THE BOARD "~ RfS EARCL,' S 
fo\ r TUDIES f,pr' ... 
1 
01.' 
, J 
....!.-• - --:..:.;..----
.in Clw.ptel' "I it is :..~lto n ttl' t 
L ..L. I ox \ n...,.L. 11 create,., 1.' n dii'1:icultiou fOl' i;i1p 10 y. e-
)1,lO'O eau:.; ::. nc i80 l; ['0 )ic 
un}. TOl' :..> ' . he cc) G LU . .Jion 1.., ! th' t ;. U _~i,;:.~ C .110 I. 0 
. U..L'D!r;d. d ~ a r:]...;ull. of th' "..1' O:J 1.011 ) n:. LU.r.\}, ... t;io:l J Lit 1.; \10 n 
1,.)(.11;101:<.,.1 L'; L .tiO.i 1 J L.L~o i liTc:Jbi Ll~{ in t i. ... ; n l'o~{imD.-i 
'in ;h;~jtoJ. :; Tavit: tion, 1 r.' d.j.t. ci'm in n p.I1;)'l1 in uni.vflr<1.o 
J h· ' I) \ i'·l i n 
t ( i 
1. 
r (\ 
, t 
. ~ - ) (, .' 
'''' 
v 
'" 
o · t 
; 
v 
c o 
" 
1 l ' 
,0 
(; . 
• 
o 1 
1 
k <::. 0 
. . 
. 
i 
i 
1 
1 ' . 0 1 .. 
t 
i 
o 0 
.1. i u 0-
.. 
o '. 
1 o· 
o 
r 
• 1 
1.' ') 
o 
1 
f ). .1. _ .... hell') ,.n :.....dviec LUI'in....., 1. Y pGriCL of resel. reil . 
I \!ould '-.100 lil~G to t,ll' nl:: r . ~ . J .trGel.' '- nd 
.I.) L • 
, , 
v. J..: . I l. Lt 
l n d.elYLe ~ "(,10 ... { . Go clJcnu. ,fl: .. 11 1'OI tho c' .lcul(tion or' 'Lilo 
15io.ncili .Lucnti Gic£; in v t' p"iicr ? 
. 
) . 
The Hoyle-Narlika r Theory 
of Gravitation 
1. I n t rod uction 
The success of Maxwell's equations has l ed to 
electrodynamics being n orma lly formulated in terms of fi e l ds 
that have degrees of f!eedom independent of the partic l es in 
the m. However, Gauss suggested t hat an a c t ion-at-a-distanc e 
t heory in which the action travelled at a finite v e locity 
might b e possible. This idea was developed by ~hee l er and 
Feynman (1,2) who derived their theory from an action-princip l e 
that involved only direct int e ractions between pairs o f part-
icles . A feature of this t heory was that the 'ps eudo'-fie l ds 
introduced are the half-ret a rded plus half-adva,nced fi elds 
clacula ted from ·~he world-line s of the particles. Howev er , 
Whee ler a nd F eynman, and, in a different way, Hoga~th ( 3) 
were ab le to show that, provided certain c osmolog ical 
cond itions were satisfied, these fields could combine to 
3 ive the observed field. Hoyle and Narlika r (4) extended the 
tlleory to g eneral space-times and obtained similar the ories 
for their 'Cl-field (5)and for the g ravitational f~eld (6). 
It is with these t heories that t his c hapter is conc e r n e d . 
. . . . . . . . . . . . / 
It will be ;3 hovJn tha t in an expi .. nding unive rse t -!le 
advanced f ields are infinite, a nd the retarded fields f init e . 
This is because, unlike electric cha r ges, all mas ses haVA the 
same s i gn. 
2. 'Ithe Boundary Condition 
Hoyle and Narlikar derive their the ory from the 
action : 
A :: 2. Z (( Cl (0., b ) cL a cl b) 
0../ 6 J) 
where the integration i s over the world-line s of particles 
0- b. - -- . 
J 
In this expression ~ is a Gre en fu nction . 
that s atisfies the wave equation: 
where 3 is the determinant of 3.j . 3 inGe the double 
in the action A is symmetrical between all pairs of 
particles Q) b , only tha t part of C (Cl. b) that i s 
sum 
sy~metrical between ~ gnd b will cont ribute to the action 
i.e. the action can be written 
be \o,Jri ttien : 
~f fS c ~ (~.-b) cLQ~ d b 
i G ( Q, b ) I } C (b , 0.). 
be the time-symmetric Green function, and can 
q~ = * qre-i: ,.1 q o.dv where Cit-eX 
and a re the r e tard ed a nd advanced Gre en f unctions . 
By requirinz that the action be stationa r y und er v a ri a tion s 
o f the g~) Hoyle a n d Narlika r obtain the fie ld-equ a tions : 
[££ ~fVl('"){X)M(b)(X)] (P"I<- ig"K R) 
o.-#b 
",mere 
c onseq u e nc e of the particular choice of Green function, the 
c ontraction o f the f ield-equations i s satisfied i d e n tica l l :: . 
~here a~ e thus only 9 equations f or the 10 components o f Q' " J l-j 
a nd the s y stem i s indeterminate. (fA) 
Hoy l e and Narlikar there fo re imp o se Z /"Y1, = Mo =CO Ds t ., 
-~s the tenth equation. By then makin~ the ' am both-fluid' 
, , L 4 AA (0.) J'l" (b) r"'> IV) 2 
approXlma tlon, tha t i s by putting L~"v ,.~ ~ - , .~ Q':/;b -. - CJ ) 
they obtain the Einstein field-equati ons : 
..!.... . ~ (R. -.!- f{ ') ., .' ~ (lA 0 " K :2. 3 (,. i'\ = - ~'K 
There i s an important difference, howeve r , b e twe e n these 
field -equ a tions in the dir~ct-particle i n tera ction t ce ory 
and i n the u s ual g enera l t he ory of r elat ivity. In the 
g eneral theory of r e lativity, a n y metric that satisfies t he 
the field-equations is admissib l e , but in the d ire c t - particl a 
interaction theory only t hose solutions of the fie ld-equati on s 
are admissible t hat satisfy t he additiona l r equirement: 
.[ ) C/oc/ ()..) cL CA 
i 
:l 2: f G (Ad!/. (7:-) (A) do,-
This requiremerit is highly r e strictive; it will be sh own 
t h a t it i s not satisfied for the cosmolog ica l solutions of 
the Einst e in field-equations, a nd it appears th~ t it c a nnot 
b e s atisf ied for a ny models o·f t he univer ;: ;e tho.t either 
contain an infinite amount of matte r o£ unde rgo infin ite 
expansion. 
The difficulty is simila r to tha t occurring in 
Newtonian t ~a ory when it i s recogn1zed tha t the universe 
mig ht be infinite. 
The Newtonian potential ~ obeys the equation: 
o 1 : .-KF 
',t.Jhere i s the de nf; ity . 
In a n infinite stat ic unive r se, ~ would be infinit e , s in ce 
the source a l ways ha s the s a me Gi gn. The d ifficulty was r e s ol-
ved when it wa s re a lized thnt the univers e was expand i ns , s i n c e 
in an exu anding universe the retarded s olution of t he a b ove 
equation is finite by a sort of'red-shift' effect. The 
advanced solution will be infinite b y a. ' b lu e -s hift' effe ct. 
~his is unimportant in Newtonia n the6ry, since one i s fr e e 
to choos e t he solution of the equation and so may i g n ore the 
infinite advanced solution and take simply the finite 
retarded solution. 
Bimilarly in the direct-particle interaction t heory the 
~ - field satisfies the equation: 
Om ~ ~ RM - N 
where tJ L3 the density of \:.Jorld-lines of particles. As i n 
the Newtoni a n case, one may expect that the effect of the 
expa~sion of the universe will be to make the retard ed s olution 
finite and the adva nced solution infinite. However, one i s 
now not free to choose the finite retarded solution, f or t he 
equation is derived from a direct-particle interaction action -
principle symmetric between pairs of particles, and one must 
choos e for fV\.. half the smm of the retarded and advanced 
s olutions. Vie would expect t uis to be infinite, a nd this is 
s hown to be so in the next section. 
j . The Co smoloKical ~olu~~0ns 
The Roberts on-Walker cosmologica l metrics have the 
form 
2. ;l.. R Q. / ). { ' Q ds ::: dt - LI: (fr-
• J - tlr~ 
Sinc e t ney are conformal l y f l a t, one c a n c h oose coordinates 
It/he re 
2 ds 
is the flat-sp a ce metric tens or and 
._ . R(c) - (7) 
/ [[ i ,t ~ K ( 'f '1- f) .iJ L I -rt .1,\ ( 't - r ) ~] J 
For example, for the Einstein-de Sitter univers e 
K = 0" R(t) ~tbJ1 
. . ~ IT) ~ . 
Jl - R - (-~ ) ( G /~ C < 00), 
.':2 .!-) 
r- ~ F (~ :: ',:3 t 3 
For the steady-st a te ( d e Sit te r ) universe 
K ::. O~ R (t) ;; ' .'..t ( - 00 ~ I: LoO) 
eT 
JL f< :. .- f ( - -d!J C 0) -- <- ~ 
""" (,
.- L-
r - f ('r ~ .- Te T) -
11he Green function q*" (~) b ) obeys t he equation 
o ~ .~ (C\) b) -t t R q -If: (CA, b ) 
From this it follows t hat 
- S~' {Qlb) 
;-.=. 3---
;, _Jdx ~.- (Jl 2.7 <A b }x b Q "'}'1? (fbbJL ) JL -3 q ,r 
::: JL-'iJ4 (a, b) 
('I~= "'-( (>, If we let \..t ..) L j 
JL.L ( c. b d S' ) . :: d~~ 7 ~ 
,rrhis is simply the flat-space Gr pen funct ion equation, ancl. 
hence G * ( l, ,Dj °l l f) 0 st (w I.~ jjf-:- 't~ l' ~ I ) 
S'1f _ Jl(c1)P 
;- gL~- "C, )'l 
The 1fV'\.'~ fiel d is given by Jl ( "C;L ') ( ,. J 
. 
(l~ ( 1;.) .:- (' ~ ;( N / - Q d:.e ~ :: .L (fYL. 'r Ih ) r) J :<. I'e.t. CL.) v: . 
the Einste in-de ~itter For universes without creation ( e.g. 
N -=- I~ '- 3 n. . 
., const. :F or universe), 
universe s ' .. ·;i th creation (st eady s t ate ) f\/ ::. ~ (lv:; c on ~-j t . 
I 
fVL (" (; ) :; )1- 1 ( 1: I ;j' NJL J ('t2.) Lt TIt '1.cLl'" 
{l. ... dv, . I ~ ·,Tt"" 
wi18r e the integration is .over the fu t ure light c one . 'rhis 
will n ormally be i nfinite in an expanding universe , e ~ g . i n 
the Einstein-de 6 itter universe. 
rL (f - 'f ) cL-(; 
;l.. I ~ 
I n tbe s '~eady-state universe 
._ ( . /.\ I )-i'f 0 _ (L. (~ )3 (i:.
2
- '1, ) cl '1:1. . 
L I 1:", [. '-
.:. 
By c({l)ntrast, on the other hand, we have 
whe r e the integr a tion is ove r t he past light cone. This will 
normally be finite , e.g. in the ~in~tein-de o i t t er univers e 
i -, ':( 
- n ... ). 
whi l e in the steady-st ate univ erse 
~ I-~) )-If "r:; n.(~' )3('2. l-L. I '-co L 1. 
;;( 
Thus it co.n 'be seen thEJ.. t the solution fV\....::. con st. of tb~ 
e quat ion 
is not, i n a c osmological metric, the half-adva nc ed p lu s 
half-re tarded solution since this would be infinit e . In f act, 
in the c ase o f the Einstein- d e Sitte r and steady-st a te metri c s , 
it is the pure retarded solution. 
4. The 'Cl-F ield 
Hoy le and Narlikar derive their d irect-pa rticle 
interaction theory of the 'Cl -field from the a c tion 
where the suffixes Q 1 b r e fer t o d i f f e r entia tion of 
C;(a./b) on the world-l ines of ~/ 6 
"-G i s a Green function obeying the equation 
Dq(X,X') _ r~(X)XI) 
F3-
I've define 
.. 1'\ 
and the mat ter-current .j by 
r eG oe ctiv e l '/ . 
.. '-' 
JK(j) .~ i. S ~4( j' b)cLh '~, 
'l'tlen C Gc) . S f; (x:, 'J) J ,< ( 'j ) J K / -3 cL 'X ') 
oc 
._ 1'\ 
- J j 1'\ 
iNe thus see tbat the sources of t he I C I -field.. are the pl a c es 
where matter is created or destroy~d. 
11.8 in tile c ase · of the I ~ I -field , the Green function 
Hoyle a n d Narlikar cla im th8t if the action of the 
' C ' -field. i s i ncluded alon , ~ \"lit h the a ction o f t he I f<'\.,- ' -field , 
a u nivers e \-l ill be obt a i ned t i·l:J. t approx ima t es t o the :3teady-
st a te universe on a l a r g e scal e a lthou g h t he r e maybe local 
irre~ul arities. In this univers e , the v a lue o f C will be 
fi n i Ge and its gradient time-like and of unit magn i t u d e. 
Given t his universe, we may check it for consistency by 
cl aculatin~ t he advan~ed and retarded 'C'-fields and f ind ing 
if their s um is finite. ','ie shall not d o thi s d irect l y but 
will s how tha t the advanced field is infinit e while the re tar-
ded field is finite. 
Consj.der a re g ion in spac e-t ime bounded by a thr ee-
d imensional s p ace-like hypersurface ~ at the p r esent time, 
and t i1e past lig ht cone £.. of some point P to the future 
of j) 
By Gaus s 's theorem 
( W C j- 3 dx <; ~ (dL cl S 
J v ) 5.. ' f- J) d rL 
Let the advanced field produc e d by sources within 
l 'hen c' and ,J, cri , L ~ wili be zero on L and hence 
d~ 
- l-( 
But J is the rate of creation of matter=: ft.. (const. ) in 
)K . 
the s teady-state unive r se, and hence 
AS the point? is taken further into the future, the v olume 
of the re~ionV tends to infinity. However , the area of t l e 
hy p'ersurface J) tends to a finite ov..ring t o horizon limit de / 
dn... must be infinite . effects. Therefore the g radient 
A similar calculation shows t he g r adient of the ret a rded 
field to be finite. The i r s ums cannot therefore ~i~e the 
field of unit g r adient required by the Hoyle-Narlikar , . t: t1e ory . 
It is worth noting tha t this result was obt a i ned 
without assumptions of a smooth distribution of matter or 
of conf orIDal:~ flatness . 
5. Con clus ion 
It i ~ one of the weaknesse s of the Einst e i n t he ory of 
re l ativity tha t a l t hough it furni s h es f ield equat i on s i t d oes 
n ot p rovid e bounda ry conditions for them. Thu s it doe s n ot 
~ive a unique model f or the uni v e r se but a llows a whole seri e s 
of mod el s . Clea rly a theory th&t provided b oundary c on~i ti ons 
and t i.1.US restricted t he possible solutions would be ve r y 
attra ctive . The Hoyle-Narlikar t he ory d o es just tha t (the 
, 1- /\- I 
r e q uirement t hat fl1, == 1." fYl...,. e- t ' t )... Ir'<o...di/. i s 
equiva l ent to a b~undary condition). Unfortuna t e l y , a s we 
h a ve seen a bove, this cond ition exclude s thos e ffi od el s t ha t 
seem ~ o corre spond to the a ctual univers e , n a me l y t h e 
Rober tsol1 - vJhll:er mod els . 
