Whee ler and
Rev. Mod . Phys . 21 425 1949
R. P . Feynman
7.
) •
J. E .Hogarth
157 1945
Proc. Roy. Soc. A 267 365 1962
4. F' . Hoyle and
Pr oc. Roy. Soc. A g 77 1 1964a
5 . F . Hoyle and
J.V. Narl i kar
Proc. Roy . Soc. A _gs2 1 78 "1964b
6 . F . Hoyle and
Proc . Roy. Soc. A gs2 191 1964c
7. L.Infeld and
A. Sc hild
Phys. Rev. 68 250 1945
8 . H. Bondi
Rev. Mod. Phys . 29 423 1957
J .V. Narlikar
J.V.Narlika r
CHAPTER 2
1•
PERTURBA'I'IONS
--
Introduction
,____
~
Perturbations of a spatially isotropic and homogeneous expanding
universe have been investigated in a Newtonim approximation by
Bonnor( 1 ) and relativistically by Lifshitz( 2 ), Liftshitz and
K.halatnikov( 3 ) and Irvine(4).
Their method was to consider small
variations of the metric tensor.
This has the disadvantage that the
metric tensor is not a physically significant quantity, since one
cannot directly measure it, but only its second derivatives.
It is
thus not obvious what the physical interpretation of a given
perturbation of the metric is.
Indeed it need have no physical
significance at all, but merely correspond to a coordinate transformation.
Instead it seems preferable to deal in terms of
perturbations of the physically significant quantity, the curvature.
2.
Notation
Space-time is represented as a four-dimensional Riemannian space
with metric tens or
gab
of signature +2.
Covariant differentiation
in this space is indicated by a semi-colon.
Square brackets around
indices indicate antisymmetrisation and round brackets symmetrisation.
·rhe conventions for the Riemann and Ricci tensors are: V (;I.,; (.b")
(:
-~
R ~"
1?91.61['~
=
'"'RPo.c b V
~
~
•
R" Pb P
is the alternating tensor.
Units are such that k the gravitational constant and c 9 the speed of
light are one.
3.
The Field_Eguatiol!§_
We assume the Einstein equations:
where
Tab is the energy momentum tensor of matter.
that the matter consists of a perfect fluid.
where Ua is the vel~city of the fluid,
}IJl.
is the density .
t't
is the pressure
We will assume
Then 9
ua ua
=
- 1
is the projection operator
into the hyperplane orthogonal to
h o..b L( b ;
Ua
0,
We decompose the gradient of the velocity vector
UQ.;b
=::.
(A.)o.b
+ 0-o.b
+ t ho.be -
as
UC&. ub
is the acceleration,
where
8 :
(Jo. b
is the expansion,
Uo.; o..
-e:.
l,\. ( C _j ol)
h d.C. holb
-
h. c.. be.
3I
is the shear,
is the rotation of the
Wo.b~ U[c_:.d]h:ht
flow lines U.
We define the rotation vector
a
I
c.ot
/'\
b
w
U
We may decompose the Riemann tensor
<..t.)o.
Rab
Ua
-::.
T
10.bc.cl
and the '<'eyl tensor
R
abcd
Cabcd :
R o,bcr.< = Ca.i.ccl - 9q(d Rc.J b - ~ b[c Rcl]CI. Cabe:.~: C[.o.b](::d.J
C ':'.1.c;.o.
-::: o
s::.
into the Ricci tensor
7
·Co.[ bed]
R;3 'Jo_[,5c{J b )
cabcd is that part of the curvature that is not determined locally by
the matter.
It may thus be taken as repr e senting the free gravit -
ational field (Jordan, Ehlers and Kundt ( 5)).
into its "electric" and
EQ..b -- -
C ~~ b c d
-:
Cabp9
8 U [°'-
11
We may decompose it
magnetic 1' compone nts.
uPu 9
,
t=' lc .. , .clJ
bJ
I..A
~ ( '- E dJ
- 4- D[~
I,)
'
')
)
and Hab each have five independent components.
ab
We regard the Bianchi identities,
E
as
field equations for the free gravitational field.
Then
(Kundt and Trlimper, ( 6 )).
using the decompositions given above, we may write these in a form
analogous to the Maxwell equations .
~°'b E. bc;ol
hcd
+ ;:,
H o.bW b
"}o..bc.d U
-
b
c
Lide
<Y e n
h
= ~
,
b
Cl.
}"-;b
(1 )
J
'
_, E.
.; i..
o..b t-
h {
( o.
1 b) c:,d e
u
c
H d.; e
f
+
,.
(3)
7
.
.L
H c..b
-
h ((). -f '7 'b)
C
J. ~ \)(_
E f cJ ; e
(2)
+-
H et\,
e - H(
0.
w
b) c.
(4)
where .l.
indicates projection by
hab
orthogonal to
Ua.
(c. f. Trllinper, ( 7) ) •
The contr act ed Bianchi identities gi ve,
)
fa+
=- o
(f+~)B
0 + ~) u_
Q
+
ft.; b h \1 = 0
(5)
(6)
The definitien of the Riemann tensor is,
U o..; [ be.) :;:.
2
'R a
f' b c. ~ p
Using the decompo siti ons as above we may obtain what may be re garded
as "equations of motion",-
(7)
)
(8)
.
+
where
2 W
h p h 9.b
Ll ( p ~q)
2,
( 9)
C\
-:: W ab W
Q
6
We als o obtain what may be re garded as equations of constraint.
;,
( 1 0)
( 11 )
)
•
( 1 2)
We consider perturbations of a universe that in the undisturbed state
in conformally flat
9
that is
Cabcd
=
O
•
By equations (1) - (3 ) 9 this implies 9
Cf 0 b-::
h b
0.
Wo.b--=- o
r ; \:,. =
0
=
If we assume an equation of state of the form 9
~ ·-=-11"1. (f;)
.
-(~en by ( 6 ) , ( 1 O) ,
::.O~
Ua.
,
...
rrhis implies that the universe is spatially homogeneous and isotropic
since there is no directi on defined in the 3-space orthogonal to Ua.
In this universe we consider small perturbations of the motion
of the fluid and of the 1i/eyl tensor.
We n egl e ct pr oducts of small
g_uantities and perform derivatives with respect to the undisturb ed
metric.
Since all the quantities we are inte rested in with the
exception of the scalars, µ, ~,
e
have unperturb ed value zero, we
avoid perturbations that merely repr e sent coordinate transformation
and have no physic al significance.
To the first order the equations (1) - ( ~ ) and (7) - (9) are
E o,.bjb-=.
I
~
h· o
b
( 13)
f;b
(1l.J.)
)
)
=
.
I
8
;
QZ..
Q.
+
•
I.,(.
0
( 1 6)
1
. 0.
( 1 7)
.J
Q
( 1 8)
J
•
cr~b
=
Eo.b - ~
( 1 5)
I
6ob
8 - '"5' hc,b
•
•e,
Uc,
~
+ u c p;.cr) h
\?
c;,..
h°'
b
(1 9 )
From these we see. that perturbations of rotation or of Eab or Hab do
not produce perturbations of the expansion or the density.
Nor do
perturbations of Eab and Hab produce rotational perturbations.
_ _ __....,..=-----The Undisturbed
Metric
Since in the unperturbed state the rotation and acceleration
are zero, Ua must be hypersurface orthogonal.
u.Q_-.."l";o.
where
;
't measure,the proper time along the world lines.
As the
surfaces "t = constant are homogeneous and isotropic they must be
3•eurfaces of constant curvature.
Therefore the metric can be
written,
where
,
¥a.
d
is the l1ne element of a spaee of
zero or unit positive or negative curvature.
'\".'e define
t
by,
~
t ,:-..
d'r
Js'l.
then
=
I
TI
,
n~( -d.t2+
dr'2)
In this metric,
..
~
3>
D.'
0}·
(pr ime denotes differentiatio.n- with respect to
t )
•rhen, by ( 5 ) , ( 7)
(20)
'i
...
·3 Q
-=-
T (f
-
0
r 3 ft )
( 21 )
If we know the relation between
µ
and
fi. ,
we may determine (l
We will consider the two extreme cases,~= 0
(r adiati on) •
For
(dust) and
.Any physical situation should lie between these.
.L..
;::: 0
f..!~
c onst •
By ( 20 ) ,
...
;, £1
r1i
1..
M
..•
o
~'2.-
I
( a ) For E
0
.J.._.
-::.
2E
( cos
0
-::.
M
l'l.
t7·
( C) For E
n
~ ~~
E
E = const.
)
9
h ~-=-M
~ t
3
(b) For g =
n
"'
0
:,L
- I)
..,.
\ =-
_!....__E ( ; ·~ .
' Ef'{\
i
=> Ln
h iEM
"-' ~
t -· t ) •
.!
o,
f'/\ t
36
)
O9
{, - cos J- ;/VI t )
)
't
J
•
!
~ <~E (t -J2=-/\f\ SL l)J~
t.)
"3
-1:;
•.
E represents the energy (kinetic+ potential) per unit mass.
If it is non-negative the universe will expand indefinit e ly 9 otherwise it will eventual ly contract again.
By the Gauss Codazzi cquati ons *R.,. the curv ature of the
hYP ersurf ac e
"t" = const. is
~R
-::.
:,
E >o
If
E
J
:,: 0
*·R
E <O
r~
t(
)
)
b
- s:r2
-::.
-::..
F
-- Ji"2
-
q
1Q
:B'or li: = )J.1-5
F
--
..
--n.Cc
3
r
<:>-i:.n. h
E
\L
t
5.
-:::.
I
E
s ..i.n
3
E
/
- f•
,v\
n4-
'I,.
-M - -0·· -E
-,<2.
)
I
o,
t
E_ (
)
C.c•S
t - I)
t,
l
o,
t
2
2.
-
)
"J
o,
.,;.
l.
J
::
E2.i.ill?~ . Per tt1rbat ions
By ( 6 )
'.L-
'
)
(c) Por g
Q
:::::
--
(b) Fori E -:::
(2
.
.
I
EM
2.
.,
C,
~ ~-..:I · ~ - - =
0. -
-:::
-
f-.::,
1:
( a ) P or E >
M
(-te7-+f)
4
1
E
(
C..DS
t
~I)
-D.-z_
6
)
,.
.
..•
For
1,.-= 0
w a..b
-
lt)
0.
b
ft )
( 1 e -+ JJ+F
)
fv=- E.
3
c...),.,
-:::
(.,.)
For
=
?P
)
w
::.
w C~
I
- -3 .w
=-
.."
w
-::-
e +'+
I
~)
f'
)
e
Wo
0
Thus rotation dies away as the universe expands.
This is in fact a
statement of the conservation of angular momentum in an expanding
universe.
6.
Perturbati.2._ns o:(_ Densitz
For
fl· =
we have the equati ons 9
O
-re
.
e - T e --=r ~
f
'
-;:
I
'l,
I
~
These involve no spatial derivatives.
Thus the behaviour of one
region is unaffected by the behaviour of another.
PeI1turbations
will consist in some regions having slightly higher or lower values
of E than the average .
If the unt verse as wh '1-.e ha s a value of E
greater than zero 9 a small perturbation will
than ~e ro and will continue to expand.
till h av e E greater
It wil l not cor.tract to
form a galaxy.
If the univers e has a value of E l ess than zero, a
small perturbation can contr act.
contracting at a time
8
However it will only be gi n
--r earlier than the whol e universe b eg ins
cont racting, whe re
~o
is the time a t which the whole w1iverse beg ins contracting.
There is only any real inst ability when E = 0,
This case is of
measure zero relative to all the possible values E can have.
However this c annot really be used as an argueme nt to dismiss it
as there might be some reason why the uni verse should have E == O.
For a r egi on with energy
(")
>i.::.
(t'l.· -
I
4~t::
l::: __!_
1'2. SS
For E = O,
1.<L-=-
I
'1-
-'t.
-
-8E, in a unive rse with E = O
t,412
+ • .•
(t 3 -l!20
)
+- .• • )
/
2
;,
Thus the p e rt urb ation g rows only as
This is not fast enough
to produce galax i es from s tatistic al fluctu a tion s e ven if these
coul d occur.
However, since an e voluti on ary universe has a particl e
horizon (Rindl e r(S), Penrose (9 )) diffGrent part s do not communi cat e
in the early s t ages .
Thi s makes it e ve n mor e difficult for
statistical fluctuations t o occur ove r a r eg : ·.n until li ght h2.d time
to cross the r eg ion.
e-
AS
before 9 a perturb at ion cannot contract unless it has a negative
value of E.
The action of the pressure forces make it sti ll more
difficult for it to contract.
JJ'!).
- +5 iJ i
-
hc,.c. Vc
•_:{e
- .J...
4-
h: 'v\
+
7
a.c
.
}J-
i
7
I::,
Eliminating G 9
f'
t.
•
U.. Q.; o. ::: O
)
h (hc,_µ~1,,
;c
--·-·
__ ______--_:::_}
r
to our approx i mat ion.
i s the Laplaci a n in the hype rsurface
"[
represent the perturbation as a sum of eigenfunctions
= constant.
of
this op e rator, where 9
These e igenfunctions will b e hyp e rsphe ric a l and pseudohyperspherj.c :J.l
harmonics in cases (c) and (a) respectively anC p l a n e waves in c ase
(b).
In c a se (c)
n
wil l t alrn only dj_screte
(b) it will take all po s itive v alues .
,lue s but in (a) and
...
'
m1ere
µ0
is the undisturbed density.
.. Ln')
..!..
B
f'o - 2
As 1 cng as
J-.1...
0
B (n) ,
/.J.o
will gr0w .
)>
Por
C
D ~- i
'c: -r
These perturbations g row for as long as light h as not had time to
t rave l a s i gnificant distanc e compa r ed to the scale of the pertul'.'bat ion
(~ r;: ).
Until that time p r essure forces cannot act to even out
perturb a tion s .
1}1/hen
r
~
..
n >> /Jo
B
Yl'
·3 {r,)
_ri-
l/
Cri")
'
c_.n
.,._I
2
.
B '1111 rl+
/
·t-
r2
n_'l.
-3
13 0) -a
,)
'Yl
e <-:Jt"t
e obtain sound wave s whose amp litude decreases with time.
~·.1
These
results confirm those obt ained by Lifsh it z and Khalatn i k ov( 3 ).
Prom the forgoin g we se e that galaxies c annot form as the result
of the gr owth of smal l perturbations .
