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THE BOA Rn
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Rf Sf ARCf! STUDIES
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CI-i.A.PT~rl
1
The Hoyle - Narlikar
Th eo~y
of Gravitation
1. Introduction
The success of Maxwell's equations has led to
electrodynamics being normally formulated in terms of fields
~articles
that have decrees of freedom independent of the
them .
However , Gauss suggested tnat an
in
ac~ ion-at-a-distance
t :1eory in \'Jhich the action travelled at a finite velocity
might b e possible .
This i dea was developed by 1heeler and
1
Feynman ( 1 , 2 ) who derived their theory from an action-principle
tha·t involved only direct int eractions between pairs of par·ticles .
A
feature of this theory was that the ' pseudo '- fields
int roduced are the half- retarded plus half-advanced fields
claculated from bhe worl d - lines of the particles . However,
··! :1eeler and JJ'eynman , and , in a different v-1a;; , Ho,-artli
(3)
were able to show that, provided certain c osmolo3ical
condit ions were satisfied, these field s could combine to
4
6ive the observed field. Hoyle and Narlikar ( ) extended
·tt1e ory to general space - t;imes and obtained similar theories
for thE:ir ' C'-field (5)and for the gravi·tational f:fueld (G).
It is with these theories that this chapter is concerned.
-
I-
-
,I\ I<
y
L J A~V
"A"'l /
I
...... . ..... I
It \'Jill bG ...;ho.:n that in
an
exp;.11ding universe t·!1e
advanced fields are infinite, and the retarded fields finite.
This is because, unlike electric charges, all Toasses
h[tVA
tho
•
same ...:;i.;;;n .
2. The Boundary Condition
rloyle and Narlikar derive their theory from the
action:
•
wnere the integration is over the world-lines of oarticles
a.
I
b.
• •
In this expression ~ is a Green function .
•
that satisfies ·the wave equation:
--
l((
.
/)
£ x. x
/-3
wl1ere
j
is the dei:;erminant of
A
in the action
3.j
• -=>inc:e
t~e
double sum
is symcetrical between all pairs of
pi:irticles a.,b , or1ly thc..t part of
symmetrical betv.reen c.... and
4·(0...b)
that is
b v1ill contribute to the action
i.e. the action can be writ·ten
A
b)
\vhere ~··)!(-( ~ _
'rhus
·lt
G
must be the time- symmetric Green function, and can
be i.,rrittien:
()"'
'1 ::
..!..
~
qre-t' -r 1qo..dv
v1here
are the r3tarded
Ey
requirin~
of the
g~
,
adval1ced Green f·L1nctions.
r~nd
that the action be stationary under variations
Hoyle and Narlikar obtain the field - equations:
i g"K
[ £. i t {Vl,lo.)( x) M(b) ( x )] (r<il< 0..
R)
-:i: b
::: -
-
<: < _!_ [
ft,K -r ~ c_ 3 o..fb
(a.){
/'II.
(b) r
9'vk. fVl,} r (o.);r
-
m
(b)
f'rL.
)
·t
;~K
G) (
(a)
a(.
fYl,
J
)i,
( b)
/l1.
.
; r\
(b)J
M.,
,
>
. Ho·,;ever, a.s a
w11ere
consequence of the particular choice of Green function, the
contraction of the field-equations is satisfied identicall : .
'1.'here a ::c thus only 9 equations for the 10 comnonents of Cr
· '.
,J ~J
and t;hc S,',rstem is indeterminate .
{o..)
:.. oyle and Narlikar therefore im')OSe
,As the tenth equation. By then m~kin
. t.ion' th a t l.S
. b y pu tt ing
·
L
approxima
L
Zrrt.
=
=cor:st.,
fvt0
the 'smooth- fluid'
< (o.)m (b)JI.- /11.. 2
L_ (V\,
a...:tb
CJ)
they obtain the Einstein field- equations:
~
~{R.
(, {IA..O
._v,
_.!_R_
l
gi-1'\
.)
:
-
-
/ t,
. .
•
K
There is an important difference , ho\vevor, bet\veen these
field-equations in the direct - particle ir1teraction tc.eory
and ii1 tl1e usual general tneory of re L&.tivity . In the
seneral tneory of relativity, any metric that satisfies t he
the field-equations is admissible, but in the
interaction theory only
t~ose
tlirect -particl ~~
solutions of the field - equations
are admissible that satisfy t he additional requirement :
-
£ /VL
(o..)
(x) =
..-
This requirement is highly restrictive;
th~t
it will be shown
it is not satisfied for the cosmological solutions of
the binstein field- equations, and it appears
th~1t
it
c~nnot
·be satisfied for any• mod.els of tl1e univer:;e tho.t eitl1er
contain an infinite amount of matte1· or undergo infinite
•
expansion .
The difficulty is similar to that
occurrin~
in
:r-rev1tonian t_icory \vhen it is recognized that the universe
might be infinite .
The Newtonian p otential
D
'·"'he re
obeys
th~
1 : - "f
is the density.
'
equation:
In an infini te staLic
the source
al~ays
unive~se ,
has the same
ved v1hen it v1as reoliz ed
th<~t
wo1Jld be
~ign .
~he
infini~e,
.
since
diffic ulty was
~ssol-
the uni versP. ·.-1as expa11d.in;;, since
in .:tn exoand ing univer se the retard ed soluti on of the above
is finite by a sort of ' red- shift '
eq~ation
ef~ect .
~he
advanc ed soluti on will be infini te by a ' blue- shift ' effect .
'l'his is unimp ortant in Ne\vto nian theory , since one i:..; free
to choose vhe soluti on of the equati on and so ms.y icnore ·the
infini te advanc ed soluti on and take simply the finite
retard ed soluti on .
~iffiilarly
in the direct - partic le intera ction tQeory the
rn, - field satisf ies the equati on:
Om
where
N
+-
f R. M = N
is the densit y of \·.;orld- lines of partic les .
As in
the Newto nian case, one may expec t that the effect of the
ex1)a11sion of the univer se will be to mc:1ke ·the retard ed soluti on
finite and the advanc ed solu·ti on infini te .
.
Howev er, one is
now not free to choose the finite retard ed soluti on , for the
equati on is derive d from a direct - partic le intera ction action princi ple symme tric betwee n pairs of partic les , and one must
choose for fV\.. half the sum of the retard ed and advanc ed
soluti ons.
Vie
would expect t •J.is to be infini te , and this is
shown to be so in the next sectio n .
j .
'fhe Gosmolos;ical i.:),olu-cions
'l1he .rtobertson- \'/alker cosmologic<:'l.l 1netrics ho.ve the
forin
- .
•
Since t.1ey are conformalJ.y flat, one
in \vhich
tr.Ley become
cls
2
:z
2
c~n
caoose coordinates
i
:L
J.
.
Q
-
; J1-<
7o..b
dx°'- cLx
,
•..v t1ere
io the flat - space metric tensor and
-For example, for the
k
;: 0 /
/
- t~_(c)
.c7)
tl~ I ·t i K ( 't r-f )iJ [I ~ ~ ii ( ~ - ( )~J
•
Sitter universe
Einstein-d~
R(l)-:: c 'L ~ T
T
2
3
("'C=
__ Q
JJt
( G .;..:_
"f. <. rD )
I
~)
•
For the steady- state (de Sitter) universe
(-o0.<.t:"Lo0)
K :. 0/ Rt.t); .
..t
e1
•
-F
--
R
-
oO <. L.
L_
o)
[,
(i:
.
-
·-
-(:
Te 7)
c;* (~,, b)
obeys t h e equation
t R~*(CA,b)
oc;-#('(o..,h)~
~ rom
(-
:--'!'-
f
'I.1he Green function
·- ·-I
-
=
~<t{Q b)
;-:__ 3
t h is it follows that
I
J2'-i
dX
dx.b
~ JL-'iJ 4 (a,b)
If we let
q~; 52.-t
5
.._ ,
v :1e11
'
Thi s is s imply the flat-s-oace Gr -.. en function equation , a nc1
•
ne11c e
r *( 't.,, 0 ). 'l l
'1
f ) -- Jl-'?c,)
~
g Cr)
The
1
I
/V\. ··fie l
d is given by
·=-
'~x:N/-3
If
ii~J2--,- c.)
J2. ( ti)f
--c- L - vc, ) ·
J2. (."C )._ Jf
f'
r
dx: \t
=
J
I ( ~re,t_ ·r
lh.0-dv: J_
lt'or uni verses without creation (e.g. the Einstein-de Bitter
3
,.,
-~ I~ ·- /1., ,,.
const . .e or
universe),
N
con.:;;t.
W\Gre the integr ation
~·1 ill
th~
~s
over the fu t ure light cone .
This
~
normally be infinite in an expandir13 :.ni verse , e
~ instein -de
0itter universe.
oO
-2
M. o...).; ( '-f,)
2-: . in
~
'1: I
- I
-
( fl -
vc )
I
cL 1 .2
..."
/., I
•
universe
s't:;eadystate
In the
/ll\...CL
J ./.
( '1;( )
·--
-
- I
l
-l
0
-
-r.
I
n.
·-I
3
(
,/\
l-
1.
T. ~
-
l
,)
cL c:.z .
I
•
.By
c~o»ntrast,
on the other hand , \ve have
where the integration is over the past light cone.
This will
normally be finite , e . g . in the
'L .
'
T
-2
f.
~in~tein-de 0i~cer
(
t'.2.. - 'L )
' _ /?_,
I
univers=
c{ '"'L1. ::.
0
-Q.
'
I
-(L
J.
I
while in the steady- state universe
IV1..
;~
{;
( 't, )
-- - -I
·rI
-1
'"f I
n.
~
·-cO
-
-
-I
-2..i
l;i.
·-:i.
(L
J
Thus i t cun be seen that the solution
ll'\.. =-
consi::; . of
th0
equation
is not , in a cosmological metric , the half-advanced plus
half- retar·ded solution since this would be infinite .
in the case of the Einntein- de
~it·ter
and steady-state metrics,
it is the pure retarded solution .
4.
~he
In fact,
'C '-b'ield
Hoyle and ]arlikar derive their direct - particle
interaction theory of the 'C' - field from the action
where tae suffixes
·'°'
h
Q
refer to diffqrentiation of
I
~~ (o..,b)
on the world- lines of l'-1.. 1
6 rcGoectivel~' ·
.........
G
is a Gr een function obeying the equation
DG(X,X')
,~e
define tl1e ' C' - field by
C(x)
and the
•
z
=
G (x)°-) ,~ clQ )
'""'
.I
.rhen
c &-J
oc
o
a L~, b)cLb ·~,
4
S' e(
X,
1) J '< (
lj ) J K
j - 3 cl_ 'X
4J
- I'\
-- J
JK
:~e tl1us .:>ee that the sources of the
w~ere
0....
by
mat~er -c urrent
J"'<~) • z_
1
&
1
C' - field. are the plc.ces
matter is created or destroyed .
i1.s in the case of the
' M,
'-field , the Green func·t; i on
must be timc- S;)7 mmet ric , that is
q(°', b)
--
,
I
hoyle and Narl ikar claim thut if
th~
action of the
' C' - fielQ is included al on,~ vii th the action of tl1e '
r~
'-field,
a univerue \1ill be obtained that approximates to th8 3teady-
state universe on a large scale althOUBh tncre may be local
irrecularities.
In this universe , the value of C will be
f i:ni 1..ie and its gradient time - like and of unit ma5nit;ude .
Given this universe, we may check it for consistency
claculatin6 the
advan~ed
if their sum is finite.
~how
will
and retarded ' C'-fields and
~e
fincin~
shall not do tl1is directly
~ut
that the advanced field is infinite while the
re~ar -
ded field is finite .
Consj.der a region in space - time bounded by a three dirnensional space - like hypersurface ])
und t(1e past light cone
9f
j)
£..
at the present time,
of some point
P
to the futl1re
•
By Gauss's the orem
w C /-3 dx ~
~
I/
Let the advanced field pro(luced by sources v1i thin
Then
C
l
and
will be zero on ~
t/
be
C
1
, and hence
•
..
0
.dut
- 1-\
J
is the rate of creation of matter= ft,. (canst . )
•
in
}K
the steady- state universe , and hence
--
11.S
the point
?
rv V.
is taken further into the future , the volume
of the re6ionV t~nds to infinity .
Hov1ever, the area of tlJ.e
hyp.ersurface J) tends to a finite limit owing to horizon
effects .
~herefore
the gradient
dc
dn-
I
must be infinite.
A similar calculation shows the gradient of the retarded
field to be finite .
'.I.'heir sums cannot therefore :-·iv;e the
field of unit g radient required by the
Hoyle -t~arlilcar
I
v
neor;,r .
•
It io worth noting that this result was obtained
without assumptions of
of conf ormul.~ flatness.
a~smooth
distribution of matter or
5. Conclusion
It iJ one of the weaknesses of the Einstein theory of
relativity that &lthough it furnishes field equations it does
not provide bour1dary conditions for them.
'rhus it does not
Give a unique model for the universe but allows a whole series
of rnoa.el:::> .
Clearly a theory th;:,t provided boundary conc...L "Gior"s
and. t.1ut:> re.stricted the possible solutions \..,rould l)e
attractive.
The Hoyle- Narlikar t11eory does just
l."'equirement that
m
I
= ;:- mt~t
't
J._ M
).
equivalent to a boundary condition) .
)
1,0
.
Unfortunutely, as .. e
tho~e ;~ odels
th&t
correspond to the actual universe , namely the
R obertso)1-~·J al~::er
Th~
th~..t(the
l.S
O...d ./
have seen above , this condition excludes
seem
VP>r\r
y
.,
models.
calculations given above have considered the
univct·s~
as being .f 'illP.d \•1ith a 11niform distribution of m<:.:.tter.
is
le ~ itimate
.
'-."his
if we are able to make the 'smooth-fluid '
approximation to obtain the
~ins tein
equations.
~lternativel--
if this approximation is invalid, it cannot be said that the
tneory yields the Ei nstein equations.
It mi.5ht possibly be that local irre_;ularities co11ld :nal{e
finite , but this has certainly not been demonstrated
'Yta..dV
and seems unlikely in vievJ of the fc..\ct th1 . . t , in t11e Tio~rle 1
Narlikar direct - particle interaction theory of their 'C 1 -field,
which is derived from a very similar action- principle, it can
be shown
advanced
without assuming a smooth distribution that the
1
C'
field \-.rill be infinite in an expandin3 uni verse
'IJi th crea·t ion .
~he r eas on that it is possible to f ormulate a dire ctparticle interaction theory of e lectrodynaffiics that does ~ot
encounter this difficulty
mf
havin3 the advcincea solution
infinite is that in electrodyncJJics tl1ere are eo.ual n11mbers
of so\1rces of nositive and negative sie;n .
Their fi.elds cu.r
c~ncel each other out and the total field can be zero apart
fron1 local irregularities .
This sug:est that
i:..
1)ossible
•:Jay
to save the Hoyle - Narlikar theory would be to allow masses
•
s
ign
.
both p osit ive and negative
0·1.~·
The action would be
G~~, b) clo. cL b
are gravitational char.~es a~al o~ous to
electric char ·es .
1
fi1
Particles of positive
' -field und particles of negut ive Cj_
q,.
in a r>osit i ve
in a ne .....,.ra·t i ve
fie ld W()\lld h&ve the noi. .mal ~ravi tational properti~s,
1
rr1
t;t'i.:.,:.t
·t11e;f would have positive gravitational and inertial masses.
'
I
-
is,
A particle of
ne~ ative
s·till J'ol lov1 a geodesi c.
a particle of positive
1
in a positive
'M
'-field
•:J(;u ld
Therefore it wo11ld be 0.tt;racted :)y
•
Its own gravi tat:;ional effect
howeve r would be to repel all other particles .