The calcula tions g iven a bove ha ve cons idere d t he univcr3 ~ 
as be i ng f ill ed \·, i t h a l1 n i f orm d i stribution of ma t t e r. 'llhis 
is l eg itima te if we are a b l e to ma k e t he 'smooth-f l uid ' 
approx ima tion to obt a. in the .U; i n s tein e qu a tions . i.. 1 ternativ ely 
i f t his a p p rox ima tion i s invalid , it cann ot be s a i d that t h e 
.t ne ory y ields the Einstein equations. 
I t mig ht possibly b e tha t loca l irr egula rities c ou l d make 
fi n ite, but this has cert a i n l y not be e n dem onstrated 
a nd se ems unlikely in vie w of the f a ct th0t, i n t he Hoy le -
Na rlik a r d ire ct-particle i n teraction t h eory o f t h e i r IC I - f i e l d , 
which i s derived from a very similar a ction-princip l e , it c an 
be s hown without assuming a smooth distribution that the 
advariced 'C' field will be infinite in an expand inG universe 
with creation. 
The re a son t ha t it is pos s ible to formulat e a d irect-
pa rticle interaction theory of electrodynamic s t hat does not 
encounter this difficulty ~f baving the adv8nced solution 
i nfinite is that in ele c trodynamic s there are equal numbers 
of sources of positive and n e gative sie;n. Irheir fi.s l d s can 
c ancel each other out and the total field can b e zero apart 
from loca l irregularities . Thi s sug~es t that ,::, p ossible i'Jay 
to save the Hoyle-Narlikar theory would be to allow ma sses o f 
both p os itive and negative s ign. The action would be 
are gravitational charg es a naloz ou s to 
ele c tric cl1ar .; es. Particles of positive ~ in a positive 
'Ill " -field a nd particles of negative ~ in a ne6 ativG 'r/\ ' 
field wou ld have the normal g ravitational propertio.s, t tl[!t is, 
tne:! ~'l ould have positive gravitational and inertial masses. 
A particle of n egative t in a p os itive IM I - fie ld wou ld 
still f ollow a g eod esic. There for e it would be &t t r a ct ed by 
a part icle of positive Its own gravit ~t ional effec t 
how ever would be to repel all other p a rticles . Thus it wou ld 
hav8 t he properties of the negativ e mas s described by Bondi(~) 
tha t is, n egative gravitat ional ma SB and n egat ive inertial 
mas s. 
S i nce t here does not seem to be a ny ma t t e r having 
t aes e properties in our reg ion of space ( where (V) .-:!:::::- CO DS t . ;:> 0 ) 
there must clea rly be separa tion on a very lare e sca le. It 
would n ot be p os s ible to identify particles o f negative t 
wi th ant ima t t er, since it is known that antimatt er has positive 
ine rtial mass. Hwever, the introduction of negative mass e s 
wou ld p rob ably raise more difficulties than it ~uld s olve. 
REFERENCES 
1. J . A. \vMeeler and 
R.P.Feynman 
2. J.A.Wheeler and 
H. P . Feynman 
7-7- J.E.Hogarth 
4. F . Hoyle and 
J .V. Narlikar 
5 . F . Hoyle and 
J.V. Narlikar 
6 . F . Hoyle and 
J.V.Narlikar 
7. L.lnfeld and 
A •. Schild 
8 . H.Bondi 
Rev. Mod . Phys . 12 157 1945 
Rev. Mod.Phys . 21 425 1949 
Proc. Roy. Soc. A 267 365 1962 
Proc. Roy. Soc.A g 77 1 1964a 
Froc . Hoy .Soc. A ~82 178 '"1964b 
Proc . Roy. Soc. A g82 191 1964c 
Phys. Rev. 68 250 1945 
Rev. Mod . Phys . 29 423 1957 
CHAPTER 2 
PERTURBA'I'IONS 
1 • Introduction 
-.-- - - ---~ 
Perturbations of a spatially isotropic and homogeneous expanding 
universe have been investigated in a Newtonic..n approximation by 
Bonnor(1) 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 directly measure it, but only its second derivatives. It is 
thus not obvious what the physical interpretation of a given 
perturbation of the metric is. Indeed it need have no physical 
significance at all, but merely correspond to a coordinate trans-
formation. Instead it seems preferable to deal in terms of 
perturbations of the physically significant quantity, the curvature. 
2. Notation 
Space-time is represented as a four-dimensional Riemannian spac e 
with metric tensor gab of signature +2. Covariant differentiation 
in this space is indicated by a semi-colon. Square brackets around 
indices indicate antisymmetrisation and round brackets symmetrisation. 
'rhe conventions for the Riemann and Ricci tensors are: -
V Ib ') ~ '"'RP V f;1.-;L <: - ~ o.cb p. 
R ~~ = R~ P bp 
1')~6C"~ is the alternating t ensor. 
Units are such that k the gravitational constant and C 9 the speed of 
light are one. 
We assume the Einstein equations: 
where Tab is the energy momentum tensor of matter. We will assume 
that the matter consists of a perfect fluid. Then 9 
where U
a 
1s the vel ~city of the fluid, 
jI.Il is the density. 
~ is the pressure 
U Ua 
a = - 1 
is the projection operator 
into the hyperplane orthogonal to Ua : 
h l'1.b U b '; 0, 
We decompose the gradient of the velocity vector Ua as 
. 
UI'1.;b:::' (..Jo.b + CSo.b + t h().b e - Uo.. tAb 
where is the acceleration, 
e -:; (Aa j 0.. is the expansion, 
,...,.. hC hd. I ho..be· vob -.:; l-\(C;ot) d.. b - -S is the shear, 
Wo.b -:' U[c~dJh:h~ is the rotation of the 
flow lines U. We define the rotation vector 
a 
Wo.. as 
I 1'\ c.ol b 
(..\)0. ":;. "l:/O-bc.cl wU 
We may decompose the Riemann tensor 
Rab and the ''-:'ayl tensor C abcd : 
R 
abcd into the Ricci tensor 
R OIbc~ : CQ.~ccl - 9q(d Rc.] b - '3 b[c Rc\JG\ - R/3 Cjo..L"-5ct16 ) 
Ca6c.~ : CLo.b]\::...I.J 7 
C~. be.G!.. -:: 0 --:::. ·C 0.[ bed] 
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(S)). We may decompose it 
into its "electric ll and IImagnetic ll components. 
E ab 
C cd o.b "; I I t- le .' .dJ c. ((. E d] 8 V' [t>.. C bJlA - 4- O[~ b) , 
each have five independent components. 
We regard the Bianchi identities, 
as field equations for the free gravitational field. 
? 
Then (Kund t and TrUmper, (6) ). 
Using the decompositions given above, we may write these in a form 
analogous to the Maxwell equations. 
b E. cd H b b c u de \ h b ho- pc;oI. h + ~ o.h W - '? o..'oc.d U (j e n = '5" Cl )-'-; b 
- I E· h { C H d.; ~ .~ ab t- (0. V; b) <,d e lA f +-
- EC,aC""b)c - YjOc.de YJ bp<=fr u. C uP (J~'J EQ.r 
.. . 
-t 2. H cia ~ bed e LA c:. 0. e =- - ~ (f -i ~) () ab 
'J 
. 
.L H h ~ c. E d je H I I c. 
c.:b - (0.. t')'b) c de \). - f +- O/.b e .,... , (0. W b) c. 
_ He. er cl 11:. (0.. b)c. - ~pr;.de '/bp9r" (;..t uP er ~ H r 
2:H d c..' t o.1hc.rJe().· t,A€ :: 0 
where ~ indicates projection by hab orthogonal to Uao 
(c.f . Trllinper, (7)). 
The contracted Hianchi identities give, 
(R Ab - t ~,d~R)ib = 
ft + (f +~ ) B ::. 0 
- ·b I c-b' -= 0 
0+ ft) u. Q 4- ft.;b h ha :: 0 
The definitien of the Riemann tensor is, 
) 
J 
, 
(1 ) 
(2) 
(4) 
(5) 
(6) 
Using the ' decomposi tions as above we may obtain what may be regarded 
as "equations of motion",' 
) 
(8) 
. h P h~ 
+ LA ( P ~~) Cl, b 
where 2, Q b 2 w =: W ab W 2. () 2.. = crab () o.b 
We also obtain what may be re g arded as equations of constraint. 
J (10) 
) (11 ) 
(12 ) 
We consider perturbations of a universe that in the undisturbed state 
in conformally flat9 that is 
Cabcd = o • 
By equations (1) - (3 ) 9 this implies 9 
h 0. b r ; b' :: 0::: 
If we assume an equation of state of the form? 1"- , -:::. if'l. (tJ.) I 
. 
:::. o~ u.o. ft.,b h\'o -(~en by (6), ( 1 0 ) , .. 
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 perturb at ions of the motion 
of the fluid and of the ;i/eyl tensor. We neglect products of small 
q,uantities and perform derivatives with respect to the undisturbed 
metric. Since all the quantities we are interest ed in with the 
exception of the scalars, f.L,~, e have unl?erturbed value zero, we 
avoid perturbations that merely represent coordinate transformation 
and have no physical significance. 
'1'0 the first order the equations (1) - ( l,t ) and (7) - (9) are 
E )6-::. o-b I h b ~ ' o. f;b 
) 
Ho (I (', \t( ~r= dje -_ Q.b +- -AbD - 1 <>- ')b) cde U. -f 0 
. 
e I 02-"3 Q + 
• • tl-
v<. .I 
0. 
J 
1 
. E ~ At!- e I h ··C· ? Of O'~ b -:: co. b - ""S U ob -"5' c. b LA c. I + U ( P;' '1) h C\. h b 
) 
(13 ) 
( 14) 
(16) 
(18) 
(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 perturhations. 
The Undisturbed Metric 
-.---...".,~~----
Since in the unperturbed state the rotation and acceleration 
are zero, U
a 
must be hypersurface orthogonal. 
where ~ measure,the proper time along the world lines. As the 
surfaces "t := constant are homogeneous and isotropic they must be 
3~eurfaces of constant curvature. Therefore the metric can be 
wri tten, 
where 
r,'e define t by, 
then 
In this metric, 
-.. 
, 
is the l1ne element of a spaee of 
zero or unit positive or negative curvature. 
~ t ;:::. I 
dT TI ' 
ots'l ~ (l"t.( _d. t 2 + 0)"2) 
(pr ime denotes differentiatio.n- with respect to t) 
Then, by ( 5 ), (7 ) 
... 
351 -=- -r (u+-3ft) 0. { 
If we know the relation between ~ and ~ , we may determine 0-
We will consider the two extreme c ases , ~ = 0 (dust) and 
(radiation). Any physical situation should lie between these. 
tor 11~ 
By (20), 
• 
. . 
( a ) 
. " ;. O. 
MO 
For E 
N\ = c onst • 
E = const • 
J 
o 9 
(20) 
(21 ) 
O. ..l- e Cos h ~ f-f t - ! ) "'- 2E ~ ~E (/ -~ ?Ln h IEM t -- t) · \; =- z. tEfY\~"'!> ./ 
(b) For Il~ = 0, 
n ::. M t'2.. 1'2.. ) 
-_ fV\ t ~ .0 
l 36 , 
(c) For EO, 
f2 ~ ~~ ( I - COS J- ~M t) ) 
E represents the energy (kinettic + potential) per unit mass. 
If it is non-negat ive the universe will expand indefinitely? other-
wi se it will eventually contr ac t again. 
... 
By the Gaus s Codazzi equations ~~R,.the curvature of the 
hYpersurface l::: const. is 
l< F< -;::. 
If E >0 / S 
*. R -e:. - n c. 
, 
",0 j( t= ) r~ -::.. 0 ;; 
E < 0 ) )': b f:<' _ . .:-n~ 
...!,4.! 
Flor ;;f\ ::: )J./5 ~......: " =-==-='""'=-
. 
. 1- ~2 F - D. , 
• 0 
3 CL :::: 
- t-
---51 {\I\ 
F .- (1'1-
Ca) Por E > 0 9 
(l t ,:>..L.n.h t - f ) 
(b) For E 
-
0 9 
~L :: t ) 
(c) For :fI: 0 9 
CL I s . .t Yl ..;.. -:::. l.. E 
By (6) 
J 
) 
I l. -::. t 
'2-
"" 
M -:: 
(Y\ ;: 
2-
) 
2- ( I e"- +f) 
-3" 
-
2- EM 
D:L-
3 
E .7 
) 
-"} 
.. 
. ft 
.. W «.b :: - W 0.10 ( ~ e .... ~) 
For it -= o ) 
0J -:: 0.)" ?P 
For 1'1.. --;. ~. 3 J 
(~ e I k ) w ~ - W t- ) It f' 
I 9 =. - -.w 3 
.. W Wo . , -:::. 
0 
Thus rotation dies away as the universe expands. This is in fact a 
statement of the conservation of angular mome ntum in an expanding 
universe. 
For 1\. = 0 we have the equations 9 
, -~e )J- -;:: 
. 
I '1 I e ~ - '3 e -1: f' 
These involve no spatial derivatives. Thus the behaviour of one 
region is unaffect ed by the behaviour of another. PeI'turbations 
will consist in s ome reg ions having slightly higher or lower v a lues 
of E than the average. If the untverse as wh '~e has a value of E 
greate r than zer0 9 a small perturbation will till have E greater 
than ~ero and wil l continue to expand. It will not cor-trac t to 
form a galaxy. If the lli1i verse has a value of E less than zero, a 
small perturbation can contract. However it will only begin 
contracting at a time 8 7:' earlier than the whole lli1iverse begins 
contracting, where 
""t 0 is the time at which the whole universe begins contracting o 
There is only any real instability when E = O. This case is of 
measure zero relative to all the possible values E can have. 
However this cannot really be used as an arguement to dismiss it 
as there might be some reason why the lli1iverse should have :IT. :.:: O. 
For a region with energy -SE , in a lli1iversc with E = 0 
11 I (t'l._ tit ) 
.H::' 4~t: - --;2 t . , . 
(SE)'2-~ '1. ) + ~__ l"' 17 -t ••• 
2<1~ 
For E = 0, . 'I- - 2. 1·"-.;:;. --::;- L . _/ 
Thus the pe rturb ation grows only as This is not f ast enough 
to pr oduce galax ies from statistical fluctuation s e ven if the se 
could occur. However, since an evolutionary universe has a particle 
horizon (Rindle r(8 )9 Penrose(9» diffe r ent parts do not communicate 
in the early st ages. This make s it even more difficult for 
statistical fluctuations to occur over a r eg : '·n ul1til light h2.d time 
to cross the r eg ion. 
e - I 2, "'3"9 -jA+ 
. 
U-.Ct= 
AS before? a perturbation cannot contract unles s it h as a negat ive 
value of E. The action of the pressure forces make it st ill mor e 
difficult for it to contract. Elimin at ing 6 9 
• \ 0 b . • Cl.. 
.l!. 0.:.; ~ ~ lA 0. i b"l t- u 0.. lA 
__ .J... 
4-
a.C· b ) /, (hc.M~ \"; c 
----.-.---~----.- ----
fA 
to our approximation. 
I C<.t.. \7 \ b '\. 7 '1 Vc I c;L v b i s the Laplacian in the hypersurf ace er = cons tant. 
'.7e represent the perturbati on as a sum of eigenfunctj.ons S (n) of 
this operator 9 where ? 
These e i genfunctions will be hyperspherical an d pseudohyperspher j.cul 
harmonics in cases (c) and ( a ) respective l y ane p l ane waves in case 
(b). In case (c) n will tal{e only dj.screte 
(b) it will take all po si tive values . 
,lues but in (a) and 
(I ~ BW; 50:" ) fJ- -=- f).· t L . I 
,On. .-I 
where )..Lo is the undisturbed density. 
As 1 cng aB fA. 0 :;> will g r 0 w. 
por 
These perturbations g row for as long as light ha.6 not ha d time to 
travel a significcmt d ist ance compared to the scale of the perturb ation ( '" r;; ). Until that time press ure forces cannot act to even out 
perturbations. 
