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The Hoyle - Narlikar

Th eo~y

of Gravitation 1. Introduction The success of Maxwell's equations has led to electrodynamics being normally formulated in terms of fields ~articles

that have decrees of freedom independent of the them .

However , Gauss suggested tnat an

ac~ ion-at-a-distance

t :1eory in \'Jhich the action travelled at a finite velocity might b e possible .

This i dea was developed by 1heeler and 1

Feynman ( 1 , 2 ) who derived their theory from an action-principle tha·t involved only direct int eractions between pairs of par·ticles .

feature of this theory was that the ' pseudo '- fields

int roduced are the half- retarded plus half-advanced fields claculated from bhe worl d - lines of the particles . However, ··! :1eeler and JJ'eynman , and , in a different v-1a;; , Ho,-artli

(3)

were able to show that, provided certain c osmolo3ical condit ions were satisfied, these field s could combine to 4 6ive the observed field. Hoyle and Narlikar ( ) extended ·tt1e ory to general space - t;imes and obtained similar theories for thE:ir ' C'-field (5)and for the gravi·tational f:fueld (G). It is with these theories that this chapter is concerned.

I-

,I\ I< y L J A~V

"A"'l /

...... . ..... I

It \'Jill bG ...;ho.:n that in

exp;.11ding universe t·!1e

advanced fields are infinite, and the retarded fields finite. This is because, unlike electric charges, all Toasses

h[tVA

tho

•

same ...:;i.;;;n . 2. The Boundary Condition

rloyle and Narlikar derive their theory from the action: •

wnere the integration is over the world-lines of oarticles

• •

In this expression ~ is a Green function .

•

that satisfies ·the wave equation:

l(( . /) £ x. x

/-3 wl1ere

is the dei:;erminant of

in the action

3.j

• -=>inc:e

t~e

double sum

is symcetrical between all pairs of

pi:irticles a.,b , or1ly thc..t part of symmetrical betv.reen c.... and

4·(0...b)

that is

b v1ill contribute to the action

i.e. the action can be writ·ten

A b)

\vhere ~··)!(-( ~ _ 'rhus

·lt

must be the time- symmetric Green function, and can

be i.,rrittien:

()"' '1 ::

..!..

qre-t' -r 1qo..dv

v1here

are the r3tarded Ey

requirin~

of the

adval1ced Green f·L1nctions.

r~nd

that the action be stationary under variations Hoyle and Narlikar obtain the field - equations:

i g"K

[ £. i t {Vl,lo.)( x) M(b) ( x )] (r<il< 0..

-:i: b ::: -

<: < _!_ [ ft,K -r ~ c_ 3 o..fb

(a.){ /'II.

(b) r

9'vk. fVl,} r (o.);r

(b) f'rL.

) ·t

;~K

G) ( (a) a(. fYl, J

)i,

( b) /l1.

. ; r\

(b)J

M., ,

. Ho·,;ever, a.s a

w11ere

consequence of the particular choice of Green function, the contraction of the field-equations is satisfied identicall : . '1.'here a ::c thus only 9 equations for the 10 comnonents of Cr · '. ,J ~J and t;hc S,',rstem is indeterminate . {o..)

:.. oyle and Narlikar therefore im')OSe ,As the tenth equation. By then m~kin . t.ion' th a t l.S . b y pu tt ing · L approxima L

Zrrt.

=cor:st.,

fvt0

the 'smooth- fluid' < (o.)m (b)JI.- /11.. 2 L_ (V\,

a...:tb

CJ)

they obtain the Einstein field- equations: ~

~{R.

(, {IA..O

._v,

_.!_R_ l

gi-1'\

/ t,

. .

•

There is an important difference , ho\vevor, bet\veen these field-equations in the direct - particle ir1teraction tc.eory and ii1 tl1e usual general tneory of re L&.tivity . In the seneral tneory of relativity, any metric that satisfies t he

the field-equations is admissible, but in the interaction theory only

t~ose

tlirect -particl ~~

solutions of the field - equations

are admissible that satisfy t he additional requirement :

£ /VL

(o..)

(x) =

..-

This requirement is highly restrictive; th~t

it will be shown

it is not satisfied for the cosmological solutions of

the binstein field- equations, and it appears

th~1t

c~nnot

·be satisfied for any• mod.els of tl1e univer:;e tho.t eitl1er contain an infinite amount of matte1· or undergo infinite •

expansion . The difficulty is similar to that

occurrin~

:r-rev1tonian t_icory \vhen it is recognized that the universe might be infinite . The Newtonian p otential

D '·"'he re

obeys

th~

1 : - "f

is the density.

equation:

In an infini te staLic the source

al~ays

unive~se ,

has the same

ved v1hen it v1as reoliz ed

th<~t

wo1Jld be ~ign .

~he

infini~e,

since

diffic ulty was

~ssol-

the uni versP. ·.-1as expa11d.in;;, since

in .:tn exoand ing univer se the retard ed soluti on of the above is finite by a sort of ' red- shift '

eq~ation

ef~ect .

~he

advanc ed soluti on will be infini te by a ' blue- shift ' effect . 'l'his is unimp ortant in Ne\vto nian theory , since one i:..; free to choose vhe soluti on of the equati on and so ms.y icnore ·the infini te advanc ed soluti on and take simply the finite retard ed soluti on . ~iffiilarly

in the direct - partic le intera ction tQeory the

rn, - field satisf ies the equati on:

Om where

f R. M = N

is the densit y of \·.;orld- lines of partic les .

As in

the Newto nian case, one may expec t that the effect of the ex1)a11sion of the univer se will be to mc:1ke ·the retard ed soluti on finite and the advanc ed solu·ti on infini te .

Howev er, one is

now not free to choose the finite retard ed soluti on , for the equati on is derive d from a direct - partic le intera ction action princi ple symme tric betwee n pairs of partic les , and one must choose for fV\.. half the sum of the retard ed and advanc ed soluti ons.

Vie

would expect t •J.is to be infini te , and this is

shown to be so in the next sectio n .

j .

'fhe Gosmolos;ical i.:),olu-cions 'l1he .rtobertson- \'/alker cosmologic<:'l.l 1netrics ho.ve the

forin

- .

•

Since t.1ey are conformalJ.y flat, one in \vhich

tr.Ley become

cls

c~n

caoose coordinates

; J1-<

7o..b

dx°'- cLx ,

•..v t1ere

io the flat - space metric tensor and

-For example, for the

;: 0 /

- t~_(c) .c7) tl~ I ·t i K ( 't r-f )iJ [I ~ ~ ii ( ~ - ( )~J • Sitter universe

Einstein-d~

R(l)-:: c 'L ~ T T 2 3

("'C=

__ Q

JJt

( G .;..:_

"f. <. rD ) I

• For the steady- state (de Sitter) universe (-o0.<.t:"Lo0) K :. 0/ Rt.t); .

..t

•

-F

oO <. L.

(i:

·-

-(:

Te 7)

c;* (~,, b)

obeys t h e equation

t R~*(CA,b)

oc;-#('(o..,h)~ ~ rom

:--'!'-

'I.1he Green function

·- ·-I

~<t{Q b)

;-:__ 3

t h is it follows that

J2'-i

dx.b

~ JL-'iJ 4 (a,b) If we let

q~; 52.-t

.._ , v :1e11

Thi s is s imply the flat-s-oace Gr -.. en function equation , a nc1 •

ne11c e

r *( 't.,, 0 ). 'l l '1

f ) -- Jl-'?c,) ~

g Cr) The

/V\. ··fie l

d is given by ·=-

'~x:N/-3

ii~J2--,- c.) J2. ( ti)f

--c- L - vc, ) · J2. (."C )._ Jf f'

dx: \t

I ( ~re,t_ ·r

lh.0-dv: J_

lt'or uni verses without creation (e.g. the Einstein-de Bitter 3 ,., -~ I~ ·- /1., ,,. const . .e or universe),

con.:;;t.

W\Gre the integr ation ~·1 ill th~

over the fu t ure light cone .

This ~

normally be infinite in an expandir13 :.ni verse , e ~ instein -de

0itter universe. oO

-2

M. o...).; ( '-f,)

2-: . in

'1: I

- I -

( fl -

vc ) I

cL 1 .2

..."

/., I

• universe s't:;eadystate In the

/ll\...CL

J ./.

( '1;( )

·--

- I l

-l

-r.

·-I

(

,/\

T. ~

cL c:.z .

•

.By

c~o»ntrast,

on the other hand , \ve have

where the integration is over the past light cone.

This will

normally be finite , e . g . in the

'L . ' T

-2

~in~tein-de 0i~cer

(

t'.2.. - 'L )

' _ /?_,

univers=

c{ '"'L1. ::.

-Q.

-(L

while in the steady- state universe IV1..

{;

( 't, )

-- - -I

·rI

-1

'"f I

·-cO

-I

-2..i

l;i. ·-:i. (L

Thus i t cun be seen that the solution

ll'\.. =-

consi::; . of

th0

equation

is not , in a cosmological metric , the half-advanced plus half- retar·ded solution since this would be infinite . in the case of the Einntein- de

~it·ter

and steady-state metrics,

it is the pure retarded solution .

~he

In fact,

'C '-b'ield

Hoyle and ]arlikar derive their direct - particle interaction theory of the 'C' - field from the action

where tae suffixes

·'°'

refer to diffqrentiation of

~~ (o..,b)

on the world- lines of l'-1.. 1

6 rcGoectivel~' ·

.........

is a Gr een function obeying the equation

DG(X,X') ,~e

define tl1e ' C' - field by

C(x) and the

•

G (x)°-) ,~ clQ ) '""'

.rhen

c &-J

a L~, b)cLb ·~, 4

S' e(

1) J '< (

lj ) J K

j - 3 cl_ 'X

- I'\

-- J

:~e tl1us .:>ee that the sources of the w~ere

0....

mat~er -c urrent

J"'<~) • z_ 1

C' - field. are the plc.ces

matter is created or destroyed . i1.s in the case of the

' M,

'-field , the Green func·t; i on

must be timc- S;)7 mmet ric , that is

q(°', b)

hoyle and Narl ikar claim thut if

th~

action of the

' C' - fielQ is included al on,~ vii th the action of tl1e '

'-field,

a univerue \1ill be obtained that approximates to th8 3teady-

state universe on a large scale althOUBh tncre may be local irrecularities.

In this universe , the value of C will be

f i:ni 1..ie and its gradient time - like and of unit ma5nit;ude . Given this universe, we may check it for consistency claculatin6 the

advan~ed

if their sum is finite. ~how

will

and retarded ' C'-fields and ~e

fincin~

shall not do tl1is directly

~ut

that the advanced field is infinite while the

re~ar -

ded field is finite . Consj.der a region in space - time bounded by a three dirnensional space - like hypersurface ]) und t(1e past light cone 9f

ÂŁ..

at the present time,

of some point

to the futl1re

â&#x20AC;˘

By Gauss's the orem

w C /-3 dx ~

Let the advanced field pro(luced by sources v1i thin Then

and

will be zero on ~

, and hence

â&#x20AC;˘

.dut

- 1-\

is the rate of creation of matter= ft,. (canst . )

â&#x20AC;˘

the steady- state universe , and hence

11.S

the point

rv V.

is taken further into the future , the volume

of the re6ionV t~nds to infinity .