Thus it would
hav~ the properties of the negative mas J described by Bondi(e)
tt1at i s , negative gravitational 1110.so and n egat ive inertial
mass .
3ince tnere does not seem to be any ma tter having
toese propert ies in our region of space (
v1
1ere M
1
JL
coi:.st . / 0 )
there must clea rly be separat i on on a very larc e scale .
would not be possible to identify part i cles of
'-'.l. ,,
ne~~ltive
with ant imat t er , since i t is known that antimat·t er has positive
inertial mass .
liwev er , the i ntroduction of nesat ive masses
would prob ably raise more difficulties than it wuld solve .
'
REFERE1~CES
1 . J . A. ~ilieeler and
R. P . Feynman
Rev . Mod . Phys . 17 157 1945
2 . J . A. \Jheeler and
tl . P . :E' cynman
Rev . Mod . Phys • 21 425 1949
1
.
J . E. Ho 6 arth
Proc . Roy . ooc . A 267 365 1962
4 . F . Ho.1le and
J . V. Narlikar
Proc . :::<oy •...,oc . A 277 1 1964a
5 . F . tloyle and
J . V. Narlikar
Proc . ~oy . Soc . A
6 . F . ..ioyle and
J.V . Narlikar
Proc . Roy . boc . A 282 191 1964c
7 . L. Infeld and
A. Schild
Phys . Rev . 68 250 1945
8 . H. Bondi
Rev. Mod . Phys . 29 423 1957
./
282 178 19&+b
CH\P':'ER 2
PERTURBAT IONS
1•
-- .-.---
Introduction
Perturbations of a spatially isotropic and homogeneous expanding
universe have been investigated in a Newtonic'Jl approximation by
1
Bonnor( ) and relativistically by Lifshitz( 2 ), Liftshitz and
Khalatnikov(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 dire ctly 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 seerns 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 tensor
gab
of signature +2 .
in this space is indicated by a semi - colon.
Covariant differentiation
Square brackets around
indices indicate antisymmetrisatio n and round brackets symmetrisation.
The conventions for the Riemann and Ricci tensors are: -
1/91.b cet
is the alternating tensor.
Units are such that k the gravitational constant and c, the speed of
l ight are one .
Ul'\I VtlC>' I Y
Ll.:il'/•~Y
CAMBRIDGE
3.
The
Field_.~gua~.io_ns
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
~
is the density .
ft
is the pressure
h G\b :
Then,
a
Ua U
of the fluid,
We will assume
-
- 1 :
is the projection operator
Sob t Ua.. l..tb
into the hyperplane orthogonal t o Ua .·
hAb
uh
::0,
We decompose the gradient of the velocity vector
UQ.;b:. Wob-+ <5'"o.b
+ t h~b 8 -
Ua
as
•
Uc.. Ub
is the acceleration,
where
is the expansion,
<r'o. b
hJ
h d. b
C.
~ IA ( c j <l)
wo.b ~
flow lines U •
a
u [c;.d]
- -S
h:. l1t
is the rotation of the
We define the. rotation vector
I
Wo.
is the shear,
I
-:;. -;: 7}o.bc.cl W
c.ol
Ll
b
•
We may decompose the Riemann tensor
Rab
E1..nd t11e •:.'cyl tensor
u
C ·. b'o.
'=='
0
-::.
R
abcd
into the Ricci tensor
Cabcd :
R cbco( = Ca.~ccl - 9q(ct Rc.J I:;,
Cabe.et~ C[.o.b)~d.J
Wo. as
7
Co.[ bed]
•
-
~
b[c
l~cl]~ -
R;3 'JG\.[,5olJb
)
Cabcd is that part of the curvature that is not determined locally by
the matter.
It may thus be taken as representing the free gravit-
ational field (Jordan, Ehlers and Kundt ( 5)).
'Ne may decompose it
into its "electric" and "magnetic 11 components.
'
C o.b
cd _
-
8
u [o.. t
b]
tc.-· ..,tJ
\.A
cC'- Ed]
- 4- O(o..
!,)
'
')
,
)
E
Q...
-=--
Ha. o.
= o
().
)
E
each have five independent components.
ab
We regard the Bianchi identities,
as field equations for the free gravitational field.
Then
(Kundt and Trfimper,C 6 )).
Using the decompositions given above, we may write these in a form
analogous to the Maxwell equations.
b
1.
n (j.
E. bc;ol
hcd
+
~
Ho.h W
b
-
'? o..~c.d
U
b
Lide
c
<Y e n
h
\
= '5"
b
l'A
(1 )
l
}"-;"
J
'
- E c(o. gb)c
•
+ 2.
H
Y/ o c.c:le
-
d
Q
YJ bcc=I. e
•
L H c:o..b - h (o. .f ') 'b) c d. e \). c.
..
-
--
E -f o~e
.. H c. (P- o-b)c. - "1 p.r; cl e
•
E
e.r
ac:l'f
uP
YJ bp~r Uc
•
Uc ue
t(fi ~)
+
--
() ob
..-
H (o. w
u p <Y cl ~
1,.4.c.
Y) b p q ,.
t- 2H("J o. ~ he. ,,I e U. '".
tA e
8
Hott.
0
(2)
(3)
'J
b)c.
H~ . .
(4)
•
where .1.
indicates projection by
(c.f. Trfunper, (7)) •
hab
orthogonal to
Ua.
The contracted B·ianchi identities give ,
-
fa+
(f+~)8
(f.+ ~)
Gl +
0
=. 0
r.;
'p
h ba
::
T cob'·b -=
0
)
>
(5)
0
(6)
The definitien of the Riemann tensor is,
U o.;[bc.]
=
2.
'R
Ol
f' be. uP
•
Using the decompositions as above we may obtain what may be regarded
as
11
equations of motion" ,-
e = zw
•
2.
'2.
n2.
I
-2a- -3l7
+
(7)
)
'
(8)
•.
..·'..
~
(9)
'
\Vhere
2 W
2.
=:
W ab W
06
'
\/Ve also obtain \\rr1at may be regarded as eq_uati ons of constraint.
( 1 0)
( 11 )
)
L-t
= - n (o
~e
n
·1b)c.cJ.e
c.(
u \.. wf
d;e
•
( 1 2)
consider perturbations or a universe that in the undisturbed state
in conformally flat, that is
0 abcd
By equations (1) -
0
•
(3), this implies,
•
~
If \Ve assume an equation of state of the form,
1:Jien by ( 6) , ( 1 0) ,
:::. o
•
= Uo.
e.
it"\. ( ~) ,
..
This implies that the universe is spatially homogeneous and isotropic
since there is no direction defined in the 3- space orthogonal to Ua.
In this universe we consider small perturbations of the motion
of tl1e fluid and of the '.ifeyl tensore
Ne neglect products of small
1
quantities and perform derivatives with respect to the undisturbed
metric.
Since all the quantities we are interested in with the
exception of the scalars, µ, ~' e have unperturbed value zero, we
avoid perturbations that merely represent coordinate transformation
and have no physical significance.
To the first order the equations (1) -
E ci-b
j
b
-
-s.h
b
I
0
f;b
(4) and (7) - (9) are
( 13)
'
( 14)
)
•
r-
-
c;a..
b+E 0 L8tr,:>
)
'
•
e -- ..
--
I
-3
- -3
'l.
e2. +
Ll0-b
•
lA.
8
Q
• Q.
.I
+
( 1 5)
( 1 6)
( 17)
J
( 1 8)
( 19)
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.
Metric
4. The Undisturbed
-l"i····-Since in the unperturbed state the rotation and acceleration
are zero, Ua must be hypersurface orthogonal.
Ua...-s~;o.
'
measure, the proper time along the world lines.
where
As the
surfaces 't = constant are homogeneous and isotropic they must be
3-surfaces of constant curvature.
Therefore the metric can be
written,
where
,
dy
1
is the l1ne element of a space of
zero or unit positive or negative curvature.
.-.'c define
t
1
by,
d. t
clT
I
:0.
TI
,
then
•
In this metric,
'
.
'
..••
(prime denotes differentiatio.n - with respect to
t )
Then, by
(5), (7)
(20)
••
·3
G -:. (2
±- (f
1" ;
ft)
( 21 )
•
If vve know the relation betvveen
µ
and
fi. ,
'vve may determine (l_
\Ve will consider the tv10 extreme cases, 1'"\. = 0
(radiation).
For
'b ;:
•
rl.\
-
:. 0
Q
f'(\
•
1.. D'2,-
•
M
(a) For E
n
I
M
I '2.
T;'
t:.1
=:
rt
0
2 'r- (cos h~
-
(b) For
--
= c onst •
••
;, r1
••
n
%
0
•
~
}1 ~
physical situation should lie between these.
.Any
By (20),
•
(dust) and
J
E
E
)
= const.
9
-
E: t
-
1)
i
- i E(~
-
1t -
1
.
~ll'l
h~r-M
-~ t -· t ) ;
'
-- o,
tz.
)
l
--
f'I\ t3
36
•
'
E represents the energy (kinet·ic + potential) per unit mass.
If it is non-negative the universe will expand indefinitely, otherwtse it irvill eventually con tract again.
•
By tl1e Gauss Codazzi equations '"R; the curvature of the
= const.
bypersurface ' l
J
is
-
2.
EM
o~
E
If
)>O
[=O
)
~
R- -
F'or
.?\
M -=
E
/Y\
-3
,::::
•
•
)
:>
E <O )
..,.__.....,:/
)
3
b
. n.c.:.
-2
)
...t
-=---)J./"S--
~
-
•
•
~2
- t-
n
••
3 (2
-
-
)l.
)
j-J· -::I~
(a) For
(1
--
-E
I
~.inh
(b) For E
-,
>
t
-
- -E'
-- -2.
I
fY\
t - I)
( C.c•S·h
-
I
0. ...
-E
(:,
.,. R -- - 0.'l..
)
,•
•
t
'.
•
2.
)
<(
0~
t
.)
-- ~
(c...ost
E
RotFttional
Pert11rb at ions
_,,,_ _ , ···--.-- ·- ... ·-·.------.. ·------·_..-=-•=ire-.-.
By (6)
3 (2
- o,
( c) Fo1"' E
5.
'L
)
)
S ..i. YI
•
'\..
o,
L
-F'
•
•
)
,..,..
t
.
-1)
)
-<. -I
0
,
]
•
•
••
For
w
Q..b
•
)
For
1ri.-:.
1
(&),.,
(2 '2.
)
•
-- -
w
-... -
w
et e +--'+ ~)
f'
I
I
- -'} w
e
)
)
--
w
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.
Perturbations of__Densi.!il.
For
fl·=
0
we have the equations,
-
-t-8
fJ --
-Te
•
fJ.
•
I
'a
I
--r: fA
These involve no spatial derivatives.
Thus the behaviour of one
region is unaffected by tl1e behaviour of another.
PeI'turbations
'
will consist in some regions having slightly higl1er or lovrrer values
of E than the average.
If the untverse as wh "1..e has a value of E
greater than zero, a small perturbation will
than
~ero
n.nd will continue to expand.
till have E greater
It will not cor.trsct to
form a galaxy.
If the universe has a value of E le8s than zero, a
small perturbation can contract.
contracting at a time
However it will only begin
8 1" earlier than the wl1ole universe begins
contracting, \l\There
--·
..
'(:"
0
is the time at vvl1ich the wl1ole m1iverse begins contracting.
There is only any real instability vvhen E
= o.
This case is of
measure zero relative to all the possible values E can have .
Ho-Yrever this cannot really be used as an arguement to dismiss it
as there might be some reason why the universe should have E =
F·or a region with energy
- cE , in a universe with E
·~E
(
t'l.- t't
4
12
r"l -
>l -
T =
I
1'1.
t
=
o.
0
... )
se
ti E )'l.~
(----·
'""t'" '1./~
f- :::
T •• •
)
2 ~~
For L: = O,
Thus the p e rturbation grows only as
'l'3 .
This is not fast enough
to produce galaxies from statistical fluctuations even
could occt1r.
i~
these
Hovv0ver 9 since an evolutionary universe has a partic1.e
horizon (Rindler(B)
9
in the early stages .
Penrosc( 9 )) different parts do not communicate
This makes it ov0n more difficult for
statistical fluctuations to occur over a rc;g: -.n until light had time
to cross th0 region.
-•
n
0
f ..s bef'ore~
I
i_
•
,,.,
_o
,,,,:
...
= - .., 0 - )A +
V'- ... '
a perturbation cannot contract unlesG it has a negative
value of E.
The action of the pressure forces Iilake it still more
difficult for it to contract .
Eliminating 6,
••
. .
l-l
0..: 0. -=
.
t,..l
I
<>-i b 1
o.b
..
+ u 0..
•
~
I.A
to our approximation.
I1
·.~re
OA.\7
I
Ve I
b :;:/
o.
vb
c-r =
is tl1.e Laplacian in the hypersurface
represent the perturbatior1 as a sum of eigenfunctions
this operator, v1i1ere,
S
(->)
constant.
S(n)
of
<:.
,•cU.
-;-.O
- - Q'2.
Yt.7...
s
{f.o)
These eigenfunctions will be hyperspherical and pseudohyperspher ical
l'1armonics in cases (c) and (a) respectively anC plane \vaves in case
(b).
In case (c)
n
\ll!ill take only dj.screte · ,lues but in (a) and
(b) it will take all positive values.
where
•
•
f'..'3
µ
0
•
1 cng as
For
is the undisturbed density •
fA 0
'7
\Vill gr0v1.
'
f'o >>
These perturbations grow for as long as light has not had time to
travel a significant distance compared to the scale of the perturbation
( ~ Q:. ).
Until that time pressure forces cannot act to even out
perturbations.
1.1/hen
...
r
B
I/
(11)
+
'
B
I
(t1J
rf.
0
I
. ..,.,
e {_ .1:3" t
· !e obtain sound v-1aves whose amplitude decreases vii th time.
T11ese
results confirm those obtained by Lifsi1i tz and Khalatnikov(3).
From the forgoing we see that galaxies cari.not form as the result;
of the growth of small perturbations.
''Ve may
~ x:pec t
that other non-
gravitational force· will have an effect smaller than pressure equal
to one third of the density and so will not cause relative perturbations
to grow faster than
l: .
To account for galaxies in an evolutionary
uni verse we must assume tli.ere 11ver·e finite, non-statistical, initial
inhomogeneities.
7.
Tpe
Steady-stat~_!.!D-jverse
To obtain the st0ady - state universe \Ve must add extra terms to
the energy-momentum tensor .
where,
c
c ·~
0.. .c.
'CA
J
Hoyle and Narlikar( 1 0) use,
'
-
'
J
Sin.ce
-r
·b
1 c..b' =-
o
)
( 21 )
•
lJ-
,/,,
Cl.+ I '-)
h ~,
\- b
b n
CL -
Q.
c c cl ;d..= o .
b
(22)
There is a difficulty here, if we require that the
11
Ci? field
should not produce acceleration or , in other words, that the matter
created should have the same velocity as t11e matter already in
existence.We must·tJ;'=tr.> l1ave
..'.I
(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.r. Raychaudhuri and Bannerjee( 11 )) . In order to
have a deterrninate set of equations we will adopt ( 23) but drop the
requirement that
Ca
is the gradient of a scalar.
The condition (23)
is not very satisfactory but it is difficult to think of one more
satisfactory. Hoyle and Narlikar( 12 ) seek to avoid this dj_ff~iculty
by taking a particle ratl1er than a fluid picture.
Hovvever this has a
serious drawback since it leado to infinite fields (Hawking( 13 )) .
From ( 17),
•••
c ea-~ ~
-= -
c;
= -
e f1
4
1~) -
(.µ-1 1'l) e
•
-
\.
•
For
f'
*ft
Tht1.s, small perturbations of density die avvny.
Moreover equation ( 1 8)
still 11olds, anc1 therefore rotational perturbations also die away.