, 
II I I B (\'l) + 12 I Y)) 1'] n'l.. 12. 0-) 
u -+ '3 u ::.0.; 
r2 When 
1:1e obtain sound wave s whose amplitude d e creases with time. These 
results confirm those obt a ined by Lifshitz and }(h a l atnikov(3). 
Prom the forg oing we see that galaxies ca:rmot f orm as the resul t 
of the growth of small perturb a tions. We may ~x:pect tha t other non-
gravitationa l forc e ~ will have an e ffect smalle r than pressure e qu a l 
to one third of the density and so will not cause relative pertul'bations 
to grow faster than l To ac count for galaxies in an evolutionary 
unive rse we must assume there were finite, non-statistical, initial 
inhomogenei ties . 
To obtai n the steady-state universe we must add extra terms to 
the energy- momentum tensoro Hoyle and Narlikar (1 0 ) use, 
(20) 
where, 
, 
? 
) 
(21 ) 
(22) 
There is a dif:ficulty here~ if we require that the "Ca field 
Should not produce accelerat ion or, in other words, that the matter 
created should havE; the same velocity as the matter already in 
existence ~We must "tl;i#n have 
.. : .. 
( 23,) 
However since C is a scalar, this implies that the I'otation of the 
medium is zero. On the other hand if ( 23 ) does not hold, the equations 
are indeterminate (c. 1'. Haychaudhuri and Banner jee( 11 ) ) • In order to 
have a determinate set of equations we will adopt (23) b,ut drop the 
requirement that Ca is the g r adient 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 difficulty 
by taking a p article rather than a fluid picture . However this has a 
serious drawback since it leads to infinite fields (Hawking (13)). 
From (17), 
. 
. ~ 
C -;: - u.. r CL Cl L 
ThuS, small perturbations of density die away . Moreove r equation (18) 
still holds , and the r efore rot a tional perturbations a l so die away. 
Equa t i on (1 9 ) now be c ome s 
e ==- - ~ [) 1. - t: (;-.t + 3 1'1.) + 1. 
The se result s conf1rm those obt ained by Hoylo and Narlikar (14) . We 
see theref or e tha t galaxies c anno t be formed in the steady-s t ate 
universe by the growth of small perturbations. However this does not 
excl ude the possib ili ty that there mi ght by a self- perpetuating 
system of finite perturb at ion f:l which coul d produce galaxies. 
(Se iama (1 5), P..oxburgh and Saff man (1 6 ) ) • 
We now cons i der perturbations of the Weyl tensor that do not 
ar i se from r ot at i onal or density psrturbations, that is, 
·b 
LI \ ) = 0 
, I OD 
Multiplying (1 5 ) by lA C Vc. and (1 6) by 
we obtain, after a lot of reduction, 
" 
7 1-::' e" +-"3 - c<b 
o· 
In empty space with a non- expanding congruence Ua this reduces to 
the usual form of the linearised theory, 
0 '3. F- I. "'-0 
- "'-.1:) " 
The second term in (24) is the Laplacian in the hypersurface 
"'r = constant, ac t ing on E . ab We will wri te 
eigensfunctions of this operator. 
where :::. 0 } 
( c:. h cA I E'"'\ k L t I ~_ \ /cJje h .(. ~ '/k)ji. h ho.rt. yJ -
V :,6 (] L, -:: 0 a \/ -::- 0 
0<-
.. /)., I cri) (YI ') 
Then E CL~ ::;:. L- (2 \ID.~ 
t AO(P] !i{')() , ] (r1 ' E Cl 10 ~ L L - \/ " n~ n ;~ -=, 
E ab as a sum of 
similarly, '" DCI'l) \/ In) o ",,10 '::! L ab 
Then by (1 9 ) A
l!') DLnl r2 I (2 - 2 -D 
substituting in (24) 
","lI'l1 6 (2' A I (Yl) f- A lYl) [ -z. n" n'"<' t n -+ )- t 6-yrr (L n 
Dll"I} [ rL '(t- t /'L) I (2 2(~ i fL) J =-0 +- ..,. r 
We may differentiate a g ain and substitute for D' 
For \'1 :» I 
and rJ .J..., ~ L >'/ n 2. 
A (]I) _"-
i ~ fr+ 3ft) nj 
so the gravitational field Eab decreases as Cl - 1 and the "energy\! 
l.(E Eab H Hab ) rl -6 
2 ab + ·ab as ~ L • We might exp e c t thi s as the 
Bianchi identities may b e written9 to the linear approximatj.on, 
Thepefore if the interaction with the matter could be neglected 
f""\ rJ -1 Cabcd would be proportional to ~ l.. and Eab ,Hab to 1 J.,. 0 
In the steady-s tate universe when ~ and e have reached their 
equilibrium values, R~6 ~ I ~ + 1'1-)~cJ, 
Je<-bc-=' Rc[aibJ -t S~lCA'R;bJ 
c.. 0 
ThUS the interaction of the !lC 'I? field with gravitational radiation i s 
equal and opposite to that of the matter. There is then no net 
inte raction, and Eab and Hab decrease as Q -1 • 
The 11 energy" ~ (EabBab + HabRab ) depends on se.ond deri vati ve s 
of the metri~. It i s therefore proportional to the frequen.y squared 
times the energy as measured by the energy momentum pseudo-tensor9 in 
a local co-moving Cartesian coordtna te system , which depends only on 
first derivatives. Since the frequency will qe inverse ly proportional 
to Q ,the ene rgy measured by the pseudo-tensor will be proportiona l 
n -4 to ~L as for other rest mass zero fields. 
9. Absorption of Gravitational Waves 
.. .• .-._.o::II::.~~~-"*--=-.. . - - -- - '-'=-:=-.=00 
As we have seen, g r avitational waves are not absorbed by a 
perfect fluid. Suppose howe ve r there i s a small amount of viscosity. 
We may represent this by t he addition of a term J.... CJ 0..'0 to the 
energy-momentum tensor, where '). is the coefficient of viscosity 
(Ehle r s 9 (1 7) ) • 
c· u lnce 
we have 
(26) 
Equations (15) (16) become 
- - 1z. y! -t fL) OOl b 
( E Clb - ~ 00..1, e) 
~:L A HCJb (28) 
The extra terms on the right of eQuations (27), ( 28 ) are similar to 
conduction terms in Maxwell 's eQuations and v~ill cause the wave to 
decrease by a factor e - ~t. Neglecting expansion for the moment, 
suppose we have a wave of the form, 
E E L'V'r ab:=:' -C(bE. 
o . 
This will be absorbed in a characteristic time "2.../). independent of 
freQuency. By ( 25 ) the rate of gain of rest mass energy of the 
matter will be ~A if 2. which by (1 9 ) will be "" E'2.. -2-~ )). 0 Thus the 
available energy in the wave is if F2 .-2 0'::' )) • This confirms that the 
density of available energy of gravitational radiation will decrease 
as (2 -4 in an expanding universe. From this we see that 
gravitational radiation behaves in much the same way as other 
radiation fields. In the early stages of an evolutionary universe 
when the temperature was ver'y high we might expect an eQuilibrium to 
b ( ) set up be tween black-body e lectromagnetic ~ diation and black-body 
gravitational radition. Since ,they both have two polarisations their 
enepgy densities should be equal. As the universe expanded they woul (. 
both cool adiabatically at the same pate. As we know the tempepatu~8 
of black-body extragal actic elec~~ ·:> omagnetic radiation is less than 
5°}C , the terrper a ture of the blac;K-body gravitational r adiation mus J0 
be also less than this which wou~.J be absolutely undetectable. Now 
the enepgy ef g ravitational radi 2 ~ 10n does not contribute to the 
ordinary energy momentum tensor 'r ab ' Neverthe le ss it will have cm 
active gravitational effect. By the expansion equation , 
, 
fJ ::0- ~ e'l. -2..d""'L--k (/;A+3~) 
For incoherent gravitational radiation at frequency v 9 
But the energy density of the radiation is 4-
Z. -2.. E V o 
whe1 8 I-LG is the gravitational 11energy" density. Thus g ravitational 
radiation has an active attractive gravitational effect. It is 
intere sting that this seems to be just half that of electromagnetic 
radiation. 
It has been suggested by Hogarth (1 8 ) and Hoyle and Narlikar (10) , 
that ther e may be a connection betwee n the absorption of radiation 
and the Ar'row of Time. Thus in "Lmive rses like the steady-state, in 
which all electromagnetic radiation emitted is Jventually absorbed by 
other matter, the Abs orber theory would predic retarded solutions of 
the Maxwell eq.uations while in evolutionary universes in which 
electromagnetic radiation is not completely absorbed it would predict 
advanced solutions. Simj.larly, if one accepted this theory, one WOlJ.ld 
expe c t r e tarded s olutions of the Einstein equations if and only if all 
graVi t ati onal r ad i a tion emitted is e ventually ab sorbed by other matter. 
Clearly this is so for the steady-state universe since "- will be 
constant. In evolutionary universes ~ will b e a ftmction of time. 
We will obtain compl e te absorption if \ >t d.z.- diverges. 
u 
Now f(",:, a ga s ? 
~ oC. Tt where '1' is the temperature. For a monat omic gas, T c:f:.. n -2 , 
therefore the integral will diverge (just). However the expression 
used for viscosity assumed that the mean free path of the atoms was 
small comp ared to the scale of the disturbance. Since the me an free 
- 1 r'I -3 path QC. I-l ~ J. t. and the wavel eng th cA (1 - 1 , the mE: 8.n free path will 
eventually b e greater than the wave l ength and so the effective vi scos ity 
n - 1 • will decrease more rapidly than ~ l Thus there will not be cOTIwl c t e 
absorption and the the ory woul d not pr edict retarded solutions. 
Howeve r this is slightly academic since gravitational radiation ha s not 
yet b een de tect e d? l e t a lone invest i gat ed to se e whet he r it corresponds 
to a retarded or advan c ed solution. 
Re:[§.rences 
---(1) Bonnor W.B.; M.N., :L11, 104 (1 957 ). 
(2) l,ifshitz E.M.; J.Phys. V.S.S.R., iQ, 116 (1 946 ). 
(3) Lifshitz E.M.? Khalatnilcov Io N.; Adv. in Phys.? .1£, 185 (1 963). 
(4) Irvine W.; Annals of Physics. ~ J ~'2 2 (lQ6'f) 
(5) Jordan P., Ehlers J., Kundt W.; Ahb.Akad.Wiss. Mainz., No.7 (1960). 
(6) Kundt w., Trtimper M.; Ahb • .Akad.Wiss.Mainz., No.12 (196~).. 
(7) TrUmper M.; Contributions to Actual Problems in Gen.Rel. 
(8) Rindler W.; M.N., 116, 662 (1956). 
(9) Penrose R.; Article in 'Relativity Topology and Groups', 
Gordon .and Breach (1964). 
(10) Hoyle F., Narlikar J.V.; Proc.Roy.Soc., A, 211., 1 (1964). 
(11) Raychaudhuri A., Banerjee S.; Z.Astrophys., 2§, 187 (1964) . 
(12) Hoyle F., Narlikar J.V.; Proc.Roy.Soc., A, g§l.., 184 (1964) . 
(13) Hawking S.W.; Proc.Roy.Soc., A,<:-ifl p~. ~ ;1';> !//ttJ) 
(14) Hoyle F$' Narlikar J.V.; Froc.Roy.Soc., A, 273,1 (1903) 
(15) Sciama D.W.; M.N., 1.12, 3 (1955). 
(16) Roxburgh LW., Sattman P.G.; M.N., )~ 181 (1 965 ) . 
(17) Ehlers J.; AhboAkadoWiss. Mainz.Noll. (1961). 
(18) Hogarth JoB.; Proc.Roy.Soc. 9 A, 267, 365 (1 962). 
C HAPI' ER 3 
Gravitationa l Radi a tion In An 
Expanding Unive r se 
Gravitationa l rad iation in empty asymptot ica lly fl &t 
space ha s b e en examined b y me a ns of a s ymptoti c expans ion s 
by a number of authors. (1 -4) '~hey find tha t the d ifferent 
components of the out <~oing radi a tion fi e ld Ilpeel off ", t h a t 
is, they g o a s different powers of the affine rad ial d istance. 
If one wishes to inve s tigate how this behaviour i s modi f i pd 
by the p r es enc e of matter, one is f a ced with a dif f iculty 
tha t do es n ot a ri s e in the case o f , say, e l e ~ t roma3ne tic 
r a di a tion in matt e r. For thi s one can consid er the radiat i on 
travellin~ throug h a n infinite uni f orm medium t hat i s stat i c 
a p a rt f rom the disturba nce created by t he rad iat ion. In t h e 
cas e of gruvit ~tional radi ~tion thi s i s no t p08di bl e . Hor , 
if the med ium we re initia lly s t a t ic, its own s elf 3ravitat ion 
would cau se it to c ontra ct in on itself and it would cease t o 
be static. Hence one is force d to investigat e g r avita ti onal 
r a6 i a tion i n matter that is either contrac ting or expand ing . 
A S in Chapter 2, we identify the Weyl or conforma l 
tensor ~- ~,\. b u,L 
o 
Ricci-tens or KQb 
vdth the fre A gravitational field and the 
with the contribution of the ma tt e r to the 
curva ture. Instead of considering gravitational r a d iation i n 
asymptotically f lat space , tnat is, space that ~pproache s 
flat spa ce a t l~rge rad i al distances , we c ons ider i t in 
a s:f mpt ot ically conforma l l y fl at space. As it i s only 
con forma lly flat, the Uicci-tensor and the density of me tter 
n e ed not be zero. 
To avoi~ essentially non-gravitat iona l p henome na suc h 
as sound waves, we will cons i~er g r avitationalradiation 
travelling through dust. It was shown in Chapter 2 that a 
conformally flat universe fill ed with dust mu st have one of 
the metrics: 
cl, < .. S'J..." (C( 1; " - c!'-f' - ~ ,,' 'f (,L El \- \", " (U f)) (a) 
(b) 
(c) 
SJ... ;. A (1'- CoS'. c) (I . I J 
d~ 2 > 12 ~ ( dtl .- cL f J - f 'I (c{()'- -d'A '-() cL ~ l~ 
Ls- '1 C 
.J~ 
_Sl _ ~At2. ( /,.2 ) 
Type (a) represents a universe in which the matter 
expands from the initial sinBula rity with insuff iciant energy 
to reach infinity and so falls back a g aj_n to another 
singul a rity. It is therefore unsuitable for a disc~ssion of 
gravit a tiona l rad i at ion b;y 3. method o f 6.Sympt otic expansions 
s i n c e one c unnot ;;et an infini te ci i stance from th~ source . 
Type (b) is the ~instein-De Sitter un ivers e i n which ~ he 
matter has j ust suffi c ient ene r gy to re a ch i nf i n i ty. It i s 
thu s a s pecia l case. D. Norman (5) h a s investigated the 
" pee ling off" behaviour in this c ase using Penro se ' s confor mal 
techn i q u e (6). He was however forced to ma k e c ertain assumpt -
ion s about the movement of the ma tter whi c h wi ll be s hown to 
be false. Moreover, h e was mis l ed by the s pecia l n a tur e o f 
the J~instein-De Sit ter universe i n which affine ari d lumi nosity 
d i s tance f:3 differ. Another reason for not cons iderin;; r ad i '3.-lion 
i n the Eins tein-De Sitter univers e is that it i s unstable. 
Th e passage of a g r avitational wave will cause i t to contract 
a :?:; a i !l eventually and develop a s i n ;:;ul a ri t y . 
~ e will therefore conside r radiation in a universe of 
t JPe ( c) which corre sponds to the g eneral c ase where the 
matter i s expanding with more th-=. n e nough energ y to avoid 
contra ctinb again. 