Hov1ever, the area of tlJ.e

hyp.ersurface J) tends to a finite limit owing to horizon effects .

~herefore

the gradient

dn-

must be infinite.

A similar calculation shows the gradient of the retarded field to be finite .

'.I.'heir sums cannot therefore :-Âˇiv;e the

field of unit g radient required by the

Hoyle -t~arlilcar

neor;,r .

â&#x20AC;˘

It io worth noting that this result was obtained without assumptions of of conf ormul.~ flatness.

a~smooth

distribution of matter or

5. Conclusion It iJ one of the weaknesses of the Einstein theory of relativity that &lthough it furnishes field equations it does not provide bour1dary conditions for them.

'rhus it does not

Give a unique model for the universe but allows a whole series of rnoa.el:::> .

Clearly a theory th;:,t provided boundary conc...L "Gior"s

and. t.1ut:> re.stricted the possible solutions \..,rould l)e attractive.

The Hoyle- Narlikar t11eory does just

l."'equirement that

= ;:- mt~t

J._ M ).

equivalent to a boundary condition) .

)

1,0

Unfortunutely, as .. e tho~e ;~ odels

th&t

correspond to the actual universe , namely the

R obertso)1-~·J al~::er Th~

th~..t(the l.S

O...d ./

have seen above , this condition excludes seem

VP>r\r y .,

models.

calculations given above have considered the

univct·s~

as being .f 'illP.d \•1ith a 11niform distribution of m<:.:.tter. is

le ~ itimate

'-."his

if we are able to make the 'smooth-fluid '

approximation to obtain the

~ins tein

equations.

~lternativel--

if this approximation is invalid, it cannot be said that the tneory yields the Ei nstein equations. It mi.5ht possibly be that local irre_;ularities co11ld :nal{e finite , but this has certainly not been demonstrated 'Yta..dV and seems unlikely in vievJ of the fc..\ct th1 . . t , in t11e Tio~rle 1

Narlikar direct - particle interaction theory of their 'C 1 -field,

which is derived from a very similar action- principle, it can be shown advanced

without assuming a smooth distribution that the 1

field \-.rill be infinite in an expandin3 uni verse

'IJi th crea·t ion . ~he r eas on that it is possible to f ormulate a dire ctparticle interaction theory of e lectrodynaffiics that does ~ot encounter this difficulty

havin3 the advcincea solution

infinite is that in electrodyncJJics tl1ere are eo.ual n11mbers of so\1rces of nositive and negative sie;n .

Their fi.elds cu.r

c~ncel each other out and the total field can be zero apart

fron1 local irregularities .

This sug:est that

i:..

1)ossible

•:Jay

to save the Hoyle - Narlikar theory would be to allow masses • s ign . both p osit ive and negative

0·1.~·

The action would be

G~~, b) clo. cL b

are gravitational char.~es a~al o~ous to electric char ·es . 1

fi1

Particles of positive

' -field und particles of negut ive Cj_

q,.

in a r>osit i ve

in a ne .....,.ra·t i ve

fie ld W()\lld h&ve the noi. .mal ~ravi tational properti~s,

rr1

t;t'i.:.,:.t

·t11e;f would have positive gravitational and inertial masses.

is,

A particle of

ne~ ative

s·till J'ol lov1 a geodesi c. a particle of positive

in a positive

'-field

•:J(;u ld

Therefore it wo11ld be 0.tt;racted :)y •

Its own gravi tat:;ional effect

howeve r would be to repel all other particles .

Thus it would

hav~ the properties of the negative mas J described by Bondi(e) tt1at i s , negative gravitational 1110.so and n egat ive inertial mass . 3ince tnere does not seem to be any ma tter having toese propert ies in our region of space (

1ere M

coi:.st . / 0 )

there must clea rly be separat i on on a very larc e scale . would not be possible to identify part i cles of

'-'.l. ,,

ne~~ltive

with ant imat t er , since i t is known that antimat·t er has positive inertial mass .

liwev er , the i ntroduction of nesat ive masses

would prob ably raise more difficulties than it wuld solve .

REFERE1~CES

1 . J . A. ~ilieeler and R. P . Feynman

Rev . Mod . Phys . 17 157 1945

2 . J . A. \Jheeler and tl . P . :E' cynman

Rev . Mod . Phys • 21 425 1949

J . E. Ho 6 arth

Proc . Roy . ooc . A 267 365 1962

4 . F . Ho.1le and J . V. Narlikar

Proc . :::<oy •...,oc . A 277 1 1964a

5 . F . tloyle and J . V. Narlikar

Proc . ~oy . Soc . A

6 . F . ..ioyle and J.V . Narlikar

Proc . Roy . boc . A 282 191 1964c

7 . L. Infeld and A. Schild

Phys . Rev . 68 250 1945

8 . H. Bondi

Rev. Mod . Phys . 29 423 1957

282 178 19&+b

CH\P':'ER 2 PERTURBAT IONS 1â&#x20AC;˘

-- .-.---

Introduction

Perturbations of a spatially isotropic and homogeneous expanding universe have been investigated in a Newtonic'Jl approximation by 1 Bonnor( ) and relativistically by Lifshitz( 2 ), Liftshitz and Khalatnikov(3) and Irvine( 4 ).

Their method was to consider small

variations of the metric tensor.

This has the disadvantage that the

metric tensor is not a physically significant quantity, since one cannot dire ctly measure it, but only its second derivatives.

It is

thus not obvious what the physical interpretation of a given perturbation of the metric is .

Indeed it need have no physical

significance at all, but merely correspond to a coordinate transformation.

Instead it seerns preferable to deal in terms of'

perturbations of the physically significant quantity, the curvature .

Notation Space-time is represented as a four- dimensional Riemannian space

with metric tensor

gab

of signature +2 .

in this space is indicated by a semi - colon.

Covariant differentiation Square brackets around

indices indicate antisymmetrisatio n and round brackets symmetrisation. The conventions for the Riemann and Ricci tensors are: -

1/91.b cet

is the alternating tensor.

Units are such that k the gravitational constant and c, the speed of l ight are one .

Ul'\I VtlC>' I Y Ll.:il'/â&#x20AC;˘~Y

CAMBRIDGE

The

Field_.~gua~.io_ns

We assume the Einstein equations:

where

Tab is the energy momentum tensor of matter.

that the matter consists of a perfect fluid.

where Ua is the

vel~city

is the density .

is the pressure

h G\b :

Then,

a Ua U

of the fluid,

We will assume

- 1 :

is the projection operator

Sob t Ua.. l..tb

into the hyperplane orthogonal t o Ua .· hAb

::0,

We decompose the gradient of the velocity vector UQ.;b:. Wob-+ <5'"o.b

+ t h~b 8 -

•

Uc.. Ub

is the acceleration,

where

is the expansion, <r'o. b

hJ h d. b C.

~ IA ( c j <l)

wo.b ~ flow lines U • a

u [c;.d]

- -S

h:. l1t

is the rotation of the

We define the. rotation vector I

Wo.

is the shear,

-:;. -;: 7}o.bc.cl W

c.ol

•

We may decompose the Riemann tensor Rab

E1..nd t11e •:.'cyl tensor

C ·. b'o.

'=='

-::.

abcd

into the Ricci tensor

Cabcd :

R cbco( = Ca.~ccl - 9q(ct Rc.J I:;, Cabe.et~ C[.o.b)~d.J

Wo. as

Co.[ bed]

•

b[c

l~cl]~ -

R;3 'JG\.[,5olJb

)

Cabcd is that part of the curvature that is not determined locally by the matter.

It may thus be taken as representing the free gravit-

ational field (Jordan, Ehlers and Kundt ( 5)).

'Ne may decompose it

into its "electric" and "magnetic 11 components.

' C o.b

cd _ -

u [o.. t

tc.-Âˇ ..,tJ \.A

cC'- Ed]

- 4- O(o..

!,)

' ')

)

Q...

-=--

Ha. o.

= o

().

)

each have five independent components.

ab We regard the Bianchi identities,

as field equations for the free gravitational field. Then

(Kundt and Trfimper,C 6 )).

Using the decompositions given above, we may write these in a form analogous to the Maxwell equations. b

n (j.

E. bc;ol

hcd

Ho.h W

'? o..~c.d

Lide

<Y e n

= '5"

b l'A

(1 )

l }"-;"

' - E c(o. gb)c •

+ 2.

Y/ o c.c:le

d Q

YJ bcc=I. e

•

L H c:o..b - h (o. .f ') 'b) c d. e \). c. ..

E -f o~e

.. H c. (P- o-b)c. - "1 p.r; cl e

•

E e.r ac:l'f uP

YJ bp~r Uc

• Uc ue

t(fi ~)

() ob

..-

H (o. w

u p <Y cl ~

1,.4.c.

Y) b p q ,.

t- 2H("J o. ~ he. ,,I e U. '". tA e

Hott.

(2)

(3)

b)c.

H~ . .

(4)

•

where .1.

indicates projection by (c.f. Trfunper, (7)) •

hab

orthogonal to

Ua.

The contracted B·ianchi identities give , -

fa+

(f+~)8

(f.+ ~)

Gl + 0

=. 0

r.;

h ba

T cob'·b -=

)

(5)

(6)

The definitien of the Riemann tensor is, U o.;[bc.]

f' be. uP

•

Using the decompositions as above we may obtain what may be regarded as

equations of motion" ,-

e = zw •

'2.

n2.

-2a- -3l7

(7)

)

(8)

•.

..·'.. ~

(9)

\Vhere

2 W

W ab W

\/Ve also obtain \\rr1at may be regarded as eq_uati ons of constraint. ( 1 0)

( 11 )

)

L-t

= - n (o ~e

·1b)c.cJ.e

c.(

u \.. wf

d;e

•

( 1 2)

consider perturbations or a universe that in the undisturbed state

in conformally flat, that is 0 abcd By equations (1) -

•

(3), this implies,

•

If \Ve assume an equation of state of the form,

1:Jien by ( 6) , ( 1 0) ,

:::. o

•

= Uo.

it"\. ( ~) ,

This implies that the universe is spatially homogeneous and isotropic since there is no direction defined in the 3- space orthogonal to Ua. In this universe we consider small perturbations of the motion of tl1e fluid and of the '.ifeyl tensore

Ne neglect products of small

quantities and perform derivatives with respect to the undisturbed metric.

Since all the quantities we are interested in with the

exception of the scalars, µ, ~' e have unperturbed value zero, we avoid perturbations that merely represent coordinate transformation and have no physical significance. To the first order the equations (1) -

E ci-b

-s.h

f;b

(4) and (7) - (9) are ( 13)

( 14)

)

•

c;a..

b+E 0 L8tr,:>

)

' •

e -- .. --

-3

- -3 'l.

e2. + Ll0-b

•

lA.

• Q. .I

( 1 5)

( 1 6)

( 17)

( 1 8)

( 19)

From these we see. that perturbations of rotation or of Eab or Hab do not produce perturbations of the expansion or the density.

Nor do

perturbations of Eab and Hab produce rotational perturbations.

Metric 4. The Undisturbed -l"i····-Since in the unperturbed state the rotation and acceleration are zero, Ua must be hypersurface orthogonal. Ua...-s~;o.

measure, the proper time along the world lines.

where

As the

surfaces 't = constant are homogeneous and isotropic they must be 3-surfaces of constant curvature.