:t;q,uation (19) now becomes
e= -
~
•
:~
9 _,,
I
e -2
'Z.
I
(
f-+ 3 r. +
)
l
.-----·
~ 3 ( 1:- (1)
These results c onf j_rm those obtained by Hoyle and l'iarlikar ( 1 4).
Vie
see therefore that galaxies cannot be formed in the steady-state
uni verse by the gro,•rth of small perturb at ions.
However• this does not
exclude the possibility tha·c there migl"rt by a self-perpetuating
system of finite perturbations which could produce galaxies .
(Sciama( 1 5), Roxburgh and Saff'man( 16 )).
8.
-----------
1
Gravitational
·
raves
•
~·-..
... ----~
~Ve no\1 consider perturba.tions of tl1e
~Veyl
tensor that do not
arise from rotatj.onal or densi·cy p8rturbations, that is,
-Multiplying ( 1 5) by
LI
f I
lA c
·b
0.
b
I
V<:.
=O
~
and ( 16) by
we obtain , after a lot of reduction,
r:
••
=
In empty space with a non-expanding congruence
~b
0
17
Ua
this reduces to
the usual form of the l i nearised theory ,
The second term in (24) is the Laplacian in the hypersurface
"'r
= constant,
acting on
We will write
Eab •
as a sum of
Eab
eigensfunctions of this operator .
where
.
V
(1-1)
I
O~>
=
0
}
v
_ yt~
Q
V
)
Then
••
-
·-
•
0
Cl'-
~o
•
'2.
(1"1)
Q.
b
Similarly ,
Then by (19)
Substituting in (24)
Al"') [
D(~}
[ r2 (j-tt f'L)
t
i
n ....
3
n''
(2
t
2
0 (f fi-)]
i
::. 0
We may differentiate again and substitute for D' ,
For
\'l
>>I
and
f2
>/
J
t1 2.
1
and the "energy"
Eab decreases as Cl r....
6
We might expect this as the
as 1 t..
•
so the gravitational field
t(Eab Eab + Hab Hab)
Bianchi identities may be written, to the linear approximation,
Therefore if the interaction with the matter could be neglected
cabcd would be proportional to n and Eab ' Hab to D -1 •
In the steady-state universe when
.
and
µ
e
have reached their
R~~ ~ /~ + fL ~
equilibrium values,
•
••
J o..b c ~ R c La j bJ :::: 0
i
S ~ L°" 'R; b J
Thus the intera ction of the
11
c 11
field vii th gravit ationa l radiat ion is
equal and opposi "i e to that of the matter .
intera ction, and
of the me·tr i Q.
decrea se as
Eab
The "energ y"
Ther'e is then no net
and
.1. (E
,.,ab
2
· ab-'-'
n -1 •
depend s on se• ond deriva tives
It is theref ore propo rtiona l to the freque nt.y square d
times the energy as ineasur ed by the energy moment um pseudo - tensor , in
a local co- moving Cartes ian coordi nate system .which depend s only on
first deriva tives.
to
r2 ,
to
ll
~L
9.
Since the
~requency
will qe invers ely propo rtiona l
the energy measur ed by the pseudo - tensor will be propo rtiona l
-4
as for other rest mass zero fields .
AQ_~orption
of Gravit ationa l
lilf~es
As vre have seen , gravit ationa l waves are not absorb ed by a
perfec t fluid ..
Suppos e howeve r tl1ere is a small a1nount o1' viscos ity.
We may repres ent this by the additi on of a term
energy- momen tum tensor , where
·A
). CJa..b
to the
is the coeffi cient of viscos ity
( El1le r s , ( 1 7) ) •
Since
we have
(25)
(J
;
c.. b
b
h
e.
Ol.
-:::: 0
(26)
Equations (15) (16) become
H~
~;e
=
(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
e-
decrease by a factor
"'t
~
.
Neglecting expansion for the moment,
suppose we have a wave of the form ,
E o.b
::::.
0
E- ~be
~V"t
.
This will be absorbed in a characte:r.iatic time
frequency.
"-/). independent of
•
of rest mass energy of the
By (25) the rate of gain
matter will be
~ A ()
2.
which by ( 19) will be
available energy in the wave is
?.
Fz )).-2 •
4- .:>.:.
A- Ei. v -2. .
0
Thus the
This confirms that the
density of available energy of gravitational radiation will decrease
as (l
-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 universe
vvhen the temperature was ver•y high we migl1t expect an equilibrium to
bo sei up between black-body electromagnetic
gravitational radition.
~
diation and black- body
Since they both have ·ciivo polarisations their
'
energy densities should be equal.
the uni verse expanded they wouJ. ':.
.As
both cool adiabatic ally at the same rate.
of black-bod y extragala ctic elec1
5°:< ,
As we know the
~omagnetic
temperatu~c
radiation is less than
the terrperc.:.t ure of the blac1c-bod y gravitati onal radiation n1us'..;
be also less ·than this which
the energy c ·f gravitati onal
vvou~.
l be absolutel y undetecta ble.
does not contribut e to the
radi~· :~ion
ordinary e11ergy momentum tensor
e.ctive gravi tatio11al ef'fect .
Now
TI
... ab •
I'reverthe le ss it will have an
By the expansion equation,
•
For incoheren t gravitati onal radiation at frequency
~
2.
::.
0
E'l.
-i
'))
But the energy density of the radiation is
•
e
f
••
\vhe1'3
µG
I
2
v ,
I
-:; - 5 8 - Z }J- G
- {
(f
t-
2.
E v
4
-2.
0
3 }1.)
is the gravitati onal 1tenergyn density.
Thus gravitati onal
radiation has an active attractiv e gravitati onal effect.
It is
interestin g t11at tl1is seems to be just half that of electroma gnetic
radiat ior1.
It 11as been suggested by I-Iogarth ( 1 8 ) and Hoyle and Na'Y'likar ( 1 O),
that there may be a connectio n between the absorptio n of radiation
and the Arrow of Time.
Thus in universes like the steady-s tate, in
which all electroma gnetic radiation emitted is :ventuall y absorbed by
other matter, the Absorber theory would predic
retarded solutions of
the Maxwell equations while in evolutionary universes in which
electroinagnetic radiation is not completely absorbed it would predict
advanced solutions.
Similarly, if one accepted this theory, one would
expect retarded solutions of the Einstein equations if and only if all
gravitational r·adiation emitted is eventually absorbed by other ma.t ter.
Clearly this is so for the steady - state universe since /\
constant.
In evolutionary universes
V1e ',ii/ill obtain complete absorption if
1
~ oc. T2
\'\There
T
is the temperature.
will be
will be a function of time.
r>t o1.r
Now f (" '.' a ga:-;,
"
For a monatomic gas , T d:. fl - 2 ,
therefore the integral will diverge (just).
diverges.
However the expression
used for viscosity assumed that the mean free path of the atoms was
small compared to the scale of the disturbance. Since the mean free
path oc µ- 1oe
-3 and the wavelength 0::. (2 - 1 ' the me 8.n free path will
n
eventually be greater than the wavelength and so the effective viscosity
will decrease more rapidly than 1()L -1 •
Thus there will not be conwlcto
absorption and the theory would not predict retarded solutions.
However this is slightly academic since gravitational radiation has no·::;
yet been
d~tectcd,
to a retarded or
let alone invest.igated to see vvhether it corresponds
adva.~ced
solution.
References
--
(1)
Boilllor W.B . ; M. N., :L1.Z, 104 (1957) .
(2)
r.ifshit z E. M.; J.Phys . u.s . s . R., 10 , 116 (1946).
(3)
Lifshitz E. M. , Khalatnikov I . N. ; Adv. in Phys . , 12 , 185 (1963) .
(4)
Irvine
(5)
Jordan P. , Ehlers J ., Kundt W. ; Ahb . Akad . Wiss . Mainz ., No . 7 (1960) •
V"J . ;
Annals of Physics .
~
J
~·z 2.
(iq6'f)
.
( 6)
Kundt \V. , Trllinper M. ; Ahb. Akad . Wiss . Mainz . , No . 1 2 ( 1 964).,
(7)
TrUmper M. ; Contributions to Actual Problems in Gen . Rel .
(8)
Rindler W.;
(9)
Penrose R. ; Article in ' Relativity Topology and Groups ',
Gordon .and Br each ( 1 964) .
M.r~.,
116 , 662 (1956) .
(10) Hoyle F . , Narlikar J . v.; Proc . Roy . Soc ., A, 'fil, 1 (1964) .
(11) Raychaudhuri A., Banerjee S.; Z. Astrophys., .2§., 187 (1964) .
(12) Hoyle F ., Narlikar J . V. ; Proc.Roy. Soc ., A, ~8.f., 184 (1964) .
(13) Hawking 8. vv. ; Proc . Roy. Soc ., A, ...:in prj,nt. ~ ~/3>
(/~.£)
(14) Hoyle F. , Narlikar J . V. ; Proc . Roy . Soc ., A, 273 , 1 (196 3 )
(15) Sciama D . ~v. ; M. N., 1J5-., 3 (1955) .
(16) Roxburgh I.'1V., Sattman P. G. ; 1'11. N. , J.12 181 (1965).
(17) Ehlers J. ; Ahb.Akad.Wiss . Mainz.Noll . (1961 ).
(18) Hogarth J.E. ; Proc . Roy . Soc . , A, 267 , 365 (1962) .
CHl\.Pl'En
3
Gravitational .Liadi&tion In An
Expanding Unive1·se
Gru.vi tational radiation in e:npty asymptotically flat
space has been examined by means of aoymptotic exoansions
by a nL.mber of &uthors . ( 1 - 4 )
·'.Chey find that the different
components of "che out,soing radiation field "peel of'f
is , they so
vS
that
11 ,
different powers of the affine raJial distance .
If one v1ishes to inves·tigc:.:.te hoi'l this beha•riour is :nodi .i:-iPd
by the pr~sence of matter , one is faced with a difficult~r
that doAs not arise in the case o.r, say ,
ele~tromagnetic
radiation in matter .
For this one can consid.er the radiation
trc.Lvellint; thro11gh
infinite t1nii'orm medium that is static
a11
apart :from "t;he disturbance cr,3ated by the ra\.i.iat;ion.
case oi' cro.vi tcttional
if che
me~ium
racli~tion
In the
t;hi:J ir;; no·t pos,_;ible .
were initially static, its own self
'
.!!Or,
5ravitatio~
1r1ouia. (;ause it to contr a ct in on itself and it v1ould cease to
be static .
raQi~tivD
i~s
Hence one is forced to investigate
gravi·tatio·~al
in matter that is either contracting or expandin;.
in 1Jhapter 2 , we identify the Weyl or conformal
·tensor
with the freA gravitational field and the
1~icci-t ensor Rctb
curvature.
with the contribution of thA matter to the
Instead of considerin; gravitational radiation in
'
acymptotically flat space , tna.t is, space that
flat sc; ...t.ce at large
~pproach~s
radial distances , \ve consider it in
rts,11n:pt o tica.lly conformally flat space .
conformally flat , the
~icc i-te nsor
.1\.s it i.;.; only
and the density of
m~tter
need not be zero .
:'o avoid essentiall;yy non- gravitational phenon1ena. sue h
as sound wavgs , we will conbiner
travellin6 through dust .
gravitationalra~iation
It was shown in Chapter 2 that a
conformally flat universe filled v1ith dust must have one of
the metrics:
\ Ls- i
(a)
I
i
2
4
S-2. ( cl {; - cl.f - s· ,,\ f
i
.:
; A ( i ·I
(b)
C(
~
'2.
J2 'l.
;..
_\)_ ;.
(c)
cLS' 'l
__J<j_
(
1
L
f4 l
.
:t /l l ri ) )
c D 't' s'r. CJ c 7' .
cas: c)
. ·1
cl t:
~ At
.
(;.
:i
I~
~
l.
1. )
).
(I, 2 )
2.
---
fype(a) represents a universe in which the mutter
expands from ·the initial sinJularity with insu fficient ener3y
to reach infinity and so falls back
si11gularity .
a~aj.n
to another
It i s therefore unsuitable for a dj.scussion of
•
ravi·tationril radiation b;y- a method of .::..sympto·tic expansions
si11ce one cu.11no·t <_;et an inf ini ve C...:.istun(~e from -eh~ source.
0
Type (b) is the ~instein-De Sitter universs in ~hich ::he
mat·ter has just sufficient energy to reach infinity .
thus a
S 1)ec i
o.l case.
It is
D. Norman ( 5) has investit;<J..ted the
"peeling off" behaviour in this case usi11c; Penrose ' s co11f ormal
6
technique ( ). He was however forced to make certain assumpt ior1s about the movement of the matter .·1hich vrill be sho~·!n to
1
be false .
;:oreover , he \vas misled by the soec i al nc.ture o""'
the ~instein-De Sitter universe in which affine ard lumir:osi .. y
d is·tance:_, differ .
Another reason for not considerin;:; ra.::li<:...tion
in the t•:instein-De Sitter uni verse is that it is unstable.
fhe passage of a gravitational wave will cause it to contract
a~ai·i e vent ually and develop a s in.;ularity .
de will therefore consider radiation in a universe o!
t,1 pe
( c) ~1hich corresponds to the general case vJhere tn.e
rna·tteI' i.3 expanding
\·Ji th
more th.: n enough ener6y to avoid
contructin;; again .
2.
fhe Newman-Penrose Formalism
~·Je
employ the notation of l{ewman and Penrose. (3)
tetrad of null vectors ,
l r--/
nf-) t'I
rn f-/.
('(}L
1·
,
f'-.
A
is introduced
lf (L
vve
label
2
()._
p.-
nf'
L,,.,
lf'
v1nere :
=
f-
=
th~se
(/,r\ Mp 1iil1v.: lf- (Y)i~ -- (lt' /l?rl;
·- f.J.
.
Mr M
l)
('L: fl~ P1'~ M ru)
c~b
uq ?
--
-- -
l
0
0
(
0
0
0
0
0
0
0
-I
L
pv
)
-~
..
-
O.b
--
.
an~ lo~ered
0
l o
have
;; -}
vectors with a tetrad
tetrad irdices are raised
vie
[ ,
7
l
( ?__:__!)
•
-1
0
2p.
().-
2v
h
fl
pL
IA f'L +
11
\lith vhe notric
II
·-
rr1 ,~ m
v
·fA
v __ fV1 fYl
(2- j_)
Ricci rotation coefficients are defined by :
( 2
-
3)
In f'ac·t it is more convenient to
complex combinations of
ro~ation
':.:01:l{
in
t ·~ rms
of 1J-;.,'el 1e
1
coefficients defined as
f ollO\'/S :
(
--
/\
I~ I
- -y
·-
2. lt
I
'I
'I
- ( )'l I
•
l
•
-
I
•
·-....
··-
-
·-
·-.
-
--
--
I
.'.:l
3~1
Lr:
v
0
-
fA -
1'l!J
fl1
.
L
n /Al. v~ 1Yl
- {v<.; v I
I
1
v
L
//.__,.;..;
1iVif l v)\
3 . Coordi.:1atcs
--.Li lee i:·l e \fJrnan a nd Penrose ,
\J e
u..( =xj)
lL ·v
I
take
vte
introduce a nu ll coo1:-dir1ate
<~ . i)
~ o
L~
Thus
•
3eodes i c and irrotati onal .
v
(l
(
Lr
':Till be
This implies
-- 0
-
~
f-
cc- ::: - £
:L + ~
vie take
along
1~s
•
~o
be parallelly
IJ.'b.is gives
transporu~d
-~ - s)
•
a se concl c oordina·te \':e talce an affine parameter
l f"
r:· f L f" ~ I
long the t:;eodesics
X
1
and
X~'
i n the surf v.ce
arc
t\·10
X : l). L
f.i.
(-:?