2. The Newman-Penrose Forma lism 
i:Je employ 
tetrad of null 
the not a tion of Newman and Penrose.(3) A 
fA-- I~ -nf"'- ~ rz 
vectors, L J n) {It; (1l is introduc e d 
6r avit ational radiation b;y a me t hod of ~ympt otic expans ions 
s i nce one c&nnot ;.;et an infini t e cL i s t anc e from th~ source . 
lJ.'ype (b) is the ~instein-De 8 i tter univers e i n 'iJ L1ich i; h e 
matter has just sufficient ener gy to rea ch i nf i ni ty . I t i s 
t hus a s pecia l case. D • Norman (5) ha s investigc1.ted the 
"peel i ng off" behaviour in this case using Penros e ' s conformal 
techni que (6). He was however forced to make ce r tain a s sumpt -
ions about the movement of the matter 1,.,rhi c. h v,i 11 be 8ho\"1n to 
be fal se . Moreover, he was misled by the specia l nature of 
the gins tein-D e Sitter universe in which affine arid lumi nos i l; y 
di .s tance :3 differ. Anothe r reason for not cons i deril1!"; r ad i ;:l.tion 
in the Eins t e in-De Sitter univers e is that it i s unst able. 
The passage of a gravitational wave will c ause i t to contract 
a~ ain eventually and develop a s in6ularity . 
~ e will therefore cons i de r r adiation in a univers e of 
t ype Cc) which corre sponds to the general case where the 
matter i s expanding \·Jith more th",n enough energy to avoid 
contractin~ again. 
2 . . The Newman-Penrose Formalism 
We employ the not ation of Newman and Penrose.(3) A 
tetrad of null vectors, L~ ~ r0~ ('L f<. is introduced 
we l abel the se vectors with a tetrad index 
~ fA :: (L !~ n.~ r~ I~ Pt fU' ) ~ ~ I 2 '5 4 \ I , 
tetra d indices a re r a ised an~ lowered with ~he metri c 
(o,b 
~q 7 ..: I 0 C) 0 ;. {~~ 0 Q 0 
I : 
I 0 0 0 ~ j I I I I 
lO -I 0 I Q I J 
vIe ha ve 
Ricci r otation coefficient s are defined by: 
be. 
'/ ~ -
In fact it is more convenient to wor k i n t e rms of ~w elve 
comp l e x combinations of rotation coefficient s def i n ed as 
follows: 
f\ = Y 
I ~ i 
1] 
I 
U 
( --
~l 
-
-
(lC, ) 
· I 
Like Newman and Penros e , we int roduc e a null coordina te 
Gl(= Xi) 
o 
we take Thus ~'lill be 
3eodes i c and irrotat ional. This i mp lies 
v - 0 ( ~ 
-( ~ f 
CC :: -[ 
l' .-
Vie t ake 11 t:', fV\ ~ A ~ d-+~ 
along L tv'- 1'his gives 
11.: [; 0 
A S a ae cond coordinate we 
long the ge odesics L fA 
~ [fA;;. 
t ake 
to be parallel l y trans port ed 
GJ) 
an affine paramet er rr (-- Xl) 
(-S .!!) 
J r /f 
X and X ~\ a r e t\'lO coord i nat e s tha t l abel t he gco c.l. e :..: i c 
in the s urfac e 
'l'hus 
In these 
The Field E~ u ations 
11,1e may calculate the Ri c c i and ~ie jl tensor c omponents fro m 
the r e l a tions y l t d' 
I b I " o:..c.L e..hc (I. c e c 0. bp ( C <: ;)) ( , ) 
G-.. b veL ~ 0 c : (L CA a.. C .~ V V _ ... / ." _ -'V - () J, Y R ~ (( '- (( 100 (} .rO 
-e e... Q.. E... 
(~) 
Using the combina tions of roi.; 2,tion coefficients a lready 
defined a nd with 1< ':; TT::: £. -;: 0 Vie have 
Yt 1 ~oo ( 3_, i 0) - f ~ 0'6' 1-j) 6' 
- .? \6' -t- Yo 6 It) ~O\ p vc '1:(' f- ib ~ t. 1- (~~ i 2) -
V d. :: o(f + Bc ,.;- ~IO (] , 13) , 
+-~- .. - ... j)? - ~f r ol..C -+ ~, (~ , lif ) - fl ,- 1\ -t ~'I ,----PO' - ~ i5) to<. -t V" , (~ , - 1... •• 
;\ F -----)/A - -t fb 'r 1 (5, I b ) - 2.0 
P t-l ;;: ~\F 'r ~ b' -I ~(f"' :l. -r 3. ,/\ (~~_!l) ~~I j)v = L\ + 'f' (v- ' -r f'3+ (] __ ~_{r) 
j;\ -- S v ~ J :;z v -r (y - ~ Y - fA - f );\ - ~ (l .' ~ ) 
~'f ~ ~~ c- ~- ( (? r;: ), -t (~ - sO\ ) b'- ti -r pOi Cs ' 'lei ) 
-~ cL - ~j~ ~, f F - ,-\ G -- 2()( F 't cicz '-r ~? - Lt;'1- /\ --r ~II (L L I) 
n - f,~, ~ ( cl. -t ~ ) /''-..,. ( J - 5 ~) . - lh T p 1/ (:s 2 ~) 
a v - Lj f'-: Y t - :2 V flT i r -r f' 2., ~ It? n. C "-__ ~ s) 
~'r - i',? ~ l/,,"- <> v r (r - 0' 'r ? ) (3 r ~ d, T P, J. (~~!f) 
r I- Lk = ,h ~ -r (i r f' - J 1 )., .,. if -r 1 () 2 U t?) 
. 
Af-h = ( rri·-f )f -:Ju.'L- ~-Y; -2/\ Cl·<i:) 
11",- df 0 (V - d-.Aft (') __ 21) 
. y- (i'-r- r ) 01.- rJ 
wher e YJ 0 L fA Vr ~ ~~~ 
L\ ~ nf' Pr ~ v f.:; 
POG K - (~. 3 i) I too . - .-- -- 2 -l/ 
--
~ 1I - -~ (I~ 1- f{ ) - ~ II (J . r~ L) -
I.r j Q 3 1.f '-
~" , .:: - .l g - 11 0 (5. 53) J. 13 
- -'-
-1- R - (3. 'j y) 91'1 .- cr :t. 1 - -2 13 --
.- J R --It) .- CP'J-O (Ll~) - J 'D -0'1. 
-.1 r ~ ~J.~ .::. - ~ ;;.~ (1 . j ~) -< ~~ -
1\ .:. f< ~~2) -LLr 
to - - c = .- C L q(r? rs L t fY'l ~' (~ . ss) I"s/3 (j( ~(.r 
r 
r( -::. - C ~ - ( L -<n~L ¥ (11 9 (3 . 31) 1'2. 13 o(PO~ 
~"--
r~ .::. - J C C '1 C ) " 1 j'l./'l ):.}. 3 4 - )e f "2 ol.~ 0 
x (e "'/LfSZ ( fL s; -r t"-}1f f\ 1fi1 ~) (i ~ 0) 
- C L.(. (L~ (2Y fit 
vi- P o~ 
Express i~; the rotat ion coefficient s in terms of th~ metric, 
\1e have: 
'j) W " P P T 6 > W - (0/. -7-r ) ~~:0.) 
J)X L = z:f' -r i( 0 , 0':;) 
]) U 0 1, ~ or >~ w- (0 + if' ) (1 ' ~ 6 ) 
~)<~-L'lS:; (r- "-i - O)( T)§" (? ' ~7) 
{ $ ~ _ S (~ (~ - <X. H L >, ( 0( - ~) ~ L (1 it iJ ) 
g ,;j- &-; w;o (~ - "') vJ+ (;; - (5 ) W >, (r -/~ ) ~ 
gU - l1w = (t<r ~ >- r) w or ,A oJ- V 6 ~><v 
As in Chanter 2 we u se the Bianchi identities as 
-'-
fielQ equations for the Weyl tensor. In the Ne wm an-Penrose 
formalism they may be written: 
(1 am indebted to R. G. McLenaGhan for these) 
g -ro -}) 1(, 4- ]) cP o-r- ;; %J' 'f "'- to - 4 F 'V,- (2;,.-td/l ) too 
-t- '2F 10;"- 26' (il ,O . (), 5 j ) 
6 t - ~,y <t ])~O~ - ~CPo, :: (~o ·· ~ ')yo - :2 (2 L- -t- f~ ) Lf7 
<>t- s6tJ. - J~oo - 2~1o/ ·~ ~bCPilcrt 10Q (], S~) 
S( $ 'f; - Dt. ') ,. ;) (j)~" - ~ ~I()) 'r ~ ~bl - Ll 000 :: 1'\ tu -9r t 
T {;.,( 1(, -I (t - 2 ,v. - :2 r - 2 i ) 0/ bO 'f (2 d-.~ Q-E) Po I 
. -y J. (T - 2 ~ ). El) .} qt1 rh t tLb rb - C rh ( 5. 53) 
'-f/o r 'fli I ~o /,01-
·~(tJr ~ ft,)-I 2(D~IL - ~ ~J<f (fTo': i1~ol) o 3rta r 
_<>t ~ ( r - f') t, - q"t tl. 0} (,6' '(1 - v'too 
t- 2(r--/~-r) 10 1 - .2/\110'" ~0::rf" 
. or (2 J.. -t 1::- J. p) cb 0' 1- .2 G' (h (. ) - \ 
_ T J.. T ol/ ..) 'J ({ ) 
1 (S '1j\ - j) \jI; '> 'r (J) 4 '- I - b' <P AD) t J (r 1 " -!j 1 i J = 0 . r 
6CfJ - jr- ~oo <>j- :r,\ ~Ol T :) (f .- f'- - j; )r, G 
--t- 0"[ 1" T (2~<Y:1 'c- 2'J.) r)'o 96- fl2 (] 55) 
3 (jr2 - S tJ ) + (D ~'L 2-- ~ ~'l-I)"t- ? (ff f, ?-~ /j 9"l 
, " (, V V. - 9 f" t "- .. b (rs - 1:;) 1(3 -. ~ b ,t 'f , 
- .2 V cp Cl 1 - ) V 1 I tJ I :I { 2 F -r ) cp" r 2. A 102 , 
I ' - J12.o r ~(v - ;<rf)1'2 -r 2{Jn c7~"-I-P ~J~ U~-6) 
r ~ t-~ - /)r~ -r- ~ 1'L'- 111'>-0 ::: 1XY; -- :)~ '1'3- ft~ 
- 2 v cP 01 --t r:l)' f I. 'I' ( P r - :2 r 'T fA) cf20 
T 2 (t-C - 0<.) 1~ 7 - 6' 1<l. ~ (J. S i ) 
/j t(3 -- ~-r4 -t t 1,,- 2 - jj Cp<-, " 3v f~ - 2 (f T ~f'-)YJ 
, t (~~ - l;) Y't - :2 v 1" - 17 110 t Q >- ~. Q 
'I ~ ( '( r p) 1:1./ -t ('i - :2? - 2 ~ ) CPa) 
.- . - ( 3. ~ s-) 
, p11~ - r Pit - S' ~<:J~ <f !JcPOI -.- 3 SI! == (, r-f - 1F ) 1\. ~ v too 
,- ).. ~(} - :} c ~, -r (2~- 2 ~ .- r) % ~ <¥- 3f ~, .2.'" 6>42., () <)1) 
I ]JJ, - r; ~l ~ ~) rh ;- 6 cil ~ :fnl) :: (2 r-f' -;-2.r--;e:) rh -- z~ 1- i )" rh 'r., 'f'O 10 I Too Y , l oo TUI 
- .2 (; <; ~) ~'O + ftp 1" 'j- 0cpD;l 'f'<;;"'12-6 (~ . {; 0 
D tb _ S' ,l - S rh -j Ll rh i- 3L111 '" v JJ r i/ (~ -- 2 ('v ( j-; ). cj) jL:/. 't''), / 11'1. I" r D/ riD / / / '1 
~ ,\ 40:J. - A cp 'J.O 'I- (~~ - -'[ ) cp,?. ;- ( :2 f - l:) cA-I 
et- :2 r et ~ ~ (~ . b !J 
I, 
I 
. Lj-. The Undisturbed Metric 
put 
then 
The undisturbed metric may be written . 
tLs '2, SL '1. L cL ~ <. - c~ f 1. - S i"- P P ( cl../} 2 r'S i f\ :2 d cp 2 ») 
5L -::. (.) l CO~ h t - ) ) 
tL -:: b ._ p 
I 
IJ~ o c a lcula te t', the affine parameter, we note thi., t C 
is an affine p a rameter for t he metric within the square 
br t·~ c kets. Therefore t -= J.Jl. 'l cL L- T 15 ( u-, G-" ~) 
will be an 8.ffine para meter for (4- ,. / ) 
15 is const~fit along t he hull g@odesic. Nor mal l y i t 
would be taken so that r.:: 0 when t ~ u.. . Hovleve r , 
i D our case it will be more convenient to make it zero and 
d efine I(' as 
r " ft'Jl2JI::' 
o 
Ifhi s means that surfaces of c onstant r are surfaces o f 
constant t This may seem r~ther odd, but it should 
be pointed out tha t the choi~e of R will not affect the 
a symptotic dependence of quantities. That is 1 if 
Then 
r -- r -t \$ 
r 
It proves e ~sier to perform t ne c a lculations with t hi s 
c hoic e of r but all results could be transformed b a ck 
to a more normal coord ina t 0 system. 
l"rom r 
The mat ter in the univers e i s a ssumed to be dust so it s 
e n ergy t enfJ or may be wri t e en 
For the 
NOVJ 
whe re 
undisturbed c a se, from Chapter 
~ :;. bR 
-52 3 
V ::;. 52 C;) 0-. 
0-
5)- = hs<>tA -#~ 
~'l ::. t 
2 
Th ere fore if vJ e try to expand f>' a s a s erie s ,in po\V ~r o f S; 
the r esult 
ttle form 
will be 
L 't') ~ 
. 08 <.J 
S fl. 
very messy a nd will involve terms of 
* 
*It s hould be p ointed out tha t the expansions us ed wi ll 
only be a ssumed to be v a lid asy mptotic a lly. They will not 
be assumed to converge a t finite d istanc e s nor will the 
, quantities concerned be a ssumed analytic. (see A. Erdely i: 
Asymptotic Expansions - Dover 
This (ioes not invalidate it as an asymptotic expansion but 
it makes i t tedious to handle. For convenienc e therefore, 
I'le will perform the expansions in terms of .52 (r; vI/hich 
will be de fined in general as the same func tion of r as 
it is in the undisturbed c a se. 'rhat is 
~vbere 
then 
51. -=- () (cos--J. t- ~ i) 'J' 
., ::. rr). [t S'f'-h2t - Qr;/,J, t " % t_ 
cL ·52 
.-
'-
-
For the third and fourth coordinates it is more c onvenient to 
use stereog raphic c oordinates than spherica l polars. 
Since t he mat t er is dust its energy-momentum te ns or 
and hence the Ricci-tensor have only four independent 
components . v/e \vill take these as A I ~o 0 I cp OJ 
( Dince CPOI is c omplex it r eprese nts two components) 
In te rms of thes e the other components of the Ricci-tensor 
may be expressed as: 
. ~o~ cP0/ 
~oo 
.~ ~r 
t 11 10 0 
-
..... 