Therefore the metric can be

written,

where

, dy

is the l1ne element of a space of

zero or unit positive or negative curvature. .-.'c define

by,

d. t clT

:0.

then

•

In this metric, '

. '

..••

(prime denotes differentiatio.n - with respect to

t )

Then, by

(5), (7) (20)

•• ·3

G -:. (2

±- (f

1" ;

ft)

( 21 )

•

If vve know the relation betvveen

and

fi. ,

'vve may determine (l_

\Ve will consider the tv10 extreme cases, 1'"\. = 0 (radiation). For

'b ;:

•

rl.\

:. 0

f'(\

•

1.. D'2,-

•

(a) For E

M I '2.

T;' t:.1

rt 0

2 'r- (cos h~ -

(b) For

= c onst •

••

;, r1

••

•

}1 ~

physical situation should lie between these.

.Any

By (20),

•

(dust) and

)

= const.

E: t

1) i

- i E(~ -

1t -

. ~ll'l

h~r-M -~ t -· t ) ;

-- o,

tz. )

f'I\ t3

•

E represents the energy (kinet·ic + potential) per unit mass. If it is non-negative the universe will expand indefinitely, otherwtse it irvill eventually con tract again.

•

By tl1e Gauss Codazzi equations '"R; the curvature of the

= const.

bypersurface ' l

is -

EM o~

)>O

[=O

)

R- -

F'or

.?\

M -=

/Y\

-3 ,::::

•

)

E <O ) ..,.__.....,:/

)

. n.c.:. -2

)

...t

-=---)J./"S--

•

- t-

••

3 (2

)l.

)

j-J· -::I~

(a) For (1

-E I

~.inh

(b) For E

> t

- -E' -- -2. I

fY\

t - I)

( C.c•S·h

0. ...

-E

(:, .,. R -- - 0.'l..

)

,•

•

2. )

-- ~ (c...ost E

RotFttional Pert11rb at ions _,,,_ _ , ···--.-- ·- ... ·-·.------.. ·------·_..-=-•=ire-.-. By (6)

3 (2

- o,

( c) Fo1"' E

)

S ..i. YI

•

'\..

-F'

•

)

,..,..

-1)

)

-<. -I

, ]

•

• ••

For

Q..b

•

)

For

1ri.-:.

(&),.,

(2 '2.

)

•

-- -

-... -

et e +--'+ ~) f' I

- -'} w

)

Thus rotation dies away as the universe expands.

This is in fact a

statement of the conservation of angular momentum in an expanding • universe.

Perturbations of__Densi.!il. For

fl·=

we have the equations,

-t-8

fJ --

-Te

•

fJ. •

--r: fA

These involve no spatial derivatives.

Thus the behaviour of one

region is unaffected by tl1e behaviour of another.

PeI'turbations

will consist in some regions having slightly higl1er or lovrrer values of E than the average.

If the untverse as wh "1..e has a value of E

greater than zero, a small perturbation will than

~ero

n.nd will continue to expand.

till have E greater

It will not cor.trsct to

form a galaxy.

If the universe has a value of E le8s than zero, a

small perturbation can contract. contracting at a time

However it will only begin

8 1" earlier than the wl1ole universe begins

contracting, \l\There

--· .. '(:"

is the time at vvl1ich the wl1ole m1iverse begins contracting.

There is only any real instability vvhen E

= o.

This case is of

measure zero relative to all the possible values E can have . Ho-Yrever this cannot really be used as an arguement to dismiss it as there might be some reason why the universe should have E = F·or a region with energy

- cE , in a universe with E

·~E ( t'l.- t't 4 12

r"l -

>l -

T =

1'1.

... )

se ti E )'l.~ (----· '""t'" '1./~

f- :::

T •• •

)

2 ~~

For L: = O,

Thus the p e rturbation grows only as

'l'3 .

This is not fast enough

to produce galaxies from statistical fluctuations even could occt1r.

these

Hovv0ver 9 since an evolutionary universe has a partic1.e

horizon (Rindler(B)

in the early stages .

Penrosc( 9 )) different parts do not communicate This makes it ov0n more difficult for

statistical fluctuations to occur over a rc;g: -.n until light had time to cross th0 region.

-• n 0

f ..s bef'ore~

I i_ • ,,., _o ,,,,: ... = - .., 0 - )A + V'- ... '

a perturbation cannot contract unlesG it has a negative

value of E.

The action of the pressure forces Iilake it still more

difficult for it to contract .

Eliminating 6,

••

. .

l-l

0..: 0. -=

. t,..l

<>-i b 1

o.b

+ u 0..

•

I.A

to our approximation.

I1 ·.~re

OA.\7

Ve I

b :;:/ o.

c-r =

is tl1.e Laplacian in the hypersurface

represent the perturbatior1 as a sum of eigenfunctions

this operator, v1i1ere,

(->)

constant.

S(n)

<:.

,•cU.

-;-.O

- - Q'2. Yt.7...

{f.o)

These eigenfunctions will be hyperspherical and pseudohyperspher ical l'1armonics in cases (c) and (a) respectively anC plane \vaves in case (b).

In case (c)

\ll!ill take only dj.screte · ,lues but in (a) and

(b) it will take all positive values.

where

•

f'..'3

•

1 cng as

For

is the undisturbed density •

fA 0

\Vill gr0v1.

f'o >>

These perturbations grow for as long as light has not had time to travel a significant distance compared to the scale of the perturbation

( ~ Q:. ).

Until that time pressure forces cannot act to even out

perturbations. 1.1/hen

... r

(11)

(t1J

rf. 0

. ..,.,

e {_ .1:3" t

· !e obtain sound v-1aves whose amplitude decreases vii th time.

T11ese

results confirm those obtained by Lifsi1i tz and Khalatnikov(3). From the forgoing we see that galaxies cari.not form as the result; of the growth of small perturbations.

''Ve may

~ x:pec t

that other non-

gravitational force· will have an effect smaller than pressure equal

to one third of the density and so will not cause relative perturbations to grow faster than

l: .

To account for galaxies in an evolutionary

uni verse we must assume tli.ere 11ver·e finite, non-statistical, initial inhomogeneities.

Tpe

Steady-stat~_!.!D-jverse

To obtain the st0ady - state universe \Ve must add extra terms to the energy-momentum tensor .

where,

c ·~

0.. .c.

'CA

Hoyle and Narlikar( 1 0) use,

Sin.ce

-r

·b 1 c..b' =-

o )

( 21 ) •

lJ-

,/,,

Cl.+ I '-)

h ~,

\- b

b n

CL -

c c cl ;d..= o . b

(22)

There is a difficulty here, if we require that the

Ci? field

should not produce acceleration or , in other words, that the matter created should have the same velocity as t11e matter already in existence.We must·tJ;'=tr.> l1ave ..'.I

(23)

However since C is a scalar, this implies that the rotation of the medium is zero.

On the other hand if (23) does not hold , the equations are indeterminate (c.r. Raychaudhuri and Bannerjee( 11 )) . In order to have a deterrninate set of equations we will adopt ( 23) but drop the requirement that

is the gradient of a scalar.

The condition (23)

is not very satisfactory but it is difficult to think of one more satisfactory. Hoyle and Narlikar( 12 ) seek to avoid this dj_ff~iculty by taking a particle ratl1er than a fluid picture.

Hovvever this has a serious drawback since it leado to infinite fields (Hawking( 13 )) .

From ( 17),

•••

c ea-~ ~

-= -

= -

e f1

1~) -

(.µ-1 1'l) e •

•

For

*ft

Tht1.s, small perturbations of density die avvny.

Moreover equation ( 1 8)

still 11olds, anc1 therefore rotational perturbations also die away. :t;q,uation (19) now becomes

e= -

•

9 _,,

e -2 'Z.

(

f-+ 3 r. + )

.-----· ~ 3 ( 1:- (1)

These results c onf j_rm those obtained by Hoyle and l'iarlikar ( 1 4).

Vie

see therefore that galaxies cannot be formed in the steady-state

uni verse by the gro,•rth of small perturb at ions.

However• this does not

exclude the possibility tha·c there migl"rt by a self-perpetuating system of finite perturbations which could produce galaxies . (Sciama( 1 5), Roxburgh and Saff'man( 16 )).

-----------

1 Gravitational · raves • ~·-.. ... ----~ ~Ve no\1 consider perturba.tions of tl1e

~Veyl

tensor that do not

arise from rotatj.onal or densi·cy p8rturbations, that is,

-Multiplying ( 1 5) by

LI f I

lA c

·b 0.

V<:.

and ( 16) by

we obtain , after a lot of reduction,

••

In empty space with a non-expanding congruence

0 17

this reduces to

the usual form of the l i nearised theory ,

The second term in (24) is the Laplacian in the hypersurface "'r

= constant,

acting on

We will write

Eab •

as a sum of

Eab

eigensfunctions of this operator .

where

. V

(1-1) I O~>

}

_ yt~

)

Then

••

·-

•

0 Cl'-

•

'2.

(1"1) Q.

Similarly ,

Then by (19)

Substituting in (24)

Al"') [ D(~}

[ r2 (j-tt f'L)

n ....

n'' (2

0 (f fi-)] i

::. 0

We may differentiate again and substitute for D' , For

\'l

>>I

and

J t1 2.

1 and the "energy" Eab decreases as Cl r.... 6 We might expect this as the as 1 t.. •

so the gravitational field t(Eab Eab + Hab Hab)

Bianchi identities may be written, to the linear approximation,

Therefore if the interaction with the matter could be neglected cabcd would be proportional to n and Eab ' Hab to D -1 • In the steady-state universe when .

and

have reached their

R~~ ~ /~ + fL ~

equilibrium values, •

••

J o..b c ~ R c La j bJ :::: 0

S ~ L°" 'R; b J

Thus the intera ction of the

c 11

field vii th gravit ationa l radiat ion is

equal and opposi "i e to that of the matter . intera ction, and

of the me·tr i Q.

decrea se as

Eab

The "energ y"

Ther'e is then no net

and .1. (E ,.,ab 2 · ab-'-'

n -1 •

depend s on se• ond deriva tives

It is theref ore propo rtiona l to the freque nt.y square d

times the energy as ineasur ed by the energy moment um pseudo - tensor , in a local co- moving Cartes ian coordi nate system .which depend s only on

first deriva tives. to

r2 ,

ll ~L

Since the

~requency

will qe invers ely propo rtiona l

the energy measur ed by the pseudo - tensor will be propo rtiona l

-4

as for other rest mass zero fields .

AQ_~orption

of Gravit ationa l

lilf~es

As vre have seen , gravit ationa l waves are not absorb ed by a perfec t fluid ..

Suppos e howeve r tl1ere is a small a1nount o1' viscos ity.

We may repres ent this by the additi on of a term energy- momen tum tensor , where

·A

). CJa..b

to the

is the coeffi cient of viscos ity

( El1le r s , ( 1 7) ) • Since

we have

(25) (J

;

c.. b

e. Ol.

-:::: 0

(26)

Equations (15) (16) become

~;e

(28)

The extra terms on the right of equations (27), (28) are similar to conduction terms in Maxwell's equations and will cause the wave to

decrease by a factor

"'t ~

Neglecting expansion for the moment,

suppose we have a wave of the form ,

E o.b

::::.