Lt )
coordinates that label the geo6.e...:.ic
L.l = cons t .
l
1_
=
·u.
L':
I\ ,' ,vY't ,,
0
i1hu s
1
(1. b)
In these
(
l =
{ ~ f)
The lield
~~u~tions
Je 1nay calculate the Ric ci and ;1Je.1l tensor com1ion8n't;s f r om
1,1
the rela·tions
a.._b c..-.:.L
o-. bc: (L
{.-{
't ·-
~
,-
-
•
")
.._L b
Using thA combinations of rotation coefficients
defined <ind with
/"C -= TT
=£ = 0
v1e have
p ()
( J.
10)
j) G'
~
I!)
]) 'f
/)d.
j)~
-
( ·s..
···-
i2)
(1.1.3)
..,_ ( 1 I~)
( :s . /~)
Pr
y ,\
(s.
Pr
(~·- tJ)
vv
- - --
I b)
{l . IY)
"~) c, J'
--
~· f ~ ~, c;-
~j_
J~
-
-
-
fv
r~
-
l
--
( {? r ;_ )
-'
f" f ->. G
-t ( .~ -
·-
J C\)
b' ·-
f, r po '
J<X ~ 7 O(~ ,- ~? -
y;
('5 . 'l c'/)
f 3 f1
-4r- tr-2vf·ir'f'?._,~IT ?J.2
r;/t,' , ( oi. .,. fe )f-
..,. (;} - ~ ~) ). -
~"
·t- /\ r
T
6 '2::_
1)
J
-
(
!.. ")
(? fA - J f )" f .,(I,
11 r - sl. = cr ;- i ·- f ) f - 2""' "'L - ,\,,-. - --r; - ). )\ ~:, ,_ t).
fI
-
1J b ~ J l. (S
-r
T
.
°I" ).
0
i
•
il °' - J''(
=
rv
-
L~
-
).
ft _
(J__L f)
J
•
D
i.·1here
LJ
d
L {" Vr "
--
- rit'
-
vr
(!
~ '/,..
v t;.)
--
~r-
·-!..
).
R -- fo
-_!_
(I~
<r
jQ
,_ K
31..f
-
-
·-I
J.
1]
R
J ':G
- J.
--
-- c/> ;J.o
--.
/\
l
l
Jxl (3- ·29)
(~ 3o)
R
.< J.~ = 1J.~
..-
-1
--
~
l~
--
.
11
(3. ) I)
0
--
•
)
<.f)
.
•
-
g
·-~~I
1,o
·-- -J_ R ·-cf, ·1
1~I
2 13
f
J
.5 JXL
Ii
1>01.
)\ i
.' ~ J_
Poo ·.:
t ·--
·--
-;-
(.A.
J --
--
()
·J
?_
--
cp,,
(1.
'~L.)
(s. s3)
( 3 . ·5 "!)
(.s
3 s-)
--
(1 . 10)
<~. 3 f)
2
~
4
-c
) ·2 I
1.
3
2
3
-i'l.l'l
J
'.l.3 4
2
cl. .
0
x ( l "'fl- 13 l ( n_ r.,.. Ld/Lf;n. ~ .P
(}
~-c)
(~. y /)
: - c
•
'
5
Expres s i11.~
the rotation coefficients in terms of th8 metric,
we huve :
l
-;t.
L
·-
"L
]Ju
I
•
•
.
g
£.,
,.
(1.y6)
&W ... 'iw- (o+f)
=
- ~x~-Llg·; {r_,_f-o)~~T>- §" (s.1ir)
\
,
5- $& ·~ S~ L ~
Si:;
-
(~-
-
I
~) $'- "' ( ; - ~) ~ '- ( 1 V0)
~ w = (~ - at) w ·+
(; -
~ ) w ·t <t- -/;:,.) 3 0 ~
j~s
in Chauter 2 we use the Bianchi identities as
-"
fiel<.1. equations for the
;,~eyl
te11sor.
In the t,ewman- Penrose
formalism they may be writ ten :
(I am indebted to R. G. McLena~han for these)
g lf'o - Y f, .,_]) cj;c:rl - $' foo"'
i·
6 'fa
- ~·y <t D~o~ - ~101::
-t- S6f;. - °Jo/
<r "'-
'fo - ~ f 'f, -( ')_:; ~!]) ~oo
'2f cf
0
1 .,.-
16' 1,o
(·4~ - ~·}'-('o - 2 {2-r_
.
-t-
(3. S 1)
.~) lf7
?oJ. (l.S~)
·3(5115; - oy,),. 2(J>1. -s ~,o) $' <P", - Li ~00 :: 1> t - 9r t
1f, <t - '1 ,v- - J r - 2 i ) cp (J<l-t- Q"£) ~O
00
-
2P101 .,_ ~b1,,<>1-(
·t
-t {, ol.
-I
00 't
J ( T - ~ I{ ) · lrh t 0 .:J.. ~ 1-1.1 'f
rA -r ~b' rb - if rh
I :2 0
I 0 1..
'f
II
I
(
s 5.3)
·~ (t)f, - f~ ); i(D~ 11. - d ~") -t (f o/o.: 11 fo,) , 3r 'f;;r
_-t _b ( o- f'-) f, - q'1 tl. °I b€f1 - 7 foo
r
- {f 11'3 -t-
(,
"Z
1,,
T
~0" <>;- :L ,,\ ~ v ( ..,. 2( f .~ f'- - j i )1- c
( ~~ .,_ ). 'c·- 9,J_ )
~ 6- f, :(
{j SS)
j
Y'
t ;_()
-if, - 1/A 'f2 .., b ( ~ - (:,) lf3 .. ] b t '{
- ~vcfo ,- )V~,o 7 2(2F-r)1.,r 2)102 .
- Jcji,_o + 1(75- ~ff)1,2 <f 2(ft'r -c.Jf-;_,-f ~J.). (f)-6
.
= {, v
~ t~ ·- Vt~ ,_ ~ f , _, - A1,,_o ::: ]:A-lf'; -- :)a., Y's - f <f'~
- 2 v 1,,..,
2 ~ f ,, ,. ( 2 r - 2 f 'T ;:;.. ) <f2u
-t 2 ( t-E - o<.) ~:;, - G c/J~
c·
J.
5"-t)
7
'I
<"J_
LJ f 3
--
~1f'4
-t
ll <f,_, " 3v t'l.·- 2 ( y t 21'-)</5
-r;) 01./r "-/ - 2 ·v · - v 10 t Q. >- t,'l
f </>.,_
"/- ( 4 t->R -
?_ -
~ <r +- p) '97,_, ., (-t - ~? - 1 v(.) cf:;.2
·.
.
( 3. <j8-)
]) f,'.<- r p,, - S' <f>oa ~<Pa,-.- 3 ~II =<~ 0- f - 2;;.,) o, • vtf
·
11
1
"1-
0
2C ~. -r { 2~- 2.;. ·- l) </; ~"i-Jf ?,,,_<>t 6' </12-i (.~ )7)
JJ"' ~ f; ,i, - ~ r1,lo ..,. 6 rJiI + .1 ])Ii = t 2. r- t- -t-.2i -~) rrA - 2~ ~ i) Toi
rh
- )_ (.;:<>r --c.) ~,o + ctp <P .. '!- ?~o:i. .,.-~fie)
( -S- 6 ri)
Df
r ~,_,- t <f,,,_-t 11 cp,, -t 3L) /] = v 0/ r ;; ~ID 2 rp.,-?- )et,,
0
- ).1,o-
'"
0
'+'10
I
00
OD
L). -
-
'i-
2p
1
'J. ;i.
(
~. (,, !)
4.
The Undisturbed Metric
The undisturbed metric
rLs '2
SL 'l l Cl, I=-
=
.5 L
put
tL
b
'"7.
·=-
(.} (_
f 1. -
ci.,
-
be written
co~ h. t -
S ; f\.
),,,
2
p ( c(,&
2
r
S i "- :Lc.{
~
2
)
J)
·-p
-cLtA?.. 2cLv--cl t
cl ~'2. ~ 5L l.
""Ghen
2
m~y
't"
- S 1 /t; ~ L..
C"t·- v..}(cl&;~,·ll Gclfri.)
2
( 4 1)
is a null coordinate
ro co.lculate t', the affine parameter ,
1
\•re
note th,_t
C
is an affine parameter for t he metric within the square
br<~ckets .
t
Therefore
::
S5L
'l
ci L-
T
"f5 ( u., g..,
cp)
(it· 2
\vill be an e:.:..f fine parameter for ( "'- ·· /)
ls
is constant along the null geodesic .
would be taken so that
r ·.: ;
when
0
t -: ;.
u._
No~mnlly
it
• Hov.rever ,
i11 our case i ·t will be more convenient to raake it zero and
,,
as
define
r --
l:-
.Jl 2 cJ_ (;- I
- ---
0
i'his means that surf aces of constant
1
constc.nt
t
•
This may seem
r~ther
be pointed out that the choice of
asymptotic dependence
Then
1$
of quantities .
0 ( t' ·-ft,)
( 4. 3)
r
are surf aces of
odd, but i t should
will not affect the
That i s, if
)
It proVeb
e~sier
to perform t.•e calculations with this
but all results could be transformed back
choice of
to a more normal coordinat8 system .
J.!'rom
r --
The mat ·t er in the univers e is assumed to be dust so its
e ner gy tenbor may be
writ ~en
'
7 ~b = f I/Gv v b
For the undisturbed c a se, from Chapt er 2
~
bR
Sl
V
0-
N O\r.f
5)_
v1here
j
~ /3 s "'t A -
i/J'lfJ ~
==
I
°'-
fE.S-s
0 (s ·- •)
s
S' 1. .:. t
'l 'he1·efore if v-1e try to expand
t, l1e
V V0-
::. 5LcJo.
/J'"
as a series ·i n po\v:r of S"'
r esult v1ill b e very messy and will involve terms of
the form
If\ S
L
.05
s
*
n.
"'It should be p ointed out that the expansions used \vill
only be assumed to be valid asymptotically.
They will not
be assumed to converge at finite distances nor will the
' quantities concerned be assumed analytic. (see
Asymptotic Expansions
- Dover
A. Erdelyi:
Tnis
t•
oes not invali date it as an asymptotic expansion but
i t makes
i~
tedious to handle.
For convenience therefore ,
.J2 ( r)
\\'ill perform the expansions in terms of
\-.re
will be defined in general as the same function of
i ·t is in tt1e undisturbed c a.se.
1,vae re
the n
I ~
r
as
rhat i s
1
AJ. [ i $1f'-~.2t - Q~,r-~ t
1"
%t-_
Jt-r JL
'lf}
cl5l ·--
•
which
cl r
A 5_ s !J4r
I
- _f2_
-
252 ~
-
1-1?~ ~,.
<B:.15l ~
-
For the third and fourth coordinates it is more c onvenient to
use stAreographic c oordinates than spheric o.l polars .
Since ·b he matter is dus t its energy- momentum te nsor
and hence the Ricci-t ensor have only four independent
componenL.s .
(;:,ince
\vill take these as
\·J e
q> 01
I\ 1'foo
,-\)
1
rti
lO(
•
is c omplex it represents t\vO components)
In te.cms of these the other components of t he Ric ci-tensor
-
may be expressed as:
7,,
~ ~2
c/Jo1 cP01
~ /\
--
~00
.3
full
l
-
......
ci> 0
0
•
I
·r
•
-
~Oi ~CJ•
t 11 ~oo
"l
•
·•
cp, ~ --
t
-- b /\ ~()/
2 I
~oo
·-
.
l +
·-·
cf i o 40~
-
-</J.o •- .~Qi
b (\ -.rp0 0
Q.
~
o/01
(er. 10)
•
cp
(.:>
0
For the undisturbed univers e with the coordinate system
/\
.
given :
--
--
~oo
4 cp ~:t
cf
Q
I
'S
4..)'1 ·3
,
R
Sl~
1fl_
-
t1
A
-#=- -·2..~
452. 3
--,,
lE
452
~01
=:
0
( ~ . 11)
Using these values and the fact that in the undisturbed
universe all the
y "s
are zero, we may integrate equations .
(j. 10- 50) to find the values of the spin coefficients for the
'
. .
..
-
•
' '\Z
·G
- 2
•
--
(-t
25l
-
--
'
i,,.-
5L 2.
--•
s· ~
I
I
-1
-
2Al7~e.
4
.
T
.
•
Jl'
Jq 'l.( { ..- e. Q..v.)SL- 2t (]_3(1
-
~
4-
-
(-)
-
+
:2 52.
J J}
A'L Jl --i-
A~
-
.
..,..
t-
2. e
1.
)--Jt -s+- .
•
•
•
4Jl.3
2
J
5. Boundary Conditions
~e
wish to consider radiation in a universe that
asympt otically approaches the un i isturbed universe :;iven
•
above .
et>Or>
and
/\
will then h&ve the values
•
o.bove plus term3 of' smaller order.
61.Ven
order o.11d the order of
•
v1hich
4:>01
may proce ed.
and
%
To de·termine this
'
t l1ere · are tv. 0
1
may talce the smallest oraers
ways
1.Il
·t;hc:~t
\<Jill perini t radiation, ths.t is
•
order t erms than these in
anC..
Lar~er
cP01
\'le
••"e
~
derivative s
turn out to have their
dependent only on themselves and not on the
of
, the radiation field .
r
-J coefficient
They are thus
not produced by tl1e radiation field and will not be
disturb~nces
consi~.t.ered .
Alternat ively we may proceed by a methoa of successive
•
•
....
approximations .
We take the undisturbed values of the s~in
cosfficients and use them to solve the Bianchi Identities as
field equations for the conf ormnl tensor using the flat
boundary cor1di tion that 1-;;- : 0 ( v-~
llJ'l S"
these r
in equations ( 3. iO - ~ (- )
soacP,
• Then substituting
calculate tl'1e dist1)rb-
ance3 inctuced in the spin coeffisients and substituting -che..3e
back in tne Bianchi Identitie5, calculate the disturbances in
the
'l)l''S
I
l!'urther iteration does not uff ec·t the orders
•
of the d isturbances .
Both these methods indicate that the boundary conditions
should be:
/l
.....
--
5L~
+ () (JJ_ -<i)
-- 01-52-=t)
~,
--
(see next section)
0(51-=t)
~x
(....
d
•
~XJ-
. . ..
'
•
d
dx (.,
~
(~-.
3)
(S-~)
\•/ e also assume 11 unif orm s inoothness" , thic,t
d
( ')_ 2)
•
J.S:
-- 0(-51-r)
--
etc ••• I
'
..L "'
vi..1...L..L.
be shown that if these bounda ry conditions
c~ =
hold on one hypersurface
const . ) they will hold on
succeeding hypersurfaces a nd that these conditions are the
most severe to permi t radiation .
6
Intee;~ation
As Newman and Penrose , we begin by integrating the
equations
(3 .
10 &11)
= f
:Vp
06
-f-
:: ?re-
j) er
where
~
f()
3 fl
-...
~00
-t"
411 r ·t
-
•
Let
-
--
•
-
('
<.) 0
Oo
·op
then
·p
let
? ··2 ., ~
-(» 'I ) '-/ ·-1
~
--
0·. I)
(6.2J
D1. y ~ -CR!
s ince S, . Cf clr ..t{. cD
(LJJ
then
•
V L(
f- T O{t)
;;..
"'-/
.:
r
F
r
where
0 (r )
}Io 11ever
ce ::. o(r -1)
therefore
l)l. 'f " -r<.j>F .,. 0
i;herefore
])
1
Y ::
F i s constar~t (6 · '+)
(-r -~) ~
C
(r -;r )
(6 ~)
(r - ~)
f-rO
y
c onstant
p
Let
-Then using:
'l) :
J
Jt
(\ -; (
- .) ,_
-
I
t
n ·-~
A.) J_
A
~.f).·).
A?.f)_ 1..