0/1 ~ ';:; 12 I .- b 1\ ~()I (I -:~a'_ ~-M ) Q - + 
~oo b !\ '\PcJo 
4o~ 1~o to; ~ -. -- (er . 10) 
100 
For the undisturbed universe with the coordinate system 
g iven: 1\ = 
-#=- {-J 
-
- "3 i4 4~~ 
90 0 ;. lE S2--; 
4·, ;:; 31i 452 3 
4~'l - 1fj - 452 
~ (J I - ~Ok - 0 ( ~ . ,,) , - ...:.-
Using the s e values and ·the fact that in the undisturbed 
unive rs e all the r "s are zero, we may integrate equations 
(3 . 10-50) to find the values of the spin coefficients for the 
unperturbed universe: Q _ n (. A 1.. ~ A':l. ;:z u.) ))-4-() _ ~ -r - _ e. Jt:-
I 52 . .523 J. . 2 . 
R3U - e Q",)52.-S- "I A"(f- ~e Q~ ~ e- 4U)Jt£ 
v .: ) :: Xv : 0 . 
, . , 
G - 2 S' ~ iA_ I - . ~ - ---- - 1- If)V~<2-4 
-
, 
'r 
..5L 7-
-..52. 1 
H Q. . e. '2'vI.) SL-2t (13 ( Ir 2 e 2;.12 -si-. I~ -- - Jt (i et-
-252. ... 4- ~ 
-(-) 
..J... ft'l. A~ et-
.2 -
:152. L SL "2- 4J)..~ 
./ S2. 2 (~) J 
5. Boundary Conditions 
~e wish to consider radiation in a univers e that 
a s ymptotically approaches the undisturbed universe ~iven 
above. cPOl> and i\ will then have t he va l ;Jes 
6 iven above plus terms of smaller order . . To d.ete r.mine this 
order a.nd the order of rpC>1 and % , t Clere· are t V!O 
wo..ys in which \'le may proce ed. ":i e may take the s mall e s t order s 
t hat will permit radiation, that is 
Lar~er order t erms than these in and 
cPD1 
dependent only on thems e lves and not on the r-J coefficient 
turn out to have their derivatives 
of , the radiation field. They are thus disturb~nce s 
not produced by the radiation field and will not be consid e red. 
Alternatively we may proceed by a method of succes sive 
approximations. We take t he und isturbed values o f the sn in 
cosfficients and us e them to s olve the Bi a nc h i Identities a s 
fi eld equat ions for the conf orma l tensor using the f l a t suaCA 
boundary cond ition that ·~ o :: 0 (V ~ '? . Then substituting 
\J.r' <; these r in equ a tions (J . iO - ~ r ) calcula te the dis t u r b -
ance s inCLuced in the spin coef f i s ients and subs t ituting the 3 G 
b a ck in the Bianchi Identities, calculate the disturbances i n 
'11 ('IS the I 1urther iteration does n ot affect the orders 
of the d isturbances. 
Both these methods indicate tha t the bounda ry condi tions 
should be: 
0(.51 - 1) 11 · A 'f (~' ) ~--
'" &.52 ] 
~oo -:. 3A + () (JJ_ -<7) (~) ---". 5l~ 
tOI - D(52~-}) - (see next secti on) (~~) 
yo :: O(51 -=r) (~J) 
\1e also a ssurne"uniform s moothness", thc-:l. t is: 
L J d 1\ ::. 0(51 -=t-) 
. ~! . , ~ ... --~X (... d XV 
~A -:::. -.sf-l "+ O(Jl -~) )51 ~Sl4 
etc .•• / 
.J..ij vi.i..J...L be shown that if these boundary conditions 
hold on one hypersurface (~= const.) they will hold on 
succeeding hypersurfaces and that these conditions a re the 
I most , s evere to permit radiation. 
/' 6 Int esr ation 
As Newman and Penrose, we begin by integrating the 
equations (3. 10 &11) 
Pp f"L .- 100 "t- 06 '"t :::: 
fla :: Jf 6' t- to 
where 3r:l 
O('t -3) ~oo :; 411 r 'f y-
Let 
..: 
then 'OP ~ P ",2 'r <\J (t~/) 
l e t p - -(JJ 'I ) \_(- I ((;2) 
-
t hen f) ':( . '1 ~ - CRI ((~ 
s ince S'r C(J cL, ..tf cL) 
P Lt ;. r"r 0(,) where F i s cons t ant (6 ' ~) 
"-I - r F -r 0 (, ) 
Howeve r ce :;. 0(7· -1) -] _ J. ) 
therefore D'J.. y _ - , cpF .,. 0 (, ~ ) ":. 0 (., .~ (~S) 
therefore -p Y - Fro (r - "~ ) 
'V .:. T r r 0 (T ~ )Tt, (is c onstant 
p ~ - f - I.r T 0 (r -1 ) 0. b) 
if ~ i s non- singula r (The case ~ s ingular corre s p onds to 
a s ymptot i c a lly pla ne or cylindrical surface s a nd will not b e 
cons i d e r ed he re). " 
.- I ( .-1.. ) C) 5) - ? ""'/ ri-.s) 
T h us f -:. -. r I 0 r ~ :-.: .... J.. . ~ ." ~ L 
6' := 0 (r -"1 ) _ 0 .( 52. - "3) 
~ - 2 Jl -:J.. r ~52-:5 
~52 -3 
Let f 
6> 
where ~ I "- J :: o( i ') 
Then u s ing : 
'U : ) 
Jt 
{\ - j ( (J '-.!.. A '.l <1')., A ~.' \l_ ~ 
- ...J Lit A.) L "_ J 1- cf .J /~ 
. ..-- . . 
J J. 
.. )2 
)Jl 
therefore 
.B' or I-L I 
li(Jl~o(,)\ ~- ~TO(5L - I) 
d52 J . 
therefore 
r 
Hep8 o.t the process with 
p ._ -:2 52. - 2_ {'-} Jl -3 T 
\6' 7;. A,52-~ 
wh e re 
)'\,.. 
cl1. (J'- -r 6(1) 0 0 (52 - ') 
J.SL It- ;; 6' 0( u-, ')I. <- ). -r 0 (31 - ') ((0-3) 
the n 
~ (~51~ 6 (/)) '-' 0 (St-I ~S5l ) 
J51 
~ :; ( (lA, X ~ ) 'T 0 (rr 16:,552. ) (t; . / It ) 
aD Unlike Newma n and Unti, we c annot m~ke r 
I 
zero b y 
the trans formation tt'/ :::.. 'T" - fJ <.:I , sinc e thi s would 
I Ft, K 
a l te r the b ound ary condition /\ -::. 52':3 . -r ('\ '-:=r 
(; .J1-
Coritinuing the above p rocess we d e r ive: 
--:.2 .- S 0 fL'f · ( .l '- 2 r-t O))l -s-f ~ - 252 "- A Jl or r· I • ..J 1" 2 A - · f . 
-2. A 't,. 4 p 0._ t- - G' 6' 52 I 0(52. - l§~ (
'"" (, A'J.. 0 '.)' 0 -0) -6 7) i:. ,('\ 
~ V I ~ 2 
, , 0 .:; ()~, - ~ ._ (? A 6" 0.,. 10 ~ ) Jl-- ~ , 
o 1 .~ (0) 
i.. 
'110 determine t 'ne asy mptotic b ehaviour o f ~t j 0/.. JP / 5' 
a nd u...:J we use the lemma p roved by Newman ~ nd Penros e: 
The (\ x. I) matrixBa nd the column vector b a re 
g iven f unctions of X such thc,t: 
B -=: O(X-Q.) 
I 
(0i) 
The () x ~ matrix ,r.} is independent of X and h a s n o 
eigenv~lue with positive real part. Any e i g envalu e with 
v ani shin g re a l part is regular. Then all solutions of : 
J i 
.. (A >( ._ / i R ) T b 
'- - J dX 
are bounde(i as X .-) cD • j is a c olumn 
vector . 
ll'or reaso; s to be explained below, vie will C3.ssume for 
the moment tha t 10 1 .-
~ 4,,1 
)51 
~ 
d xJ 
\. 
c(SL -S) . 
CJ (SL -b ) 
::. 
O(51 - ~) 
We take as j the column ve c tor 
~~) 
(S)' S- ,lr JL J. A 51"<' A J)l(\ 52 <., e 352.'2 e :3 nJ.C it.52 C 1-&J CJ] L.. - 11) J ) , r'" ) -..) ) ~ .) .)) JI I .LJ, 
(~) 
r 
By e qu at ions 3. S J 
J 
3,/3 ) 's . (~ 
I 
3 . ~s ,3· y: V 
A- ~ . 
. -3 o (, R 6ft 000 0 0 i 5.ft 0 
0 0 0 0 00000 0 0 
0 0 
0 0 
0 0 (:.JJ) 
() 0 
0 0 1 
0 0 
00 0 f)00G 0 0 0 0 
o 0 .... i -/0 0 0 0 0 -2. C) 
(J ~ I 0 o - 100 0 0 a -2 
E and b a re 0(51-1) expressions i nvolvine; 
F J Cf) * ' ~"fo 'lL to ' 4 Cl and)i ?Oi 
'i 'hus r: ;:: oC 5L -» 
tA, ~ I 5' J S ~ '" 0 (51. " 2- ) 
W :: 0(1) 
0 inc e ~ ;; J.. 1-8 
o (SL. - C).) . 
lJsinG thi s VIe i n t egrat e e qu at ion (s., 'J...) b y the same 
met hod as ab ove . 
r 
~e may make a null rotation of t he t etrad on each null 
ge odes ic L' 
-= Lj-t fJ'- _., {fA (Llf- fl.f ,. Cl fYt/...J- .- o...fVLjA- i Q.O-.. :: 
M ";v. :- fo1l~ .,... a. Lt-
0- is constant a lone the ge odesi c since the tetra d i s 
paralle lly tr~nsported. 
By taking 
iD 
'/;. ~ 0 we may ma ke 
0 nde r a null rot a tion 
~;o, = ra. -t D. 1'00 
Phus until we h ave specified the null rotation we c ann ot 
i m!_J o se a boundary condition on ~~ 
. '/Oi 
more sev ere than 
We will specify the null rot e .. tion by 
vc Q ::::- 0 and in tha t tetrad system will imp os e the 
boundary condition that ch ..:: 0(51-=1-) and is uniformly smooth. 
'ro/ 
Then by using this condition on rh and -L:..::: 0 (Jl-j 
rOI 
by equation 
w 
r 
'-
using this in eQuation ~. 5 i) 
~ :: 0 (-n--6) 
then by equat Lm h. . /'1) (. ~ \.i)' \.J VC .; 0 . ..l'L 
putting this bac k in equation 6·4' 1.{) 
w :::. 0 (5L - '1 [05 ·52-') 
by equation (-g.sq ~I = 0(52 -1 LOj 'SL) 
b y e quat ion (1 . (2) ~t ~ 0 (SI.- -~ LoS ·)2 ) 
by equation 
by equation (1. ~ /) 
, if' .~ 0 (.5J- -7) 
'(, 
by equation 
by equat ion (3 ." "I) 
51-1 \\ - J) w :::; v...J 0 i (j(...JL 
(6, 15) 
Cb.2.!J 
i.-
By differentia ting the equations used with r espect t o X 
one ma y shoH tha t ''/1: ) 0<. I )3> I <-t I g l-) W a.re 
uniformly smooth. 
Addin~~; equations a nd: S , (;, 0 : -
I !t - j)~1. <f Pt .. ·-g~ol't DJ! , ). to -3(' t~ t '10. t - t too - '2 J.r. o 
-r .i F ~" 9- 6' T'lo Cb 21() 
We may use t he lemlna a gain ~:l i th J 
Jllf' ~l J - J1'2. -
-f1_ ~ --
By equ 3.tions S . /G' '3 . 11-" t;;. 2. g-. I ( 
- 2 
o 0 0 
o 0 - I 
:E and b -are O(5L -2 ) 
'~1. :: 0 (J1. '- ~ ) 
~ '-' 0 (52 - ~) 
There fore 
O{fl - l) 
and are unifor mly . smooth 
From (r;, . 2 q) and Cl . It) \..;e may show 
f-" ~ t n -I 'r o(n -2. foJ 52..) 
using this in (6 . 2.~) 
LJr~ ~ 0 CSL·-5 fDS JL) 
then by C~. 11-) 
f'- ~ f Y2.. - l
t 
r C(t!v, xL)Jl - 2r o(n-3 0.,S fL) (~o) 
and Y<"J- :;. 0 (A - ~) 
r 
Adding e quation 6. ss) 1- :2 X (5.:,. ';9 ") 
d 'to -j) t of j) ~ 1./ - ~ 9>"0 1- J. S n ~ 9. ), r, -9 f t~ 
- 3 ( f- 1" 2 f' ) 1, 0 1" 2 ((S .- 01-) ~ 1.0 -f ~ f et 21 (£ J ;y) 
Therefore 
r 
By equation (1. /0) 
V ~ vo - it(; J1-~ .. 0(52 - 5) (G.~O) 
By the orthol}orma li ty relations ( 2 .1.) 
'!1~ . ' -j 0 X "-(fw T S"cJ) 
_ XI-°i- O(J-~-ltJ 
:3 cj- - - (r Y, r SJ) L, d' cS , 4 
- - (s"} JO ~ tS) O~ jC'-2 1151-<;) y 0(52 - G) (r,0}) 
By making the coordinate transformation 
U( 
- LA-
-
yl 
- I 
-I 
, (LL; XL) 1& L- Cl-X .- X ...,.. 
where 
.gC( 3) X \.f-°C 3 
- X I -r C.; 3 . '- . J l(-'3 C 
-J I. 
XI\-O ::: 0 
We still have the coordinate freedom 
xL ;:;. j)~(xj) 
\ve may u s e this to reduce the lea.ding term of 3 LJ (~,} ;. ~) t.) to 
a c onforma lly fl a t metric (c. f. Newman and Unti) , t hat i s : . j 
9 ci ~ I] "pP S'1 (.)C~- '2. fJ5c.-') r d51 -C) (/,~ 
vlhere 
7. Non-ra d i a l Equ at ions 
r By comp&rine; co effi c i ent s of t he v a riou s p O'.rV ers of JL 
·s] i n the n on-rad i a l equa ti on s of ) , r e l a t ions b e twe e n 
the int egr a tion constants o f the rad i a l equations may be 
ob tai ned : 
In e quation ·S. 23 the term in JL -I i s 
'3 A ( tJ r - 0) - S A 4" Q 0' l; 
t herefore ({ 0 ;- i 6 - - J 
t h ere for e by (to. '.) '}) 
J 'l. U ;J 1-' 0 (51 - j ) U ... 3 ... D, "t-
I n equation (3 _ SU) , the constant t e rm is 
\/0 
t here f ore \/0:: 0 
By the .)1. - 2. term i n . (s , ~ 7-) 0 . 
P ._ P.. ;: ( ? 0._ 0' ) y 
) , 
<~ By the ::fl te r m 
P,. ~ 0 
.I I 
t l.1.el'efore i 
-
_ {;1 
~ 0 - 0 1. 
if 
r By making a spatial rotation of the tetrad 
r 
I 
d j\ 
J~~ 
""here 
If -
-
By making a coordinate transformation* t: 1-1 cL 1/ er ( --
the r efore 
By the 
t rlerefore 
"' This transformation does not upset the boundary conditions 
on the hypersurface 
o 
u .. - ::. CL 
r 
By t he Jl'-Y . term in (? 2 5) 
>" ~ ~ ( 0' 0 .- 6' 0 ') (7-: i I) 
;) . jl .-
By the n - Y term in 0. 22) 
t -~ 0- ( 6 :,- () 0) EO L<- VS' - ~ El. "- S V (6~ ,O -6 0) (t-~ 
By the J2.. -2 term in 0, S-o) 
2 cv 0 .- L...J IQ :;: i li}'"\ Cl ), T '3 
.'. w
D .~ .~~(SV6 °-~ 26°\/S)-r K(X L) ~Q.~ 
4-
By the J1-~ term in (J .2 o ) 
<AS !! 0 / 0 L.t\,,\i7 0 _ J/\'0'·7("'0 ....... 
e, V r -r l{ W .- e ...) V t:' ~ ~ V J ~ 
t herefore C:;.. K -:.. 0 
the r efore J u :;. 0 
"l. '1...""- ) 
o fL (1 '- e F ~ 2 
Using (t . (it.'b), in (6, 2f[) 
'1"L ~ o(.n - 'g.,~ 51) 
(' 7- IV) 
(:r . /5) 
(r . !~) 
Bv (S , S J ) - 0· ~ I ) 6. GO) Q. G i) - v 
.2 ~01 - o CJL - :r) J l.,-
r d ~bO 0 (n - '1) -- -J~ 
~ !\ ./" o (52 - :r-) 
J ~ (?_ (cl) 
(~ -~) ~ t" :;.. 0 J~ 
The r e fore if the b ounda r y c onditions (~. i ~ <-t ) l10ld on 
one null hypersur f ace, t hey wi l l hold on succ eedin~ hypersur-
fac es . 