E- ~be

~V"t

This will be absorbed in a characte:r.iatic time frequency.

"-/). independent of

• of rest mass energy of the By (25) the rate of gain

matter will be

~ A ()

which by ( 19) will be

available energy in the wave is

Fz )).-2 •

4- .:>.:.

A- Ei. v -2. . 0

Thus the

This confirms that the

density of available energy of gravitational radiation will decrease as (l

-4 in an expanding universe .

From this we see that

gravitational radiation behaves in much the same way as other radiation fields.

In the early stages of an evolutionary universe

vvhen the temperature was ver•y high we migl1t expect an equilibrium to bo sei up between black-body electromagnetic gravitational radition.

diation and black- body

Since they both have ·ciivo polarisations their

energy densities should be equal.

the uni verse expanded they wouJ. ':.

.As

both cool adiabatic ally at the same rate. of black-bod y extragala ctic elec1

5°:< ,

As we know the

~omagnetic

temperatu~c

radiation is less than

the terrperc.:.t ure of the blac1c-bod y gravitati onal radiation n1us'..;

be also less ·than this which the energy c ·f gravitati onal

vvou~.

l be absolutel y undetecta ble. does not contribut e to the

radi~· :~ion

ordinary e11ergy momentum tensor e.ctive gravi tatio11al ef'fect .

Now

... ab •

I'reverthe le ss it will have an

By the expansion equation, •

For incoheren t gravitati onal radiation at frequency ~

2. ::.

E'l.

-i '))

But the energy density of the radiation is

• e

••

\vhe1'3

µG

v ,

-:; - 5 8 - Z }J- G

- {

E v

-2.

3 }1.)

is the gravitati onal 1tenergyn density.

Thus gravitati onal

radiation has an active attractiv e gravitati onal effect.

It is

interestin g t11at tl1is seems to be just half that of electroma gnetic radiat ior1. It 11as been suggested by I-Iogarth ( 1 8 ) and Hoyle and Na'Y'likar ( 1 O), that there may be a connectio n between the absorptio n of radiation and the Arrow of Time.

Thus in universes like the steady-s tate, in

which all electroma gnetic radiation emitted is :ventuall y absorbed by other matter, the Absorber theory would predic

retarded solutions of

the Maxwell equations while in evolutionary universes in which electroinagnetic radiation is not completely absorbed it would predict advanced solutions.

Similarly, if one accepted this theory, one would

expect retarded solutions of the Einstein equations if and only if all gravitational r·adiation emitted is eventually absorbed by other ma.t ter. Clearly this is so for the steady - state universe since /\ constant.

In evolutionary universes

V1e ',ii/ill obtain complete absorption if 1

~ oc. T2

\'\There

is the temperature.

will be

will be a function of time.

r>t o1.r

Now f (" '.' a ga:-;, " For a monatomic gas , T d:. fl - 2 ,

therefore the integral will diverge (just).

diverges.

However the expression

used for viscosity assumed that the mean free path of the atoms was small compared to the scale of the disturbance. Since the mean free path oc µ- 1oe -3 and the wavelength 0::. (2 - 1 ' the me 8.n free path will

eventually be greater than the wavelength and so the effective viscosity will decrease more rapidly than 1()L -1 •

Thus there will not be conwlcto

absorption and the theory would not predict retarded solutions. However this is slightly academic since gravitational radiation has no·::; yet been

d~tectcd,

to a retarded or

let alone invest.igated to see vvhether it corresponds

adva.~ced

solution.

References

(1)

Boilllor W.B . ; M. N., :L1.Z, 104 (1957) .

(2)

r.ifshit z E. M.; J.Phys . u.s . s . R., 10 , 116 (1946).

(3)

Lifshitz E. M. , Khalatnikov I . N. ; Adv. in Phys . , 12 , 185 (1963) .

(4)

Irvine

(5)

Jordan P. , Ehlers J ., Kundt W. ; Ahb . Akad . Wiss . Mainz ., No . 7 (1960) •

V"J . ;

Annals of Physics .

~·z 2.

(iq6'f) .

( 6)

Kundt \V. , Trllinper M. ; Ahb. Akad . Wiss . Mainz . , No . 1 2 ( 1 964).,

(7)

TrUmper M. ; Contributions to Actual Problems in Gen . Rel .

(8)

Rindler W.;

(9)

Penrose R. ; Article in ' Relativity Topology and Groups ', Gordon .and Br each ( 1 964) .

M.r~.,

116 , 662 (1956) .

(10) Hoyle F . , Narlikar J . v.; Proc . Roy . Soc ., A, 'fil, 1 (1964) . (11) Raychaudhuri A., Banerjee S.; Z. Astrophys., .2§., 187 (1964) . (12) Hoyle F ., Narlikar J . V. ; Proc.Roy. Soc ., A, ~8.f., 184 (1964) . (13) Hawking 8. vv. ; Proc . Roy. Soc ., A, ...:in prj,nt. ~ ~/3>

(/~.£)

(14) Hoyle F. , Narlikar J . V. ; Proc . Roy . Soc ., A, 273 , 1 (196 3 ) (15) Sciama D . ~v. ; M. N., 1J5-., 3 (1955) . (16) Roxburgh I.'1V., Sattman P. G. ; 1'11. N. , J.12 181 (1965). (17) Ehlers J. ; Ahb.Akad.Wiss . Mainz.Noll . (1961 ). (18) Hogarth J.E. ; Proc . Roy . Soc . , A, 267 , 365 (1962) .

CHl\.Pl'En

Gravitational .Liadi&tion In An Expanding Unive1·se

Gru.vi tational radiation in e:npty asymptotically flat space has been examined by means of aoymptotic exoansions by a nL.mber of &uthors . ( 1 - 4 )

·'.Chey find that the different

components of "che out,soing radiation field "peel of'f is , they so

that

11 ,

different powers of the affine raJial distance .

If one v1ishes to inves·tigc:.:.te hoi'l this beha•riour is :nodi .i:-iPd by the pr~sence of matter , one is faced with a difficult~r

that doAs not arise in the case o.r, say ,

ele~tromagnetic

radiation in matter .

For this one can consid.er the radiation

trc.Lvellint; thro11gh

infinite t1nii'orm medium that is static

a11

apart :from "t;he disturbance cr,3ated by the ra\.i.iat;ion.

case oi' cro.vi tcttional if che

me~ium

racli~tion

In the

t;hi:J ir;; no·t pos,_;ible .

were initially static, its own self

' .!!Or,

5ravitatio~

1r1ouia. (;ause it to contr a ct in on itself and it v1ould cease to be static . raQi~tivD i~s

Hence one is forced to investigate

gravi·tatio·~al

in matter that is either contracting or expandin;.

in 1Jhapter 2 , we identify the Weyl or conformal

·tensor

with the freA gravitational field and the

1~icci-t ensor Rctb curvature.

with the contribution of thA matter to the

Instead of considerin; gravitational radiation in

acymptotically flat space , tna.t is, space that flat sc; ...t.ce at large

~pproach~s

radial distances , \ve consider it in

rts,11n:pt o tica.lly conformally flat space .

conformally flat , the

~icc i-te nsor

.1\.s it i.;.; only

and the density of

m~tter

need not be zero . :'o avoid essentiall;yy non- gravitational phenon1ena. sue h as sound wavgs , we will conbiner travellin6 through dust .

gravitationalra~iation

It was shown in Chapter 2 that a

conformally flat universe filled v1ith dust must have one of the metrics:

\ Ls- i

(a)

I i 2 4 S-2. ( cl {; - cl.f - s· ,,\ f i

; A ( i ·I

(b)

'2.

J2 'l.

;..

_\)_ ;. (c)

cLS' 'l __J<j_

(

1 L f4 l . :t /l l ri ) ) c D 't' s'r. CJ c 7' .

cas: c) . ·1

cl t:

~ At

(;. :i

1. ) ).

(I, 2 )

---

fype(a) represents a universe in which the mutter expands from ·the initial sinJularity with insu fficient ener3y to reach infinity and so falls back si11gularity .

a~aj.n

to another

It i s therefore unsuitable for a dj.scussion of

•

ravi·tationril radiation b;y- a method of .::..sympto·tic expansions si11ce one cu.11no·t <_;et an inf ini ve C...:.istun(~e from -eh~ source.

Type (b) is the ~instein-De Sitter universs in ~hich ::he mat·ter has just sufficient energy to reach infinity . thus a

S 1)ec i

o.l case.

It is

D. Norman ( 5) has investit;<J..ted the

"peeling off" behaviour in this case usi11c; Penrose ' s co11f ormal 6 technique ( ). He was however forced to make certain assumpt ior1s about the movement of the matter .·1hich vrill be sho~·!n to 1

be false .

;:oreover , he \vas misled by the soec i al nc.ture o""'

the ~instein-De Sitter universe in which affine ard lumir:osi .. y d is·tance:_, differ .

Another reason for not considerin;:; ra.::li<:...tion

in the t•:instein-De Sitter uni verse is that it is unstable. fhe passage of a gravitational wave will cause it to contract a~ai·i e vent ually and develop a s in.;ularity .

de will therefore consider radiation in a universe o!

t,1 pe

( c) ~1hich corresponds to the general case vJhere tn.e

rna·tteI' i.3 expanding

\·Ji th

more th.: n enough ener6y to avoid

contructin;; again .

fhe Newman-Penrose Formalism ~·Je

employ the notation of l{ewman and Penrose. (3)

tetrad of null vectors ,

l r--/

nf-) t'I rn f-/.

('(}L 1·

f'-.

is introduced

lf (L vve

label

2 ()._

p.-

nf'

L,,., lf'

v1nere :

th~se

(/,r\ Mp 1iil1v.: lf- (Y)i~ -- (lt' /l?rl; ·- f.J.

Mr M

('L: fl~ P1'~ M ru) c~b

uq ? --

-- -

(

-I

pv )

-~ ..

O.b

an~ lo~ered

l o

have

;; -}

vectors with a tetrad

tetrad irdices are raised

vie

[ ,

( ?__:__!)

•

-1

2p. ().-

2v h

fl pL IA f'L + 11

\lith vhe notric

·-

rr1 ,~ m

v ·fA v __ fV1 fYl

(2- j_)

Ricci rotation coefficients are defined by :

( 2

In f'ac·t it is more convenient to complex combinations of

ro~ation

':.:01:l{

t ·~ rms

of 1J-;.,'el 1e 1

coefficients defined as

f ollO\'/S :

(

I~ I

- -y

·-

2. lt

'I 'I - ( )'l I •

•

·-.... ··-

·-

·-.

.'.:l

3~1

Lr: v 0 -

fA -

1'l!J

fl1

. L n /Al. v~ 1Yl - {v<.; v I I 1

//.__,.;..;

1iVif l v)\

3 . Coordi.:1atcs --.Li lee i:·l e \fJrnan a nd Penrose ,

\J e

u..( =xj)

lL ·v I

take

vte

introduce a nu ll coo1:-dir1ate

<~ . i)

~ o

Thus

•

3eodes i c and irrotati onal .

(

':Till be

This implies

-- 0

cc- ::: - £

:L + ~

vie take along

1~s

•

be parallelly

IJ.'b.is gives

transporu~d

-~ - s)

•

a se concl c oordina·te \':e talce an affine parameter

l f" r:· f L f" ~ I

long the t:;eodesics

and

X~'

i n the surf v.ce

arc

t\·10

X : l). L

f.i.