We may ~x:pect that other non-
gravitational force ~will have a n effec t smaller than press ure equal
to one third of the density and so will not cause relative perturbations
to grow faster than
t
.
To account for galaxies in an evolutionary
universe we must assume there were finite, non-statistical, initial
inhomogeneities.
To obtain the steady-state universe we must add extra terms to
the energy-momentum tensoro
Hoyle and Narlikar( 1 o) use,
(20)
where,
Ctx.""' C
'Cl.
.I
'
?
)
•
f"-
'j-
~+
'h.) e
+·
r .1 b
u. a. C ll. ...__
b
-=- o
( 21 )
(22)
There is a dif:ficulty here, if we r equire that the
11
ca
field
should not produce acceleration or, in other word s, that the matter
created should have the same velocity as the matter already in
existence ~We must · (1; ~fo
I, ave
( 23,)
However since C is a scalar, this implies that the rotation of the
medium is zero.
On the other hand if ( 23 ) does not hold, the equations
are indeterminate (c. f'. Raychaudhuri and Banner jee.( 11 ) ) •
In order to
have a determinate set of equat ions we will adopt ( 23) b,ut drop the
requirement that
C8
is the gradient of a scalar.
The condition (23 )
is not very satisfactory but it is difficult to think of one more
Hoyle and Narlikar( 12 ) seek to avoid this d iffic ulty
satisf actory.
by
taking a p article rather· than a fluid pictur e .
However this has a
serious drawback since it l eads to infinite fields (Hawking ( 13 )).
From ( 1 7) ,
C
.
.
~
-;:-l.,l
CL
Cl
rL
Thus, small perturbations of density die away .
Moreover equation (1 8)
still holds, anc.l therefore rot at ional perturbations also die away.
Equation (1 9 ) now becomes
e ;: : -
~
e
7-. -
~ (;J.. t
3
f"'-)
+ 1.
These results confirm those obt ained by Hoyle and Narlikar ( 14 ).
We
see therefore that galaxies cannot be formed in the steady-state
universe by the growth of small perturbations.
However this does not
exclude the possibility that there mi ght by a self-perpetuating
system of !'i ni t e perturbation s wh ich coul d produce galaxies .
(Sc iama ( 1 5 ) , Roxburgh and Saffman ( 1 6 ) ).
We now cons ider perturbations of the We yl tensor that do not
arise from r otational or density p 8rturbations , that is,
·b
LI
JI
Multiplying ( 1 5 ) by
LA c
I
;
-:::.
0
O.".)
Ve.
and (1 6) by
we obtain, after a lot of reduction,
In empty space with a non-expanding congruence
Ua
this reduces to
the usual form of the linearised theory,
D ""
F·
-c... I.
i:, -:..o
.
The second term in (24) is the Laplacian in the hypersurface
"'t' = constant, acting on
We will write
Eab •
eigensfunctions of this operator.
where
=O
V v6 :,b
Then
\I
D
/)., I
"
E
-::,
)
CL b
-::;:.
L-r
E
Cl
1.o
- 2-
-::.
D
(Y1J
(_(IJ
r2
Cl
o<-
\/o.b
A.''<1,')
A_'"1r;J I J \/
-r/2
n :,
L
(Yl \
.
.;,
E
ab
as a sum of
similarly,
0 ~lo -=
~
L
DL~) \ / (_Y) )
ab
Then by ( 19)
1
substituting in ( 24)
We may differentiate again and substitute for D'
For
n :>> I
and
A (!I)
-"-
so the gravitational field
HEab Eab + Hab Hab )
as
f2.
E
U -1
decreases as
ab
-6 •
and the
11
energy'II
We might expect this as the
Bianchi identities may be written~ to the linear approximation,
n
l'-
·0
(
-,
9. eol ~"'
f2
Cc.bcol
)C. ....
)
(I
-=
J o.bc
The ref ore if the interaction with the matte r could be neglected
Cabcd
I"'"\
would be propor tional to ~ L
and
In the steady-state universe when
equilibrium values,
R~6 ~ / ~
J a-In
+
-=
K
::.
0
Eab , Hab
µ
and
e
r'l -1
to 1
-
t
•
have reached their
J-i.)~
c [ ci.; bJ
L,.
:1 c La. 'R ; bJ
ThUS the interaction of the
11
C11 field with gravitational radiation is
equal and opposite to that of the matter.
interaction, and
Th e
11
Eab
energy 11
of the metri~.
and
.i
(E· abD
,:.-:ab
2
Hab
There is then no net
decrease as
+ R ab Rab )
(\ -1
~ l
•
d epen d son se•on d d eriva
· t ives
·
It is therefor e proportional to the frequen•y squared
times the energy as measured by the energy momentum pseudo-tensor 9 in
a local co-moving Cartesian coordinate system , which depends only on
first derivatives.
to
Q
Since the frequency will
qe inversely proportional
, the ene rgy measured by the pse udo -tensor will be proportional
n
to ~L
-4
as for other rest mass zero fields.
9.
As we have seen, gravitational waves are not absorbed by a
perfect fluid.
Suppose howeve r there is a small amount of viscosity.
We may represent this by the addition of a term
energy-momentum tensor, where ').._
A 0-a..b
to the
is the coefficient of viscosity
(Ehlers 9 ( 1 7 ) ) •
Since
I
we have
(25)
( 26 )
Equations (15) (16) become
•
Ea.lo
T
-
-
;_
~ -1 fL)
( Ea b
'-1
·:J-1
e1
b +
H 1, e
0
-
'"
h ( ct Y/ b) c cJ e.
OOl b
- ; 0 0 __1,
g)
( 27;-
E. ( J.;e.
LA c.
(28 )
The extra terms on the right of equations (27), (28) are similar to
conduction terms in Maxwell's equations and will cause the wave to
. ,._ t
e- ~
.
decrease by a factor
Neglecting expansion for the moment,
suppose we have a wave of the form,
E a.b
E
t'V't
::::. a - ab E.
.
This will be absorbed in a characte:ristic time
frequency.
"2../?i,
independent of
By (25) the rate of gain of rest mass energy of the
matter will be
~- ~
er 2,
which by ( 1 9) will be
available energy in the wave is
F 2.
4 o.=.
.-2
V
'2.
.
~
Ei.
0
V
-2.
•
Thus the
This confirms that the
density of available energy of gravitational radiation will decrease
as
(2 - 4
in an expanding universe.
From this we see that
gravitational radiation behaves in much the same way as other
radiation fields.
In the early stages of an evolutionary univer se
when the temperature was very high we might expect an equilibrium to
b,:; set up b e tween black-·body ele ctromagnetic :. diation and bl ack-body
gravitational radi tion.
Since . they both have ·two polarisations their
·,
'i
energy densities should be equal.
.As the uni verse expanded they wouJ. 1~,
both cool adiabatically at the same rate.
As we know the tempe rattr'.'G
of black-body extragalactic elec t-'.'omagnetic radiation is less than
5°K , the terrper a ture of the blacK-body gravitational radiation mus,,.;
be also les s than this which wou~.'1 be absolutely undetectable.
Now
the energy cf gravitational radi F tion does not contrib u t e to the
·r ab
ordinary energy momentum tensor
active gravit ational effec t.
l7 -; /::.
~
I
e - 2. ()
<?..
'l.
•
Ne vertheless it will h ave s.~1
By the expansion equation,
-
I
~
(
1'.A. -+ ,
4~)
,,
For incoherent gravitational radiation at frequency
<i- 2.
v ,
:::.
But the energy density of the radiation is
.,
Q •
whe:i."8
µ0
is the gravitat ional
11
energy" density.
Thus g ravitational
radi a tion has an active attractive gravitational effect.
It is
intere sting that this seems to be jus t half that of electromag netic
radiation.
It has been suggested by Hogarth ( 1 8 ) and Hoyle and Narlikar ( 1 O) ,
that there may be a connection between the absorption of radiation
and the AI:row of Time.
Thus in lmiv8rses lik.e the steady-state, in
which all electromagnetic 1°adiation e mitte d is JVentually absorbe d by
other matter, the Abs orb er theory would predic
retarded solutions of
the :Maxwell equations whi le in evolutionary universes in which
electromagnetic radiation is not completely absorbed it would predict
advanced solutions.
S imj_larly, if one accepted this theoI'Y, one would
expe c t retarded s olutions of the Einstein equations if and only if all
gravitational radiation emitted is eventually absorbed by other matter.
Clearly this is so for the steady-state universe since
constant.
In e volutionary universes
\.
~ oe. Tt
where
'I'
J
is the temperature.
will be
will be a function of time.
\ :>t dzr di verges.
We will obtain complete absorption if
/\
Now
For a monat omic gas,
therefore the integral will diverge (just).
fr· '.'
a ga D,
T
..0.. - 2
cf:..
,
However the expression
used for viscosity assumed that the mean free path of the atoms was
small compared to the scale o:f the disturbance.
Since the mean free
path oc: µ - 1a-:. .1.r-i1.. - 3 and the wavelength ~ (1 - 1 , the mE: 8.n free path will
eventually b e grea t e r than the wavelength and so the effective visc os ity
will decrease more rapidly than~r-.l -1
•
Thus there will not be conwl c t e
absorption and the theory would not p r edi ct retardud solutions.
However this is slight ly academic sinc e gravitational radiation ha s no~
yet been detected, let a lone invest i gated to see whether it corresponds
to a retarded or advanced solution.
&l,.t§ renc e s
(1)
Bonnor W.B.; M.N., :LU, 104 (1 957 ).
( 2)
Lifshitz E.M.; .I.Phys. u.s.s.R., 1Q, 116 (1 946).
(3)
Lifshitz E.M.
(4)
Irvine W.; Annals of Physics.
(5)
Jordan P., Ehlers J., Kundt W.; Ahb.Akad.Wiss. Mainz., No. 7 (1960).
(6)
Kundt W., Trlimper M.; Ahb.Akad.Wiss. Mainz., No.12 (196~) .
(7)
Trlimper M.; Contributions to Actual Problems in Gen.Rel.
(8)
Rindler W.; M. N., jj_§, 662 ( 1956).
(9)
Penrose R.; Article in 'Relativity Topology and Groups',
Gordon .and Breach ( 1964).
9
Khalatnilcov I.N.; Adv. i n Phys.,
~
7 '}'Z 2.
(10) Hoyle F., Narlikar J.V.; Proc.Roy.Soc., A,
(11) Raychaudhuri A., Banerjee
s.;
1£.,
185 (1 963 ) .
('Cf 6f)
ill,
1 (1 964).
Z.Astrophys., 2.§., 187 (1 964),
(12) Hoyle F., Narlikar J.V.; Proc.Roy.Soc., A, g§1.., 184 (1964).
(13) Hawking 8.W.; Proc.Roy.Soc., . A , ~ .
~ ;/'!> {/'ft.£)
(1 4 ) Hoyle F., Narlikar J. V.; Proc.Roy.Soc., A, 273, 1 (196 3)
(1 5) Sciama D.W.; M.N., 1_12, 3 (1955).
(16) Roxburgh I.W., Sattman P.G.; M.N., ~ 181
(17) Ehlers J.; Ahb.Akad.Wiss. Mainz.Noll.
(1965).
(1961 ).
(18) Hogarth J.E.; Proc.Roy.Soc., A, '?:&_7, 365 (1962).
CHAPI'ER
3
Gravitationa l Rad i a tion In An
Expanding Universe
Gr ~vit a tiona l rad i at ion in e mpty asymptotic a lly fl at
spac e has b ee n exam ined by means o f asympt oti c exp a ns ions
by a n Llmber of a uthors.( 1 - 4 )
~hey find that the d ifferent
comp one nts of the out c~ oing radi a tion fi e ld
11
pe e l off 11, tha t
is, they g o a s diffe r e nt powers o f the affine r ad ial Qist anc e.
If one wishes to inve s t i gate how this beha viour i s mod i f i Rd
by the p r e senc e of matter, one is f a ced with a dif f icult y
th a t d o es n ot a ri s e in the c as e o f , say, e l e ~t r oma3n e ti c
radiat i on in matt e r.
For this one can cons i d er th e rad i at io n
tr a vellin~ throug h an infi nite unif orm med ium t hat i s st a tic
apart
f rom t he dist urb a nce cre ated by the rad iat ion .
In the
cas e of gruvit Mtional r adi &t ion thi s i s not pos dibl e .
Hor,
if the me~ ium were initi a ll y stat ic, it s own s elf 3 ravi tat io n
would c a u se it to c ont r a ct in on itself and it would cease t o
be stati c .
He nce one is f orced to investigate g ravitati onal
rai i a ti on i n matter tha t is either c ontracting or expandin 6
AS
.
in Ghapter 2, we identify the Weyl or conforma l
<1. bcr.L
tensor
• t
1R. icci-
•
with the fre A gravit a tional field and the
()
-ensor t\.ctb
curvature.
with the contribution of the matt e r to the
Instead of c onsidering gravitati onal rad iation in
asymp totic a lly f l a t space , tnat i s , space t hat a ppr oa che s
f l at
space at l ~rge
rad i al dist a nce s , we c onsider i t i n
;_,ls y mpt o t i cally conforma l ly flat space.
1\.s i t i s only
con form a lly flat, the Ricci-t ens or and the dens it y of me tter
need n o t be zero.
To a voi ~ essentially non-gravitati ona l phenomena su c h
ac sound wav e s, we will con ~i ~ er g r a vit ationalradiat i on
travelling t hrough dust.
It was shown i n Ch apter 2 that a
c on for mall y flat universe filled with dust mu st have one of
the metrics:
(a)
J. I . ·.1.
~
L
('
i
_
_().
( cl I:
l -"
SL ;. A ( 1·(b)
d~ 2.
;.
~
- cl.f. - ':i ,,.1.
-:i
p cLu
(
jj 1.
:..
_,. .
$ ,r\
cor. c)
(I.
l ~'1
C
._f)__
c· ri,
7
l)).
1. )
12i( d.t1 - cL(1 - / 'i(cJ,_()'-.,.-,,, J..9cL1
_Q - 3 At 2.
(c)
,1
0 c
( '·
i~
.2 )
(it, -~L1, - q," /, r(c1.,{) s',,,"(lLvv
.D ,
A ( co S' IL. t; -
'L_,.