By CB , <;1-) 
o (5L - ~) tf't - ( }20J 
1].\ he "peelinG off" behQviour is therefore: 
·t~ - CJ ( t -i) 
-lf3 ::. () (i -1) 
f~ - o (T -]) (1- 2/) .-
L -
Yt - o(r -~) -
'fo ::. o (r - -i) 
As mentioned be fo re , this asymptotic behaviou r is ind e p end e n t 
of t h e zero of r and will hold f or any affine ~arameter r 
To perform the rem~ining int egr a t ions we will -assume 
r 
'l'hen : 
fA) "' ~ " (C; \-;; 6' 0_ 2 6' °V :; ) 52 ,;{ I fi ~ "'( S V 0 0_ Ye 'I} s) 
~ , '- ~ 
r j(?O> y,O))J2 -3-r O(fl -'T~5J2) &')6 ) 
r ( 0 -l -~ J-r I '~'2-51- ~ {l3 J2 - ~-r ~ {f 1l"-1<')5r Y 
1- 0 (5L -<; ) C'i-, 2 ? ) 
f - 11 52 -; -1 1-(I-t- e :/ lA) 5l - '2. -r /-1 '> ( i -t- '2 e ~ 0-) Jl -Cl 
) ~ ( 
I / ' - () \J ( ' 5 _ ,2: <2 CA ,f- t: Cj (.l \ I 6) (0. -'- C') -;- j', t , \)........ j. - .~ /7 l c- t '. e I 8' J er- ,_ 6' -'- 6 7- /' J. ._ 0 Y Q ) , . .:{) 
' ~ . -
x 52 - 'f..- O{ Jl - ) ( 7- '7. 8) 
A ~ ~ (G)~ - 6° )5[ -: Jt6)~ S2- 2-ro (SL -~ (1_~!) 1 
:2 
v - - j ((6' ~ __ er 6) e cc ps - 1 e '-cs 17(~~- 6'0) ))"r-'- I 
'1 0 (51 -!,) (J ]0 
v( - i~iA\lS'SL-;2_ff}e~ 'VS51-~ 
0i- ~ [If 'le LA \75' (f., ~ C CC ) --r -e. ti;tv s) ()~ 0) ]2- 't 
., 0 (SL -~) (?: 'S i.) 
~]" e "-S' 52 -:. fle "S52 -Sy ~ [f) 2 e. ""s(fr ~ s) 
~ekSuOJJ2-Y'r 0(52-') (J- -SC) 
U" 1n'2- ttcf" 7 r ~6 )5L-2-,-O(JL -3) (r :5 '5) 
J. . 
. ~ (0 _ 
~( . -
w~et~~ _ 
I '-f a - VF(;) :::. [e- '2-'-'(f VV6°-S'(VSYV6 °) 
I 9. f'l. - . 
. .J- 6 "CV' S) )~ 6' 0::; V V S}-~ 6" Q 6:i~7-[C 
o !J D , ( r "L "T 7/1 'l.. I 5 L-ll1. deJe. 7'>(l.A. I IL e ~ . . -7-. j 2) 
L{, :0 t 1 0 J2 - r1' t f 'SL -'( LOj'.J2 T r ~JL 8', () (-T 9 b5.52 ) 
e- k [S 11 (r: + J§ Cr){J r ~,~<; ( 6) P)'j 
___ JS rA c> 
'rOJ ' (t·;J) 
~ -ek (S'(j -2V~)(5f1y-;·- t; T / );C)6') 
+ v 1"'// b VO) 
lAV\J€-~ rn,l ~J 
. Llr i ""> '3, d.0 I 
[ ')/ <:.. r, (t-. ~ r) 
'rhU3 che LA- derivative of r I; , n epenCls on l ;;: on it s elf 
and not on the radiation field. It therefor e re ur e sents a 
type of disturbance unconnected with r adiation. If it i s 
zero on one hypersurfa ce, it will re main zero. In this c ase 
it is possible to continu e the ex~ansions of all quantitie e 
in ne gative p owers of J2 without any log terms a ppea.rins . 
The metri c has the form: 
i'y o 
,. '2.. 
) 
The asym~totic group ts the group of coordinate transformat ion s 
that leave the form of the metric and of the boundary c ondit ions 
unchang ed. It can be derived most simply by considering the 
corresnonding infinitesimal transformations: 
---h rA " _ ~DO ~ J/ <{ rt -r J i 
To obt ain the asympt otic group v-!e de l!l Cl.nd 
By 
~ 0 (5L-- j 
o (5L -1) 
o (SL~' 
r 
1-1 I. _ /t01(lA
J
X{.,) 
f(vI) (t?) 
(?~) 
(~~V) 
B;y ~;1 
.:::- (j ( fL'- ':; ) 
f(~ ::. (j ( S2 - 2 ) 
f 1"2- '2 c - 0 f( f-' ] -
-
\ ) 2-
f(( 
) ; 15 J. > ' fr , ) 5 
14 ' I /-( ) v 
(~) 
j '- ~ /-r j i t-
'-; 
.~ T -I p ? J\ " T C) /SL .. ~ \ ! er / 0 \ ) 3 (L /l-. ) 
TO (J2 - ~ 
Gt;;~ 
('tt. I'l) 
(~(;) 
(e. '12) and~ , 1'3) imply that k°c:. is an anal yt ic funct ion of 
X'3 --t- ~ 'K,1 . 'rhis is a cons e que n ce if the f a c t tha t we hi:~tVe 
, I 
r educe ti the lea ding t e r m o f ~~ to a conf orm~lly f l at 
f orm. fj.'hlW t he only a llo';/ ed transf ormations of x ~ a r e the 
conf ormal transformation s of the form: 
~ : 4 
'X 'r L- X ;;;: 
be i 
s ix parameters r\ b c.. cL t.-A..... I ,J ) a re g iven 
. j 
K i s un iouel v ~ .J 
~etermined by (~/~) .K ~ i s a l s o un i quely de t ermined. . fJ.'hu ,::; 
t he asymptotic g roup i s isomorphic to the con forma l g r oup i n 
t wo d i me n s ions. SachsC~) ha s s hown tha t this is i s omorphi c 
to t il e tlOmozeneou s Lorentz Group _ It is als o howe ver il3olilOl"prl.i 
to t he g roup of motions o f a 3-space of cons tant negat ive 
c u r vature which i s the g roup of the unperturbed Roberts on -
J a l k er s pace. Thus the asymptot ic Group i s the same &s tb~ 
group o f the un ci. i s turbed space. I t i s n ot e n lars ed by t he 
p r es ence of r adi a tion. Thi s i s i n te r esting b e cause in t he 
case of g r av it ationa l radiation in emp ty , asymp t ot i c a l l y f lat 
spac e , it turns out tha t the asymptotic g roup c on t a i ns n ct 
on l y the 10 dime ll sional i nhomogeneous Lorentz group, t he g roup 
of motions of fl a t spac e , but al ~ o i nfinite dimens ional 
1I supertrans lations" • It has been s ugges t;ed that; th('s-e 
Bupe r t r ans l a tions mi g h t have s ome phy s ica l s i gni f ican c e in 
e l e me nt a r y p a rticle phys ics. The a bove result would s eem 
to indi c~te that t his i s probGbly not the c ase si nce our 
un iverse i s cc l mos [j certai ;'l l ;y no t a s ympt otic Lilly f l at t hough 
it mc;.y be asywpt otica lly Robert son-,Jalker. 
9. What an ob ~ erver would me a sure 
The velocity v ector V of an observer movin~ wi th the 
du st 1:Iill be: 
V 
I 
V 
1. 
V 
'~ 
V 
4-
::: 
M 
o (n-~) 
o (Jl-1 ) 
, the projection of the ",.J ave v ector (
CL. 
in t he 
M 
observer ls r est-space ( the apparent direction of t he wave ) 
will be: :2 
~ . 
J) VV 
- I 
M (h 
M 
,- 2 00'-5:) 
c~ - JL 
.,. 
-
o (fL -Lt) t 
-- T J.. 
-1 
).. 
t ~ 
.3 
-0 -l-
If. 
The ob s erver 's orthonormal tetrad may be compl 8t 8d b y two 
space-like unit vectors 
S" .::; 0 CJL -4) 
S - O{Jl '-Z) -
'). 
f 
., 
- J1 
-3 
J 
) I - .-
-
-12 LJ 
0( 
.. 8 Itlri t e e-
o.. 
and t 
M 
...: 
• I 
-L 
JT 
, 
L 
-. ..--/ 1 
~v measurinc the relutive acc~l erat i ons of nei ~hbouring dust 
.J 
p a rticles, the observer may dete r mine the ' electric' 
cOiO]bone nts of the gr avitat ional Vlav e ~ 
E . .- - C V? V <t 
O-.b 0... f . b '1-
In the 
r--
t: .:: 
.-j-
ob s erver ' s tetrad this ha s components 
...... 
0 '-0 0 (JL.-it ) -+ 0 0 0 0 0 Cl 0 0 
0 
0 
C) 
o 
o 
0 
-L 
1. 
- J (:) 
o 
i 
-( L 
0 
-
-
.- i 
0 0 
0 0 
0(51- 4 ) 
o (Jl-5) 
.' . 
.~ 0 (Jl-6) 
11..) , 'I 
I .~. 'y ) 
0 0 -1 
This SllOUld be compared to the b ehaviour for a symptotically 
fl a t s pace f or \vhich 
=-- 6 OCr-I) ~ oCr-i) f= 0 () l' ~ 0 0 (j Q 0 0-1 
+ {: ~] C) er -2) o -~ -
( --1) ( 
of- () I -I o r -r- I 6 1° ~ / 0 0 0 o --1 ? 
0 -- I 0 0 0 o-~ (c[S) 
1 . I . oncli , -, . G. J . v-<.n '_0: ... ~ur~ 
U,llu. . , . :..1. . 1,e tznc r 
roe 0 tOJ . ,"; OC . 1 • 2 
') 
r; • • 
) . I J . ,1 . Ii(' ·,J ._L..11 
:11 ' . Pc'nr'o.Jc 
L& 
• 
i 
. ,1-' , ~ ~ _" .9'1 1 "', . \~) f . 1 h .. ,, \i ll ., I 1y~..J 8 ./ ... _ ~/\J __ 
~ . 
c. , .lon.co",o 
.' .. 
'I . l. • .l:.. . I ' U.C1:1~) lY'''' O~T 1) '-' r; '1 "1 ') \,'_.l '~. ..J u v. ~ (",....". .,I 
1. I . ondi , , . • G . J . v~n '_0: )ur~ 
aucl • • • : . ... . 1 le lizncr 
') 
( ... . 
;.; . ~I ~r . f C\.J _~_ 
: l.wl ~ . :P ,IlI'O dO 
,I . 0 L . e ~l. 1 
t'cl _. ~ . J . :1 i 
'r C)O ' 0 ~ -I " c) 0 l .I . ') C • . H,oS-
r oe . :-t0 J . iJOC • l\ 0 '- '1lj -103 'I C) 2 
1 tl . l E' '106-::'_) '.0 j 1L. 1 0 nyc u '.I ;J.)) :J ,_ 
-' • ) • .!.J 0 _'_4-..1.11 lllOS.L.J ,Lonuon U -d EH'si ty <JG-) 
c. (. Pon.co .0 
'I . l • .1.:.. . ,.:)O.CL1S 
S in;;ul ari t i e s 
If the Einstein equations 1:!ithou t co s rnolog ic a l const ant 
a re s at isfied, a Robertson-Walker model c a n 'bounce' or a v o i d 
a singu l a rity only if the pres s ure is less tha n mi nu s one-
t hird t h e den sity. This is clea rly n ot a proper ty p osses sed 
b y n ormal matter though it might be pos s ess ed by a fi e l d of 
negative e n ergy density like the 'C' field. However t here i s 
a g r a v e qu antum-mecha nic a l d ifficulty associa ted wi th t h e 
, 
existence of negat i ve ener3 Y dens i ty , for there would b e 
nothin~ to p revent the creation, in a given volume of spa c e-
time , of an infinite number of qu anta of the negat ive energy 
fi e ld a nd a corresp ond ing infinity of p a rticle s of p o s itive 
e n e r gy. If \ve therefore e x clude such f ields, all Robert son-
VJa l .ker mod el s must be of t he 'big -ba n g ' type, t h a t i s they 
have a singularity in the p a st and m:::ybe one in the f uture 
a s well. 1 It has been sU6 3 e sted that the oc c urrenc e of 
t hese s ingularities ' is a consequence of the hig h degree of 
symmet r y of the Robertson-Walker models which r e strict s t he 
e xpans ion and contraction so that they are purely r adi a l and 
tha t more realistic mod el s w~th fewer or no e xact symmetries 
would not have a singularit y . Thi s chapter will be devoted 
to an examination of this question and it will b e shown t hkt 
provided certain phys ically reasonable condit ions hold , ~ny 
model must have a singularity, tb;:,t is , it c annot be a 
geodesically complet e C~, piecewise C2 ~anifold. 
2. The Funda mental Equation 
CA 
'.£111e expans ion f) ::. V '6-
I 
c ongruence with unit tangent v ector 
of a time-like g e od e s ic 
V 0... obeys equat i on (7) 
of Chapter 2: 
e V 0- = 
;0. 
(i) 
A point will be said to b e a s ingular point on a Geodesic 
(( of a time-like ge odes ic cont:;;rue:::lce iffh'or the congruence 
i s infinite on y at ~ A point '} will be s a id to be 
conjuGate to a point f a long a geodes ic t if it i s a 
singular po int on ( of the congruence of all time-like 
g eodesics t hrough p 
to a space-l ike hypersurfaca . 
~ will be said to con ju3ate 
if it is a singula r point 
lJ~ of the congruence of ge odes ic normal s to I . .Lin 2,1 ternnt i ve 
description of conjugate points may be g iven as follows: 
le t K~ be a vector connecting points corresponding dis tanc es 
o.lon;; t\..;o neighbouring e;e ode sics i n a congruence 'Hi th uni t 
tane;ent vector V~ Then tio... is I dragg ed I along by the 
congruence, t nat is 
j 11~ .::. 0 
"- 1) re Cl. V 1< b :: , , 
-
0.' h J)<J , 
p21\a.. 
-
R 0. ° V b V c /1 cL 
- bccL 
-----P <> 2. 
0... 
Introducing a n orthonormal tetrad e parallelly transp o~ted 
<l. 0.. VC\. ;1') 
v].on6 'V with °e -:: we have 
u 
cL "1 I{ rv I{ (4) Cl. M 
fV\ t'\.. 
d s<J. 