(-:?

Lt )

coordinates that label the geo6.e...:.ic

L.l = cons t . l

·u.

L': I\ ,' ,vY't ,,

i1hu s

(1. b) In these

(

l =

{ ~ f)

The lield

~~u~tions

Je 1nay calculate the Ric ci and ;1Je.1l tensor com1ion8n't;s f r om

1,1

the rela·tions a.._b c..-.:.L o-. bc: (L

{.-{

't ·-

•

.._L b

Using thA combinations of rotation coefficients defined <ind with

/"C -= TT

=£ = 0

v1e have

p ()

( J.

10)

j) G'

I!)

]) 'f

/)d.

j)~

( ·s..

···-

i2)

(1.1.3) ..,_ ( 1 I~) ( :s . /~)

Pr y ,\

(s.

(~·- tJ)

- - --

I b)

{l . IY)

"~) c, J'

~· f ~ ~, c;-

~j_

r~ -

( {? r ;_ )

f" f ->. G

-t ( .~ -

·-

J C\)

b' ·-

f, r po '

J<X ~ 7 O(~ ,- ~? -

('5 . 'l c'/)

f 3 f1 -4r- tr-2vf·ir'f'?._,~IT ?J.2 r;/t,' , ( oi. .,. fe )f-

..,. (;} - ~ ~) ). -

·t- /\ r

6 '2::_

( !.. ")

(? fA - J f )" f .,(I, 11 r - sl. = cr ;- i ·- f ) f - 2""' "'L - ,\,,-. - --r; - ). )\ ~:, ,_ t). fI

1J b ~ J l. (S

-r

°I" ).

•

il °' - J''(

ft _

(J__L f) J

•

i.·1here

L {" Vr "

- rit' -

~ '/,..

v t;.)

~r-

·-!.. ).

R -- fo

-_!_

(I~

,_ K 31..f

·-I J.

R J ':G

- J.

-- c/> ;J.o --.

Jxl (3- ·29) (~ 3o)

R .< J.~ = 1J.~

..-

-1

~ l~

(3. ) I)

•

)

<.f)

•

g ·-~~I 1,o ·-- -J_ R ·-cf, ·1 1~I 2 13

.5 JXL

1>01.

)\ i

.' ~ J_

Poo ·.: t ·--

·--

-;-

(.A.

J --

() ·J

cp,,

(1.

'~L.)

(s. s3) ( 3 . ·5 "!)

(.s

3 s-)

(1 . 10) <~. 3 f)

-c ) ·2 I

-i'l.l'l

'.l.3 4

cl. .

x ( l "'fl- 13 l ( n_ r.,.. Ld/Lf;n. ~ .P

(}

~-c)

(~. y /)

: - c •

Expres s i11.~

the rotation coefficients in terms of th8 metric,

we huve : l

-;t.

·-

]Ju

•

£.,

(1.y6)

&W ... 'iw- (o+f)

- ~x~-Llg·; {r_,_f-o)~~T>- §" (s.1ir) \

5- $& ·~ S~ L ~ Si:;

(~-

~) $'- "' ( ; - ~) ~ '- ( 1 V0)

~ w = (~ - at) w ·+

(; -

~ ) w ·t <t- -/;:,.) 3 0 ~

j~s

in Chauter 2 we use the Bianchi identities as -"

fiel<.1. equations for the

;,~eyl

te11sor.

In the t,ewman- Penrose

formalism they may be writ ten : (I am indebted to R. G. McLena~han for these)

g lf'o - Y f, .,_]) cj;c:rl - $' foo"' i·

6 'fa

- ~·y <t D~o~ - ~101:: -t- S6f;. - °Jo/

<r "'-

'fo - ~ f 'f, -( ')_:; ~!]) ~oo

'2f cf

1 .,.-

16' 1,o

(·4~ - ~·}'-('o - 2 {2-r_

-t-

(3. S 1)

.~) lf7

?oJ. (l.S~) ·3(5115; - oy,),. 2(J>1. -s ~,o) $' <P", - Li ~00 :: 1> t - 9r t 1f, <t - '1 ,v- - J r - 2 i ) cp (J<l-t- Q"£) ~O 00

2P101 .,_ ~b1,,<>1-( ·t

-t {, ol.

-I

00 't

J ( T - ~ I{ ) · lrh t 0 .:J.. ~ 1-1.1 'f rA -r ~b' rb - if rh I :2 0 I 0 1..

(

s 5.3)

·~ (t)f, - f~ ); i(D~ 11. - d ~") -t (f o/o.: 11 fo,) , 3r 'f;;r _-t _b ( o- f'-) f, - q'1 tl. °I b€f1 - 7 foo r

- {f 11'3 -t-

1,,

~0" <>;- :L ,,\ ~ v ( ..,. 2( f .~ f'- - j i )1- c ( ~~ .,_ ). 'c·- 9,J_ ) ~ 6- f, :( {j SS)

t ;_()

-if, - 1/A 'f2 .., b ( ~ - (:,) lf3 .. ] b t '{ - ~vcfo ,- )V~,o 7 2(2F-r)1.,r 2)102 . - Jcji,_o + 1(75- ~ff)1,2 <f 2(ft'r -c.Jf-;_,-f ~J.). (f)-6

= {, v

~ t~ ·- Vt~ ,_ ~ f , _, - A1,,_o ::: ]:A-lf'; -- :)a., Y's - f <f'~ - 2 v 1,,.., 2 ~ f ,, ,. ( 2 r - 2 f 'T ;:;.. ) <f2u -t 2 ( t-E - o<.) ~:;, - G c/J~ c· J. 5"-t) 7 'I

<"J_

LJ f 3

~1f'4

-t

ll <f,_, " 3v t'l.·- 2 ( y t 21'-)</5 -r;) 01./r "-/ - 2 ·v · - v 10 t Q. >- t,'l

f </>.,_

"/- ( 4 t->R -

?_ -

~ <r +- p) '97,_, ., (-t - ~? - 1 v(.) cf:;.2 ·. . ( 3. <j8-) ]) f,'.<- r p,, - S' <f>oa ~<Pa,-.- 3 ~II =<~ 0- f - 2;;.,) o, • vtf ·

"1-

2C ~. -r { 2~- 2.;. ·- l) </; ~"i-Jf ?,,,_<>t 6' </12-i (.~ )7) JJ"' ~ f; ,i, - ~ r1,lo ..,. 6 rJiI + .1 ])Ii = t 2. r- t- -t-.2i -~) rrA - 2~ ~ i) Toi rh - )_ (.;:<>r --c.) ~,o + ctp <P .. '!- ?~o:i. .,.-~fie) ( -S- 6 ri) Df r ~,_,- t <f,,,_-t 11 cp,, -t 3L) /] = v 0/ r ;; ~ID 2 rp.,-?- )et,, 0

- ).1,o-

'+'10

L). -

'i-

'J. ;i.

(

~. (,, !)

The Undisturbed Metric The undisturbed metric

rLs '2

SL 'l l Cl, I=-

.5 L put

'"7.

·=-

(.} (_

f 1. -

ci.,

be written

co~ h. t -

S ; f\.

),,,

p ( c(,&

2 r

S i "- :Lc.{

)

·-p

-cLtA?.. 2cLv--cl t

cl ~'2. ~ 5L l.

""Ghen

m~y

't"

- S 1 /t; ~ L..

C"t·- v..}(cl&;~,·ll Gclfri.) 2

( 4 1)

is a null coordinate ro co.lculate t', the affine parameter ,

\•re

note th,_t

is an affine parameter for t he metric within the square

br<~ckets .

Therefore

S5L

ci L-

"f5 ( u., g..,

cp)

(it· 2

\vill be an e:.:..f fine parameter for ( "'- ·· /)

is constant along the null geodesic .

would be taken so that

r ·.: ;

when

t -: ;.

u._

No~mnlly

• Hov.rever ,

i11 our case i ·t will be more convenient to raake it zero and ,,

define

r --

l:-

.Jl 2 cJ_ (;- I

- ---

i'his means that surf aces of constant

constc.nt

•

This may seem

r~ther

be pointed out that the choice of asymptotic dependence

Then

of quantities .

0 ( t' ·-ft,)

( 4. 3)

are surf aces of

odd, but i t should will not affect the That i s, if

)

It proVeb

e~sier

to perform t.•e calculations with this

but all results could be transformed back

choice of

to a more normal coordinat8 system . J.!'rom

r --

The mat ·t er in the univers e is assumed to be dust so its e ner gy tenbor may be

writ ~en '

7 ~b = f I/Gv v b For the undisturbed c a se, from Chapt er 2

bR Sl

N O\r.f

5)_

v1here

~ /3 s "'t A -

i/J'lfJ ~

°'-

fE.S-s

0 (s ·- •)

S' 1. .:. t

'l 'he1·efore if v-1e try to expand t, l1e

V V0-

::. 5LcJo.

/J'"

as a series ·i n po\v:r of S"'

r esult v1ill b e very messy and will involve terms of

the form

If\ S L .05

"'It should be p ointed out that the expansions used \vill only be assumed to be valid asymptotically.

They will not

be assumed to converge at finite distances nor will the ' quantities concerned be assumed analytic. (see Asymptotic Expansions

- Dover

A. Erdelyi:

Tnis

t•

oes not invali date it as an asymptotic expansion but

i t makes

tedious to handle.

For convenience therefore ,

.J2 ( r)

\\'ill perform the expansions in terms of

\-.re

will be defined in general as the same function of i ·t is in tt1e undisturbed c a.se.

1,vae re the n

I ~

rhat i s

AJ. [ i $1f'-~.2t - Q~,r-~ t

%t-_

Jt-r JL 'lf}

cl5l ·--

•

which

cl r

A 5_ s !J4r

- _f2_ -

252 ~

1-1?~ ~,.

<B:.15l ~

For the third and fourth coordinates it is more c onvenient to use stAreographic c oordinates than spheric o.l polars . Since ·b he matter is dus t its energy- momentum te nsor and hence the Ricci-t ensor have only four independent componenL.s . (;:,ince

\vill take these as

\·J e

q> 01

I\ 1'foo ,-\)

rti

lO(

•

is c omplex it represents t\vO components)

In te.cms of these the other components of t he Ric ci-tensor

may be expressed as:

7,, ~ ~2

c/Jo1 cP01

~ /\

~00

.3 full l -

......

ci> 0

•

·r

•

~Oi ~CJ• t 11 ~oo

"l •

·•

cp, ~ --

-- b /\ ~()/

2 I

~oo

·-

l +

·-· cf i o 40~

-</J.o •- .~Qi b (\ -.rp0 0

o/01

(er. 10)

•

(.:>

For the undisturbed univers e with the coordinate system

given :

~oo

4 cp ~:t

4..)'1 ·3

Sl~

1fl_

-#=- -·2..~

452. 3

--,,

452

~01

( ~ . 11)

Using these values and the fact that in the undisturbed universe all the

y "s

are zero, we may integrate equations .

(j. 10- 50) to find the values of the spin coefficients for the

. .

• ' '\Z ·G - 2 •

(-t

25l

i,,.-

5L 2.

--•

s· ~

-1

2Al7~e.