I)
(
I . .{ )
Type (a) re pre sents a universe in which t he matter
expands from the initi a l s i ncul a rity with insu ff ici e nt e ne r gy
to r each infinity a nd so falls b a ck a g ain to a nothe r
singularity.
It is therefore unsuitable f or a di sc0ssi on o f
g r a vitationa l rad iation b;y a me t h od of <CSy mptot ic exna n s i ons
.
.
s i n ce one CcJ.nnot ;;et an inf i n i Ge d.is ta:nc e f rom t h~ sou r ce .
Type (b) is the ~ ins tein-De S itter unive rs e i n wh ic h che
ma tter h as j u st sufficient ene r g y . to re a ch infi n i ty .
thus a s p eci a l case .
D. Norme.n
It i s
(5) h a s inve stigate d t h e
" pe e l ing off" behaviour in this c a se using Penro s e 's con f o r mal
te chn i q u e ( 6 ).
He was howe ver fo r ced to ma k e c e r t a i n as sumpt -
ion s a bout the mov ement of the ma tter whi c h wi l l be s hown to
b e f a l s e.
More ove r, he was mi s led by the s pe cia l n a tu r e
o:
t h e ffi instein -D e 0itter univers e in which affine a n d l umin osi t~
d i s t ance ::3 diff e r.
Anothe r reason for not con siderin ;; r a d i a tion
i n the ~ ins t e in-De Sitter univers e i s tha t it i s u nst a bl e .
Th e pas sage of a g r a vitati ona l wa ve will c a us e i t to contra c t
a g a i n eventually and develop a s in~ ul a rit y .
~ e will therefore consid er radi a tion in a uni v e r s e o f
t J p e (c) which corre sponds to the g eneral c a se whe re the
ma tte r i s exp anding with more th.:.n enoug h e n erg y to a voi d
contra ctinb a gain.
2.
The Newman-Penrose Forma lism
We e mp loy the not a tion of Newman and Penro s e.( 3 )
tetrad of nu11 vectors,
LY; n.f") f0~ f1 !A..
,
A
is i n troduc ed
expansions
6 ravitational radiation b;y a method of .8..Sympt otic
.
s ince one Cc.>.nnot ;_; et an infini te ci.istance :from th~ source .
'I'ype (b) is the ~ins tein-De Sitter univers e i 0 whic h 1; he
matter has just sufficient energy. to re a ch i nfi nity.
tb.us a s p eci a l case.
It i s
D. Norman (5) h a s inve stiga.ted the
"peel ing off" behaviour in this c ase using Penro s e ' s confo r ma l
techni que ( 6 ).
He was however forced to ma ke c e rtain assumpt-
ion s about the movement of the matter whi c h vri l l be s ho1·m to
b e false.
Moreove r, he was misled by the s pe cia l n at ur e of
th e 8instein -D e 0itter universe in which affine and lumi ~osi t~
d i _s tance r:3 differ.
Another reaso n for not cons iderin;-~ r ad i a. tion
i n the Einste in-De Sitter univers e is that it i s unst able.
The p assage of a gravitational wave will cause it to contract
a 3; a i !j eventually a nd develop a s in,:;ul a ri t ;y .
~e will therefore consid e r radiation in a univ erse of
t yp e (c) which corre sponds to th e g eneral case where the
matter i s expanding with more th;:,n enough energ y to avoid
contra ctinb again.
2 • . The Newman-Penrose Forma lism
We emp loy the notation of Newman and Penrose.(3)
tetrad of null vectors,
l ~ nr-) IrnI/
r-_
1
I
rnL ft
1·
,
A
is i ntroduced
we label t h ese vectors with a. tetrad index
t
f)- ::
L!~ n.~ r1 t;" M ~ I
(
a_
~
1
2
I
3 4\
'
tetrad indices a re r a ised an~ lowered with ~he me tric
c,,b
~O ?
1
C)
0
0
0
0
0
0
0
- j
lo
Q
- I
;
0
;;.
I
I
( ?_~
;
I
0
I
I
I
I
J
v,e have
Ric ci r otation coefficient s are defined by:
b u.
Zr1- ;v 2
0-
(..1-.
(\_ bc.,
y
I
·2l-
V
In fac t i t i s more conv eni ent t o wo~k ·i n t 2rms of ~welve
c omplex comb inations of rotat ion co effic i e n ts defined a s
fo llows :
K
-
{
I~
l]
I
u
v
(
~l
-.
--
i
Coordinates
·-3.--------Like Pewrnan a nd Penrose , we int rodu ce a nu ll coordinate
u_(·=x;)
{L I. V
we
L~
take
(Li)
0
Thus
3eodes ic and irrotational.
(_ r
i
':Ti ll be
This i mplies
- 0
~
::
/l ~
we t ake
al on g
L~
(
-[
t: ::
fv\ ~
.
A("-
d.+ ~
11.' his g ive s
[I.:[ ~ 0
to be para llelly tra ns p ort ed
~)
'.l \
X )
As a se cond c oordinate we t ake an a ffine parameter
long the ge odesics
Lf-""
[I"' ::.
)f
X'-t~ arc t\·1 0 coord i nates that l a bel the gc ocrn:._: i c
&'
1
X
r
and
.rhu s
1
i
l -
(r,)
The Fiel d E~ ua tions
1J'ie may c a lcul a te the Ric c i and \ie yl tensor c omponents f r om
t he r e l a tions
h C ;rL
o.-b 0cL
C,..
R
=
y
t
a' y - I
r -~
~
~
b I , , ,.... d
().. ~ C ·r. . . .
t ·-
a'
~/)c
l\ C
e.
l
C
0.
4-
b.v ( '
o..hc.cL
d; /
V
).
(
] .
,-
(L5 )
Using th0. combina tions of ro tat ion coef ficients al ready
,,,, --
defined a nd with
Pt
-
l
+-
\.
oG
=
T[
-I-
e
-;:
-
])
-
'1:
=
~r
+-
rs~
·+-
f ,o
--
~{
r
o>.. 6
.+
V,
-
'f o1..
1) o(
j)~
J)y
yA
Ptl
7)
V
-:::
::
f'
·-
YC)
-c (,
+-
-f
I
-t-
t.
-r
6
~0\
·r
>i
+
Y:"f ·
b. -l
-r
(1 _._13)
(1·- . --/it )
( 3i . i5)
----
(s . I b )
C~ 2-0
r~
I!)
C-L_i2)
'f.e •• f2 ·- ;\ -t 1,1
;\ F .,. fb ·r
~{
we have
( J_. i O)
J) 6"
-t-
0
?oo
f
216·
'-[
y, )
Q,
_, ~- R +
o._(! (,J b
C
CC
T
rs+
J.A
cp11
( L__}l)
(l_:_!_t)
.2 ~ v
j) _ ~-v -:.
-r { '( -
~·r ~tc- ;- ((?
~
r~ -· r F - >i
r ;:_ ) ,
-
d_ - ·
f1
f_\ -
gV
1,,
,
(
!J.;v- ,
-
~'r - L1 ~
rl
-
Lk
O
.
.A f - h
Ll "' -
o
tt
=
-t ( -~
6' ·-
ol. -t ~ )
t
Sy - )A - f );\ ·-
s{X')
20( Fy
0(-~
b ·-
·,
-ri
(]
Po ;
T
~? - lt;
f' --,. ( J - 5~ ):A -
f
·t- ; \
3T
.'11)
(s . 'lc1 )
·-r
?" 6-~~)_ ,J
(S 2 2-)
'P J I
yt - 2vjs ir ' f' 2., X5. 1:1 ')_ (~ ?:_ 5)
'fr - r (;u - o' ? )(3 ~ f' ~ ('12j)
J (3 , <1 f' - j 1) 5f "" 1
0 t ~)
cr r ·- f ) r - :}_
k - y; - ). /\ \
t
T
6 V
-i
T
T
=
·r
T
<> °r
CX
0
T
2
ft)_
,:I._ "'- -
r" - .d. <1- -)oft - I" > rJ
oi. -
c:s__2 t)
rh oo ·.:
'1
·-+"' R ~
11
;/o o
(s. 3 i)
(J. -~ L)
·-
----
(usJ
(1 . s l)
(I_~_!_)
fa -
- C == ·- C
1"$ I 3
r
t',
-::.
- C -)"2
'3
C(
f tf
l q/Yl Bl t fY'I f
-c«Po~
L -<-nrL r t1 9
(s.sff)
(33 1)
- ·-·- -
Expressing the rotation coefficients in terms of t he me tri c ,
we have:
I
f
] XL
])u
·-
o
=
-r:f i(
(},/,))
-iW ... ~w- Co +j )
(3-~6)
"T
~/-L\(; {r- ~t-o )~~ T)§" (sq)
ff~ S( ~ (~ - cx.)f-i (" -~) t L (1 vo)
s
<t -"'-)w -(5 )w -i-<r- f) ~
su - LI <t.,. .:r- r) ._; ..- -A w - v- 6 ~
,;j - &-; .,_) ;
w =
+ (;;_
As in Chanter
2 we u s e the Bianchi identities as
.c
fiel Q e qu a tions for the Weyl tensor.
In the Ne wman-P e n ro se
for malism they may be writ t en:
(I am indebted to R. G. McLena 0 han for these)
g 1/5'o - }) f, +]) c/>
o-( -
~ o/o;;-
'f "'-
i·
too
t:, - 4f 1/f,
'2f 9)oi
- ( 2:,,_-14(] )
1.6' (/\o
. C-3. Si )
-r
- ~·,r er D1ol - ~q>o, :: (ltcr ·- ~Y'f' 2 (2 l1+7
s6 tJ. - l1oo - 2~1o, ·1- c2€"cpi,~t ?0 Q(1. s2 )
6 'fa
0
-t- ,~ )
-
at-
3($1/5: - D'f'/) ,-;) (J> cp,. - ~~/CJ 't ~ ~ o 1 - J 1 J), '(' 9(f;_
-t {, 1f, -I <t - 2 ,v- - 2 r - 2 i ) et '1 (1 ,;_r Q "i) Po I
· '1 J ( T - 2 ~ )· rl> ~ 1o rA
+- ~b cb
rh 0 J...
( :S 5 3)
'"f I
'f
I j O - C 1'
00 '.:-
o(
·
-
00
O
·l
0
\-
II
tor
(1:Jf. - rt, )-t :i(D<p 1.L - ~ 1J-r Cf To: Li fo,) . 3r
-~t _b ( 0 -- f- ) t, - q'1. t_1- °f (; € f 1 ·- v ~ o o
T J(f'-/'·-r) 1o, - Jl,,\~10+ ':J'-1::rj;,,
0
-
1 Oi
°r (
J, ~
.
'"C :-
7
'fi -P\f;) .,.
ip) (b o·
T
c,-
/; ~[:_
?,,
)
G. rh
T
(
::2/\
2~ "r 2 'c ·-
(. )
~
TJ..t
(D4,., - S'<p,.o/ -r
·- tri - J,~ fo·o
"I'
-t-
J.
::2
(r1 . -LJ1J = 0
1o, ~
').J_ )
2(f .~IV-
t ./.Q ~ 6' f, 2
- j
· \
::> Ct)
{
i )r
(s )5)
0
3 (j
.
f'?- - ) 'P:i) + (D 10.2-- S 1,J .,. 2 (S'f, 0-~ <l 9,, J
" (, v y, - 9f" t -i -, (; ((?- C ) 1f'3-. ] ety .
- -< cp
V 1,
2{ 2F -r ) Cf,, t 2 ). o/o 2 .
. - X?1-o 'r q_ ( V - :irn 9, 2 (ft .. C 7t:i,-P ~).~ u :,6)
V
CJ I -
0
)
/
1 .,
r
(J tl"' · ·-
0
l
·?:i
/) J:,
7 1.J
~ f'h
_ A(h
-::; 3 ~ ·1J~
T·2. ,
T'2
T '<-
,-
CJ
- .2 V f,-,
..,. J. (
LJ
-t
t (
1
]) 1, rf,, - r <f>D'J.
2- -
-- A 1'0- 2C ~'
'I'
::
-t
.
"1-
-I" {
3Y ~ 2 ( f
(i -
l:i<Po,.,. 3 sII =('2 o- f
d. .-
T
9/') -f;
Q>-
t,
.
'l
:z? ·- ) ., . ) c/J~2
{ 3. <:;s-)
_
2r 2
I
f1
(
t 2 V 1,, - V 110 t
L] </J,_ 1
2 -
f tIn<f
f - J f .T /7' ) </2 0
G' cp ,)._ ~
( J. s i)
1,
~ ~ - •;) 'f. 'I ~ ( r.,.. P.) t,.,
·-
I
p
o<.) 1; _ 7 -
t-E -
lf3 ·- r-yr4 t <jJ.,_
.
f}),
't
-- ) a._ 1 Jr: ·7 "3
-
t,) 1a ~c;. 3f f,_""I
2;;,) <fo, T V 'P
6'
00
12., (l s-c;)
Pl1t
tt., - r ~) ~ ~) (h -,- L rli ~ J'nI) ::: t 2 r - t- 1-J. 1- -~) rA -- z~ i ) rA
'f,o
Jo
Too
Y
/cm
{ 01
- 2. (,;:°'" ~)'?,a+ Vp cp" 1- Ccfc,:i .,-c;-· f?.ci
( 5· 6 0
Df u. - f 1,.,- J: Cf,.,_-t 111,, ,- 3LJ /] = v ~01 r ;; f,a -2 0rj,c )t,,
t-
I
~
)1
er-
2r
a1 -
Acp
et~~
<J.O
'1- {
~ %- "i) (f, 2. T
(
(~
:Z
f- 't) f 'L'
0:!)
I
. 4.
The Und isturb ed Metric
The undisturbed metric ma y be written
/1,s 2
put
,
SL 'l. l cl,(:- 2 - ci, (L
,5 )_ ·~ (,} Cco 'S h t - I )
S ii''- P
f ( cl.,& 2 r 'S; " \ ( 12 ))
b ·- (-)
·L,c -::
I
then
t", the affine param eter, we not e th,,t
J~ o c &lcula te
1
C
is an affine parameter for t he metric within the squ are
b r c1.c ke ts.
Therefore
t
-
S.Jl.
'1.
cL t"
&( u.., G.,
T
cp)
(~)
will be an 8.f fine parameter for ( 4- ·• /)
l]
is constant along t he null g@odesic.
would be taken so that Y..:: 0
when
{;
-:c.