W ~1ere n ... (\.. <>-- 6 R VC Vd. 0-- - e, e bel 
-
<A.Q, M (VI., 
1 ... solution of (~) vlill be c alled a Jacobi field. 'J:here are 
clearly eight independent solutions . Since V 0.. and!{V 0... are 
solutions , the other six indeuendent solutions of &) m~y be 
t aken ort h03 0nal to VG.. '.:ehen i- is conjugate to p along 
a ge odesi c t if, and only if, there is a Jacobi field alon§ ( 
which v anishes a t p and ~ 11his illay be Sh01:111 a s fol lo' [s ; 
the J°C!..cob i fields which vanish a t p may be regarded a s 
g eneratinc; neighbouring geodesics in the irrotationa l congruence 
of al l time-like g e odesics through F . Therefore they obey 
'v 
n.-
·t/ \ 
They may be written 
'.-vhere etA 
MY\ 
"--
cl g 
, A CS) L~'ear p 
/hr\ 
a J acobi f ield 
But A (s) 
M I\. 
I lherefore (9 -
-
and cL~f-) 
AA" 
ot ~ :t 
I 'here fore et 
oLs 
t\.. 
d 11 I de p 
will b e positive definite. 
I ~ (M (A») 
M(A) cLs (\1)'\ 
1\\" 
r-
'::' -0... A (\I\r h-
(ft ) is finite 
Hence f), is infinite where a nd only where 
The r e \'fil l be 
Thus the two definitions of conjugate points a re equivalent . 
(rhi s als o shows that singula r ,)oints of c ongru ences. 0. ~c· e 
points where neighbouring geodesics intersect. 
}'or null g eodesic congruences with par a ll e lly transport ed 
I. ~ t a ngent v ec tor C we ma y define the converg e nc e f as 
~ e d e fine a s ingular p oint of a null ge odes ic congruence as 
one where f i s infin ite. 
~he cond ition t hat the pre ssure i s g re a ter t han minus 
one-t hird. t tJlH: d ensity may be stated more g enera lly a s 
c ondition (a ) . 
(a ; f") 0 ) 
~ 
observer \·; i th 4-veloci ty t,.,) , where 
i s t h e ener~y d ensity in the rest-frame of the observer and 
T .- _ . 4.. I i s the rest-ma s s density . 
Condition (a) will be sat i sf i ed by a perfect fluid with 
densi ty f'- > 0 and pressure p"/ - 3 J--l' • 
·VC>.. any time-like or null v e ctor 
It i mplie s ~ c'vto for 
Therefore by equati ons 
(1) and ( 5 ) any t ime -like or nul l irrot a tional g eod esic 
congruence mu st have a singular p oint on each c eodes ic wit hin 
8. f i n ite affine distance. Obviously if the f lo\'! -lines form 
~n irrota tiona l g eod esic cong ruence, t here will be a phy sical 
singularity a t the singu lar p oints of the cong ruence where 
the density and hence the curvature a r e infinite. This will 
b e the c a se if the universe is filled wit h non-rot c!.ting dW3t 2 , 3 
Howev er, if t he fl ow-lines are ,n ot geodesic (ie. n on-vani s hing 
p r essure Gradient ) or are rotat i ng, equation (1 ) c annot be 
a p p lied directly. 
b Spat i ally Homoe;eneou s Ani s otropic Univers es 
II'he Hobertson-';Jal ker models are spatia l ly 110iJOc;elle ous 
and i s otropic, that is, t hey have a s i x parameter ; roup of 
moti ons trans itive on a sp hlce-like surface . If we reduc e the 
symmetry iby ccmsiderin1.5 models t h,:; t are spat i a lly l!omoger: eous 
but anisotropic (tha.t'! is, t hey h2..v e a three par ameter gr oup 
of mot ions transitiv e on a spac e-like hypersurf~ce) tnen 
the matter fl ow may hav e rot a tion, ac c e l erat ion ~nd shear . 
Thus there would seem to be the possibility of n on - singula r 
models. L. Shepley4 has invest i gated one particula r 
homOGene ous model conta ining rot at ing dus t and has shown that 
there i s a lways a singulari ty . Here a general result will be 
prove d . 
The re must be a singularity in every mode l which satisfies 
condition ( a ) and, 
(b) there exists a q .... of motions on the space or on 
universal coverin~ space * , r~3 which i s transitive on at 
Gee section 5 
least one space-like surface but space-time is not stationary, 
(c) the energy-momentum tensor ' i s tha t of a p erfect fluid, 
--
. la..b Lt, L\... is the ta',-; r;ent t o 
f low-lines and is uniquely defined as the time-like e i gen -
v ector of the Ricci tensor. 
PHOO}l' 
R , t he curvature scalar nrust be constant 
like sur face of transitivity tt 3 of the gr oup. 
on a S'OclC8 -
,~. 
'':: herefore Ko 
J D.... 
ltlU s t be i n t he direction of t he unit time-like normal VIA. 
\'J ~1 ere RIo..... g:c.. :; f ) C) 
I 
( () \ i s an indicator e le. ;CL- ) 
rl'hen '\{a.: b] =0. 
= +1 if 
= -1 if 
R,G. i s 
J 
pc.st directed 
R,c,. i s fu t ure direct ed. 
) 
Thus U'\. is a congruence of geod sic irrotational time - lilce 
v e c t ors . By condition (a), Ro...b vo... V h > 0 
Therefore the con~ruence must hav e a sin~u l ar p oint on each 
geodesic ( by equ a tion 1) either in the future or in the nast . 
Furt~er , by the homogen e ity, the distance a long eac h geodesic 
J fr om U to the s ingula r point mus t be the s ame f or each c eo~e3ic . 
~hus if the surfaces of trans i tivity r emain s pa ce-l ike, t hey 
must d.egenerEl.te into, at the most, a 2-s urface 0 2 v/hich Vlill 
be uniquely defined Let M 
of the matter which intersect 
be the subset of the f low- lines 
C2 • Let L b e L:11e n 0 11-
3 
empty s ubse t of H intersected by Since ther e i s a 
3 
e; roup transi ti ve on 'If ,/... must be 11 itself. Thu s al l t ue 
f lo \"J -lines t ilr ough 1t 3 must inte r sec t t he 2 - s urfiice C2 . Thu s 
the dens i t y I,'f i ll be infini t e t here an\). there will b e a 
ph~s i c8.1 s i ngul arity. Alternat ively i f the s u rfaces of 
transit ivity d o not remain space-like, ther e mu st be at 
1 28.s t one .:mrfa ce which i s null c a ll t hi s 8 3• At ·-.3 F 0 , -- ~ , = 
R.'f',. -1 0 ( if R. is z ero, 'vI e can take any othe r scal':-.J_r ) ,0.. 
9 01ynomi al in the curvatur e tensor and it s covariant de r iva t i ves . 
'':''hey c annot a ll be zero if space- t ime i s not s t at ion:.:\ ry ) . ':: e 
i ntroduce a g eode s ic irrotat ional null congruence on 3 3 vri tl1 
t angent v ector L 0", where [ _ Q '.rhen by eouation ( 5), 
c... : 0.. 
there \>li ll be a s ingul a r point of eac h nul l ge odes ic in 
within f i nite affine d i stance either infue f uture or in t ~e 
past. i he 2- s urfac e of t hese sincular p oints wi l l b e uni0uely 
defined. 'Ihe sa.me ar gumen-0 used before sho"\'I s t ha t the dens i ty 
become s infinite and the re is a phys ic a.l singularity . I n fac t 
as 8 3 is a surfa ce of homog ene i t y, the whole of 83 will bs 
s ingul a r and it is not me aningful ~ to ca l l it nul l or t o 
distinguish t his c c se from the case where the surfaces of 
t r an s itivity rema in space-like . 
'£he c ond itinns ( a ), (b), (c) may b e we akene d in t\'JO vrays. 
Conditi on Cb) tha t there is a group of motions thr ouGhbut 
spac e - t ime may be replac ed by (b/) and (d). 
I 
( b ) Thffi"B is a s pace-like hypersurfa ce 
.'V CA are t hr e e independent vector fields A 
i4 ~ ~ 0) . '1 lZ 
XfA b c.. ~l; bed e 
f1, one homoGent'ous space 
on 
section. 
in vlh i ch there 
such t h2:t 
. 'l'hat i s , t he r e 
(d) 'f her e exist equations of stat e such that t he \Jauc hy 
development of H' 3 is determinate. 
Then succe~ding space-like s urfaces of cons t ant R 
a re homoGeneous a nd much the s ame proof c an b e g iven t hat ~he re 
I 
are no non-singula r models satisf ying (a), (b ) , (c), (d ). 
The only property of perfect fluids that has been u s ed 
in the ab ove proof is tha t they hav e well defi ned fl ow-line s 
intersection of which implies a physical s ingular i ty . Obviously , 
hO!l/ever, t lli s property \'lill be possessed by a much more ge rc·..)}"'c,.'" 
cl ass of fluids. For these, we define the f low v ector as che 
t i me-like e igenvector (assumed uni que) of t he ene rgy- momentum 
tens or. ~hen we can repla ce condition (c) on the nature of 
the matter by the much weaker condition (e). 
(e) If t he model is s ingul arity-free, the flow-lin es f or~ 
a smooth time -like congruence with no singular point s ~ith 
a line t hroug h each point of space-time. 
Cono. i t ion (e) \'/ill be s atisLed automatically if conditions 
(a) and (c) are. 
This proof rests s trong ly on the assump tion of 
homog eneity which is clearly not s atisfied by the p hys i cal 
u n iverse loc ~lly thoug h it ffi c y bold on a l a r ge enough 3c ule. 
Howe ver it ';Jould seem to ind icate that l a r g e s c a l e e :f.f :ects 
like rotation cLUlnot prevent t he s ingul a rity. 
It is of interest to examine the nature of t he s i ngula rity 
i n the homogeneous anisotropic models since th i s i s more 
likely to be representative of t be g eneral case than that of 
the isot ropic models. It seems t h a t in general the collapse 
will be in one direction~5 that is, the universe wil l co l l aps e 
d own to a 2-surface. Near the singularity, the volume will be 
p roportiona l to the time from the singul a rity irrespective of 
the pre cise .nature of the matter. It also ap~') e ar s t hat the 
nature of the particle horizon i s different. There will be 
a p a rticle horizon in every direction except tha t in whic h 
the collapse is taking place. 
4. Singularities in Inhomogeneous Models 
Lifshitz and Khalatnikov6 claim to have proved tha t a 
g eneral solution of the field equations will not have a 
singul a rity. Their method is to contract a solution with a 
singula rity which tbey claim i s representative of the 
g eneral solution with a singularity, and then show that it 
has one fewe r a rbitrary function tha n a ful l y g ene ral solution . 
Clearly their whole proof rests on whet h er their solution 
is f ully representative and of that they g ive no proof . 
Indeed it would seem tha t it is not representat ive since it 
involves collapse in two d irections to a 1-s urfa ce whereas 
in g enera l one would e xpect collaps e in one d i rection t o El 
2-surface. In fact their cla i m has be e n proved false by 
Penrose 7 for t h e case of a collapsing star usipg the n ot i on 
of a 'c los ed trapp ed surfa ce I • A simila r met hod will be 
u s ed to prove the occurrence of singularities in lopen I 
universe mode ls. 
5. IOpen l and 'Closed
' 
Models 
Th e met hod used by Penrose to p rove the occurrence 
of a physic a l s ingularity depends on the existence of a 
non-compact Cauchy surface. A Cauchy 'Surface will be t aken 
to mean a complete, connected space - lik@ surface that 
intersect s every time-like and null line once and once only . 
{ o t all spaces p ossess a Cauchy surface: examples of those 
that d o not include the plane-wave metrics,8 the Godel model , 9 
and N. U.T. space:O However none of these have any physic &l 
signific CtJlce . Indeed it would seem reasonable t o demand of 
any phys ically realistic model that it possess a Gauchy 
surface. If the Cauchy surface is compact, t he mode l is 
commonly said to be 'closed ' if non-compa ct, it is said to 
t o be 'op en'. The surf&c e s, t = consta nt, in the Robert s on -
\rJa l lm r s olutions for normal matt e r a re example s of Cauchy 
s urf ace s . If K = -1, they h ave negative curva t ure a nd i t i s 
fre quently st a ted that they are non-compac t . Th i s i s not 
nece s s a rily s o: there exist possible top olog i e s f or which 
they a re c omp a ct. However, t he following sta t ement s ma y b e 
ma d e ab out the topology of the surfaces t = constant. 
If the curvature is n egative , K = -1, t h e u riiversal 
covering s p ace is non-compact and i s diffeomorp hic t o E3 . 
Any other topology can be obtained by identifica tion of 
1 1 
p oints. Thu s any other topology will not be simp l y c onnec t e d 
and, if compact, must h a ve element s of inf inite orde r in t he 
f und a ment a l group. Further ~f compact, they c a n ha v e n o 
f t · 12 g ro up 0 mo lons. 
If the curvature is zero, K = 0, the universal cove r ing 
s pace is }i; 3. rh . . bl . t 1 - ' 13 I ere are elghteen POSSl e op o 061es. I f 
compact they ha ve a G3 of motions and Betti numbers, £1 = 3 , 
'12 
. B2 = 3 . 
If the curvature is positive, K = +1, the universal 
cove ring spa ce is s3. 'rhus all topologies are compa ct. 1' h e 
Betti numbers are all zero. 12 
S ince Cl ~ingularity in the univers a l covering space 
imp lies a singularity in the s pace covered, Penrose's m8 t hod 
is a pplicable not onl y to spaces tha t have a non-comp a ct Cauc hy 
surface but a lso to spaces whose universal covering space 
has a non-cOJ-:1pact Cauchy surface. J:1hus it is a pplic8.ble to 
mode ls which, at the present time, are homog eneous and iso-
tropic on a large scale with surfaces of app roximate homo-
g eneity which h a ve negative or zero curva ture. 
6. Th e Closed Trapned Surfa ce 
Let T3 be a 3-ball of coordinate radius r in a 3-surfuce 
&3 et = const.) in a Robertson-Walker metric with K = 0 or -1. 
Let qa be the outward directed uni t normal to T2, the boundary 
of T3 , in H3 and let Va be the past d irected unit normal to 
H3. Consider the outgoing family of null g eode s ics which 
int ersect T2 orthogonally. At T~ f ,their conver gence will 
:; ~ ( V
o
,-; h f q 0..; h J() C\. ~ b +- t 0. t- b ) be: f 
where fO'- to.. are unit snace-like vectors in H3 orthos onc.l 
to a and to each other, q 
1. [(~ J kr~ J tt,.erefore f ~ I i·· eo - k , R 
If J\J.- 7 0 and K = 0 or -1, by tak ing r large enoug h, we may 
. 2 
make1 nego.tive at T. Therefore, i n the lane;uag e of renro se , 
T2 is a closed trapped surface. 
Another way of' seeing this is to consider the diagram 
in which the flow-lines are drawn at their proper spat i al 
d i s·tance from an ~server. ~hey al l meet in the singulari ty 
a t t = o. I f the past light c one of the ob s erver i s drawn 
on this d i agr am , it initial ly diverg~ s from h i s world-l i ne 
) • It reaches a maxi mum p rop er r adius ( t :: 0 ) 
and then converges again to the singularit y ( f > 0 ) • '.Llhe 
intersection of the converging libht cone and t he s urface ~{3 
g ives Cl c losed trapped surfa ce T2. If the red-sh i ft of the 
qu a si-stellar 3c9 is cosmologic a l t hen it will be bey ond the 
point f ~ 0 if we are living in a Robertson-0alker type 
universe with norma l matter. However, t he assumptions 6f 
homog e ne ity and isotropy in the l a rge seem to hold out to the 
distance of 309 . '.rhus there is good reason t o believe t h(;!. · 
our universe d oes in fac t conta in a clos ed trapped surf2ce . 
I t should be pointed out that the posses s ion of a closed 
trapped s urface is a larg e scale property that does not den end 
on the exact local metric. Thus a mod el th&t had loc a l irregul-
a ri t ies, rotation and shear . but Has similar on a l arge s cale 
at the pre s ent time to a Rob~rtson-Walker model . wou ld h av e a 
c los ed trapped s urface . 