•

Jl'

Jq 'l.( { ..- e. Q..v.)SL- 2t (]_3(1

(-)

:2 52.

J J}

A'L Jl --i-

..,..

2. e

)--Jt -s+- .

•

4Jl.3

5. Boundary Conditions ~e

wish to consider radiation in a universe that

asympt otically approaches the un i isturbed universe :;iven •

above .

et>Or>

and

will then h&ve the values

• o.bove plus term3 of' smaller order. 61.Ven

order o.11d the order of •

v1hich

4:>01

may proce ed.

and

To de·termine this

t l1ere · are tv. 0 1

may talce the smallest oraers

ways

1.Il

·t;hc:~t

\<Jill perini t radiation, ths.t is

•

order t erms than these in

anC..

Lar~er

cP01

\'le

••"e

derivative s

turn out to have their

dependent only on themselves and not on the of

, the radiation field .

-J coefficient

They are thus

not produced by tl1e radiation field and will not be

disturb~nces

consi~.t.ered .

Alternat ively we may proceed by a methoa of successive

•

....

approximations .

We take the undisturbed values of the s~in

cosfficients and use them to solve the Bianchi Identities as field equations for the conf ormnl tensor using the flat boundary cor1di tion that 1-;;- : 0 ( v-~ llJ'l S" these r in equations ( 3. iO - ~ (- )

soacP,

• Then substituting calculate tl'1e dist1)rb-

ance3 inctuced in the spin coeffisients and substituting -che..3e back in tne Bianchi Identitie5, calculate the disturbances in the

'l)l''S

l!'urther iteration does not uff ec·t the orders

•

of the d isturbances . Both these methods indicate that the boundary conditions should be:

.....

5L~

+ () (JJ_ -<i)

-- 01-52-=t) ~,

(see next section)

0(51-=t)

(....

•

~XJ-

. . ..

•

d dx (.,

(~-.

(S-~)

\•/ e also assume 11 unif orm s inoothness" , thic,t

( ')_ 2)

•

J.S:

-- 0(-51-r)

etc ••• I

..L "'

vi..1...L..L.

be shown that if these bounda ry conditions c~ =

hold on one hypersurface

const . ) they will hold on

succeeding hypersurfaces a nd that these conditions are the most severe to permi t radiation .

Intee;~ation

As Newman and Penrose , we begin by integrating the

equations

(3 .

10 &11)

= f

:Vp

-f-

:: ?re-

j) er where

f()

3 fl

-...

~00

-t"

411 r ·t

•

Let

•

<.) 0

·op

then

·p

let

? ··2 ., ~ -(» 'I ) '-/ ·-1

0·. I) (6.2J

D1. y ~ -CR! s ince S, . Cf clr ..t{. cD

(LJJ

then •

V L(

f- T O{t)

;;..

"'-/

where

0 (r )

}Io 11ever

ce ::. o(r -1)

therefore

l)l. 'f " -r<.j>F .,. 0

i;herefore

])

Y ::

F i s constar~t (6 · '+)

(-r -~) ~

(r -;r )

(6 ~)

(r - ~)

f-rO

c onstant

Let

-Then using:

'l) :

(\ -; (

- .) ,_

n ·-~

A.) J_

A ~.f).·). A?.f)_ 1.. - et ~

..) )

JJl

Integrating ,

3 ~ Jl-; 0(1) a...,.

:;. - A

the ref ore

.i!'Or

fB Â·r0~1-')) .lfJ- -r O(i)_)dJ;_ Jl,.- 0(1) -

"t-

o (

.n_

~ -9)

1-L /

d~ J1

dfl

-i-

0 (,)

-= -

Iv T 0 (SL - I)

the r ef ore

l'tepoo:t the process vii th

- 2 52.

::.

/\ \l-3 1-t _; L I

•

l .52- ~

:;..

d k ( J)_ _,. 6 (1) ~ o (SL - ') J.Sl /t_ :; 6 o( {fa ~ -r 0 (J1 - I) I

~ (-51-r ~Sl

Unl ike

(6.12)

\vhere

then

-2

'- ) .

0 ( 51~-I

(t)) .:;

(.J 0

tf e111man and Unti, we c a nnot m:· .ke

the transf or.:natio11 'r

1 :::.

't" -

alter the boundary co11di tion

P I

0 ,

A .I\ - (; 52 :! -

lD552.)

zero by

since this would i'

K <\ -=r .) J_

Cont i nuing the above p rocess \ve de .:·ive:

f ~ - 2Si-2_ A 51 -;_ .,_.

<; Ll 4

-li ' '

'l"

n <J..

j .SY'(.,. (iA -z._ J .4 f 0

o 'J.

( .- I ;i..

6. o . ~

0 )

5l ->

\)- -

~ Q(..n - 7) f~

J ;.

J '-

c--' J & . ' ·:J

(t.1l)

•

To determine the asJyrnptotic behavio11r of' "\lr ol.. ,8 t ... l '.> II "~ and u.:> \'Je use the lemma "Oroved. by Newman L:nd Penrose : ':I1he

matrixBand the column v ector

• given funct ions of x

B-= The

()

2 o{ x ) 6 -

...

o(x-'l) I

•

J.S

real part is r egul a r.

}(

are boundei:i us

(6. , .i) and has no

Any eigenvalue v1ith

Then all solutions of:

(A )( _, 7 R ) J

••

i ndependent of

ei.:;env;;.:.lue i-1i th p ositive rea l part. vanishin~

are

such that:

matrix

ft..

-) cl)

•j

is a c olumn

vecto r . J.!'O r reaso, s to be explained below, v1e will assume for the u1oment that

101 d

J __a_ )

)( v

We take as

rI~n)_~ ')'jlr

1 )

...

d xj

·-

)5l

()L-S)

o (SL-b)

~O I

\ ~' . I J

-- o(SL-~)

the column ve ct or

JL J. A 51~A Sl. i(\ 52. ~. e3 52.").. I

)

)_, )

.:.) )

:e~ )3 nic.) l-).J2 JIc~vJ ~ I

(6'. }_O)

A-

' ) ·- .J

s J)

By equat ions

A f:, A- o o o o

0 fo

S, 13) "l. r4

000000

3. ~ s I 3. l.f y i 5.ft 0

£·. ]

0 •

•

0Qe) 0

-I

oo0

-2

-I

a -2

C) .._,

CJ ~1 0

CJ 0 0

and

o(Sl- 1)

are

•

expressions involving a nd

J r'h • Tot

)Sl ,:hus

</..,

t: .:

0(SL->)

~, ~', S"

'° o (5L-2.) :: 0(1)

{;.)

•

>.Jlnc e

'-[ ;; --

-oL

CJ (51..- <)_)

('3 . /·i)

Usin0 this vie i11tegrate eouatior1 method as above .

't 0

obtain }<'

i.A.

(

l) 51- 1

the same

. 3

~ (j ( .5l . )

v:e rr1C1.y ma.ke a null rotation of the tetrad on each null

geoa.esic

-is constant alone the geodesic since the tetrad is parallelly

tr~nsported .

we may make

By taking Jnder a null rotation

9'o, = fo1

'l1hus

until

\·1e

im] ose a boundary condition on

= u

vc :;.

<in-·- s) 0

<foo

o. have s-pecified -che null rotation we cru1:not "t

•

A /01

more severe than

We v1ill specify the null rotc-;.tion by

and in that tetrad system will impos n the

uniformly smooth. boundary condition that eh 'ro, ~ 0(51--:t) and is and ~; o(. Jl Then by using this condition on

by equation

w ·;:

•

·1

using this in eq_uation

Yi= then by

~. 5 i) 0 0--6)

A. 1·1)

equati~>n

\"'

'f ;

(

_;'l_

-~·)

putting this back in equation (? · C.f ~) w :::. 0 ( 5)_ -1. lo5

('5 .St)

b;y equation

(1.

b y eauation

0 (

S2. - 1- lo 5 .SL)

11)

~~ o(SL -~ ~'5

JL)

equation

by equation

52-)

·:;. 0 ( _))_ - 1-)

equation

by equation

"-1)

•

By differentiating the equations used \Jith resuect to X ol R one r.. a ,i sho :1 thc:..t ."'J•" T1 ' / /-'/ uniformly smooth. 1

<-t

e re

a nd:

Ad,lin,_; equations

~t - j)~;_ '1- J>f.. ·-b~Oi't DA ~ ).. to - 3f' t~ t

--;- jf cp,,

"'t

~. b O : -

')d.

t- t

'l,0

foo - '2J1. o (l2i)

may use the lemma

a~ain

with

-3. /G. 3.

By equ t..ti ons

11-,.r G. 2.8'.

and

'.i'heref ore

-I

o(s'L-2) ·v.1- :: o (n·- ~) are

~ ·:: . 0 ( 52 - ~)

CJCJl. -1) and are uniformly smoctb

From

(t,

2 q)

and

f-' ~ using this in

(0.

(·1 _It)

#5)_ -

and

0(.51. - 2 {oJj(__)

0 (Sl. ·-5 Co5 JL)

(1. 11-)

y"l- : .

I 'r-

may shov1

2.. ~)

Lf.,_ "' then by

\-le

0 ( J2. -

S-)

?. i 31 1 (V) 1. I ~J.K':>,, 5. <t ~ 's. ~ S"

Integrating the radial equations o

·-

3. ~0

o< Jl - fl ~ °Sl- s.,. ~ ( S A1ci '?._ f ol 7 ci °b- (>)_)'L- ~ o {J2 - )@.sl 2 0 ~ JS:~ -{-f() J2- ST ~ (3 !9(-? o_ f orsO~ .~o6~ )n-<tcy 0 (Jl-S"~,3<; ~ 0 ·- /1 J2. - I -f- j)_1_n - 2.1 0 ( 52. - S ) ~' '$]) 0(

•:-

Y. 4 /\ .,, _A o52_ - '2._ ft ( ,\ o"!- 6' "') __,'l,. - ST Q (.JL - 't)

(6 . 5 V)

f lJL- ':-ASl3f "°-.- ~ (3fj""'S""-10SL'.:_ g'-6 0) 52- <r., O(fl-~ ':.x ~ : : X + o (.52. -5) 0 s&) •

·o=

1-0

·-

~- 31-)

Adding equation

~11r _ T~

v~~ 111'

SS) t

Drfi~I -

~ ri

2 X ( J.:. <; 9 )

TLo

et-

sn =-

~), ('(\ -

•

Jp I

,,~

·rherefore

&(59)

(~ · Jo)

By equation

v ·= v -0

±y ; Jl

x -(5ic:J ;-

;

.: x(,

0 ( ..)'l -

0 ( J2

- 5)

{ 2 .1) .

By the orthor:ormali ty relc:tions

J iJ.4

-!( 1'

~ LW)

- 4J

- (S .. S ~ g· _sJ) 1

By making the coordinate transformation

Ll (

II I

x ' waere

c3 ..J

·: le

l,.o

( (;

·--

' {,

( u,

_ x· 30( I -r

XL) 3 ) _

c_;3

x ~o (

) y-

:. 0

still have the coordinate freedom

Xi..::.

j)L(xj)

1.ve may use this to reduce the leading term of

3~J, (~,)'; 3.; <-)

a conformally flat metric (c.f. rTewman and Unti), that is: I

~i ~

~PP r1 (·si-~.- 2 Rn-<;),, d.51 -(,)

lt J

•

v1here

•

..._

Ecuat ions

Non-r~d ial

By comparing coefficients of the various •uo :Jer.s o.f 1

·{ 3

in the non-radi a l equations of

, relations between

the integ r ation constants of the radial equations may be obtained:

3. 23

In equat ion

- l4 A ( 0 Q

the ref ore

.,.

the term in 0

0 -

therefore by

( f-· I)

0 (5J.-I)

u ·-:. In equation

J_ A 4

(

)

•

Jl-' is

(3 , Su) \/o

, the constant term is

therefore

p - p . ~ (( )

By the

n~:i

·-

-l.