LL
Not'mally it
. Hovrnv er,
i n our case it will be more convenient to make it zero and
d efine
-f .
If" a s
r -
i:Jl2 cL
C
t;-'
( 4 - 3)
0
_0hi s me ans tha t surf aces of c ons tant
1
cons t ant
t
r
are surfa ces of
This may seem r ~ther odd, but it should
be point ed out that the choibe of
a symptotic dependence
R
of guanti ties.
will not a f fe ct the
Tha t is , if
·f ·~
Than
f
~
r ' -- r -r \$
It prove s e a sier to perform t ~e c a lculations wi th t hi s
c h oice of
r
but all r esu lts could be transformed b a ck
to a more normal coordina t~ system.
li'rom
(l . ]) r
The matt er in the unive r s e is a ssumed to be dust so its
e n ergy t ens or may be writ t e n
F or the undisturbed c a se, from Chapt e r 2
r"
V0Now
:;:.
-
bA
Sl
::..
3
52.. c;J
0-
5L
where
'I1herefore if we try to expand
/J-
as a serie s •i n p ow :;r o f
~
t he r e sult will be very messy and will involve terms of
t tie form
*I t
l_05 iris
s
*
11.
s hould be pointed out tha t the expansions u s e d will
only be a ssumed to be v a lid asymptotically.
They will not
be assume d to converge a t fin ite d ist anc es nor will the
q u a ntities concerned be assume d analytic. (see
Asymptotic Expansions
- Dover
A. Erd e l yi :
This
d oes no t invalidate it as an as ympt otic e xpansion b ut
it makes i t
tedious to h and le.
For convenienc e therefore,
.J2 ( r)
we will perform the expansions in terms of
will be de fined in general as the same function of
-=-
{}
r : . A). [
wi:ler e
the n
r
as
'rhat is
it is i n t he undisturbed c a se.
51
wh ic h
(cos-J. t' ~ i )
i s11'-~Jt- Qr;,~ t
cL -52. ·-et T
·-
·J·
·r-% t_
As_ s fl(i -r
132~ if)_~
For the third and fourth coordi nates it is mor e c on v enient t o
use stereog raphic c oordina tes than sphe ric a l p ol ars.
S inc e the mat t er is dust its energ y-mo me ntum t e ns or
and h enc e the Ricci-tensor have only four independ ent
components .
( ti i nce
q>
01
~/e will take these as
I\
1
(?
0 0
,
q>
0
1
is c omplex it r epres e nts two comp onent s )
In te rms of the se the other components of the Ric c i-te ns or
may be expressed as:
· q}o I
cp6/
~00
....-
+~r
900
_tll
~I~
12
-=
b I\
·--
I
1()/
-----
foo
40~ 9io
-·-
(1
+
-rL
~~o,_ ~-fil
b /\
· ,~c)O
~
-
o/01
(er. ,o )
100
For the undisturbed univer se with the coordinate sy s tem
g iven:
(\
-#=-- i.~
:;
(foo
::.
q
O
I
t.;- ...)<')_ '3
1E
J7_ '>
4.,
4~
(-;
1fJ_
~
452 3
--
'J.
-,
18
452.
~oi
-- 0
( '-( . II)
~
Using the s e values and ·the fact that in the undisturbed
unive rse all the
(3.
1( '-
5
are zero, we may integrate equ at io ns
10-50) to find the values of the spin coef f icients for the
- G --
cl-- --
2
•
....
p
-
252.
--
t
u
.'.l
-
·I
:2
y__ s- e
I
J)_ '2..
-
_ft
4
J2
.
4
+
--511
\
- (-):152..
-
1AV~ e.
(A-
i
4-
<-t""
t
?.._ i.;,)5L-'·\
(l 3 (
I t-
2e
2/ ..J2- ~
::i.;- _
~
fJ '2.
A~
L J2. '),
4-
4-Jll
2.
(~)
5. Boundary Conditions
We wish to consider rad iation in a univers e that
asymptotically approaches the und isturbed universe 3 iven
(pOD
ab ove .
6 iven
and
/\
will t hen h8.ve the va l ;.1es
ab ove plu s term3 of small e r ord er.
order G.nd the order of
4>01
wuys in which we may proce ed.
and
·:·i e
%
To d ete r.mine this
'
t t.1ere · are t v.' O
may t ake the small e s t order s
that will permit radiation, that is
and
Lar~er order terms than these in
c/Jo,
derivatives
turn out to have their
dependent only on thems e lve s and not on the
of
, the radiation field.
r
-J c oef fici ent
They are thus disturb ~n c es
not produced by the radi a tion field and will not b e consid e re d .
Alternatively we may proceed by a method of successive
..,
approxima tions .
We take t h e und isturbed values of the sn in
coefficients and us e them to s olve the Bi a nc hi I d enti t i es a s
field equations for the conf orma l tensor using t h e f l a t snacP,
bo und a r y cond ition that
llf",;
these
[
·y
0 (
:
0
in equations ( 3 . ;O -
v-9
~
r )
. Then subst ituting
calcul at e the dis tu r b -
ance s inCLuced in the s p in coef f i c: ients and sub st ituting the ;;e
b a ck in the Bianchi Identities, calculat e the disturb anc es in
the
"'l l r' s.
~urther itera tion does n ot a f fe c t the orders
1
of the d isturba nces.
Both these methods indicate tha t the bounda r y conditions
should be:
/1
100
to,
Yo
;,;
-=.
A
-r 0(51 -1 )
(~~/)
3A + () (SJ_ ·-~)
SL~
(G)
-,,,,,.....,
&,52 ]
-
= 0(52"-~)
(see next secti on)
0(51 -=t-)
::
(~_:_])
(~_:J)
We also a ssume 11 uniform smoothness", tha t is:
J
dX 'Yi?
J_
(,..
_d_ A
)S2.
-:::.
. .......
--d /\
JX
1PI
=-
0 (5)_ -=,-)
t.,
't
0(5L -~)
~Sl ~
et c .•• /
~~
w~ii be shown that if these bounda ry conditi ons
hold on one hypersurface (~= Const.) they will hold on
suc c eeding hypersurf a ces a nd that these conditions a re the
most . severe to permit radiation.
6
Intesra tion
As Newman and Penrose, we beg in by integrating the
equations
(3.
10 &11)
"j) f
J) er
where
100
f
:::.
7-
Jf(J
::
-- -r foo
fo
t- 0 6
t-
3~
:;
4/i rJ
-;-
.,..
Let
O(tt- 3)
then
l et
then
I_
·op - p··2., ~
p -- -(p 'j ) \.f ·- I
D'2 y : _ -C/?7
f r (() cLr
since
V Lt
..t{.
the refore
((:.2)
; . FT o(,)
ce ::.
therefore
(l12 )
oCJ
"'-I - r F -,However
(6_.,)
v~ 'y :.:
f.,
CJlr
0
-$.)
~~
where
'F
i s const a nt
Cr )
cpF , 0 {,
p 1 - f ·,O (r - ·'5.)
-,
)
~ ) -:. 0 (, -3.
1
-1
(t 5)
i f } is n on-sinGula r (The case~ s ing ular co rres p ond s to
asymptotic a lly plane or cylindrical surfaces and will not be
conside red he re).
Thus
Let
f
-:. -·o/
6'
-
\6)
where
~I
.
·- I
1
(
·-.l
Qr~
)
C)
5) - ?
:-.:--J... . ,
.,.
,,..,/ri··f)
'1!L
(r -·t ) - o.{52- 3 )
- 1 Jl-:l. r gJ2-:5
0
~
h, 52 -3
hi.
J
= O(i)
Then using:
-J
Jt
-
(t · (?)
("\ - i (
_) L
I
t
. A n_ ~
1- "'f
A.)n)_ ·-..!.. -A !l n').
·---- .J . .
3._.
1~
J
J
2
~---
Integ rating,
Q
j
::
I
l,,,.,
_ 0.
"t"
ff _ l
ft ·t O(,<]_ 1 ,) )
Jl-; 0(1)
.JJ_
1:
J)_ ,.- 0(1)
therefore
i or
h_,
li (31-r oc,)\
JS)_
therefore
).
~
- "---r
0(52_ -')
0
( ) d J)_./.
i ) '-
Hep 8 D.t the p roc es s with
fe
·--
- 252. - 2 ·- ,4Jf2. - 3 552. - y
l52-~
(6. 12)
T
:::..
r
I
0 ( &>5 .JL )
5 --
wher e
k
0
=-
~
(J)- -r
J.51
A-
~
(~ .n~
then
(,)
0(1)
o (SL -
;.
6 6( ().
-
0 (, '))
I
)c.
<.. ) .
...,
1
)
o(n -')
t
(c0._s)
o(5)..-1~552)
J.51
~
Unlike
::; ( ( .;., x' ) '\'" 0 (52 -,~3 5l)
(.) 0
Newma n and Unti, we c a nnot m~ke
th e transf ormat io n
'r
1 :::..
'r -
f)
0
I
a lte r th e b oundary cond ition
/\ -=-
ft
(6. Iv)
zero by
\-
sinc e this would
,
. K
52 ':3 . -r
(;
Continuing the above p rocess we der ive:
-~ :l
.- 3
0 fl- If
5) -:::,..
.)
-
( .l
'- 2 A51 - sf .;. - 252 ·- As2. °" 1.. ~
-:i.A - , f ·
( ·if/ 4.,. 4A~Po·- f~1._ 6.o 6"') f2 -; 0(52. - 7) ®'· ,8
0 )
"t"
~
r·" .:.:
o
v
(Y ~
I
. -lt ·-
~
(1 A6"
1
0
..,-
1
lr: )Jl--~.
1 -~
((}J)
determi ne ti.1e as,yrnpt oti c b ehaviour o f ~
1
1 10
and
> ol..
1
,P , 5
i..
we us e the lemma p roved by Newman ~nd Penros e :
u.;)
The
r
1
matrix Sand t he column v ector
(\ X I)
b
ar e
given f unctions of x such thE.tt:
B-=
The
n
o(x - 2 )
matrix
x ri.
.(.}-
I
is inde pendent of X
eii!;e nv ;:;.lue with positive r eal part.
va nis hing real part is r egula r.
and ha s n o
Any e i genvalue with
Then all soluti ons of :
..
,,.
X
are bounded. as
·-) cl)
. .J
is a c.: olumn
vector .
J!' or reaso: s to be expl ained. below, we will assume f or
the moment tha t
(t . I 5)
We t ake as
(S)
'
j
the column ve c tor
~ '") lr S2. 1 A/ Sl")., A) 51 l(\
S2. <.. e 351-:i.
. )"""')
j )
L. - 1 I )
e3 Jl.')_C.) 'f)_J2 jlC1'(;j CJ]
~ )
I
.t-J.
(t_2?)
3. S / J 5, 13 J ·1. {(I, S.
By e quations
A- ~··- J')
00
r
A b /f o o O o o
O (o
000000
0
O
0
0
0
0
0
0
0
o/
0
0
0
C'JOO
D0Q00
0 o "-i
-/ 0 0 0 O O -?..
and
b
-Io O O 0
are
0(51-1)
d><.
'f.
-=
r/.., ~/ (',
0(SL-')
S" "'
(;.)
u inc e
0
0
%j,d~ to , ~
1· , er; Vo ,~
cJfL
'i'hus
3. (f y
i 5. fr O
0
o
B.
~ S1
o (5L-2.)
::
~ ;; o/.
0(1)
+--
C)
8
en-()_ ) .
0
CJ
O
() -2.
expressions involvin~
·a nd
0 ,.
J ri
)si Toi
(1 . ,' J...)
LJ sine, thi s we i ntegrat e eq u at ion
b y the same
\-.J e obt ain .
met h od as ab ove .
_ 3"
't ° CLA. X t, ) 51-
'.l.
(@ )
+ 0 ( 5L )
)
r
r
je
may ma ke a null rotation of the te t r a d on each n ull
L' /J-
ge od esic
(LI
f-
M ~ /u
0-
Lf-
-:::
fl.f ,. a.. n,-"-
:::
Mt-
::-
r
-·'
o.....
nf<-
i
Q CA.
{ r-
,- o. L f-
is constant a lone the ge odesic since the tetr a d i s
parall e lly tr&nsported.
we may ma ke
By takin g
Jnd e r a n ull rot a tion
1'
0 ,
= c/>o,
-t
o.
~
,o
::: 0
?oo
1:hus until we h a ve specified the n ull rotation we c ann ot
imp o se a boundary condition on
~v
JO/
more seve re than
We will speci fy the null rot at ion by
'"'Ca /
o
and in tha t tetrad system will i mp o s 8 the
'r'o, .:. () ( 51-·-1) a nd is uniformly smoot h .
Then by us ing thi s condition on
rh
a nd '1:.. :: 0 (Jl"j
b oundary condition tha t
rh
fo1
by equation
w ·-
using this in eQuation
then by equati [m
r
'-Yi' ::
(5 1·1)
i; '
p utting this bac k in equatio n
o (SL.")
0·i./ {{)
o(5L-J.Lo552.J
w =by equation
-
O(
5l - 1 lo 5 _
5)_)
by e quation
by equation
by equation
{1 .e; t)
I
j/"'
:;_
(6, 15)
0 ( .5)_ - 7-)
'( i
by equation
by equation
(_b . 2__§)
(g .'( '1)
j,
By differ e ntiating the equations used with res p ect t o X
one may shm-1 tha t
-~
J
o<-
1
~/
Lt;
I
5
.
L-
w
a.re
)
uniformly smooth.
Add in::; equations
'-
I~t - .D'f;_
I
<t
a nd:
S , (,
O :-
Df. -tcpO/~ D;L ) to - 3r t'l t '1 .. t - f- P
,- jf 1. 6'cp?.O
00 -
9-
'2 d. f o
(_UY}
we may u se the lem1na again
-
J
r
.Jl If
.JL '2.
Jl.
}]
fU- -
r
3 . /G · 3. It-" ~. i8'.
J3y equ a tions
I
(
AR
J
;,·1i th
b
and
--
·~ 1 .
0
0
0
0
0
0
-I
-
are
Therefore
~A "11
-2
O(SL- 2 )
o (.n.. ·- ~)
::
O(J1-~)
~ ·=
-.
f
o(n_ -')
.::-.
and ar e uniformly smo oth
and
From
~
u sing this in
(&.
(l _I?}
f 5l -, .,,.. o{.n -
c~.