- et
~
J.
..) )
JJl
Integrating ,
3 ~ Jl-; 0(1) a...,.
:;. - A
the ref ore
.i!'Or
fB ·r0~1-')) .lfJ- -r O(i)_)dJ;_
Jl,.- 0(1)
-
1
"t-
o (
lo
.n_
~ -9)
1-L /
d~ J1
dfl
-i-
0 (,)
-= -
Iv T 0 (SL - I)
the r ef ore
6,
IC
l'tepoo:t the process vii th
- 2 52.
--
::.
-
/\ \l-3
1-t _; L
I
h,
•
l .52- ~
:;..
d k ( J)_ _,. 6 (1) ~ o (SL - ')
J.Sl
/t_ :; 6 o( {fa ~
-r 0 (J1 - I)
I
~ (-51-r
~Sl
Unl ike
(6.12)
--
\vhere
then
-2
6
'- ) .
0 ( 51~-I
(t)) .:;
(.J 0
tf e111man and Unti, we c a nnot m:· .ke
the transf or.:natio11 'r
1
:::.
't" -
alter the boundary co11di tion
P
I
0
,
A
.I\ - (; 52 :!
-
lD552.)
\.
zero by
since this would
i'
K
<\ -=r
.) J_
Cont i nuing the above p rocess \ve de .:·ive:
f ~ - 2Si-2_ A 51 -;_
.,_.
<; Ll 4
-li ' '
'l"
I.
n <J..
~n
o
j .SY'(.,. (iA -z._ J .4 f
0
o 'J.
( .- I ;i..
-
6. o
. ~
6
o
0
)
5l ->
\)- -
~ Q(..n - 7) f~
J ;.
,
J '-
c--'
J & . ' ·:J
(t.1l)
•
To determine the asJyrnptotic behavio11r of' "\lr ol.. ,8 t ...
l '.> II "~
and u.:>
\'Je use the lemma "Oroved. by Newman L:nd Penrose :
':I1he
(\
x
matrixBand the column v ector
I)
•
given
funct ions of x
B-=
The
x
()
2
o{
x
)
6 -
...
o(x-'l)
I
.4
•
J.S
real part is r egul a r.
}(
are boundei:i us
(6. , .i)
and has no
Any eigenvalue v1ith
Then all solutions of:
(A )( _, 7 R ) J
••
x
i ndependent of
ei.:;env;;.:.lue i-1i th p ositive rea l part.
vanishin~
are
such that:
matrix
ft..
b
-) cl)
T
b
•j
is a c olumn
vecto r .
J.!'O r reaso, s to be explained below, v1e will assume for
the u1oment that
101
d
J __a_
)
)( v
We take as
rI~n)_~ ')'jlr
1 )
j
...
d xj
·-
4
(?
)5l
()L-S)
'
--
i
o (SL-b)
~O I
'
c)
f1
\ ~' . I J
-- o(SL-~)
the column ve ct or
JL J. A 51~A Sl. i(\ 52. ~. e3 52.")..
I
)
)_, )
.:.) )
:e~ )3 nic.) l-).J2 JIc~vJ ~
I
(6'. }_O)
A-
~
'
)
·- .J
s J)
3
By equat ions
A f:, A- o o o o
0 fo
00
S, 13) "l. r4
I
G
000000
0
3. ~ s I 3. l.f y
i 5.ft 0
0
0
0
0
0
0
0
0
0
0
£·. ]
0
•
0
0
•
f)
0Qe) 0
0
0
0
-I
oo0
0
0
-2
0
-I
o0
0
0
a -2
C) .._,
CJ ~1 0
B
a
CJ 0 0
and
b
o(Sl- 1)
are
•
0
expressions involving
a nd
J
r'h
• Tot
)Sl
,:hus
</..,
t: .:
0(SL->)
~, ~', S"
'° o (5L-2.)
:: 0(1)
{;.)
•
>.Jlnc e
'-[ ;;
--
\
-oL
t-
fS
CJ (51..- <)_)
0
('3 . /·i)
Usin0 this vie i11tegrate eouatior1
method as above .
~e
't 0
obtain
}<'
i.A.
(
l) 51- 1
by
the same
.
3
~ (j ( .5l . )
J
v:e rr1C1.y ma.ke a null rotation of the tetrad on each null
--
geoa.esic
-
-
-is constant alone the geodesic since the tetrad is
parallelly
tr~nsported .
we may make
By taking
Jnder a null rotation
9'o, = fo1
'l1hus
until
\·1e
im] ose a boundary condition on
~o
i
=
u
vc :;.
<in-·- s)
0
<foo
o.
have s-pecified -che null rotation we cru1:not
"t
•
A
/01
more severe than
We v1ill specify the null rotc-;.tion by
and in that tetrad system will impos n the
uniformly smooth.
boundary condition that eh
'ro, ~ 0(51--:t) and is
and ~; o(. Jl
Then by using this condition on
~O
I
by equation
w ·;:
•
'
·1
using this in eq_uation
Yi=
then by
~. 5 i)
0 0--6)
A. 1·1)
equati~>n
\"'
'f ;
0
(
_;'l_
-~·)
putting this back in equation (? · C.f ~)
w :::. 0 ( 5)_ -1. lo5
('5 .St)
b;y equation
(1.
b y eauation
by
~
=
0 (
S2. - 1- lo 5 .SL)
11)
~~ o(SL -~ ~'5
JL)
equation
by equation
by
1
52-)
·:;. 0 ( _))_ - 1-)
equation
(g
by equation
l(
"-1)
.
•
By differentiating the equations used \Jith resuect to X
ol R
one r.. a ,i sho :1 thc:..t ."'J•"
T1 '
/ /-'/
uniformly smooth.
1
<-t
e re
I
a nd:
Ad,lin,_; equations
~t - j)~;_ '1- J>f.. ·-b~Oi't DA ~ ).. to - 3f' t~ t
--;- jf cp,,
'
"'t
6'
~. b O : -
')d.
t- t
'l,0
foo - '2J1. o
(l2i)
~e
may use the lemma
a~ain
with
-3. /G. 3.
By equ t..ti ons
I
11-,.r G. 2.8'.
-
and
'.i'heref ore
)_
0
0
0
0
0
-I
o(s'L-2)
·v.1- :: o (n·- ~)
are
~ ·:: . 0 ( 52 - ~)
-
CJCJl. -1)
and are uniformly smoctb
From
(t,
2 q)
and
f-' ~
using this in
(0.
(·1 _It)
#5)_ -
and
0(.51. - 2 {oJj(__)
0 (Sl. ·-5 Co5 JL)
(1. 11-)
y"l- : .
I 'r-
may shov1
2.. ~)
Lf.,_ "'
then by
\-le
0 ( J2. -
S-)
?. i 31 1 (V) 1. I ~J.K':>,, 5. <t ~ 's. ~ S"
Integrating the radial equations
o
·-
3. ~0
1
I
·
.
o< Jl - fl ~ °Sl- s.,. ~ ( S A1ci '?._ f ol 7 ci °b- (>)_)'L- ~ o {J2 - )@.sl
2
0
~
JS:~ -{-f() J2- ST ~ (3 !9(-? o_ f orsO~ .~o6~ )n-<tcy 0 (Jl-S"~,3<;
~ 0 ·- /1 J2. - I -f- j)_1_n - 2.1 0 ( 52. - S )
~' '$])
0(
•:-
::
0
0
-
5
0
r
Y.
4
/\ .,, _A o52_ - '2._ ft ( ,\ o"!- 6' "') __,'l,. - ST Q (.JL - 't)
(6 . 5 V)
2
f lJL- ':-ASl3f "°-.- ~ (3fj""'S""-10SL'.:_ g'-6 0) 52- <r., O(fl-~ ':.x ~ : : X + o (.52. -5)
0 s&)
•
~
·o=
1-0
-
·-
~- 31-)
Adding equation
~11r _
T~
v~~
111'
(3
SS) t
Drfi~I -
.
~ ri
2 X ( J.:. <; 9 )
TLo
et-
~
sn =-
~), ('(\ -
T
'
•
1
Jp
I
,,~
T~
·rherefore
&(59)
-
(~ · Jo)
By equation
v ·= v -0
±y ; Jl
x -(5ic:J ;-
;
.: x(,
L
0
0 ( ..)'l
-
i-
0 ( J2
- 5)
{ 2 .1) .
By the orthor:ormali ty relc:tions
J iJ.4
-!( 1'
~ LW)
- 4J
- (S .. S ~ g· _sJ)
1
-.
By making the coordinate transformation
Ll (
--
II
I
x
'
waere
c3
..J
/
X
·: le
l,.o
( (;
--
I
--
I
·--
x
'
{,
'
T
c~
( u,
_ x· 30( I -r
XL)
3 ) _
c_;3
x ~o (
3
) y-
:. 0
still have the coordinate freedom
Xi..::.
j)L(xj)
1.ve may use this to reduce the leading term of
3~J, (~,)'; 3.; <-)
to
a conformally flat metric (c.f. rTewman and Unti), that is:
I
9
~i ~
-
~PP r1 (·si-~.- 2 Rn-<;),, d.51 -(,)
(t
J
lt J
•
v1here
JO
•
..._
-
7.
Ecuat ions
Non-r~d ial
-
By comparing coefficients of the various •uo :Jer.s o.f
1
·{ 3
in the non-radi a l equations of
JL
, relations between
the integ r ation constants of the radial equations may be
obtained:
3. 23
In equat ion
- l4 A ( 0
Q
the ref ore
0
.,.
the term in
0
0
-
-
J
-
therefore by
( f-· I)
0 (5J.-I)
u ·-:.
In equation
J_ A
4
~
6
(
)
0
•
Jl-' is
(3 , Su)
\/o
, the constant term is
therefore
p - p . ~ ((
)
By the
n~:i
0
·-
-
-l.
0 )
I
t er:m
p - ~i -
0
.....
ther.'efore
~
0
--
0
-0
~
I
(:;.. · ~).
if
By
makins a spatial rotation of the tetrad
(\/\ I
\4/e
make
1'
S
{J--
I
':::
rea l.
e.
- L cp
f'-
fVl
\~e take 5::
{
{
i l I --rt )(
stereo-.:;ru.,nic project factor for a
B;/
f
2- sphere .
( 1.
the Jri_- lt- term in equation
z.)
-t ;(' j, , the
~ 8')
~
\I ~ e. u.
~<> "-i VSev.
r}.
· where
By
the
0
·::.
-.J-
\]
JJ--~
')
:
_cl_
(.,
I
'\
cJX
X;)
l(
(5. r2-. I)
in
·iJ er·m
f- . 5)
1 e ~u ( S V \! S r S V 'i):7S')
/vl
I
0
·:;.
fl ~ - c:; 1.. v v
2
l/
·::
By
-
-A
<)_ [.
i
+
Oj
~ 0
" - 'J. ;v-- lo s e,Q,U.
e. Q ~ J
Lt (·~ · '5 f ) ~ . 60) (s. G t)
2
0
J
-
cn-
1)
00
•
i.-1nere
t3y m<.-i.i:eing a c oordinate transformation*
cy (
i.J..
--
(1-. <t)
cLv. I/
1-1
'r -
•
•
t1
therefore
I
v
I
:: o
I0
flL.
:::
A2
-
By tne
Jl.
term in
-
lo
0
- '-I
e
'2.. c.....
-f.
0 '
3
'
(J . 2..b)
2
'
C}.U..
f~ tnerefore
p
~
o
J '
-
1- A~
o
-s- f + b
i
l)
2
- lo ,~ ~
2u... :: 0
-.,,,
I
-t-
7- Li..
'.
C(x~ e, 3
-
~This transf ormat i on does not upset the boundary conditions
on the hypersurface
(z.1o)
Jl - Y
.JY tue
A
S2_-<.f
3 ( (3· ~I
=
0
By the
(?. 2 5)
term in
the
JL-2
0
../. \.A.)
o
.-
~
)
By the
~
'-A
;
':2...
'3
the r·ef ore
2 6°fjS) T K(x. c) e, Q.t.
-
I
"T
~ w
C :;.
o
C+- I 5)
(3.20)
term in
S 17
o
vr
the ref ore
s VG
0
t.,.
..J1,-~
(t-~
Llf-c
--
g,_ '-'-<
(T- I I)
Q. S-o)
,I I
. . t-J"
)
e '-- V ~ - ~ e.'-'- SV(o~ ."'-6: "')
0
term in
(....J
6' 0
Q.22)
term in
~ -~ -- ( 6 ~' - cS
By
-
·-
e
k -;. 0
t.L
SV- ~
o
J
o r7 r
.....
:.:. - CJ v ,) e
CJ-
IC/)
(·::;- . IS)
(t= .ff;_)
Using
Then by
Using this in
B~r
v
(5 ' s / ). 6. ~ 1)
.
J
f}.60) d -b t)
·-- () (.FL - :r)
101
1b0
J L-._
d
d~
--
(S2 -9)
0
_J_ /\
o (SL -:r)
_,
-~
c) v--
d
---
J~
0
l .? . I~)
( 5L -:;)
Thercf ore if the boundary conditions
{ S"".
i- Y)
hold on
one null hypersur!ace , they will hold on succeedinG hypersurfaces .
By
(JL-2_)
0
~'he
"peelin~
off" beho..viour is therefore:
CJ (
-
-
lf 1
f~
•
'f
.
'f
o
L
t
(j ( i
--
0
--
-1)
-1)
(r
-1
]_.
--
o(r - i)
O
·2
{r - 2 )
f.
21)
As mentioned before , this asymptotic behaviour is
r
of che zero of
and \·;ill hold for c..ny affine .)&rarneter r
rem~ining
To perform the
for definiteness :
J1- 1-
.. 9
••
inde~en~e~t
. <:>
01
' y: 11··/-
rfi i
I o;
..,..
•
intesrations we will &ssume
JL. -8·r 0(5l - 9)·
_
t~ f)_.1(.,. o(JL-'1)
T
' fi52 ·3 T !\ fo-1-
(Jl -j,')
-r Q
•
4-
-.
SfJ. J1
~0
-5'
T
.51-9
o( _fL-10
T
too
'l'hen :
.,.,._ [. fl
' _1r
v( 5
~
-
1
--o e
•
Q;...
-
I
<6'
0
1.
J. <..l.i..)
e
.
.L [ n ~( 3 ·7- . o0 iu
+ 3 . 11 - '? er e, -r
6
t
<> (
S Ro "'r
f' i
o
0
Q.,
~Ll
r 6'
0
-
5·0~-cl n - b
v
J ':2
..
(}
r-t
rl. e.,
\.(
u.)· - '-f
,~ 0
0 0
(J A5°-r
t-.2S)
0(52.-c;-)
--
0 "Jl,-'f__
= -j (~:,"I
•
f ~ -)- J2. -:;
(2 fJ:S
0
..,.
1(:) JL -" 7 o (52-6' ( f-. l y)
·f ," )51->.,
a(JL - ",fps
])_)
~
1( ;, ., if/)·
r
.
·3
J2 - r
-
·
-r
- :£ - (-}
fl, -
:2
/l S2
-i -
)
J.
8'
-
1.
(~ .o
1
i"
e :< u.) Sl -
2 ..,_
.If
5(
1
~
y
..,.
-
'-J JI
0( Jl - )
·-
0°
.
~
)Jl- -
~r
)1
~·
x S2 - 'f
_I
(t-. 2 i-)
1 <..(
q
.
i
~ (ffJ "_ t'7") 5l - if
- 51
-<;)
0(52
1
(! 2 0)
o(52 _,, u, 52)
5
~ f} '2, 52 - ':_ f-} 3J)_
lt
u
I
~
2 e :i.u)
J)_ - 3
<
-+ ?. 8)
•
/}6)~
2.