Fo llowing Penrose it will be shown t hat space-time has 
a s inr;ularity if there i s a clos ed trapped. surface and : 
(f ) E ~ 0 for any observe r with velocity 
( g) there is a global time orientation 
(h) the universal covering space has a non-compact CC\.1J chy 
s urface 
PROOF' 
3 H . 
Assume space-time is singularity free. Let F4 b e the 
A 
s et of points to the past of H7 that c an be joined by a 
s mooth future directed time-like line to . T2 or its int e rior 
T3 .· Let B3 be the boundary ofFLj·. Local consiierat ions show 
that B3 - T3 i s null where it is non-singul a r and is 
generated by the outg oing family of past d ire cted null geodesic 
segments which have futur e end-point on T2 and past end- p oint 
,,"here or before a singular point of the null geodes ic 
conb:cuence. Bince at T2 , the convergence, f > 0 and 
Si n ce · Ro...b L 0.. L b ~ 0 by (f), the convergenc e must 
become infinite within finite affine d istance. Thu s B3 - T3 
will be compact being generated by a compac t family of compact 
segment s . Hence B3 will be compa ct. Penrose' s method i s 
7, . 
then as follows: approxima te B7 arbitrarily clo~ely by a 
smooth s pac e -like surface and proj e ct B3 onto H3 by t he 
normals to "this surfa ce. 'l'his g ives a many-one contL-:uou s 
mappin~ of B3 into H3. ~ince B3 is compa ct, its image B3* 
must be compact. Let c1Lq) be the number of points of B3 
ma pp ed to a p oint q of a3 • cl( Q) . will change only a t the 
7. 
inters ection of caustics of the norrnals with H7 
by continuity c an on~y change by an even number. 
since this is the i dentity and 
'J his is a contradiction, thus the assumption tlls.t spc\ce-t i lJ1e 
is non-singul a r must be f a lse. An a lternative proc eedure 
which avoids the sliGhtly questionable step of app roximating 
B3 by a space-like surface is possible if we adopt c ondition 
(e) on the nature of the m~tter, then B3 may b e proj ~ ct ed 
continuously one-to-one onto H3 by the flow-line s . ~hi s 
again Le ads to a contradiction since B3 is compact a nd H3 
is not. 
J:"A t}a~ i?"bov~ p;coo,f ;ij- "'&15 n~C~55lill;'y to Q. 9rna nc. t hat 
a3 "bo a CaYQby ijurf~cg gthe rwi~ ~ t~g wbo l e gf j 3ffiiGht not 
h e,ve been projeoted onto 1-;3 u ;-/e 'dill dofino Cl cO ffi i ';; aYC L1;Y 
~urfaoe (s,G,c) ac a cOffiploto CO"A"AoctoQ. ~p~co liko £urf ace 
7-
il. 5 C 5 q .? will "b~ ~ ('&1nc };:J,;;T ' Sll"!~C~ ,fo;c ;o;int ~ 1'408'r it, 
throuGh these points. aovJ9yor, fyrthor 16.-.'e/Y thoro ma y bo' 
:!'"C ~ion6 for ',;l.''l.ioh it is not a Gayohy surfaoe. Let ,P4 b e ~he 
lil~t of !;) oi"At:a .for wbic:b. H3 is Oil C.mcb.y sllrf&icQ " "Ad lot Q S b t<' 
t h e boundary of these points. 0 3 if' I-"'f' ,,.,- i!:'", O)CiS '6S, Hill b e 
called tho Ci?Yc:b.y Horjzon relative to H3, 
• 
~QneratinG ~3 H:lust have at least ORe S '; Bgul5:r J:) oine Nhie ~1 
:Ql~ 81n~ tast tf.ley rll;~~t ge be~ndod ::'n at least OR e d irection. 
it OifflW-'±C c:maplc of s. G. c 'dith a Ca~chy horigon i~ a iS19~c e 
l~kc,surfacc of oonstant negativo ourvaturo oOffiple tel~ 
\~thin '50:c null oone of the point in ;)';in~co',rc1ci spaoe. m:J:E: 
Bull oone formc the Ca~ohy horis6ft • 
• f 88:Plflitionc (e) and (g) hold, then a model ',dth El 
&o:~'ipact s.G.s, n3 H:lust have the topology: 11.1)( t:: i • . 
buppo.:::\e tecre \lere a region VY through '({hich '(5tlere 
no flm. linec interseoting H3 , thon V:1 tho boundary of 
h ~ :re 
FfiY ii t b@ a i:;:kr.a@ ... J,j,J'L8 fHlpfa08 ;;eB@pat@d by 1'10';: lino G !.,'hiel1 ,10 
intorcooi; H). Proceeding :slong t};:].@s@ flo','=ling:s in tb.€ 
d ireotion of thoir intor~ection wit};:]. a), w@ ~y:st ro~c~ i ~ 
v3 dO@a not intor ceot 1" -' I • 
&it the e)(ict onco of thilS end-point c ont;c;;Q.ictra (@) cinc 0 i t 
:i.8tplio£ do sinGularity of tho £10'I/-1ino' c ong!f'Yonc e. 'J:'huG V 4 
mUB b be eliII' bS and e ver:r point fins a flo'I1 line througl3: it 
interseoting H). Thuc we havo a hOfsooffiOrphiEfA of the space 
by aSSigning to every point of tloc ')ace ito€ 
ds..otanoe along the flo'" lino from H3 eJnd the point of iniw~­
section of -'a~le f10\1 lific vJith oH). It oan also be GhO',in that; 
-9!Tl this c aGe iI3 ffiUst be o. Gauc hy surfaoe. :i? or GUB .} O i3o€ 
, . C 
t l:1ere ',.'Ore a Gau ohy horigon . (Q.$ this C 8 n intore @ct 
each ;fJ Oh'c] j ne OiIt l"IlOst on ce IJ,'h'.!H'€l£Or€l thoro is a :lOffiee-
Hl orphism of ~~ in:l;,Q- H3 • Eurtger 1 by (e). t? 1TO,.y f ] o ),f=l~e 
n1 1 m3 . b. , . . -J3 1: nu s . \01( l. S 0 m e 0 fllO rp Cll. c 'b 0 l .. 
~ompaet. If condition er) holde every null gp od@eic gener utop 
:3 
of ~ hao at least one end point. This must be in the 
from H3 since in the direction t owards H3 
each generator ffiUst be unbounded . This hm:evcr is iE1 ;)ossible 
5iRce ~J i~ CO~lp;ilct TbllS U3 ;i S Cl Cnucb.y sn"£iil,,"~ . 
8. ~ingularities in 'Clo s ed' Universes 
r ~here 1e a eingulurity in ~very m6de l which sat i sfi es 
(a), (g) and (i). 
(i) 'l'here exists a comp a ct Cauc hy surface H3 '''' ho se uni t 
normal VC>-. has positive expansion everywhere on H3. 
PROOF 
_.-
For the proof it is necessary to establish a cou j) l e of 
lemma s. As sume that s pace-time is singularity-free. The 
foll o,dng result is quoted VIi thout proof , it m2.y Jr'£,a;; -nJ -;;y be 
derived from lemmas proved in reference 11. 
I f ? and 't- are conjugate points along a ge od e s ic r 
and· X is a point on r not in fC),. then 1:' must have a con j ug 8.t e 
p oint in f} '. 
An immediate corrollary is that if q is the fir s t ? oint 
a lonG y conjugate to p and y is in pq then y h as no 
conjugate points in pq Also since the re sult that x 
hb.s a conjugate point in pq can only depend on the values 
of et in pq, any irrotational geodesic congruence including 
MA 
the geodesic r must have a sinGular point on 6 in pq. 
Thus if q is a point on M3 and ( is ths geodesic normal t o 
M3 through q, then a point conjugate to q along ' r can not ~ccur 
until after a point conjugate ,to M3 e 
If M3 is a complete connected space - like surfa ce which 
inters ects every time-like and null line from a point p, '.'i s 
may define a function over M3 as the square of the geodesi c 
distance from p which is taken as positive if the geodesic is 
time - like and negative if the geodesic is space-like. ~e call 
this the world function G'" \'iith respect -to p . For t he c l os ed 
set of v D.lues 6'.?-O (5 will be a continuous ( i n 
general multi- valued) function over ~13. A time - like ge o':u:; s ic 
( from p will be said to be critical if it corresponds 
to a value of 6' for \vhich cs- ~ fA. ~ 0 L i. :. i,,?., 3) 
,of' ~ 
where \ f, ~ are thr ee ~ndependent vectors in 1"1 3 . Cle (lr l y 
i-
a critical geodesic must be orthogonal to M3 . A geodesic 
which is critical will be said to be maximal if it corresponds to a 
local maximum of 
Lemma 1. 
A geodesic t cannot be maximal for a smooth M3 if there is a 
point x conjugate to M3 but no point conjugate to q on 't in qp, 
where q is the intersection of ~ and M3. 
Let f and g be the Jacobi fields along ~ which vanish at X 
m m 
and f respectively. They may be written 
n 
f = A( s) f/q , 
m mn 
n 
g = B(s)g/q 
m mn 
Then 
n 
any h 
h ( i~ I~ ~ ~~ t I~) ~ ( n must be positive for 
since if it were negative for any h by taking a = ·~ b h h 
mb m n 
beyond q, it would be possible to have a point y on 6' beyond q 
conjugate to t.. before a point conjugate to p If it were zero 
X would be conjugate to f This shows that the surface at q 
of constant geodesic distance from p lies nearer to f in every 
direction than 
does. 
the surface at ~ of constant geodesic distance from 
Since X is conjugate to M3 the surface at q of 
f 
constant geodesic distance from lies closer to 4i in some 
direction than M3 does. Hence is not maximal. 
l i iu~t b~ pOaitive d.eiinite a t x. But at ,c 
<0 
tl4 I) ib A 
'TRue;}. (a - 8) Gannot DO p Oeit ive d€Lrinit~ 2t q. ':i:horo 
. ' ~ ~ 
fce;r;:e tbe:ce Irm s t; be dj~~Q tj OD 
(\t\ 
for: ~'lb j elJ 5Qm~ t~ AA 1\.. 
cL ! fV'. i\. Ko k ~ ·14 ~ d .. 
"! ~ a~ &n4- /'.'\ 1'\ V\l~ vv K K ,/ 
"" 
I/, K 1,.) • < 
~ ~ 
~t q, J .. .' hore io) ;;i.1a t};;}.e. mlit +angent macto:c of tbe CQngru ~K(;O 
.f " t '"' b. ,'I'h . " J " • -'- • v '- ) ~ a. Geo,'l el3;bCebl:t'i;mg pillS In I~oe (1 J :ce C .. J OD u ' .. JI" SllrJ; 'CC'i 
of Gonotant 3 000:00io diotanGo from p 1i09 e~<g.@E to p t han 
~fte surface IXi 3 cl ooo 6 Therefere '( is not J.Y8.XilflaJ.. 
7-
If MJ is c ompact or i f the intersection of all t ime- l i ke 
and null limes with rV13 is comp a ct, 6' mus t have a maximu:.: 
v alue, thu s there must be a s eodesic norma l to M3 through p 
longer than We u s e this to prove a nother lemma . 
Le ll1ma 2 
If P lies to t he future (pa st) on a time-like g eode sic 
( throug h q, beyond a point z c onjugat e t o q , and there 
3 exists a comp ac t Cauchy surface H t hrough q, t hen there 
mu s t be another t ime-like g e od esic f rom p to q longer than r . 
Le t y be the last point conjuga te to q on t before p. 
Let x be the near e st p~int to p conjugate to p in pq. Let 
r be a p oint in yx. Let K3 be the set of p oint s which h a v e 
a f uture (pa s t) directed ge od esic of lenGth r q from q. Then 
K3 will be a s pace-like hypersurfa c e t hr ou Gh r. Le t F4 
be the 'set of points which have a t l ~ ast one future (past) 
directed geode s ic from q of length Gre ater than r q . 
boundar y of }i,4 , J3 c:: K3. Since p is in F4 and s ince 
'l 'hen t he 
every 
p a st (future) directed time-like and nvll line from p intersect s 
H3 h ' , . ~. 4 t; 1 . t ' J 3 - t 1 3 ' W lCll lS no LJ ln F , they mus' a so ln ersec"G . • Le ' 
be the init;®r:s:s'C t ion of J3 and these lines. Sinc e H3 is 
compact, £3 must be compact. 00nsider the function U Hith 
respect to p over K3. Its maximum must lie in the comp act 
region L3. But, by the previous lemma ( is not maximal , 
ru ora over, loc a l considerations s how that a singular point in 
the surface J3 cannot be a maximum 'of (5 'l'hus the 
maximum value of () must OC t; ur for a 2: e ode sic fro m p ort llo-
gonal to £3. This must also be a g oedesic fro m p to q of 
length creater than ( • 
Usin:; these two lemmas the theorem may be proved . .:.;ince 
the future (past) directed normals to n3 a l.' e converging 
ever Y"Jhere on H3 , there must be a point conjugate to H3 a 
finit e distance along each future (past) directed geodesic 
normal. Let ~ be the maximum of these distances. Let p 
be a point on a future (past) directed geodesic normal at 
a distanc e greater tha n ~ Consider the function cs- wi th 
A Let~ respect to p over the compac t surfa ce r .? b e t he 1 • 
b eoJesic f r om p norma l to H3 at the point q , where 6' has i ts 
maximum . 'I1here must be a p oint conjugate to H3 a long A in 
But i f there is no point conjugate to q al on g ~ i n QP , t hen 
\ c a nnot be ma x i mal by the first lem~a. If however there 
i s a point conj ugate to q a long ~ in qp , then there mu s t be 
a long er geode s ic from q top by the second leffil la . thus 
qp . 
is n ot the g eodesic of maximum length from H3 to p. ~his is 
a contrad iction which shows that the oris ina l a ssumption t ha~ 
the space \o,Jas n on-singular mu s t be false . 
~hi s proof could als o be u sed t o show the oc curren c e 
of a singula rity in a model with a ~on-compact Cauchy surf~ce 
provided t (,.a t the expansion of its norma ls was bounded av~ay 
from zero and p rovided tha~ the int ersection of the Cauchy 
s urface with a ll the time-like and null 11ines from a point 
was compa ct. 
1 . O. Heckman 
2 . I'. Raychaudhuri ,-,. . 
3. A. Komar 
4. L . C. Shepley 
~ C. Bear / . 
6. El.M Lifshitz and 
I. M. ~(halatnikov 
7. R. Penrose 
8 . R. Penrose 
9 . K.Godel 
10 . C . ~N .1\'1isner 
11. J . i"l ilnor 
12. K. Yano and 
;:). Bochner 
13. O. He ckman and 
b . Bhucking 
1L:-. H. Bondi 
I.A.U. 6yrnpo s ium 1961 
Phys.Hev. 98 1 :23 196 5 
Phys . Rev. 1 OL~ 5LI-4 1956 
Proc. Nat.Acad. Sci. 52 1Lj-03 1964 
Z. As-c rophys. 60 266 1965 
Adv. in Phys. 12 185 1963 
Phys. Rev.Lett. 14 57 1 965 
Rev. (lod . Phys . ':t!. 215 /165 
Rev. Mod. Phys. ?:1 1:t~7 i1lrf 
.J. M8.th. Phys . 4- QZ,4 r~ 
' Morse Theory' No. 51 Annal s of 
of Maths. Studies. ~rincetown 
, 
Univ. Press. 
' Curvature and l3etti i'Jumbe:cs I 'T o. 32 
~nnals of Maths. bt ud i es . 
Princetown Univ. Press . 
Article in 'Gravita tion' ed . ,vit -cen 
Wiley, New York. 
'Cosmology' Cambridge Univ. ~re s s