,1-)
I-"- f
c
and
y.,
=-
2
!o:5J'l)
2 ?)
1-f'.,_ :;
then by
we may show
J'l - 't
0 ( 32. -
0(52,"5 t15Jt)
11' o(U,, X L).52," \- 0(n.,-3 (,,5 f)_,)
S-)
Integrating the radial equations
S. i 3/ 1./~) 1. I ~J.K:,,,
·-
r
? ,~
( "
/\
(X
-
~ ·- 2 (-/
- ·
0 \l ol J L,
$
..f.. ( .7
"t
')_
..) (}
':i._
6
cl .··-
t
0
oi..
0
- 0 o( 6'
7
)s:32- 2 - Q~ JL-s.,. -~ (3;9~0- fo?
0 - ff 5), - , -,. j)_2s)_ - 2.-,- 0 (52 -s )
0 ..,. .~ 0
0
'
4
.,,_ )i oJ2. - '2._ t4 ( ,\ o')" 6'
1
u) 52, - s_,.
.f ·~ ffe-:-A52'r"°-r- H31tf"-(
)<. ~
cc
U
=-
's. ~ S"
(t 0 .
3.
o(
$. ~ ~
O)·
tr
·- l/
.J1
0
O(Si.. - ~)
y-
/
0 L'J2
)n-ycr o(Jt-~~_JJ
(1, S3)
(6 , H)
.
0
f:'- f6°)Jr\- 0(5;,-s)t~
Xco+ 0 (Jl -s)
(G , S6 )
- 4 (oo-i- j'")D} A(h-(t-t')Jl -J A,_ -1
X ( l
Qo
f o) {o J 52_ r J o -,- o(_Jl )
-t-
-57c(.{.,,_J_/./
. .,
o;-
(~Y )
·rhere fore
(1. Jo)
By e qu ation
v
r
=V
O
f_
-
'f' ; Jl -~ .,( 2
By the ort ho1:orma li ty r elat ions
j
-
sLj- -
-(f c3-,.
Xc
ill =
O(
(G ~ o)
52 - ')
.1.)
i~cJ)
o c~-(i )
X'- Oi-
-er Y, rsJ)
L-,
- - (;"°f J°-t tsJo(Jl~!) AJl -')
1"
-r
s, ~
0(51 -G)
0:!i!J
By mak ing the coordinate tr ansf orma tion
LL ' -'r
I
I
It-,
X
whe re
C
LA-
-
.-
'
X
;
'
L,
T
Cl, (LL, XL )
3
..J
1.
lj
X
/ 1.,0
:::.0
C;,
We still have the coordina te freedom
XL ~
I)~(xj)
.ve may u s e this to reduce the le a.ding term of
3 LJ (~)..: \ t.)
a c onforma lly flat metric (c. f. Ne wman and Unti ), t hat is:
9 ~i
~
_ 9 ?? g,1 (.Ji-~ -
'2 pn-') ,, d.sr')
to
.
(t_!J
J
where
7. Non-ra d i a l Eau at ions
r
B)" comparins coeffi c i ent s of the vario u s powers o.f
·cf3
)
in the n on-radi a l equations of
JL..
, r e l at io ns betwe e n
th e int egr a tion constants of the r ad i a l equat ions may be
obtained:
the term in
n-1
J
'-
·i s
therefore
ther e for e by
(to .3 '1-)
J
rn equation
u ~ '(-' 0 (51-1)
"J.
U - -:;._ ...f1
(1 ~ Su) ,
"t-
t he constant t erm is
\lo
t heref or e
By the
::fl- 1
{/0 :: 0
te r m
p
t i1er ef ore
( ] . ·1 )
D
~
0
r) i
-i
).
· CD).
if
r
By making a spatial rot a tion of the t e tra d
01\ tf ~ e,_- Lcp ('11 rtz. ,
g r eal. \fo tak e 5:;; ~ ( I -t/; {/-t ;:,/) ,
we make
st e r eo g r a ph ic pr oject fa ctor f or a
jl_·- 4' te r m in e qu a tion
By the
J.
0
f'
~
2-s p her e .
(] .
~ 8')
e.~
- {VSe"-
J
~ _d_
'J ·: -~ ·~ -r J X 4·
· wher e
By the
b fjr;
-
5)_,- ~ t e rm i n
~ - r2._
1)
1e' '-' (~VVS r ~v7s)" -2r,
r
o
-
-
-f)
lf
'<- -
2
. lf 5/ ) ~- 60) ~-fat)
c~-
2
d(A.
0
~ <.. nvv5e.'7 i.o S Q,u.
_
- -A~["/' +-~ 'i2..LLJ
·-
By
t he
-
rh
-~ 0 (52. - 1 )
TOj
-
fi\.(vPf\/ P)
)
cfo
(oo
d
J~~
r
.(U)
/\
.vh ere
1
!{ -By making a c oordina te transforma tion*
t ( --
S'' /-/ cL
I/
LLO
the r efore
By the
t r1erefore
r
*This transformation does not upset the boundary conditions
on the hypers urface
(),- =-
u...
0
Jl·-y
3y t tie
(?. 2 S)
term in
>" = 3 (o· ~.·- 6.
.
..)L - 4 term in
By the
'f.; '- (
r
6
By the
2
..
(.,..;) 0
l
'T
t heref ore
C :.
the 1:ef ore
Jv
. Then by
°--
(]. 20)
e c,._sv C'
WO -
i
0
~
-
JO' \7 S' e ~
O
k -:. 0
(r re;)
0
·::.
"l.
O
vs- - i e."' SV((,"'-6:") (7-~
2 6°fJS) T K(x c) e, Q.c.
fl_ ( I ·- e
'2..
(t . G)(-1-.16), in (6 tJJ{)
P
I
Using
e '-'$'0)
C
term in
s 'v(
)
0.
~"'( s V 6
..)1,-(:,
e., C..
0
i I Tlj"\ 3
:;
;,
co
k) C>
By the
--
a-
-
term in
._Q-2
i...-.;) 0
~I
Q. 2 2)
(tJ!)
o )
JI
'1.. LA.
(:;-. / 5)
(t- -1~)
)
-~
-
i·l.
~
0 ( J2, - t
i~ 52-)
'
I ~) ,tj 52, _ !]__2 { I + e.,'l } fl - 2-r _E_ (
Using this in
1 _
;_
l.t
+ () ( S2.. - ~ -t~5
G2&')
2
52..)
--r,_ _;.
0
(n - ")
t
t- /
')
J2 -J
(L!i)
(!- 18')
Bv
V
.
(s , '5 I ) . 0. ~ ,) @· bO) Q.(; i)
_2 10,
c) c ..
r
I
-J~ +
d
C? 0
_J_
J
/\
-
0
(.51. - :;-)
--
0
(n ·- c,)
a{52--+)
.,...
V'--
iyc
J~
0
:::;:..
{_?. ( CJ)
( 52.. -:;)
Th.eref ore if the boundary c onditions
(
5". i -=--
Y)
l1old o n
one null hypersurf a ce, they will hold on s ucc eedin5 hyper surf a ces.
By
(6 , S-?-')
<f~
'11 he
a c51·- i)
-
"peelin6 off" beh~viour is therefore:
·i~
-
·y;)
::.
f~
'-(
to
(
CJ ( t
-j)
a ( r - 1)
0 (
....-
T
-:J)
J_ .
·--
o(,-i)
o ( r - ·1)
( t-.20J
As mentioned before , this asymptotic beha viou r is independent
o f t h e zero of
r
and wil l hold f or a ny affine ~ar amete r
To perform the rem~i ning int egr a ti ons we will a ssume
r
'l 'hen:
r
/,}
"'
@, [;. (
y
-r
r
c; Vo
2 6' 'v s:) 52 · 2 ; 11 e.-"'( s 9o J6 ' Vs_)
0
0
-
0
-
i ( cp;;., ft )_752·3_,_ o( S2 -
4
r " - :i - f
.n -
! 5l - '- -{Ls
'2,
i ,
J2 - '<j ~ ('{It-
~
I ; ·-/7
f1_ \J ( '5 -. .2:
l
O
t Y,..
]. ·o
r
.
2 .,. .//
J (6'.
o
·
e>(
e <2 /
tA
Of
4
u)\ cy-I ·- 6) (o.6"
-·iQ_
;
2.
-- 6
6
)
e"'
)I
t [ f)
, O (S2...
-·- 6
(2)
-;-- )
.
.
?-
PS' -
Je
c..
2 e"'
-s:)·
t·
o. "'""; ·
/
( 1- 7. 8)
.
::__'
17- . 2 C
J
( ~--·- -
I
S /7(6', 1° - §
0 )
)52 ·-)_
(r sC?_)
o( Sl. - s)
i ~ ~ V S'-52-:2_ f
-
.i..8 e
67 6).52_ - : ;q o· ~I S2 -.:s__ o (52 - 'IJi
-
-j ( ( 6' ~
'i
J2_ - _j
( "
0( R - )
.,.
) l
V
>( u 2 e 2.1;.)
~-
J1 - 'f
2 ?)
j-.
)
X
f /') x -v
C
f - /l 52-• - i '-( 1-t- e :lc')_n., -
. A ~
(;<u )
IP5 J2)
0 ( 52, -,; )
1-
-
·- ~
vs~(-'J
Ae
<r
;
~\/Sn-~
CC
c)
.,. -e '(J;(Y7 s) 0~ 0).rr 'f
("
-~ Ji)
1
(
e '-,s 52. - ~ r; e, "S.52 - 5-.;.
C:
_ e, "'-S a- 0
U
1
r
J12 -
xs" ~:) [<f>o~
lf
f ~" )5l-'2-,- o(Jr 5)
7
c [-
b
O
'r
6'_,~
;
T
-
_J_
2
(5
7n-s-r
o
.1
)II
t FJ ~ ( - 6' CJ-;-
4
X (
6~~
'-'{
.
( ~ lA- (
/_
6
2
Y
<5__0-- 6, ~)
I
0?e ~(--2 (0 c LI
JI
'f
;
Ii 6~ 1,,.
-
0
Jt
- 6 " )\
J I
-
l{-
i 6_
(.U!J
)JC\ [--6,~
v(s (6;,
O { Jl
0
-
'Y-
0
)}
o .)
) r1
)I
)
J_ 6~
9_
rn
':L(CJ °+ 3-_ Cfo- _5 (5
L/7
r
; 3::- i !r oo_] .j l ·- Yer
o
T
( ~ 1==
6~~ 1
-- ;
rio} fl-<;i- 0(5?_ -~ ~
1-
~.
-t-
·
t~
-r
.
~b
(t- :n)
.
f :,
T
(?- s c')
CJ ( 52 - ')
9 .,,
{n'- t(f",.
cc
fF2 e. "-s (5§: -r ~ s )
!,_ [
e. c.:L (,~(5°)
e
2. ,.,1..
·1v-· (s...:)
_
'.,»
- s-)
(!.JS)
vs - ~(As v(a· rs O) ) R -v
_o --
) )
.il,'B'
u
0
)
~o;J_n_-'·r o(n-
__
-v·-s- ~ S>V (6) )~- ~Ii/ 6
6)
(J-
o ))
/
]6)
r
't·n
.,1..
f
o(
.
()
, _:s(ffr Cfo,)
- () )
6-::j (.) -o
O
j
-- 0
? I+ 6.
0
a:-
u
.'J.
(?-. s?-)
~
e, (,_ [
s \7 (f: + -1§ 0°) {2 t ,~, ;<; 0°)
.._ JS
rA
C)
Ps/
(?-. ~7)
y 01
- e "- (S' ~ - 2 v~)(5Atj'i -t; -r 1~1?6°)
+ V<fb',
lA V\
(J . 00)
J e_f:e lfl1 /
A-e__
. Lir i
..,
[
<=-
I)/
J
'~
i/.0
I
r /
(?-. CJ r)
Thu s che
0---
der ivat iv e of
fi,
and not on the radiation field .
(t epend.s only on it se l f
It there for e re ures ents a
type of disturban ce unconne ct ed with r ad iatio n .
If it i s
zer o on one hypers urf a ce, it wi ll re main
In t his cas e
zero.
it i s p o ssib l e to conti n u e the ex 0ans ions of all quantitie e
in ne rs a tiv e p owers of
J2
wit h out any log terms a ppear ing .
The metric h a s the form:
j
=
()
'2..
)
The asym ptotic g roup ts the Gr oup of coordinate tra ns f ormati ons
that l eav e the f orm of the metric and of the bound ary c onditions
unchanged.
It can be de rived mo st simply by conside ring the
corres n onding infinit esimal t ran sf ormat ions:
(~)
K") (E)
(?i)
f
rA
·-
~DO
(f~v)
-
To obt a in the asymptotic group ve demand
r- /( ; .
J-
j
f g3 22. .::
.
.__
12_
2., (...
(j (
=
' '
~5 l,d
--
._.
~
-·-
~o o
-
r rf
-
By
j
==-
0
('( t)
(5L--j
o
{SL- 1)
-- o (SL-)
O i
~J
I "f
52. -1.j)
0
-
f
·-
0 (.k-b)
-
~) 11
IS
O(SL-z)
I
~__J
~ f 5 - c95
-
H
- 0
._..
/
11
l.
--
fr()/ (
u_J
i(
l,)
(~ )
B;y
S '-f
.
/-(
I
(u )
J V
Ii r - 'p<.
(J {
.52. -- y ) =
rt 1
JI
I\
I
t--
2 v 1-(
11'0~(x:i-)
{{LJ3, -
5
.l I
1, / 3 -r
o( SL - 's) ('< ;cJ
-tO ( s;_ - ')
(U/)
(?Jd .
( 't. 'l)
( ~C; )
(e-. ·t,2)
and ~.
)('3 --j- ~
x:_1
(3)
•
imply that
k.
0
C: is a n analytic fu nct ion of
This is a conse que n ce if the f a c t tha t we hc:we
'I
re tlucea the leadi ng t e r m o f ~~
form.
to a con f o rmally fl at
x
~i.'huu the only a llo ·.-1 ed transf o rmation s of
i..
a r e the
conf ormal transformations o f the for m:
·x
l
l.
l
·t' t... X
=
be.
i·!b.en T,he s ix parame t ers
6.etermined by
('?.
/(J,)
0-.. ,
i{ :l
i
b , c., cl
a r e g ive n
K
I
i s uniquel y
i s a l s o un i quely determ ined. .