-
,._
-
v ~ -j (6~-6°)e~PS-r
0 ( J1 -5)
7- ]0)
fl S'Sl-;2- -r Ae ~.V s SL - ~
* [ P oie 17S' ( J "" -f e "') -r -e. ~(-VSJiS c Jfl - 't
±~\A
·--
'+
v.
-
•
•
•
-
e ~ s 5L - -~
- e {,_$ 0-
0
i [ f) 2 e_
(52 ->)
(} e ~S' 52 -")T
_/12
-<; "r (')
5 (.f
~
r
VL
s
(f- i c:)
.
{n,-z_ t(Cf c
~'io)SL-2.,,o(JL-S)
7
1
_c,, (As -
x·s --
.
b
~
v
CJ
--
-
-::o
- 6
""(
=>
-=..
2..
-r-
-r -- -7.
,/i
.
I
I
I
r:;:.o
u
i--
OJ
O;
o
.._)c.
)ii
2
) fI
-f- /)"- ( -
[J'"J..(<3~ ~
52-sT
o
lf
·-
-
(f 0-r-
6
]_
2
I
'f
i
6
-
)/
x ( 6~~ - 6 °) 2-
-
~
0 ,
55)
-G
\\
S2_ ~· J c
{
._
·-
a=o)I
-
-c_-,
C,_ ·3
"f
2_
S&
f"
-o
6~
JI
o .\
)f I
/
--
0
6--::
(1-
{.?.?rr)
I
-
6
<l - S"
__; G ·f-
n-2
o.
-
T _j_
() .
.
(.)
T
15
·-
...:--
~)
/II
v(s (~, 0
6")1;_)
-
s==
-e"'-(06°) \75-2'"'S7 (6°- cf ) I S2-"
;T
I
<f lie
2 ( <i
·LI- 6v--s-,_
S
'V
(<Y
c:_ ,;, 6 c: J
&'
)} ~
0
0
- -
JI
I
_.. ,
0
Vv
-
'+
-
0
c-
)
)I
i
Cfo~,_ n_-'., o(JL-
6
(t-.]6)
W~e 9-Q,:
()
.
~
.
).
-
::
I
(,.0
J
Q, 'r
t
0
J
J1
·-r
•
~
.
--:r
-
•
~
l
--
~
sv
lA..
...
s
I
l
•
+~: +
/5
oo) .
'+
ol
-
+y
I
bi
· 1fri
1 I)/
·s,+0,l
;::
-
•
'i1 hU3
derivative Of
t; 11G
r
Ii
'
ar1d not on the radiation field .
:·epenQS
On itself
0l1ly
It therefore re 1Jresents a
type of disturbance unconnected with radiation.
If it i s
zero on Ol1e hypersurf ace, it 1vill remain
In this case
zero.
it is possible to conti!'l.ue the ex· )ansions of all q_uanti tie~·
in neg; a tive po1,.1ers of
J2
without any log terms
a~pea~in3 .
.... .
·11he mevric has the form:
~
li
'2.4
~ ~[
CJ.'
3 G}
·--
i$
'J
-... ~
i'f
-
CJ )
-- J1?.. r- 0 (52 -<-)
-- 0 ( ~ -y)
-- ·- 2 f ~). g J Jl - r ·tI
/
t-
The asym)totic group ts the
~roup
I
3
'2..
- I
(
;
2 R ?'2. S(JA-~
of cooruLnate
c;(~?_- 6)
transfor~a~iors
that leave the form of the metric and of the boundary conditions
unchan~ed .
It can be derived most simply by consnering the
corresnondin~
infinitesimal transformations:
(f, I)
'l'hen
Kd.) (!c 2)
-
f /J
(?. 5)
-
f
0
'fo I
obt~in
-
~
oo
rA
Q
To
(
::;.
-
I
~ (..
'
S'3 ~J
~) 11
f.- i o<.
'?._ [_
(~. y)
• •
J I
CX
r ~OU
-
f3
--
0 (yt_,-b)
.........
0
(5L-7)
-
0
{-5G-1)
-
-
-
01
I
:S
0(51_-<-)
(j (52,-4)
2.2
~~
R.
( o
J'.J'~::
=
f :s3 .:
-~3
'
~
k
~
I
Ji
D
I\ So(
r. o<
1l
JJ I
~)f.
n
f(J ~
.i o(
f( / (
r S-)
the asymptotic zroup we deuand
:(
..._
--
--
_.
·-
-
'95
I 't
.:::. 0
~. &'
o (SL-,
dy
11 '
--
<6.
t-)
k
'""2..
:::.
/( '2
·f-
) 2.
,, k'.::.
(( ( y q
) ;
.J
0
JJ
I<
)J..
c\
I
k
•
I::
1\0/ (l(~)
j
"3 :::- f( OS (v.., 'X-
(U)
J
l.) -
( ?. 11)
\
l
'
1. 2)
I
w
'""->)
00 . I )
(~ 10)
re. t.2)
.
I
0
1"3) imply that K. '- is an
)( '3 -t- t,_ 'X.1
. ·rhis is a consec;uence
and&.
anal:-ftic fu11ction of
if t
t1e
f a~t that 1t1e h;.,~"1e
recluceu. the leadi11g term of ~
form .
·~huu
the only allowed
.,
v'
to a
'.J
co~for~~llv
., fl~t
transfor~ations
of
'
)(
/,
·'r""v
c;..
+-'~
e
utJ...
conformal transformat i ons of the form :
--
c (
x 3 + L xlf-)"I cl
{).cl - be
':!hen lic"te
six param8ters
c1.etermined. by (<if. I<+)
Q..,
..{ 1
--
b c. cl
,
I
i
•
ure c;iver:
is also uniquely determined .
~nuci
the asymptotic ;roup is isomorohic to the conformal sroup in
ciachs(~)
two uimensions .
to the
~01006eneous
has shown that this is isomorphic
Lorentz
It is
~roup .
~1lso
11o;vevo r· iso1i1orpf1 ic
to the group of motions of a 3- space of constant
ne~ative
curvature which is the group of the unperturbed rtobertson.alker space .
sroup of the
presence of
Thus the asymptotic
unc~i3turbed
r~di~tion .
space .
~roup
is the
It is not
san1e
enJ.ar~ed
&s th·;
by
This is interesting becausA in
the
·i-.
i:; '" ...
c
case of e;rc..--ri tational radiation in en1pty, asyr.:ptotically -·ic:.. t
space, it turns out that tha
a8ympto~ic
group cortains net
only the '1 O dimer1sional ir1homoge11eous Lorentz __;rol1p , the z:;roup
of
11
motions of flat space , but al8o infinite dimensional
supertrar1slations 11 •
sugges~ed
that
thcs~
.
might have some physical significance in
particle physics . The &bove result would seem
supertransl~tiona
~lementary
It has been
to
ino.i~c.-te
.
·t;hat this is nrob<..,bl ...·' not the cc..s e
sir~ce
our
ur1 i ve:rse is ,:...ln1os t certai:·: ly not asyt'lptoticu.lly flat though
it muy bG asymptotically
9.
Robertson - v~allcer .
v/hat an ob .. erver would 1neasure
of an obs erver
The velocity vector
movinr~
~
with
du s -c 1:1ill be :
-
Jl~i.,..
..
v).
--
v
v
--
·~
--
(9~)
4'
, the projection of the ;,..:ave v 0ctor
f\"'\
L
~
in t;he
observer's rest - space ( the anparent direction of the wave ·
1:1ill be :
2
q,
.
-
s - vv
/l"'I
M
M
\
•
'
c~
I
-
J1.
-- -I
)..
t;2..
-2
T
.,.
I
o(J!--~)
O(JL-~)
0 ( Sl. -- ~)
--
c:
L
.3
--
(}
l-
lf
'l'he oboervcr' s
oi~thonormal
tetrad may be
com-;il~t~d
t\•IO
C
and
space - like unit vectors
b7
M
o
4)
O{J2 -z)
<:; -.
I
~
'.l
5'J
3 -\./
-
Cu~ -
cI -
0 { ___,'2_ - Q.)
--
t
::i..
{
.,
- J5.
~
·.j 2
.I
l
.
ff
3
I
(J..
-
f;
•
:e v:ri te
0 ( ._,)<]_ ·- ~)
t
-
·-
u
L'
Ji
(~z. _5)
e.""' --
13-:r m9c:..surin._; tb.e
rel~tive
o.ccelerc.tions of neirjhbouring dust
p<...rticles , the observer ma.y determine the 'electric '
cou1ponent s of the gravitational v1ave ~
p
E.
a..b
~
-c°'-f bi
V
v
'1..
In the observer's tetrad t!lis has corrt1)0nents
-
,,,,,.
E --
,.,. . 0
0
0
0
r
CJ
I
l
I
0
0
'
0
C)
0
0
I
I
0
o(-52-4)
-
-;
•
-1
I
t
0
.l-(
0
0
O(J2-~).-
0
0 (Jl-5)
..
0
-
0 O{Jl-6)
0
•
--.L
I
0
-
,....
•
-;-
0
-J
~
•
( J1_ -'r)
0
•
0
·-r
a
C>
0
l..
I i;_ \
-I
0
0
( _1. r
).
,\
J
'"hi
J.
.;.;
' ......1ould be cor.ipared to the behaviour for asymptotically
space for \vhich
fl~.t
- --
t
-0
0
0
.
"-t-
0
,....
o(r--')
0
0
0
CJ
0
C)
I
0 - l
0
l
0
l
-
-CJ
[
I
0
0
0
l
l
0
0
·- l
l
eJ (r-2)
.
-r
-
I -1
0
0
0
o(r-;)
o(r -1)_1- 1
0
C>
i
-~i
~-~{
-~)
() v ?"
0"'
o-i
0
!q:;)
\
t
l
•
•
• ondi ,
1.
. G. J . v·n
. . .s
eh...
•
' . 1: •
4.
• .L .
d
/.
roe . oy . Soc . A. g·-··
roe . oJ . oc •• 10
-
ro...,c
1....
• •J .
ti
•
en.re
l.
.
•
i ver""i
91
J
roe . •oy • oc •
yo .
v•
1 19c,2
,9 2
10~
1)62
)
yo . ~
t1 .
• 0
•
•
e
~ur~
• • . 11etzncr
(.;rid
2.
~c
1~
2
1
1 9 'I
~1
1
J'.j'
Sin,:;ulari ties
If t;he Einstein equations ·1·Ii thout cosrnological constarJ.t
are satisfied , a Robertson- VJalker :::nodel can ' bounce ' or avoid
a singuli:i.ri ty only if the pressure is less than :::iinus onethird tne density .
This is clearly not a property ~ossessed
by normal matter though it might be possessed by a field of
ncGative energy density like the 'C ' field .
nO\vever there
.
lS
a 5rave qua11turn- mechanical d.ifficul ty associated \·Ji tn the
existence of
ne~ative
ener3y density, for there would be
nothin 0 to prevent the creation , in a given volume of space tin1e, of an infinite numb ex' of quanta of ·Lhe negative er1ersy
fiGld ~nd a corresponding infinity of particles of posi~ive
enere;y .
If \ve therefore exclu·de such fields , all Robertson-
\val \.er moC1el:.>
rnus·t be of the
1
big- bang ' type, that is the·;,r
.
11C1.ve a sin0 uiurity in the past and
as \·Tell .
m~ybe
one in the future
1 that the occurrence o=
It has been sug~ested
these si11Gt1lari ties is a consequence of the high degree of
symmetry of the Robertson- Walker models which restricts the
expansion and contraction so that they are purely radial &nd
t;hai; inore realist i c models \.,r i th fe111er or no exact sy1n1netries
v1ould not have a singularity .
rhis chapter \•Iill be devoted
1
to an exarnination of this question an:i it :1ill be sho\-1n tht.t
1
provided certain physically reasonable conditions hold, a~y
t
model must 11ave a singt:lari ty, th;.- t is, it cannot be a
2
5eodesically complete c~ , piecewise c manifold .
'
2. 'l'he
~··un<lamer1tal
Sou at ion
9
'.rhe expansion
~ongruencc
D.
v .
=-
1
of a time-like ;eodesic
6--
vii th unit tangent vector
VQ..
of Jhapter 2 :
obeys equation (7)
~
eI°' V°' =
. v ct ·v b
CAb
{t )
I
will be said to be a bingular point on a Geodesic
A uoint;
~
((
of a time - like geodesic coni:;;ruence if Of or ·the congruence
is infir1i te on
y'
at
conjur;ate to a point
singular point on (
p
fr
•
f
A point ~ \vill be said to be
along a geodesic
·~he
•
A noint
i
will be said to conju;&tc
tf3
if it is a sing~lar ·1oint
w3
. .tin altern8.ti-:re
conbruence of geodesic normals to 11
a.escription
let
if it is a
of the con0ruence of all time - like
to a sp<.,ce - like hypersurfac&,
of
t
of conjugat;e points may be 3iven as follo :Js:
1
K~ be a vector connec·bin3 points correspondins distances
alone; t'v-:o neighbouring 3eodesics in a
conGruence .-1ith u11.i-t;
1
tan...;ent vector
V°:
K°'-
Then
is ' drag,;ed ' along by the
congruence, t.1at is
J l\
"-
,
J) KC\.
'
{!1)
(/.I
R°" .
--
I
I
kb
I
J)2 K cz
\
v b'
--
JJs
f
-- C)
<>..
vbvc1~ct
(s)
'bccL
J) >2.
~
Introducing &n orthonorraal tetrad
u~on··
(J
'V
Cl.
eu
v1itn
oL 'l J\
s
ol.
0..
-
vo.
-
I\
(4)
n..
CJ,
v.1~1ere
I'\..
(\..
-...
0..
e,
1 ..
e
b
R
C\., Q,
M
fVV
solution of (!.t)
0....
~
vie have
Q_
(If\
parallelly transpo.rted
fl1
t'v
M
e
bcl
vc vd.
i'Iill be called a Jacobi field .
clet...rly eight independent solutions.
Since
Va..
·.rhere :ire
and
solutions, ~c11c other six indeuena.ent solutions of
ta~en ortho~onal
a 3eodcsic
't
to
•
i1hen
1
'f...
a::·e
(f/J m<--y ·oe
p alon~
if , and only if, there is a Jacooi fielQ alon§
1.-JJ..lich vanishes at
p
and
~
.
{
:.i1his UJ.ay be bho111n as f ollo'.lS:
ti1e J·a.cobi fields which va11ish at
e;encratin~
is conjugate to
l{:.,;o..
f
may be regarded. ss
i1eighb ourint; geodesics in the irrota·bional congrl1cnce
of all ti1ne - lilce ge odesics through
f .
Therefore they obey
•
v
-
·-
mo.y be 'N'ritten
~L'h ey
I\.
d f{
--
cL A
.. nere
~
M I'\
cl
., ear
p , A
y
A
-
~
g
{s)
will be oositive
definite .
..
lh ('\
p
a J acobi field vanishing at,
J3ut
A Cs)
m ¥\.
'11heref ore
_d..
(9 -d<l,(A) cLs
/\\I'\
cL 'l
(-)
Ml\
ol ~ 'l.
Hence ()_
-Q
Mr
1.S
infinite
if, and only
if,~(!},}"/-J); 0
(dd(A>)
f\1 f\
A
h.
•
1.S
ols
.
q,
r
ci (~}
'.:'l1e.re fore
and
·Ihe:.:·e \:ill "'.:>e
:!{\ cLs I)
exp (sf
.1
and
f
cl~
\'I' here
finite
anu- only wl1ere
cld(!)
...- 0
Thus the tv10 definitions of conju.:;ate points o.re equivale11-c .
'.l'his also shov1s that singular j)Oints of con3ruences a:r:·e
points \vhere neighbouring geodesics intersect.
•
~or
null geodesic congruences with ourallelly
a._
l
tangent vector
~ran~ported
f
vie may define the conver;ence
as
in Chapter
1~ e
define a singular point of a null Geodesic congruence as
one where
e
is infinite .