0 )

t er:m

p - ~i -

.....

ther.'efore ~

0 -0

(:;.. · ~).

makins a spatial rotation of the tetrad (\/\ I

\4/e

make

{J--

':::

rea l.

- L cp

f'-

fVl

\~e take 5::

{

i l I --rt )(

stereo-.:;ru.,nic project factor for a B;/

2- sphere .

( 1.

the Jri_- lt- term in equation

z.) -t ;(' j, , the

~ 8')

\I ~ e. u. ~<> "-i VSev. r}.

· where By

the

·::.

-.J-

JJ--~

_cl_

(.,

cJX

X;)

(5. r2-. I)

·iJ er·m

f- . 5)

1 e ~u ( S V \! S r S V 'i):7S') /vl I

·:;.

fl ~ - c:; 1.. v v 2

·::

-A

<)_ [.

~ 0

" - 'J. ;v-- lo s e,Q,U.

e. Q ~ J

Lt (·~ · '5 f ) ~ . 60) (s. G t)

cn-

• i.-1nere

t3y m<.-i.i:eing a c oordinate transformation*

cy (

i.J..

(1-. <t)

cLv. I/

1-1

'r -

•

t1 therefore

:: o

flL.

:::

By tne

Jl.

term in

0 - '-I

'2.. c.....

-f.

0 '

(J . 2..b)

C}.U..

f~ tnerefore

o J '

1- A~

-s- f + b

- lo ,~ ~

2u... :: 0

-.,,,

-t-

7- Li.. '.

C(x~ e, 3 -

~This transf ormat i on does not upset the boundary conditions

on the hypersurface

(z.1o)

Jl - Y

.JY tue

A S2_-<.f

3 ( (3· ~I

By the

(?. 2 5)

term in

the

JL-2

../. \.A.)

)

By the ~

'-A

;

':2...

the r·ef ore

2 6°fjS) T K(x. c) e, Q.t.

~ w

C :;.

C+- I 5)

(3.20)

term in

S 17 o vr

the ref ore

s VG

t.,.

..J1,-~

(t-~

Llf-c

g,_ '-'-<

(T- I I)

Q. S-o)

,I I

. . t-J"

)

e '-- V ~ - ~ e.'-'- SV(o~ ."'-6: "')

term in

(....J

6' 0

Q.22)

term in

~ -~ -- ( 6 ~' - cS By

·-

k -;. 0

t.L

SV- ~

J o r7 r ..... :.:. - CJ v ,) e

CJ-

IC/)

(·::;- . IS)

(t= .ff;_) Using

Then by

Using this in

B~r v

(5 ' s / ). 6. ~ 1) .

f}.60) d -b t) ·-- () (.FL - :r)

101 1b0

J L-._ d d~

(S2 -9)

_J_ /\

o (SL -:r)

_, -~

c) v--

---

l .? . I~)

( 5L -:;)

Thercf ore if the boundary conditions

{ S"".

i- Y)

hold on

one null hypersur!ace , they will hold on succeedinG hypersurfaces .

(JL-2_)

0 ~'he

"peelin~

off" beho..viour is therefore:

CJ (

lf 1 f~ •

'f .

(j ( i

-1)

-1

]_.

o(r - i) O

·2

{r - 2 )

21)

As mentioned before , this asymptotic behaviour is

of che zero of

and \·;ill hold for c..ny affine .)&rarneter r rem~ining

To perform the for definiteness :

J1- 1-

.. 9

••

inde~en~e~t

. <:> 01

' y: 11··/-

rfi i I o;

..,..

•

intesrations we will &ssume

JL. -8·r 0(5l - 9)· _

t~ f)_.1(.,. o(JL-'1)

' fi52 ·3 T !\ fo-1-

(Jl -j,')

-r Q

•

SfJ. J1

-5' T

.51-9

o( _fL-10

too

'l'hen :

.,.,._ [. fl

' _1r

v( 5

1 --o e

•

Q;...

<6'

J. <..l.i..)

. .L [ n ~( 3 ·7- . o0 iu + 3 . 11 - '? er e, -r

<> (

S Ro "'r

f' i

o 0

Q.,

~Ll

r 6'

5·0~-cl n - b v J ':2 .. (}

r-t

rl. e.,

\.(

u.)· - '-f ,~ 0 0 0

(J A5°-r

t-.2S)

0(52.-c;-) --

0 "Jl,-'f__

= -j (~:,"I

•

f ~ -)- J2. -:;

(2 fJ:S

..,.

1(:) JL -" 7 o (52-6' ( f-. l y)

·f ," )51->.,

a(JL - ",fps

])_)

1( ;, ., if/)·

r .

·3 J2 - r

· -r

- :£ - (-}

fl, -

/l S2

-i -

) J.

(~ .o

e :< u.) Sl -

2 ..,_

.If

..,. -

'-J JI

0( Jl - ) ·-

0°

)Jl- -

~·

x S2 - 'f _I

(t-. 2 i-)

1 <..( q

~ (ffJ "_ t'7") 5l - if

- 51

-<;)

0(52

(! 2 0)

o(52 _,, u, 52) 5

~ f} '2, 52 - ':_ f-} 3J)_ lt u

2 e :i.u)

J)_ - 3

-+ ?. 8) •

/}6)~ 2.

,._

v ~ -j (6~-6°)e~PS-r

0 ( J1 -5)

7- ]0)

fl S'Sl-;2- -r Ae ~.V s SL - ~ * [ P oie 17S' ( J "" -f e "') -r -e. ~(-VSJiS c Jfl - 't

±~\A

·--

•

e ~ s 5L - -~ - e {,_$ 0-

i [ f) 2 e_ (52 ->)

(} e ~S' 52 -")T

_/12

-<; "r (')

5 (.f

(f- i c:) .

{n,-z_ t(Cf c

~'io)SL-2.,,o(JL-S)

_c,, (As -

x·s --

. b

-::o

- 6

""(

-=.. 2..

-r-

-r -- -7.

,/i

r:;:.o

i--

.._)c.

)ii

) fI

-f- /)"- ( -

[J'"J..(<3~ ~

52-sT

·-

(f 0-r-

I 'f

6 -

x ( 6~~ - 6 °) 2-

0 ,

55)

-G

S2_ ~· J c

{

·-

a=o)I

-c_-,

C,_ ·3 "f

-o 6~ JI

o .\

)f I

6--::

(1-

{.?.?rr) I

<l - S" __; G ·f-

n-2

T _j_

() .

(.)

·-

...:--

/II

v(s (~, 0

6")1;_)

s==

-e"'-(06°) \75-2'"'S7 (6°- cf ) I S2-" ;T I <f lie 2 ( <i ·LI- 6v--s-,_ S 'V (<Y c:_ ,;, 6 c: J &' )} ~ 0

- -

_.. ,

)

Cfo~,_ n_-'., o(JL-

(t-.]6)

W~e 9-Q,: ()

(,.0

Q, 'r

·-r •

~ .

--:r

•

lA..

...

s I l

•

+~: +

oo) .

· 1fri 1 I)/

·s,+0,l ;::

•

'i1 hU3

derivative Of

t; 11G

ar1d not on the radiation field .

:·epenQS

On itself

0l1ly

It therefore re 1Jresents a

type of disturbance unconnected with radiation.

If it i s

zero on Ol1e hypersurf ace, it 1vill remain

In this case

zero.

it is possible to conti!'l.ue the ex· )ansions of all q_uanti tie~· in neg; a tive po1,.1ers of

without any log terms

a~pea~in3 .

.... . ·11he mevric has the form:

'2.4

~ ~[ CJ.'

3 G}

·--

-... ~

i'f

CJ )

-- J1?.. r- 0 (52 -<-) -- 0 ( ~ -y) -- ·- 2 f ~). g J Jl - r ·tI

The asym)totic group ts the

~roup

'2..

- I

(

;

2 R ?'2. S(JA-~

of cooruLnate

c;(~?_- 6)

transfor~a~iors

that leave the form of the metric and of the boundary conditions unchan~ed .

It can be derived most simply by consnering the

corresnondin~

infinitesimal transformations:

(f, I)

'l'hen

Kd.) (!c 2)

f /J

(?. 5)

f 0

'fo I

obt~in

( ::;.

~ (..

S'3 ~J ~) 11

f.- i o<.

'?._ [_

(~. y)

• • J I

r ~OU

0 (yt_,-b)

.........

(5L-7)

{-5G-1)

0(51_-<-) (j (52,-4)

2.2

R. ( o

J'.J'~::

f :s3 .: -~3 '

~ I

D I\ So(

r. o<

JJ I

~)f.

f(J ~

.i o(

f( / (

r S-)

the asymptotic zroup we deuand

..._

_. ·-

'95

I 't

.:::. 0

~. &'

o (SL-,

11 '

<6.

t-)

'""2..

:::.

/( '2

·f-

) 2.

,, k'.::.

(( ( y q ) ; .J

I< )J.. c\

•

I::

1\0/ (l(~) j

"3 :::- f( OS (v.., 'X-

(U)

l.) -

( ?. 11)

1. 2) I

'""->)

00 . I )

(~ 10)

re. t.2)

1"3) imply that K. '- is an )( '3 -t- t,_ 'X.1 . ·rhis is a consec;uence and&.

anal:-ftic fu11ction of if t

t1e

f a~t that 1t1e h;.,~"1e

recluceu. the leadi11g term of ~ form .

·~huu

the only allowed

to a

'.J

co~for~~llv ., fl~t

transfor~ations

)(

·'r""v

c;..

+-'~ e utJ...

conformal transformat i ons of the form :

c (

x 3 + L xlf-)"I cl

{).cl - be ':!hen lic"te

six param8ters

c1.etermined. by (<if. I<+)

Q..,

..{ 1

b c. cl ,

• ure c;iver:

is also uniquely determined .

~nuci

the asymptotic ;roup is isomorohic to the conformal sroup in

ciachs(~)

two uimensions .

to the

~01006eneous

has shown that this is isomorphic

Lorentz

It is

~roup .

~1lso

11o;vevo r· iso1i1orpf1 ic

to the group of motions of a 3- space of constant

ne~ative

curvature which is the group of the unperturbed rtobertson.alker space . sroup of the presence of

Thus the asymptotic unc~i3turbed

r~di~tion .

space .