'1.'t1u .s
t he as ymp totic g roup i s iso morp hic to the conforma l g roup i n
t wo dimens ions.
6 a chs(~)
has shown that this i s iso morphic
to t i1c t10rnoz;e ne ou s Lor entz i3roup.
It i s als o however ico111 orpr1.i
t o the g roup o f motions o f a 3-space of constant ne g at ive
c urvature wh ich i s t he g rou p o f the u n perturbed Robertson J alker space.
Thus the asymp to t ic 3 roup i s th e same &s thn
e;r oup o f the unci_ isturbed space .
presence of r ad i a tion.
It i s n ot enlarc;ed by t he
Thi s i s i n te re sting b e c a u se i n t he
case o f g r av it a tional radiation in empty, asym pt ot ical l y flat
space , it turns out tha t the asympt ot ic gr ou p c ont a i n s n et
o n l y the 10 dime r1 sional i nhomogene ous Lorentz g ro u p, t he g ro up
of
11
moti ons o f flat s pac e , b ut a l ~o i nfinite dimension al
supertran slati ons 11
•
It has b een sugg ested that th Gff-?
s upertransl~t ions mi ght have s ome phy s ic al si gni f icance in
e l e mentary p a rticle physics.
The a bove result would s ee m
to ind i c~te that this i s prob 0bly n ot the c a s e si nce our
un iverse i s a l rn os~ certa i n l y n ot a s ymptoti c ull y f l at t h ou gh
it may be asyuipt otic a lly Robert son-·,Jalker.
9.
What an ob ~erve r would me a sure
V
The velo c it y v e ctor
of a n ob s e rv e r movin s wi th the
du st 1...ri ll be:
v
Jl .. ; +
..!..
V
~
1
Jl. +
0 (J]_ -~)
V
-~
v
=
o (JL-:r)
4'
, the projection of the wave v e ctor
r'I'\
in t he
ob~erve r 's r e st-space ( the apparent direction of t he wave )
will be:
ci
M
c~
-.
J)2
'/I')
/Y)
--
.~ 2
J'L
t
__.
i
).
vv
-
2.
o(A-~)
0 (J2 - l[)
"I'
-;-
I
-=.
i
.3
--
(/
L
lf-
The ob s erv er ' s orthono rmal tGtrad may be comp l et0d b y two
and
space-lik e unit v e ctors
t
(V\
S'
s
0 (SL ·- \j)
0 (JL - (f)
.;;
C>{ J)_ ·- Z)
--
0 (52. - Q.)
..::
').
s
-
Ji
'.3
)
l./
--
• I
f
.,
·---I
f;
- l
.3
IT
Ii
L'
f;
u
.j 2
(0~)
o(
: e v1rit e
~v
·I
\j..
e.
mea surinc the re l utive a cc e l erat i ons of n e i ~hb ouring dust
p a rticles , the observ er may dete rmine the
co rn]b onents of the gr a vi tat io nal wav e ~
E . ·0-. b
~C
0...
f. b c;,,
V? V ci.
1
el ectric 1
In the ob s erv er ' s t et rad t ·:1.is h as compone n ts
E
.::
......
-a
0
0
0
0
0
0
0
i
0
0
0
.L-{
0
0
fj
J..
-
0
0
0
o(S2 -<+ ;
0
I
,.,, L
0
0
0
0
I
0
a -J
0
·-r
0
( J)_·-it ) -r
0
-
·- i~ 0 ( Jl- 5)
o r n-b)
le . \
I .~)
).
Thi s s~ould be co~par ed t o the b ehaviour for a s ympt otically
fl a t space f or ,vhich
,--
[:::
0
0
0
{::)
0
0 - i
-
+[:
(
0
0
,..
6 o(r- 1)
~] O(r-
2
)
+-
~
0
C)
-1-
0
·- {
:!
0
0
C)
0
o ( r- i)
o(r- ~
1-
7
C)
o'O(r- 1 )
~
o o-~
0
·-1
(9_5)
• l .
oncli ,
.• G. J . v ....n cc_· ~ur__;
c.J.lld
• •' .
..
')
(.
,OJ •
;oc A. 2
roe .
t OJ . ,.J O , • I • : '/0
_
••• 1Lth . 'nya .
/ •
. 1d
l[ •
~ • 1
i..l
~··
e tr .u. _
.r • l . J . . i
'l '19 2
_
·10?1
!CJ
2
) ),>f, '1962
J . ici.th . 1 : Yf' . d. , 9
'l )u2
D . ,10,.·- 1
6.
-; .
roe .
_,. . 1,e t;:.:;ncr
• .t10Il.i.'O 0
t
• .i.~ . ,.:i.:J.C.i:lS
l
' O..: • .·Io;'f .
1
,oc .
t..>
•
,,q~_::,
,.l:>:J
1yn . lcv . 1,28 2 ~-J ··
~
2
• f • oncli ,
. •G. J .
111<.J.
v-·
n cc · ,ur~
11e t; zn ~r
r ,c .
,.•1
.
••
.
.
<l
L
r> o
c.
/ .
)
•
..
L1 • 1 O~J .,
0
,.
10 -·-i.un
H, .Pcnro ,,c
L • • ~ . '·'
.:.!.Cil .. .)
. 1" t11
0
"'·
•
L1yr1
, u "
.;
,, .l ),,(·)
'105-2·_
"./
J • m t 11 • l yn • d..
l
J. J .
'l '19 ,2
• H.OJ . b oc • I • ,_"/0 ·103
:i: oc
;:,
,OJ . ..;oc 1 • c
. .. .
2;i
10G ·..;
Lo
LtO,.
U 1i ve.r.';;i +,~l
9G·;
ro.; . Jo;y . [,oc .
,. l ytj • L' ~ V,r •
..,t e-)
,,
........
r;.
.;
1 .J
c..')
1CJ
J2
S in::;ularities
If the Einstein e quations ,.-1 ithout co smolog ic al const ant
a r e satis fi ed , a Robertson-Walke r model c an 'bounce ' or a v o id
a singul a rity only if the pres s ure is less than minu s onethird the dens ity.
This is clea rly n ot a property p ossessed
by n o rmal matter thou gh it miBht be p ossessed by a fie ld of
negative energy density like t he 'C' field.
However ther e i s
a g r ave q u antum-mechanic a l difficulty associated with the
existe nce of negative
energy density , for there would be
n othin 0 to prevent the creat ion, in a given volume o f ~pa cetime , o f a n infinite number of qu a nta o f the negat ive energ y
fi e ld a nd a corresp ondin g infin ity o f part icl es of p ositive
ene r r:;y .
If
\,Je
Wa l k er mode l s
therefore exc lude s uch f ields, al l Robertsonmust be of the 'big -b ang ' type, that is the y
h a ve a s inr;ularity in the p a st and m~ybe one in the f uture
as well.
It has be e n s u g 6 e sted
1 that the oc c urrence o f
these singul ariti es is a consequence of the high de gree o f
symme t ry of the Robertson-Walker models which restricts t he
expansion and contra ction so that they are purely radial ~nd
that more realistic models w~th fewer or no exa ct symmetri es
would not have a sing ularit y .
This chapt er will be devoted
to an examination o f thi s qu estion and i t wi ll be shown th~t
provided certain phys ic a lly re as onable condit ions hold ,
model mu s t h a ve a singularity , th.:",t
is, i t c annot b e a
g eodesically complet e C~, p iecewise
c2
~~y
manifold .
2 . ~he Fundamental Eq u ation
~
V
'i 'he expansion
of a time -like ge odes ic
,·0-
c ong ruence with unit tangent v e ctor
Vo..
obeys e qu at i on (7)
o f Chapter 2 :
e
; 0.
will be said to b e a s ingul ar p o int on a s eod es ic
A p oint
c(
(i)
V °' =
of a time-like g eodesic con(:;iI1Ue:e1.ce i f Of or the cong ruence
i s i :nfi n i te on
y
at
c onjugate to a p oint
singular po int on
r
s e odesics throug h
p
fr
f
A p oint '} will be said t o be
a l on g a ge odesic
A point
t
K~
is a
tf 3
will be sai d to conjugate
if it i s a singu l ar p oint
o f the cong ru e nce of ge od e s ic normal s t o
let
if i t
of the con g ruence of a ll tj_me-like
to a spuce -like hyper s urfaca .
de scription
t
H3
• An a lt erna.t ive
of conjugate p oints may be g iven a s fo llo :J s :
1
be a vector connect ing p o ints corresp onding d i stanc2s
al onb t wo neighb ouring 6 e od esics in a
c on g ru ence 1:ii th u nit
v~
tanc ent ve cto r
;,,o...
Then
is
I
dragg ed I along by the
congruence, t nat is
'
'
Introducing a n ort h on ormal tetrad
o].on 6
V
<l.
e
with
oL "1
0..
u
-=
VQ
(Y\
(\..
--
0....
(VI.,
1-..
parallelly transp o~t ed
(\'}
we have
(4)
0.. /{
d s'l
n. .
°'-
l"v
/,(
/Y\
w~1e re
e
e,
so lution of (~)
0...
eM
6
h.,
R°'- e:.. bcl vc Vet
will be c a ll e d a Jacobi field.
cle a r ly e i g ht independent solutions.
Since
Va...
·l 'here a re
and
fv°'-
8.1'8
s olutions , the other six indeuendent solutions o f&) m~y be
taken ortho3 onal to
a g eodesic
f
Vo... .
'i1hen
't,.
is conjugate to
p al on g
if, and only if, there is a J a cobi field alon§
which v a nishes a t
f
and
9;
(
'l1his ;:nay be s hown a s fo llo 1 1s :
the J·acobi fields which v anish a t
f
may b e regard.eel as
g eneratin2; n eig hbouring ge odesics in th e irrotationa l congruence
of al l time-l ike g eodesics through
p.
The r e±ore they obey
cl If
GL
,v
-
fr\
n.,
f'.\
ff\: n..
g
'l'hey may be written
Jt
M
1.. ; ne re
-
A C<J)
-=
t\.
d f1
cl!;
/\1 n .
Ip
cl A
/'(\ Y\
d g
, A (s)
will b e p osit ive d efinite.
1h i'\
p
a J acobi f ield v anis h ing a t
Jlut
(9 --
~
I
cld(A)
cL.s
I\\"
and
cL 'l
1-1
1\1\(\
oL ~ "l.
I'heref ore
1
Hence ()_
et
oLs
q_
if, and OJ1J.y i f
oe
,~e_t (f}_}c1))::. 0
exp<s;y,,cLs')
!}}s) "-
'11he ref ore
and
Th e:ce Hil l
'j
f\1 i'\
y
- a.,
(\,\ r
(ft J
(chl ( A>')
A
h.
is fi n ite
i s infinite wher e and only where
Thus the two d efinitions of conjugate points ~re equivale nt .
Thi s ~ls o shows that singu l a r p oints of c ongru ences a~ e
points where neighbouring g eodesics intersect.
~ or n ull g eodesic c ongrue nces wit h parall e lly tr ansp ort ed
I_ ~
t ang ent v ect or l.
as
we may d e fine the conv ergence
f
in Chapter 3 .
l)
I1hi~ obeys
C>....
L ;; (
_
i r- 6' ,s -+-
La. L h
1 (.)
-.r
II(°"
(<;:)
h
~e d e fine a s inb ular p oint o f a nul l ge odesic congruence as
one where
f
i s infi n ite.
~he cond ition t hat the pre ssur e i s g re a ter than minus
one-t llird t tJJ'B: d e ns ity may b e stated mor e g enera lly as
c ondition (a) .
(a )
obse rver 1_,1 i tl1
f /
0,
for a.ny
4-veloci ty i,..J
A..
, where
G ;:. -) C;,_b
l.JQ wb
,
is t h e e nerzy d ensity in the r es t- f r am e o f thA observer and
-. ~
1-
i s the rest-mass density.
I
Condition ( a ) will be satisf i ed b y a p erfect flui d wit h
dens i ty
f'- >o
and. pres sure
p > -1 ~ .
any time- like or null v e ctor
·vc...
-1.
·
t i· mp 1 ie
· so
V c,V7~ 0 f·. or
~~
Therefore by equations
(1) and ( 5) any t ime -like o r nul l irrot a tional ge od esic
c ongruence mu st have a sing ular p oint on each c eodes ic wit hin
a f i n ite affine d ist ance.
Obvio~sly if the f lo w-lines form
un irrotationa l g eod esic conl ru ence, there wi ll be a physic al
s ing u larit y a t the singu lar p oin t s o f the congruence where
t he densi t y and hence the curvatu r e a r e infinite.
This will
b e t h e c ase if the universe is filled with non-rot c,.ting dw-,t
2 ,3
1. oweve r , if t he fl ow-lines are _n ot g eodesic (ie. n on-va ni shing
p ressure ~radient) or a re rot ating , equati on (1 ) c annot be
app lied d irectly.
b Sp a tially Homor5eneou s Ani s otrop ic Uni v erses
l'he Robertson-·;Jal ker models a re spati a l ly l1oi,1 0 6 e;-le ous
1
and i s otropi c, that is, the y have a s ix parameter group of
motions trans itive on a sp ~ce-like surfac e .
I f we re duc e the
symmetry by c·c msiderini,5 models th,::,.t are s pat i a lly 110.rn oge r: e ous
bu t anis otrop ic (that t is, t hey have a thr ee par ame ter gr oup
of mo t ions transitive on a spac e -like hypersurf~ce) then
the ma tter fl ow may h ave rot a tion, ac c e l eration ~nd shear.
Thus there would seem to be the possibility of no n - singul a r
mod els.
L. She pl ey 4 has invest i gated one par t icul a r
hom o5 en eous model containing rot ating dust and has shown t h·t
there i s al ways a singul a rity.
Here a g enera l re sult will be
prove d.
The re must be a ~ingularity in every model whi c h satisf ie s
condition (a) and,
(b) th ere exists a
qr
of motions on the spac e or on
un iversal coverin~ space * ,
Y-~3
v,hich i s transitive on a t
Gee section 5
*
least one s p ace-like surface but space-time is not stationary,
(c) the energy-momentum tensor · i s that of a perfect flui d ,
-
' lo..6
U, "-- is the ta·,; e; ent i~ o
fl ow-lines und is uni quely d efined as the time-like e i gen v e ctor of the {icci tensor.
PHOOli'
R , t he curvature scalar nru st be const ant on a suace l ike su r fa ce of tr a nsi ti vi t ,y
ft 3
of t he g rou p.