'.L't1e condition tho.t the pressure
•
l. ::,)
greater ·than
one-t11ird t'!:L'a! density may be s·tated more c;e11ero.lly as
co:1dition (a) .
(a;
observer vii tl1
,...
~
t;" /
0
'
f >
4- veloci ty
I'\
i...J
j_
:;.
C>....
,
T
for a.ny•
)
v:l1ere
G : :.
-
,,~,"\b
J <A.~ (,.,,)
~
is the enorcy d.ensity in the rest- frame of thA observer
T :::
-A-
l
~nd
is the rest - mass d.ensity .
Condition (a) will be satisfied by a perfect fluid with
density~>
It
o and pressure
any time-lil<:e or null
v ·~ ctor
v
c,vb
implies J\~
70 for
0
Therefore by equations
(1) &nd (5) any time - like or null irrotational beodesic
congruence must have a singular point on each ceodesic withi~
c. finite affine distance .
~n
I
Obviousl;y· if tl1e -10·.-!-lines f or;il
irrotational geodesic con3ruence, there will be a ohysical
sinsularity at the singular points of the congruence w~ere
the density and hence the curvature are infinite .
~his
will
be the case if the universe is filled with non- rot&tin5 dust
?
~,
llO'•Jever, if the flo\·1- lines are not seodesic (ie . 11on- vanisnins
pressure ~radient) or are rotating, equation (1) cannot be
applied directly .
3
•
2, opu:tially :1omo;;eneous . .i.nisotro.)ic 0ni verses
~he
ar1J.
dobertson - .~alker
b~atial~y no;~o;eneous
isotropic, that is, ·they have a si): para:1:eter sroup of
surfac8 . If v!e redl1ce tl1e
considering models tl1;;;t arc spati::.-lly ~!omoger: eot1s
motions trunsitive on a
f
models are
symmetry ib,y
su~.ce -lik e
but; anicotrouic (that '.is , t11ey hc....ve a ·thr·ee parameter grou_p
of mol,ions transitive on a space-like
tne
mbt~er
hypers11rfc~ce)
flow may have rotE1tion , acceleration
tnen
~n~
there would seem to be the possibility of non- singul&r
~od.els . .L . Shepley4 has in"'restit;atod. one particulo.r
~hus
l1omoc;e:neous model containing
rotatin[~
there is alv1ays a singularity.
dust and has shoi,1n thc.t
Here a seneral result will be
proved .
·.J.'hei"e
r.uus~c
be a ::;ingulari ty in ev ery model \·1l1ich satisfies
condition (a) and ,
(b) there exists a
q. -
of
mo~ions
universal coverin~ space~, Y;}3
on the space or on
v1hich is transitive on at;
..:>ee section 5
le~st
one space- like surface but space - time is not s·tationary,
(c) the energy- momentum tensor is that of a perfect fluid,
-flow - lines
•
~nd
is uniquely de fined as the time-like eigen-
vector of the ki cci tensor.
I
•
R, tl1e curvature scalar 111ust be co11st<:l.nt
,.,
...
o.
On
like sur i'ace of tra11si tivi ty ff 3 of the s roup .
f
--1~"e
bl_
c...tv .
-
R·.
vc-..
·.::here fore
111u st be in the direc t i on of the unit time-like normal
3
to f/
•
/ 0...
~
\-J11ere
e CR jU- J
hen
1
'.L
.
lS
an indicator - +1 if
- - 1 if
(( I
•
lS p:::.st
("..
J
R. c..
•
lS
J
directec.
.
1u-cure C.irecteC:
~
·\{a.: b] =0.
Tl1us V:"" is a cone;ruence of ge odesic irrotational timo-lilcr-;
v ect; ors .
.i3y condition ea) ,
-~her· efor e
the
con~ ruence
R°"b va... v h > o
•
sin~t1l<.tr
must have a
point on e <::Lc'-"
seodesic ( by equation 1) either in the future or in the
I
~ast .
£urther , by the homogeneity, the distance al ong euch 3e oiesic
1·ro1n l1 J to the singular poi11t must be the same for each ,-;eo«ce::.:ic .
~nus
if tne surfaces of
transitivi~y
r emain s·Jace - like,
2
must degener E·t e into , at the most , a 2 - surfaco c 111hich
be uniquely ciefined •
Let M be the subset of the
of the matt er which intersect
empty subset of
3
H
group transitive on
c'2- .
intersected by
1f , /. .
Let
fY .
L
flo\·1-li~;..es
be 1..11e 21on-
Since there is a
must be [--/ 3 itself .
Thus all t L.• e
flow-lines t trough
3
H
must intersect the 2 - surfuce
·bhe denoity •:: ill be infini"'ce tl1ere
r>hJJical sinGularity .
CL"1..:.
Alternatively if
c2 .
Thus
-C;h0re i::ill be a
~he
surfaces of
transi ti vi ty do not remain space-like, there must be <::.t
least one .::>urface \vhich is null - call this
P..~
J
10
s3 . }~t s3, F -
O,
•
( if~.,0.. is zero, vte can talce any other sca.J....:..r
~) olynorr:iul
in the curvature tensor and its covari.::nt d.e1· ivatives .
·.!. hey cunnot all be zero if s9ace - time is not station:.:::-y) .
..e
..,.
introduce a geodesic irrotational null coigruence ons? t:ith
t an00nG
·
vec~or
Le-.
111here ['
ti1e.L·e will be a singular
"
~ .
: ()...
poin~c
"hen by e ci ua+i·on
v
1
~
;, /~),
.
-·
.?
of each nul L geodesic in ;:_;.
vii thi11 i'i11i tG t..ffine distance si ther in ille future or in t:~e
past . ~he 2- surface of these sincular points will be uni~1;ely
defined .
·The sa1ne arr;umen·c used before cho;...,rs tha·t tl1e dc:r1si ty
becomes ini'ini te and there is a physical singulaI·i ty . In fact
US
83 is a surface of
sin~ul~:c
homo~eneity,
the whole of
s3
will bE
and it is not 1neaningfu1.· ~o cal.!. it null or to
distin~uish t~is ccse f r om the case ~here the surfaces of
tr~nsitivity
remain space-like.
:J;he conditinns (a), (b) , (c) may be v1eakened in ti·.Jo ·,·1a:.rs .
Condition (b) tnat there is a group of motions throu~hout
space - Lime m~y be replaced by (b/) and (d) .
•
I
3 .
Thei"B is a space - liice hyper surf ace !1
ir1
(b )
~re three independent vector fields X.~
ix.°' 9
~ 01
j 1Z
'
xl;
-·-
hcd. e
f?3 one homoGeneous space section .
bC..
(d)
'
~,
"".:1ere
such thlit
.
. ·ihat is , therP-
'I'here exist equations of stat;o such that the 1.Jaucl1y
develoument of
-
H1
Then succeeding
are
on
, .
~·1i.1icr1
ho.ao~eneous
is determinate .
so~ce - like
surfaces of constant
and. much the same proof can be siven t::ict vu.ere
I
are no non-sin~ul~r models satisiying (a) , (b) , (c) , (d) .
Th0 only property of perf cct fluids that has been
u~ed
i11 the a ·o ove nroof is ·that they have well defined f loi:1-li1:cs
intersection of v-1hich implies a physical .=ingulcL.Ci ty .
howe v er ,
·t~is
proper·ty will be possessed by a much ::ore
clcss of fluids .
F or t hese ,
\•Te
~e~ ;: · 2.
define the fl ov1 vector as
time - like eige11vect or (assumed unique) of the
tenaor .
Obv: Jusly,
1..:1.. :
energy-n~omentu:u
•.:.•hen \-Je can reol ace condition (c) on tl1e nature of
tl1e matter b;y· the mu c h \'lea..."l{er condition ( e).
(e)
<..-\
If the model is singularity- free, the flow - lines -f Q">"•-:""
-
.L. - -·
s;nooth time -like congruence \-Jith no s in~ulo.r poi11ta -.:iLh
a line t.:1--ough e a ci:l point o f space-time.
Conci..i tion ( e) v1ill be satisf:Bd automat ically if conditions
(a) and (c) are .
I
'l'his proof rests strongly on the ;:ssumption of
homogeneity 11hich is clearly not satisfied by the physico..l
1
universe
loc ~lly
however it
~·/Ollld
though it may hold on a large enough acule .
seem to indicate that la.r.'ge scale e.f'fects
like ro·t;at ion cannot preve11t the singularity.
It is of interest to
ex~mine
the nature of the
singul ~rity
i n the nomogeneous anisotropic models since thJ s is more
likely to be representative of tae general case than that of
~he iso~ropic
models .
It seems that in general the collapse
.!ill be in one direction., 5
that is, the uni verse \-:ill co::..ls.pse
down to a 2- surface . Near the singularity, the volume will be
proportional to the time from the singularity irre spective of
the precise
nature of the matter .
It also ap pears that the
nature of the particle horizon is different .
a particle horizon in every direction except
There will be
th~t
in which
the collapse is taking place.
4.
din~ularities
in Inho!o3eneous
~odels
~ifshitz and Khalatnikov6 claim to have proved that ~
general solution of the field equations \vill not have a
sinsularity.
Their method is to contract a solution with a
singul&rity which they claim is representat ive of the
general solution with a singularity, and then show that it
has one fewer arbitrary function than a fully general solution .
•
CleurLy tt1eir whole proof rests on whether their solutiun
is fully representative and of "cl1at ·they :_;ive no proof .
Indeed it would seem that it is not representative since it
involves collapse in two directions to a 1- surface
\~hereas
in general one woul d expect collapse in one direction to a
In fact their claim has been proved false by
2- surface.
Penrose? for the case of a collapsing star using the notion
of a ' closed trapped surface ' •
...~
similar method v1ill be
used to prove the occurrence of singularities in 'open '
universe models .
5. ' Onen '
~nd
' Closed ' Models
The method used by Penrose to prove the occurrence
of a physical s i ngulari ty depends on the existence of a
non- compact Cauchy surface .
A Cauchy surface will be taken
to mean a complete , connected space-like surface that
intersects every time- like and null line once and once only .
~ossess
exalliples of those
8
that do not include the plane - i,.1ave metrics , the Godel model ,·9
Not all spaces
and N. U. T . s pace:o
significct.i."1ce .
a Cauchy surface :
Howev er none of these have any p'iysic.:l
Indeed it v1ould seem re a sonable to deznand of
•
any physically reali st i c model that it possess a Gauchy
surf ace .
If the Cau chy surf ace is compact , tl::1e model is
commonly said to be ' closed ';
if non- compact , it is said to
\
The surfz.ces , t = constant, in the . 1.obertson.
·to be ' open ' •
•
ialker solutions for normal matter a.re exanrples of Cauchy
1
surface:..> .
If K = - 1 , they have negative curv<J:ture and it is
f'requently stated that they are non- compact; .
necessarily so :
they are compact .
:l 'his is not
topolo~ies
the r e exist possible
for which
Howev er , tt1e f ollo\vin;s statements may be
made about the topology of the surfaces t
=
constant .
If the curvature is negative , K = - 1 , the universal
:s3 .
covering space is non- compact and is di ff eo111orpl1ic to
~ny
other topology can be obtained by
points .
identific~tion
of
'l'hus any other topol ogy \·.;ill not be simpl;>r connected.
and , if compact , must have elements of infinite order in tJ:1e
Ij
fundamental t;roup . l<"u r t her if compact , they can r1ave no
.
12
group o f mo ions
.
t
If tl1e curvature is zeI'O , I\ = 0 , ·the universal coverint;;
space is ..£ 3 •
1
· h teen possi· bl e t opo 1 0 0- ies
·
'rhe r e a r e eig
. 3
compact they have a G
3
of motions and Betti numbers, 3
1
I.L.co
- 3,
B2 = 3 . 12
If the curvature is positive , K
l
coverin~ space is
= +1 ,
the universal
s3 .
Thus all topologies are compact .
12
Betti numbers are all zero.
·.i'he
Since a singulari ty in the universal covering space
implles a singularity in the space covered , Penrose's mGthod
is applicable not only to spaces that have a non- compact Cauchy
.
I
l
I
surface but also to spaces whose universal covcrins sDace
has a non-co:-:-ipact Cauchy surface .
l
hus it is applic::;.ble to
1 1
.i.
models which , at the present time , are homogeneous and isotropic on a large scale with surfaces of approximate homogeneity which have negative or zero curvuture .
6.
1
·1
he Closed
'11rapped
Surf&.ce
Let r3 be a 3- ball of coordinate radius r in a 3-surf ~ce
a3
(t = const . ) in a
Robertson-~Jalker metric with K - O or - 1 .
Let qa be 1Jhe outward directed unit normal to T2 , the boundary
of 'i 3 , in H3 and let Va be ·the past directed u!li t normal to
1
H3 .
Consider the outgoi ng family of null ceodesics which
intersect T2 orthogonally . At T~
--
be:
-, ( vo...
l
.
..
~ b
f ,
tl1eir conver~ence will
q()._ h 1( ~ ~ ~ b +I
/; ~ {: b)
I
\.,rhere fO'- to.. are unit snace - like vectors in H3 orthot;ons.l
,
to q a and to each other ,
i;teref ore
If JV-. 1 0
f
and 1(
...
=0
'1
R
(Jj_.
'3
-
-K
- -r I ;- kt
I
Q. -]
or - 1 , by taking r large enough, vre raa.y
make1 negu.tive at T2 • TheI'efore , in the lan3uage of Penrose,
T2 is a closed trapped sur face .
.;mother way o.f s e,ei ng this is t o consider the diagram
..
in which the flow- lines are drawn at their proper
distance from an ooserver .
at t
'l'hey all meet in tl1e
sp~tial
singulc::.rit:~i
J . lf the past light cone of the observer is drawn
=
on ·tais
dia~ram,
).
it initially diverges from his world- line
It reaches a raaximurn pi"oper radius ( (:> .:. 0
and then converges again to the sine;ulc.ri ty (
f > ()
)
\)
.L he
1 1
•
intersection of the convergin3 li 0 ht cone and the surf uce 3 3
gives a closed trapped surface T2 . If the red- shift of ~he
~uasi-stellar
poi11t
e
~
0
3C9 is cosmological then it will be beyond
if
\·Te
~he
are living in a Robertson- .ialker type
universe with normal matter .
However, tae assumptions of
homogeneity and isotropy in the large seem to hold out to the
dis·tance of 309 .
I
.rhus the.re is
1
our universe does in
fac~
tjOOd
reason eo believe th'-L
contain a closed trapped
sur~ece .
It should be pointed out that the possession of a closed
trapped surface is a large scale property that does not
Qe~end
.
..
on the exac-t; local metric . Thus a P.J.od.el thr t had locc..l irre0uJ..ari ties, rotation and shear but \vas similar on a larse scc:.le
at the
pi~esent
time to a Robei"tson- 'l"Jallcer model \-.rould have a
closed trapped surface .
Following Penrose it will be shown that space - time has
a singularity if there is a closed trapped surface and :
I•
.
(f)
E); O for any observer with velocity
(g)
there is .a global time orientation
the universal covering space has a non- compact
(h)
Ca~chy
surf ace H3 •
Assume space - time is singularity free .
I
Let F
L~
be the
3et of points to the past of H3 that cun be joined by a
2
smooth future directed time - like line to T or its interior
T3.
Let B3 be the boundary of F 4 •
Local cons:iierations
that ~3 - T3 is null where it is non- 8ingular and
shO\'!
•
lS
t;;enerated by ·the outgoing family of past u.irected. null geodesic
sec;ments v1hich have future end- point on T
2
and past end- point
·.!here or before a singular point of the null geodesic
2
> 0 and
, the convergence,
T
at
~ince
.
conbruence
f
si11ce
Ro..b L ~ lb ~ 0
by (f) , the conversence must
become infir1i te vii thin finite affine distance .
hus B3 - r.r3
1
'..L
will be compact being generated by a compact family of cown&ct
seg1ne11ts .
1ience B3 v1il l be compact .
7,
approximate
tnen as follows:
B7
Penrose' s metho·i is
arbitrarily clo~ely by a
smooth space - like surface and project B3 onto
normals to
~his
~his
surfuce .
H
3 by t~e
conti~uous
6ives a many- one
.
0ince B3 is compact , i·ts image B3¥
.
I
I
I
must be compact .
ma1ped to a point
'
Let ci(Q) be the number of points of B3
q
of I-13 .
ci((\))
.
will change only at the
7.
7
intersection of caustics of the normals \vi th H
by continuity
•
l\1oreover ,
can only change by an even number .
•
since this
~his
.
l S
the identi·ty D-i'1.d
is a contradiction, thus the assumption
is non- si11gular mu s t be false .
th~t sp~tce -ti1ne
al ternv..ti ve proceedure
hl1
which avoids t11e sliGt1tly questionable step of approximating
B3 by a space -like surface i s possible if we adopt condition
( e) on the nature of the mu.tter , tl1en B3 may be proj ~c·bed
continuously one - to-one onto H3 by the flow- lines .
~his
again l;eads to a contradiction since B3 is compact and H3
is not .
In tbe
abov~
n~c~ssary
proof it bras
to deQand that
~
~
£' b& a Cauchy surfac8 oth@rwis~·the w~o l e of B' might not
have been projeoted onto
li3.
We will define a somi-Jauchy
'iur:faoe (o,G,s) as a cottlplo·t;e connected epace- lika eurfc.ee
i;h.gs,t
intersect~
eve:I.'Y tii:t1@-lilce iiind null line at 1t1:os·t
O"i.1i0ie.
7.
~· s . C.s .I? will l/Q a c;'3ucti.y · sur:f'ace for. ;oi:Q.t• i'.tear it,
tim~-likQ
is:!:&i5 is, it· will intersect ove;;:-y
tl1rou,;:1 these points •
nOV.' OVOr,
anC. i1ull l:i.nie
further &>1.,,_y lihoro 1Tiay be
?e! iono for whieh it ic not a Gauohy curfaoe .
~et of uoints for which H3
I
Q3
'
. f it oxis·l; s,
l:
oallod "bhO Q!:U;Cl?:Y: liori z.on rel a ti 1re to H3,
I
J
•
f4
is a Cauchy su.rf-.i.c" e..nC.. lot
tac boundary of those points.
•"blrfaoo,
Let
CQ>
will
fft\!:18is 1':
b c- the
<Ql
-oe
8:
b ~
r.~M
.
:1/u;p·i;hO.PHl.O:Fw
if condition (f) boJ..de tho Rl:ill
e@@&@di&C
I '
at lo&ot one sxnsul &r
thftiet thoir
v iitUet
w~..:.n•
it
·s:i:m~e c::~&ltl:plo
l·~lec
ho
~~
-DOUnGo
- Au
1
poin~
tthien
· one ttirccu
-·
.... ion.
·
+
ave~G
~
•
ln
of s • J, s "ff·ith o. Cauchy l1origo n is a
~pGooe
· ou:efae c of eons·sa nt negati ve our..,la ture ooeple tel5•
\titl1in tac nu±l eone of the oeint in
1
"ia 1n:o·:; S .<:i
•
space.
r
11
B:ull oonc forao the tJauohy horiso :a.
~£ 88Bdi tioao (o)
and (g) hold, than a model 'iith a
- . s,
~mpaet o. ·G
- -EE 3 ----8
must h :\,..e
theepo
t
±ogy:
lt
4•4
1
t: .
~1100}'
there ''ere a rc5ion '/
V l.j throug h which taerc were
·8uppo~e
no flow lines iatcrs eoting S 3 , then ~I l the bounda ry of
be a t:hme- liko surfao e t;onera ted
•if",Y§t
inters ect H3 ,
t/li
__
~- -
1)y floi·:= lines ri·hioh '~ e
Procee ding •long thoso !lo .i:-linoi ;i in
1
·1;;1;;i,,e
-
d ireoti on of their intors oction with H.?, ¥JO :Q:Uat reac a 6.jji
.,
3
~4d-point of tho gon@r ator sine~
V does not inters ect l I•.
I
&:it..
the -0xiste nco of
~plios
thi~
onO.-p oint contra dicts (o)
a singu larity of tho i'lo·. x-.lin8 cone;ru once,
•
•
~inco ii;
'?hue
1/
,, ~
•
:t'lus b be em1> b;y and ewer;, point has a fle;1 line ti'lrous a i-V
interc oeting :a3,
0
I
b
,..,
vv
f)
tt
Thus wo
1 :.,/
, C. I -. huy assign
•
•
~-e
ing
hd.'l9
,
~o
a hoi:aoo morphi se of the space
every
·
+
poin~
of iho s ·J ae e i ·t .a
6:S:o=6anoe along the flow-l ino from 53 and tho point of
s·eetio n of li:1c flo\11- line •11ith H3.
f
•
intG~-
It oan also be shoi... n ·that
I
..1.x>.
=rx
s e""'"ne
+.hl:.
-~~vn--
r--1t
3
.._ '00 a
ffiUSv
,,v&Uehy
"6t1ere '!'fore a Cauohy horizo n ·
eacb !lo•·r= line a.t n1ost once .
•
GUPfao e,
(((-1
M'or
this c&n
•
'
~herefore
H? .
~'urtj;j.er,
.w:ust \i.nter~Qct .cQ 3 .
Tnus
~ 3 is h.omeo~o:r1)hic
O.irecti c-.1* 6.way from H3
e~ch
"by ( e)
If condit ion (f) h.olde every null
of ~·3 fi&s· at least one end point .
~l1ere
intor~ocv
.
L±OX00 -
every :floli!= l :i.ne
-:o H.3 and. is
g~od~sic
3Bno:» .to:F-
sine~ in tho direct ion to\tJard s
aenera tor auo=6 be unboun dsd.
oin~ularities
....
Tnio must be in the
W
.
a
ire,"}o ssiole
l:O
r
iniso· n·eve
m, .
«
8.
uuu •OcJ.1€
there is a
FBorpl:iistfl. of ~:l in"t;o ·
€01ripa ot.
'
nun
me
•
.
.
ses
univer
'
Closed
in '
.
is a singu larity in every model which satisf ies
(a) , (g) and (i) .
•1ihere exists a compa ct Cauchy surfac e H3 \·1l1ose
(i)
normal
I
V
~
.
.
l l l l l t:
3
has positi ve expans ion everyw here on h •
PROOF
the proof it is necess ary to establ ish a
~or
cou ~ le
11.Ssume that space- time is sin6u lari ty-fre e .
le111mas.
follo;,. 1ing result is quoted v1ithou t proof , it
of
'I'he
m'~-..l :a:.~ac-~]:y
l)e
derive d from lemmas proved in refere nce 11 .
if
?
and
ci,.
are conjug ate points along a geode sic (
and 'X is a point on
point in
I
J
f} ·.
r
not in fCj... then X' must have a conjugc-.;.te
iln
r
e lon3
i~mediate
corrollary is that if q is
conjugace t o p
conju5ate points in
pq
and y
•
•
•
pq
J.n
J.S
~he firs~ ~ oint
then
•
y
a.G..3
no
Also since the result that x
h o.s a conjugate point in pq ca11 only depe11d on the values
of
f
in pq, any irrotational 5eodesic congruence includin5
C\.
M f\
the geodesic
r
must have a sin.t;ular point on
.rhus if q is a point on
1
f"i3
t
in pq.
is the geodesic normal to
and (
M3 tl1roug h q , then a point conjugate to q alons ·
f
canno·t occur
3
until after a point conjugate .to M •
If 113 is a complete connected s1)ace-like surf o.ce \·1i1icli
.
inters e cts every time-like and null line from a i)oint p, '::e
may define a function over M3 as the square of the geodesic
f
distance from p which i s taken as positive if the seodesic is
time-like and negative if the geodesic is space-like.
this the world function CY \vith respect to p.
set of v<:,lues
o
:?O
,
v1here
11."
l I
3
A time-lil{e geo ... c sic
o
from p will be said to be critical
to a value of <5 for \vhich
, € f'J..
Ji'or ~he c l osed
wiJ.l be a continuous ( in
general mult1 i-v<:1.lued) function over
r
~e call
~
u,, ,AA
..
C• /.A
t,
~
0
if it corresponds
I i.
L..
:. .•, ~ 3>)
1
are three independent vectors in !'13.
Cle~· rly
...
(
a critical geodesic must be orthogonal to M3.
•
\
l
A geodesic
which is critical will be said to be maximal if it corresponds to a
local maximum of
•
Lemma 1.
cannot be maximal for a smooth M3 if there is a
A geodesic
X
point
where q
Let
f
and
conjugate to M3 but no point conjugate to q
is the intersection
of !
on
"'f
in qp ,
and M3 .
~
f and g be the Jacobi fields along
m
m
respectively.
They may be written
which vanish at X
n
f
=
A(s) f/q ,
•
mn
m
n
g = B(s)g/q •
m
mn
h1 (
Then
n
any h
e~
x;
<{,
J
°r
~ J ff¥ I ~
i
c/..s
n
r
must be positive for
since if it were negative for any h
by taking a = ... b h h
mb
mn
beyond q , it would be possible to have a point y on
beyond q
6
conjugate to
X
before a point conjugate to
would be conjugate to
f
.
)(
does .
Since
o£ i
•
If it were zero
This shows that the surface at q
p
of constant geodesic distance from
direction than the surface
p
lies nearer to
f'
in every
of constant geodesic distance from
X is conjugate to M3 the surface at
constant geodesic distance from
p
direction than M3 does .
'6
Hence
lies closer
f
to Jiil
is not maximal .
1
q
in some
of
p
,u :;.t 'be
o
":Q
4
•it i
"'~e
,.
G.O
...,(].
.. g
A• A
.!JA 1\
. . bO
£ l:Ill:
.J..
j
t.
~
M
'-:>
(.,(
t
i
s
But
*·
a•-0
~E
f
I
g f &' <
-'
MO
7\
lb
0
A
cannot be positive definite .;:.t a.
fcare there
m1Jst be somq
(v\
t<
dj recti on
1\.
&.nd
~'\
fi
10
1/
.,.
Q
1\\: •, rv
4
~""
K
vv
re
\,zbi
c:b
d
I\.
P.*t
for
i.2
A •
"'
I
'
I\.
2 i~ th~ unit tangent vector of tbc congrueLoa
of ooncbant JCodccic distanco from p lies aloser to p thaB:
~he ourfaee M3 6occ,
~horoforo
f
is not m~xim~l.
7
If r·1./ is compact; or if the intersection of all
t
and null limes with M3 is
comp<.~ct,
value, ·c hus there must be a
•
I
a rna:x:irau:..
~eodesic normal to M3 through
p
'i-ie use this to prove another lera:tiG.. •
.Leinma 2
..:>
If p lies to the future (past) on a time-like ·:·eoO.esic
(
throue;h
q, beyond a point z conjugate to q ,and
exists a compact Cauchy surface
l
6' a1ust have
, ...
.
."C ime_ll<e
7.
7
II
t~ere
through q, then there
1nust be o..nother time-like geociesic from p to q longer tl1.a1-_
Let y be the last point conjugate to q on
t before p .
Let x be the nearest point to p conjugate to p in pq.
r be a poin·t in yx.
•
Let
Let K3 be the set of •uoints which have
a future (past) directed geodesic of lensth rq from q.
-l
r.
~hen
K3 will be a soace - like
hypersurf~ce
be the set of points •..vhich have at
- t
.Le
throuGh r .
l~ast
-~4
i!'
one fu\,ure
directed Geodesic from q of length creater than rq .
·1·hen the
Since p is in FLJ- o..11d since every
l
past (future) direc ted t i me- like and null line from p intersects
4
H3 which is not in F , they must also intersect J 3 •
Bince ri3 is
be the int•ersection of J3 and these lines .
8onsider the function Cf'
compact , L3 must be compact .
respect top over K3 .
L' .
\·1ith
Its maximum must lie in the compact
~
region
T.et
_, ,3
~
But , by the previous lemma (
is not m~ximal ,
1uoreover, local considerations show that a singular ooint in
the surf ace J3 cannot b e a max imum of C>
..
t
n1aximum value of
t:~onal to L3 .
cr-
must
o c~ur
•
'fhus tl1e
for a ..:.:eode sic from p ort:.J.o-
Thi s must als o be a goedesic from p to q of
length cre~1:ter than (
•
.
0111ce
Us in.~ tt1ese two lemmas the theore1a may be urovea .
.
H3
converging
to
the future (p•-st) directed normals
eve.._"Y'·lhere on H3 , there must be a point COnJU(?;O.te to H3 a
~
a.1~e
•
finite distance a l ong eacn future (past) directed geodesic
nor1nal .
(
Let ~
be the maximum of these distances .
Let .u
be a point on a future (past) directed geodesic normal at
a distance
~reater
than
•
Consider the function G
v1i·ch
•
•.
.
l
Let~ be l-he
respect to p over the compact surface :13 .
~eodesic from p normal to H3 at the point q , where~has its
maximum .
•
l;here must be a point conjugate to li3 along A in en .
1
But if there is no point conjugate to q along
cLJnnot be maximal by the first lem!;1a .
~in
qp , then
If however there
is a point conjuGate to q ~long~ in qp, then there must be
a
lon~er scode~ic
from q to p by the second lemi.a.
is not the seoa.esic of maximum leng"t;h from H3 to p .
a contradiction which shows
·. :. hus
.
·- nis is
tha~ ~he ori~inal assu~ption t~et
the spc..ce \vas non-singular must be false .
Tt1is proof could also be used
of a singularity in a model
vJi th
L.
~o
show the occurrence
·non- compact Cauchy
sui~f ~:·.ce
'i.
l)I'OVided t tiat tl1e expansion of its normo..ls was bounded av:ay
l
from zero and provided that the intersection of the Cauchy
surf ace \vi th all the time- like and null l ·ines from o. point
v¥as compact .
l
•
•
1•
2.
••
:;
.
o.
~~
....\.
.
I . A. U.
Hecl<:1nan
J{ayc haudhuri
~hys . ~ev .
. Kornar
4.
l.i . C . t.ihepley
t::
/ •
c.
6.
~.M
'l .
l{. Penrose
uympo~iun
98
1
1961
;23 1965
Phys . 1{ev . 1 QL~ 5L1.4 19 56
Proc . Nat . Acad . 6ci . 52 1403 1964
Z. Ascrophys . 60 266 1965
Benr
Adv . in Phys . 12 185 1963
Lifshitz and
I . M. ~(halatnil<:ov
rt . Penrose
9. I( . Godel
10 . C. •.1 . l'1isner
11 . J . i·:ilnor
b.
Phys . Rev • .Lett .
1L~
57 1965
Rev . : .od . Phys .
'!!.
215
17 65
?:d 44-7 l'ftr f
Rev . Mod . ?hys .
J . Math . Phys . ~ qz4 r~
' Morse Theory ' No . 51 Annals of
of Maths . ~tudies . 2rincetown
,
Univ. Press .
'Curvature and Bet·!Ji
12. K. Yo.no and
~nnals
.::> . Bochner
~tudies
1
1'To . 32
.
Princetown Univ . Press .
l~rticle in 'Gravit;ation ' ed . .;;it·cer-1
13 . a . Heckman and
~.
of Maths .
i~umber·s
·riiley, .r;ev,r York .
i::il1ucl\:ing
' Cosmology ' Cambridge Uni v . ::.'ress
14 . H. Bondi
I L~
-
\ (.A.c•10;,
J
\
'
l
\
I
Ph.D.
51:;7