·.:. ·he ref ore
I
Jllu s t be i n t h e d irection of t he unit time-like n ormal
t o.
f.
0...
V°'-
c/ 3
-vCA..
v111 ere
~, °'--
R:o.. ::
f)
C)
I
e C,c.n jCL- \J
n ,'
1nen
A/
Vea.: b]
i s a n indicator
=
0'
=
+1 if
=
-1 if
R, c.
R, c..
i s past d irected
J
i s fu t ure d ire cte d.
)
•
':L1hu s \,{"' i s a congru©nc e Of <?'fl Od sic irrot ational time - lilc E-;
v e c t ors .
By condition (a),
R0-.b V"'- V
h
>0
·:r her ef or e t he con 0 ruence mu st hav e a s inc3;ul ar p oint on e Elc h
g eode s ic ( b y equ a tion 1) either in the futur e or in th e nast .
Furt~er , b y the homogene ity, the distance a long eac h g e odesic
f rom
J
l-1 to the singular point mu s t be the. same f or eac h ce o.:,_esic .
~hu s if the surf ac es of tran s i tivity r e main spac e -lik e, the y
mu s t de g enera te into, at the most, a 2- s urf ac e c 2 which wil l
be uni quely defined
Let
M be the subset of the f low- li n es
of t he matter which intersect
c2
l1 3
intersected by
e; rou p t ransitive on
1j , /... must be
e mp ty s ubset of
.
Let
N
!--/
3
.
L
b e L:he n on -
Since t he r e i s a
itself.
Thu s al l t i.>.e
f lovJ-line s t nr ough
/t 3
mu st intersect the 2 -surf a ce
c2 .
Thu s
tb.e de ns ity ':: ill be infini t e t here anl1- th e re \,v ill be a
ohys i cal s ingul a rity.
Alternatively i f the s urf aces of
transitivity d o not rema in space-like, there mu st be at
..,3
l 2as t one .surf a ce which is null
call this 33. At 0 ' F
-
R.~ ·1 0
J
( if fZ,,o.. is zero, vie c a n take any ot he r
SC c:l.
=
o, '
;:,.r
9 olyno mi al in the curva tu r e tensor and its covariant d e r iva tives.
·.::hey c annot a ll be zero if space-time i s n ot station:.::ry 1
i ntrodu c e a g eodesic irrot at ional null cong ru ence on s 3
0-.
t angent vector
L
where
l v..
'-': e
•
\'[j_tl1
Then by e a u a tion ( 5) ,
t her e wi ll be a singular p oint of each nul l ge odesic in S)
within f i nit~ ~ff ina distance eit her infue f uture or in t ~e
past. t he 2-surfa ce of thes e sincul ar p oint s wi ll b e uni 0ue ly
def ined .
'J: he same a.rgumen-c us ed bef ore shoi'IS t ha t th e dens i t y
become s infinite and there is a phys ic a l sin[;ul arity . In fact
a s s3 is a surface of h omog eneity, the whole of s 3 wi ll bs
s ingul a r and it is not meaningful ~to cal l it nul l or t o
disting uish t his c &se from the case where the surfaces of
transitivit y remain space-like.
'L'he c onditions ( a ), (b), ( c) may be weakened in
t\'JO
Conditi on (b) t hat there is a g roup of motion s throuGhbut
/
space-t ime ma y be replace d by (b) and (d).
v,ays.
I
in i:ih ic h the re
Thm"B is a space-like hypersu:cf a ce
(b )
O.
are t h r e e ind e p endent vector fields X
JI;.().. 9 c. ~ 01
'
b
j l<
Xa. he d. e
C>
on
su ch tha t
• 'ihat i s , there
f?, one h omoGent'ous space section.
( d)
.I·her e e xist equati ons of state s uch that t he Cauc hy
1
develop ment of
H3
is determinate .
Then succe~ding sp a ce-like surfa ces o f constant
R
are homo3eneous and much the s ame proof can b e g iven thut ~ h e re
I
u re no n on-sing ul a r models satisfying (a), (b ) , (c), ( d ).
The only p roperty o f perfe c t
fluids tha t
has been u s ed
in the ab ove p roof is that they h a v e well de fined flow -l i ne s
intersection of which implies a physical :3 ingul cLrit y .
Obvious ly ,
ho•i'lev e r, t t1i s property \,Jil l be p o ssessed by a much r:o re g e rc ,2rc,.'.
cl a ss o f fl uids.
F or these, we define the f low vecto r as
-t~
t i me -like e igenvector (assu med uni que) of th e ene rgy- momentum
tens or.
1 h e n we c an re p l a c e con di t ion (c) o n the nature of
the matter b;y the much we aker condition (e).
(eJ
If the mode l is s ing ul a rity -free, the flow-lin es f or~
a s moot h time-like congruence with no singular p oints wi th
a line t i rough e a ch point o f space-time .
Co nd ition (e) will be satis.fed aut omatically i f c ond i t ions
(a) and. ( c) are.
This proof rests strongly on the assumption o f
homogeneity which is clearly not satisfied by the phys i cal
u n iverse loc Lllly thoug h it mEy hold on a l a r ge enoug h 3c ale .
Howe ver it '.-10uld seem to indicate tha t large s c ale e.f.f·e ct s
like rotation cannot prevent the s ing ul a rity.
It is of interest to examine the nature of the s i ngularity
i n the homogeneous anisotropic models s ince thi s is more
likely to be representative of the gene ral case than that of
the isotropic models.
It seems t h a t in general the collapse
will be in one direction,5
that is, the universe wi l l co ll&pse
d own to a 2-surface. Near the singularity, the volume wil l be
proportional to the time from the singularity irrespect ive of
the pre cise
.nature of the matter.
It also ap~; ears tha t the
nature of the p a rticle horizon i s different.
There will be
a p a rticle h orizon in e v ery direction except tha t in whic h
the collapse is taking place.
4. S ing ularities in Inhomo3 cneous Models
liifshitz and Khalatnikov6 claim to have proved tha t &.
genera l solution of the field equations will not have a
singularity.
Their method is to contract a solution with a
singularity which they claim i s representative o f the
g eneral s olution with a singularity, and then show that it
h as one fewer arbitrary funct·i on t han a ful l y general s olution .
Clearly t he ir whole proof rests on whether their soluti on
i s fully representative and of that they Give no proof .
I ndeed it would seem tha t it is not repres ent a tiv e s ince i t
involves collapse in two d irections to a 1-surfa c e where as
in g enera l one would e xp ect collapse in one d irecti on to a
2-surface.
In fact their cla i m has be e n p rove d f a l s e by
Penro se ? for t h e c a se of a collaps ing star usipg the n otion
of a 'c losed trapp ed surfa ce'.
A simila r
met hod will be
u s ed to p rove the occurrence o f singula rities in 'open '
universe mo de ls.
5.
'Open ' a nd 'Clo sed ' Models
The met hod used by Penrose to p rove the oc currence
of a p hysic a l s ing ul a r ity depends on the existence of a
n on-compa ct Cauchy surf ace.
A Cauchy ·surface wil l
be t aken
to mean a complete, connec ted space-like e u rfaoe that
int erse cts every time-like and null line once and once only .
( ot all spac e s p os sess a Ca uchy s urface:
e xamples o f tho se
tha t d o not include the plane-wa ve metrics, 8 the Godel model , 9
and N.U.T. s pace~O
significance.
However none of these have any physic &l
I ndeed it would seem r e asona ble t o demand o f
any physic al ly realistic model that it possess a CJauc hy
surface.
If the Cauchy surface is compact , the mode l i s
commonly said to be 'clos ed '
if non-comp a ct, it is sai d to
to be 'open '.
The surfac e s, t
= const ant ,
in t he Robertson -
Walker s olutions for normal matter a re example s of Cauchy
surface s .
If K
-1, they have negative curva ture a nd i t
=
fre q uently stated tha t they are non-compact.
necess a rily s o:
is
Th i s is n ot
there exist p ossibl e top olog ies for which
they are c ompa ct.
However, t h e following s t a t ements ma y b e
made ab out the top olog y o f the surfaces t
= constant.
If the curvature is ne gative , K = -1, the uriiversal
11
c overing space is non-compact and is diffeomorphic to E 3 .
Any other topology can be obtained by i d entific at i on of
p oints.
and ,
'l'hu s any other topology will not be simply c onne c t ed
if compact, must have elements of infinite order . in the
f undame ntal g roup.
.
group o f mo t ions.
Further ~f compa ct, they c an have n o
12
If the curvature is ze r o, K = O, the univers al covering
space is 'E3 .
. ht eBn possi. bl e. t op o 1 0 0- 1es.
·
'13
'rh ere are eig
If
compact they have a G3 o f motions and Be tt i numbers , B1 = 3,
=
3 .
'12
If the curvature is po s itive, K = +1, the unive rsal
c ove ring space is
s3.
Thus all topolog ies are compac t.
12
Betti n u mb ers are a ll zero.
.i:he
1
Since a ~ingularity in the univers al covering space
implies a singularity in the space covered , Penro se ' s method
is applicab l e not on l y to space s that hav e a non-comp a ct Cauc hy
surface but a lso to spaces whos e unive r sal cove ring s p a ce
has a non -coi;ipa ct Cauchy surf a ce.
'lhu s i t is app lic a bl e to
mod els which, at the present time , are homog ene ou s and i s otropic on a large scale with surfac e s of a pproximate ho mog eneity which h ave negative or zero curva ture.
6 . Th e Closed Trapn ed Surfa ce
Let 'l '3 be a 3-ball of coordinate radius r in a 3- s urfc::,. ce
a3 (t
= const.) in
a Rob erts on-Walker metric with K
= O or -1 .
Let qa be the outward directed unit normal to T2 , the bounda ry
of £3, in H3 and let Va be the past directed unit normal to
H3.
Consider the outgoing family of null g eode s ics which
inte~sect T2 orthogonally. At T~
p
I
be :
where f<>...
=
t °"
l {V
o.. ;
l
h
1-
If J\1-- 7 0
their conver gence will
a ,.. , h 1( ~ °'" s> b +-J, "'- I
t
a..(;-
b)
are unit snace-l i ke v ectors in H3 orth o5 onc:.l
to q a and to e a ch other,
tei.ere fo re
f ,
[& - k
R
f
and K
Q.
3 .
=
i f i ·• kr ~ J
0 or -1 , by t aking r large enough , we may
. 2
makef ne go.tive a t T •
There f ore, i n t he l ane;u age of Penr o se ,
T 2 is a closed trapped s urface .
.Another way of se,e ing -this is to consider the d i a gram
in which the flow-lines are drawn at their proper spat i al
d i s·bance from an~serv er .
at t
=
~he y all meet in the singu l ~ri ty
J . If t he past light c one of the observer is drawn
on ·bhis d i agram , it initia lly diverges fr om his world-line
( f <0
).
It re a ches a maximum p roper radius ( (
and then converges again to the singularity (
intersection o f the convergin0 li 6 h t
give s a c losed trappe d surface T 2 •
quasi-stellar
f
p o int
~
0
3C9
f >a
:.. 0
)
).
'1'".J.e
cone and t he surf ace I{3
If the red-s h i ft of the
is cosmologi c a l then it wi l l be beyond the
if we are living in a Roberts on- 0 alker type
universe with norma l ma tter.
However, t he assumption s 6 f
homogeneity and isotropy in the large seem to h old out to the
distance of 309.
'.l.1hus there is good r ea.son to believe th~ ·
our uni verse does in f a c t cont a in a closed trap ped surf2ce.
It
shoul d be pointed . out tha t the posses s ion of a clo sed
trapped s urface is a larg e s c ale property that doe s not den end
on the exact loc al me tric. Thu s a mod el th&t had l oc a l i rre~ula ri t ies, rotation and shear . but was similar on a l arge sc &le
at the pres ent time to a Rob ~rts on-Wa l k er model . wou l d h a v e a
c lo sed trapped s urfa ce.
Fo llowing P enrose it will be shown that space-time h as
a s ingularity if there is a clo s ed trapped surf a ce and :
~
(f )
E
( g)
there is a g lobal t ime orientation
0 f or any ob server with ve l ocity
(h)
t he univers a l covering s p ace h a s a non-co mpact Ca u chy
s urface H3 .
PROOF
As s um e s pace-time is s ing ul a rity free.
7:
s et of point s to the past of H7 t hat
.
- Li-
Let .l
b e t he
c an be joined b y a
smooth future directed time-like line to T 2 or its i n t e rior
T3 •
Let B3 be the boundary of FLJ..
Local cons:iiera t ion s show
that B3 - T3 i s null where it i s non- s ing ul a r a nd is
g e nerat ed by the out g oing family of p a st d ire cte d null ge od esic
s egm ents which have future e n d-p oint on T 2 a n d pas t
end - p o int
-.v h ere or before a singular point of the null g eodes ic
Since at T 2 , the convergence,
con ~ruence.
s i n ce . Ro...b
LCl. l b } 0
f >0
by (f), the conve r g enc e mu s t
b e come inf inite vii thin finite aff ine d istan c e .
will be compact being generated by a compac t
s e g ment s .
He n ce
and
B3 will be compa ct.
7.
'l1hu s I33 - ·1 3
f am il y o f comn a ct
P e nrose' s met h od i s
'
approximate B 7 a rbitrarily clo ~e l y by
t he n a s follows:
a
s mooth s p a c e -like surface a nd p roj e ct B3 onto H3 by t h e
n ormals to -t his surfa ce.
map pin~ of
B3 into H3.
'J.'his g ives a ma n y-one con tirrn ou s
Bince
must be compact.
Let
d.lq)
ma pp ed to a p oint
q
of
a3.
n3 is compact, its im a g e B3 *
be the number of points of B3
ci((v)
will cha n g e only a t t h e
7.
int e r s ection of caustics of the norma l s with H7
b y continuity
c:L@)
•
More over,
c a n on~y c h ang e by a n even nu mb er .
since t his is the identity and
f his is a contradiction, thus the assumption that sp~ce-t ime
is non-singul 2.r mu st b e false.
An alterna tive proceedure
which avoids the slic;htly questionable step of appr oxima ting
;(
B7 by a space-like s urface i s p ossible if we adopt c ond ition
(e) on the nature of the m6t ter, then
B3 may b e proj 8ct ed
continuously one-t o- on e onto H3 by the i1 ow-lines .
~hi s
again J.:;e ads t o a contradicti on since B3 is comp a ct · and H3
is not.
,,,,;15,
J::A. tb.Q ebGVQ p;coo.f ii-
nQC~5s@ry to d 9mand. tho.t
H3 boa CaYQhy ~urfec~ otberwis~ t~9 w~ol~ of i 3 ffii 5ht not
hc.ve beeI'l projected onto l-1 3.
'd e Hill define a soffii 2auc \y
~urfaoe (s,G,s) as a aoffiploto co:A.:A.oatoa. ~p~co-liko ~urf aoe
7
.ii.
5 C 5 q ? will.
1:;>Q
;.i
C.ilncJ;:i;y · su;ci';;ilC'6 i'or r oint e 1:iowr it ,
I
t hrour; :1 these points.
aowovor, fyrthor 6<•.'e'Y thore 1;:iay be·
:!"CgioRo for wl1ieh it is not a Cauohy ourfaoe.
li.Qt
of :p oint:;. .for wti.ic:b. H3 is a C<awc:b.y snr:f'eJCQ ,md lot
.,
....
~,
,.,ounu.ary
"St1C
O
f .,_bueoe.
i~
oa.llod "t>ho C' euc:b.y Hori
-
~uPfaGo,
•
•
PO±Ilb6,
ZOP
r.R
J
~
re] ati ve
,
if ±.
:I:-
to
B3
0
I-
b
•
.
•
I-
OJC±S'bB,
ap
I
Let ,P 4 b e ;.,:cie
jftelo@;j;;
•
\!l
~
Ql
b
~
11 -oe
~
a
.• ,
al?
..L
FuPthermor 9 if condition (f) holds t~~ n~ll ge@~eaiec
~~nerating ~J FHUBt have at least oae s ·; B:gul&r ,t,oi:n:-e Nh ie h
tlas t tlaoy a:iust 1e>O bounded in at leas t on e clireetion .
:i;i.~ ..ina
.1't oifflw.,.1 e e:;~&fflple of s, C, c 1.,rith a Cauchy horigon is a gf:)3.Ge
l~kc-ourfaee oi constant negative ourvaturo ooffiple tel~
WO!i.thiB: ·l)iJ.e n1:1ll oone of the point in Viin~co'.rcki c paee.
B.ull oone
the Gauohy horisoB:.
fOPHl.G
it;f @8!-HLi.tionc
&e rnpa et o.G.s,
rr i~J:®
n3
(o) and (g) hold, thon a aodel 1.dth a
ffiUGt
have the topology:
HOOiP'
V9 through
then V 3 tho
euppo~e teere were a region
no flow lines interseoting H3,
iihich ·b ·t1ere
boundary of
HiY~t b@ a t:kme=l.ike euPfa.oe g sneJ?a.ted by flo1.·: lines whieh ,: o
intoreooi; H3,
Procoodi:a.5 :el.o:a.g tb..@~@ flo ·,·-linga i:a. tb.€
d ireotion of their i:a.toreoctio:a. witb..
~d--point o.f tho go:rH~.rstor since
a3,
V3 do@s
w@
:s:i.1.,u.t rgeic ii.
s> ~
not into r ceot
1I· ·' •
~ t the eJEictonoo of thia ond-point cont;c~dicts C@) cinco i t
::i.wtplioc a cin 0 ula.rity o.f tl:l.o .flo·,c -li:a.o· c ongr:i,.w:a.o o,
'.i:huc
114
v
nius b be em1, by a11d e oer:r point Bao a flow line thr ougl3. it
interceoting :g3,
=trn
lf~ K' E i
Thus wo l:l.avo a. hoFaoomorphisFA of the space
by aooigBing to ever:,. point of ·lho ·c ·0 ac o i-. toi!
distance along the flow lino from I-13 a.ad tho point of i-.:a.:Go~section of -'a~1e flow line ·,.;rith H3.
It oan also be cho·.,n that
4:n this cauc a 3 H1uot be a Cauc hy surface.
there
1, .1
e r e a Cauohy ·h origon ·
e·-1cb f -1O"' = J in e :.it mo s t
on ce
i:s orphei: sm o~4.l..t-G- H3.
~,,
·.i: ni.J s
(Q..s
]? or '
GUB .J O GI€
.,
thi ~ c ~n intors@ ct
'l'her efore thoro i ;; a :10.:-e e eE:\Jrt n ex , by-4-e). e v T'y f J OJil =l :i...ie
0
m3 1.· s b. ome orn. orp ,o.ic
·
·
u3
·c; o i..,
.lwl(
&)
:nd.
j
g
If oonditi~f) holds ev~:ull gP od @s ic g e ner ·_· t o:P
-Gompaot,
of ~ 3 ha s at least one end p oint.
This must b e in t h e
f~ om H3 since in t h e direction tow a rds H3
each g enerator must be unbound e d.
s.i:i:.ce
~.1 i~ co~p~ct
This hO\rnvcr is i rn ~)os sibl e
rcbvs ~13 is a Cm 1c :b.;y sn.:£1il~~ .
8. d ingularities in 'Clo s ed ' Universes
rl1 herc is a s i ngul a rity in ·@v@ry model whi ch s at isfi es
,-,
(a), ( g ) and (i).
( i)
'l1here exists a comp a ct Cauc hy s urf a ce H3 who se u n i t
n ormal
V
O,..
has p ositive expansi on everywhere on H3.
-PROOF
-For t he pr oof it i s necessary t o est ablish a cou p l e of
lewma s.
Assume that s p ace - time is singul arity-free.
The
fol lo wing result is quoted without proof, it m2.y Jr".2,a;;-n -y
1
) e
der iv ed from lemmas proved in refer en ce 11 .
If
? a nd '1,.- are conjug ate p oints along a ge ode s ic (
and · X is a p oint on
p oint in
f} ·.
r not
in fCJ,.. then 'X' mu s t have a conjug;-J.t e
An immediate corrollary is that if q is t he first ? Di nt
y
a lonG
conjugate top
conjugate points in
and y
pq
is in
pq
then
y h ~s no
Also since the result that x
h0.s a conjugate point in pq can only depend on the values
of
C\,
""
in pq, any irrotational 5eodesic congruence including
(\
the geodesic
r
must have a sins ular point
Thus if q is a point on M3 and (
011
O in
pq_.
is the geodesic normal to
M3 through q, then a point conjugate to q along · (
cannot ~ccur
until after a point conjugate _to M3.
If M3 is a complete connected space-like surfa ce whic h
inters ects every time-like and null line from a n oint p, ,_,;e
may define a function over M3 as the square of the geodesic
distance from p which is taken as positive if the geode sic is
time-like and negative if the ge6desic is space-like.
this the world function CY with respect -t.o p.
set of v,,,. lues
o 40
where
A time-like g e o~es ic
from p will be said to be critical
to a vo.lue of <5 for which
,€
"
f-A..
}'or the c l osed
(5 will be a continuous ( i n
general multi-valued) function over M3.
r
~e call
CS-
€
,'fA l
f.A.. :?.
0
if it corresponds
Li. :.
1.1
are three independent vectors in M3.
a critical geodesic must be orthogonal to M3.
::t,
;>)
Cle l1 rly
A geodesic
which is critical will be said to be maximal if it corresponds to a
local maximum of
Lemma 1.
'6
A geodesic
point
X
where
q
Let
and
Then
n
any h
f
cannot be maximal for a smooth M3 if there is a
conjugate to M3 but no point conjugate to
is the intersection of
~
and
=
n
A(s) f/q'
mn
g
m
=
n
B( s)g/q
mn
r
h( <lx;e. I°r -,- ';-;:-t Ii ) (_
'f"
'>(
before a point conjugate to
would be conjugate to
f
)(
does.
Since
X
o£
a
p
a
mb
= ·"' b h h
mn
beyond
q
If it were zero
This shows that the surface at q
p lies
of constant geodesic distance from
direction than the surface
which vanish at .X:
by taking
beyond q, it would be possible to have a point y on
'f;..
'f in qp,
must be positive for
n
since if it were negative for any h
conjugate to
on
M3 •
f and g be the Jacobi fields along
m
m
They may be written
respectively.
f
m
q
i
f in every
nearer to
of constant geodesic distance from
is conjugate to M3 the surface at
constant geodesic distance from
p
lies closer to
direction than M3 does.
'O
is not maximal.
Hence
q
f
,I in
some
of
a.ofinito a t x.
I·
g) fi
A· A
'i 1
am:i. (! - ~)
f-e;z::e tbe:ce mnst
M
j"\..
K
K
~t q, whore
io2
/IA
But at ~c1
s
i
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e annot be poieitivo [email protected]@ .:: t q .
(v\
be som~ 0 j ~~gtj OD
cl
f &'
g
!
ia tl::l.ii.
/
-,
11
tlA
for:! ~,zb j c.b
I(
~
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~
':i:horo
d~
"
.:!
J;.Lit i-augent :vector of t be congru oL~;o
of oonotant ;:; eocleoie dictanoo from p l ie s eJ:.®'S'e-~t o p thB:tt
~~c ourfaec M3 d oes,
Therefore
f
i€ noL maxi~al.
7,
If M~ is c ompact or i f the intersection of al l time-l ike
and nu ll li:m.es with
J:t3 i s compa ct, 6
111us t hav e a maximu:.:
v alue, t hus there must be a ; eodesic normal to M3 through p
lon~er than
y
We use this to prove another l emme .
.Le mma 2
If p lies to t ~e future (pas t) on a time-like ge od e sic
-(
through
q, beyond a point z c onjugate t o q , a nd there
exists a compact Cauchy s urfa ce H3 throu; h q, t l1en t here
mu s t b e another time-like g eo~esic f rom p to q l onger than
Let y be the last point conjugate to q on
f
before p .
Let x be the n earest p~int top c onjugate to p in p q .
r be a po int in yx .
r.
Let
Let K3 b e the set of point s which ha v e
a f utu re (pa s t ) directed g eod esic of lenbth rq from q .
Then
K3 will be a s pace-like hy pe r s urfa c e t hrou~h r .
be t he ·s e t of p oints which hav e a t
Let F 4
l Rast one f u t ure ( p ast)
d irect ed g eodes ic f ro m q of l e ngth g r eater t h an r q .
,oound a r y
~ 1.,
, 4
;.· ,
J 3c: K3 .
OI
c .
.
uinc
e p i. s in
1.,-.
~
i h e n t he
4 an d s ince
.
e v ery
p a st (future) directed time-like a n d n~ ll line from p interse c ts
. li'4 , they must a 1 so in
. t ersec t
'H3 wl.
11.c h.i s no ~u 1.n
b e the init®r:s:e c t ion of J3 a n d these line s .
re g ion
L' .
Le t 1,3
S ince a 3 i s
0 ons ide r t h e functio n <S'
compact, L3 must be compa ct.
re s pect to p over K3.
J 3.
with
Its maximum mu s t lie in the comp a ct
But, by the p revious lemma
f
is not m,ximal,
more over, loc a l cons iderations s how t hat a sing ul ar p oint i n
the s urf a ce
J3 c a nnot be a maximum ·o f
maximum valu e of
t~ onal to L3.
cr-
mu s t
(5
'i'hu s the
oc c: ur f or a 2:eode si c from p ort :.J.o-
Thi s must a lso be a g o edesic fr om p t o q o f
length c reater than
(.
Us in~; the s e two lemmas t he th e orem may be proved .
H3 a .::.~e c on v erg ing
e v e r y where on H3, there must b e a p oint conjugo.te to H3
oi nce
the fu t ure (p a st) d irect e d n ormals t o
a
finit e d ist a nce along eac h future (pas t) d irected g e odesic
norma l.
Le t
i
be the ma ximum of these d ist a nc e s.
Le t p
be a point on a future (p a st) dire ct e d g eodes ic n orma l at
a d ist anc e s reater tha n
~
•
Consider the f unction
CS"
wi th
7-
respect to p over the comp ac t
b eoJ. e sic f r om p normal to
maximum.
'J here must be a
surf a ce F1.? •
Let~ be the
H3 at the point q , i'ii.1e :re CJ has i ts
p oint con,jugate to H3 a long A in qp .
But if there is no point conjugate to q along~ in qp , t hen
\
c a nnot be ma ximul by the first le m~a .
i s a point conju~ate to q a long )
If however th e re
in qp, then there mu s t b e
a long er ~eodes ic from q top by the second lemma.
is not the g eodesic of maximum length from H3 top.
t hus
~ his is
a contradi ction which shows that the oris ina l a ssump tion t net
the space \vas non-singular mu s t be false.
~hi s proof could also be u sed to show the oc c urre n ce
of a singularity in a model with ~ ~on-compact Cauchy s ur f ~ce
provided t ri.a t the expansion of its norm a ls was bounde d m-my
from ze:ro and provided tha~ the int e rsection of the Cauc hy
s urface with a ll the time-like and null l 1ines from a point
was comp a ct.
o.
Heckman
,.,.,.
Raycha udhuri
A. Koma r
? •
4 • .L . C. She pley
C:
c . Behr
.,I •
6. :m. M Lifshitz and
I. M. ~(halatnikov
7. H. Penrose
8 . rt . Pe nro s e
9. K.Godel
10. C . •ii . Mi sner
11. J. Milnor
1.
2.
.
1 2 . IC. Ya.no and
ci . Bochner
13. O. Heckrnan and
1 LI..
-2: .Shucking
H. Bondi
I.A.U . 6ymp os ium 1961
Phys. tlev. 98 1 :23 1 )6 5
Phys . Rev . 1 0 L~ 54-4 1956
Pr oc. Nat. Acad. Sci. 52 11.j-0 3 1 964
Z. As -c rophys. GO 266 196 5
Adv. in Phys. 12 185 196 3
Phys. Rev. Le t t. 14 57 1965
Rev. Viod . Phys. '!I. 2.1:, i165
Rev. Mod. Phys. ~ 44-7 114-f
·J. Math . Phys. 4- qz4 f~
' Morse Theory' No. 51 i nnals of
of Maths. Studies. ~rinc et own
,
Univ. Press.
1 Curva ture a nd Bet ti i:~umbers ' No . 32
knnals of Maths. b t u d i es .
Princeto wn Univ. Pr e s s.
Article in 'Gravit a tion ' ed . ~it ~en
Wiley, New York.
'Cosmology ' Cambridg e Univ. ~r e s s