~roup

is the

It is not

san1e

enJ.ar~ed

&s th·; by

This is interesting becausA in

the ·i-. i:; '" ...

case of e;rc..--ri tational radiation in en1pty, asyr.:ptotically -·ic:.. t space, it turns out that tha

a8ympto~ic

group cortains net

only the '1 O dimer1sional ir1homoge11eous Lorentz __;rol1p , the z:;roup of 11

motions of flat space , but al8o infinite dimensional

supertrar1slations 11 •

sugges~ed

that

thcs~

might have some physical significance in particle physics . The &bove result would seem

supertransl~tiona

~lementary

It has been

ino.i~c.-te

·t;hat this is nrob<..,bl ...·' not the cc..s e

sir~ce

our

ur1 i ve:rse is ,:...ln1os t certai:·: ly not asyt'lptoticu.lly flat though

it muy bG asymptotically

Robertson - v~allcer .

v/hat an ob .. erver would 1neasure of an obs erver

The velocity vector

movinr~ ~

with

du s -c 1:1ill be :

Jl~i.,..

v).

v v

·~

(9~)

, the projection of the ;,..:ave v 0ctor f\"'\

in t;he

observer's rest - space ( the anparent direction of the wave · 1:1ill be :

s - vv /l"'I

M \ •

c~ I

J1.

-- -I

)..

t;2..

-2

.,.

o(J!--~)

O(JL-~)

0 ( Sl. -- ~)

(}

'l'he oboervcr' s

oi~thonormal

tetrad may be

com-;il~t~d

t\•IO

and

space - like unit vectors

4) O{J2 -z)

<:; -. I

~ '.l

5'J 3 -\./

Cu~ -

cI -

0 { ___,'2_ - Q.)

t ::i..

{

- J5.

·.j 2

l .

(J..

•

:e v:ri te

0 ( ._,)<]_ ·- ~)

·-

(~z. _5)

e.""' --

13-:r m9c:..surin._; tb.e

rel~tive

o.ccelerc.tions of neirjhbouring dust

p<...rticles , the observer ma.y determine the 'electric ' cou1ponent s of the gravitational v1ave ~ p

E. a..b

-c°'-f bi

'1..

In the observer's tetrad t!lis has corrt1)0nents

,,,,,.

E --

,.,. . 0

o(-52-4)

-; •

-1

t 0

.l-(

O(J2-~).-

0 (Jl-5)

0 O{Jl-6)

0 • --.L

,....

•

-;-

-J

•

( J1_ -'r)

•

·-r

l..

I i;_ \

-I

( _1. r

'"hi J. .;.; ' ......1ould be cor.ipared to the behaviour for asymptotically space for \vhich

fl~.t

- --

-0 0 0

"-t-

,....

o(r--')

0 - l

-CJ

[

0 l

·- l

eJ (r-2)

-r

I -1 0

o(r-;)

o(r -1)_1- 1 0

C> i

-~i

~-~{ -~) () v ?" 0"'

o-i 0

!q:;)

•

• ondi ,

. G. J . v·n

. . .s

eh... •

' . 1: •

• .L .

roe . oy . Soc . A. g·-··

roe . oJ . oc •• 10

ro...,c

1....

• •J .

•

en.re

•

i ver""i

91 J

roe . •oy • oc • yo .

v•

1 19c,2

,9 2

10~

1)62

)

yo . ~

t1 .

• 0

•

~ur~

• • . 11etzncr

(.;rid

1 9 'I ~1

J'.j'

Sin,:;ulari ties

If t;he Einstein equations ·1·Ii thout cosrnological constarJ.t are satisfied , a Robertson- VJalker :::nodel can ' bounce ' or avoid a singuli:i.ri ty only if the pressure is less than :::iinus onethird tne density .

This is clearly not a property ~ossessed

by normal matter though it might be possessed by a field of ncGative energy density like the 'C ' field .

nO\vever there

a 5rave qua11turn- mechanical d.ifficul ty associated \·Ji tn the

existence of

ne~ative

ener3y density, for there would be

nothin 0 to prevent the creation , in a given volume of space tin1e, of an infinite numb ex' of quanta of ·Lhe negative er1ersy fiGld ~nd a corresponding infinity of particles of posi~ive enere;y .

If \ve therefore exclu·de such fields , all Robertson-

\val \.er moC1el:.>

rnus·t be of the

big- bang ' type, that is the·;,r

11C1.ve a sin0 uiurity in the past and as \·Tell .

m~ybe

one in the future

1 that the occurrence o= It has been sug~ested

these si11Gt1lari ties is a consequence of the high degree of symmetry of the Robertson- Walker models which restricts the expansion and contraction so that they are purely radial &nd t;hai; inore realist i c models \.,r i th fe111er or no exact sy1n1netries v1ould not have a singularity .

rhis chapter \•Iill be devoted

to an exarnination of this question an:i it :1ill be sho\-1n tht.t 1

provided certain physically reasonable conditions hold, a~y

model must 11ave a singt:lari ty, th;.- t is, it cannot be a 2 5eodesically complete c~ , piecewise c manifold . '

2. 'l'he

~··un<lamer1tal

Sou at ion

'.rhe expansion ~ongruencc

v .

of a time-like ;eodesic

6--

vii th unit tangent vector

VQ..

of Jhapter 2 :

obeys equation (7)

eI°' V°' =

. v ct ·v b

CAb

{t )

will be said to be a bingular point on a Geodesic

A uoint; ~

((

of a time - like geodesic coni:;;ruence if Of or ·the congruence

is infir1i te on

conjur;ate to a point singular point on (

•

A point ~ \vill be said to be

along a geodesic

·~he

•

A noint

will be said to conju;&tc

tf3

if it is a sing~lar ·1oint w3 . .tin altern8.ti-:re conbruence of geodesic normals to 11

a.escription let

if it is a

of the con0ruence of all time - like

to a sp<.,ce - like hypersurfac&, of

of conjugat;e points may be 3iven as follo :Js: 1

K~ be a vector connec·bin3 points correspondins distances

alone; t'v-:o neighbouring 3eodesics in a

conGruence .-1ith u11.i-t; 1

tan...;ent vector

V°:

K°'-

Then

is ' drag,;ed ' along by the

congruence, t.1at is

J l\ "-

J) KC\.

{!1)

(/.I

R°" .

J)2 K cz

v b'

JJs

-- C)

<>..

vbvc1~ct

(s)

'bccL

J) >2. ~

Introducing &n orthonorraal tetrad u~on·· (J

Cl.

v1itn

oL 'l J\ s ol.

0..

vo. -

(4)

n..

CJ,

v.1~1ere

I'\..

(\..

-...

0..

1 ..

C\., Q,

fVV

solution of (!.t)

0....

vie have

Q_ (If\

parallelly transpo.rted

fl1

t'v

bcl

vc vd.

i'Iill be called a Jacobi field .

clet...rly eight independent solutions.

Since

Va..

·.rhere :ire and

solutions, ~c11c other six indeuena.ent solutions of ta~en ortho~onal

a 3eodcsic

•

i1hen

'f...

a::·e

(f/J m<--y ·oe

p alon~

if , and only if, there is a Jacooi fielQ alon§

1.-JJ..lich vanishes at

and

{

:.i1his UJ.ay be bho111n as f ollo'.lS:

ti1e J·a.cobi fields which va11ish at e;encratin~

is conjugate to

l{:.,;o..

may be regarded. ss

i1eighb ourint; geodesics in the irrota·bional congrl1cnce

of all ti1ne - lilce ge odesics through

f .

Therefore they obey •

·-

mo.y be 'N'ritten

~L'h ey

I\.

d f{

cL A

.. nere

M I'\

cl ., ear

p , A

{s)

will be oositive definite . ..

lh ('\

a J acobi field vanishing at,

J3ut

A Cs)

m ¥\. '11heref ore

_d..

(9 -d<l,(A) cLs /\\I'\

cL 'l

(-) Ml\

ol ~ 'l. Hence ()_

-Q

1.S

infinite

if, and only

if,~(!},}"/-J); 0

(dd(A>) f\1 f\

A h. •

1.S

ols .

ci (~}

'.:'l1e.re fore

and

·Ihe:.:·e \:ill "'.:>e

:!{\ cLs I)

exp (sf .1

and

cl~

\'I' here

finite

anu- only wl1ere

cld(!)

...- 0

Thus the tv10 definitions of conju.:;ate points o.re equivale11-c . '.l'his also shov1s that singular j)Oints of con3ruences a:r:·e points \vhere neighbouring geodesics intersect.

•

~or

null geodesic congruences with ourallelly a._

tangent vector

~ran~ported

vie may define the conver;ence

in Chapter

1~ e

define a singular point of a null Geodesic congruence as

one where

is infinite .

'.L't1e condition tho.t the pressure

•

l. ::,)

greater ·than

one-t11ird t'!:L'a! density may be s·tated more c;e11ero.lly as co:1dition (a) . (a;

observer vii tl1

,...

t;" /

f >

4- veloci ty

I'\

i...J

j_ :;.

C>....

for a.ny•

)

v:l1ere

G : :.

,,~,"\b

J <A.~ (,.,,)

is the enorcy d.ensity in the rest- frame of thA observer

T :::

-A-

~nd

is the rest - mass d.ensity .

Condition (a) will be satisfied by a perfect fluid with density~>

o and pressure

any time-lil<:e or null

v ·~ ctor

v c,vb implies J\~ 70 for 0

Therefore by equations

(1) &nd (5) any time - like or null irrotational beodesic

congruence must have a singular point on each ceodesic withi~ c. finite affine distance . ~n

Obviousl;y· if tl1e -10·.-!-lines f or;il

irrotational geodesic con3ruence, there will be a ohysical

sinsularity at the singular points of the congruence w~ere the density and hence the curvature are infinite .

~his

will

be the case if the universe is filled with non- rot&tin5 dust

llO'•Jever, if the flo\·1- lines are not seodesic (ie . 11on- vanisnins pressure ~radient) or are rotating, equation (1) cannot be applied directly .

•

2, opu:tially :1omo;;eneous . .i.nisotro.)ic 0ni verses ~he

ar1J.

dobertson - .~alker

b~atial~y no;~o;eneous

isotropic, that is, ·they have a si): para:1:eter sroup of surfac8 . If v!e redl1ce tl1e considering models tl1;;;t arc spati::.-lly ~!omoger: eot1s

motions trunsitive on a f

models are

symmetry ib,y

su~.ce -lik e

but; anicotrouic (that '.is , t11ey hc....ve a ·thr·ee parameter grou_p of mol,ions transitive on a space-like tne

mbt~er

hypers11rfc~ce)

flow may have rotE1tion , acceleration

tnen

~n~

there would seem to be the possibility of non- singul&r ~od.els . .L . Shepley4 has in"'restit;atod. one particulo.r

~hus

l1omoc;e:neous model containing

rotatin[~

there is alv1ays a singularity.

dust and has shoi,1n thc.t

Here a seneral result will be

proved . ·.J.'hei"e

r.uus~c

be a ::;ingulari ty in ev ery model \·1l1ich satisfies

condition (a) and , (b) there exists a

q. -

mo~ions

universal coverin~ space~, Y;}3

on the space or on v1hich is transitive on at;

..:>ee section 5 le~st

one space- like surface but space - time is not s·tationary,

-flow - lines

• ~nd

is uniquely de fined as the time-like eigen-

vector of the ki cci tensor.

I •

R, tl1e curvature scalar 111ust be co11st<:l.nt

,., ... o. On

like sur i'ace of tra11si tivi ty ff 3 of the s roup .

--1~"e bl_ c...tv .

R·. vc-..

·.::here fore

111u st be in the direc t i on of the unit time-like normal 3 to f/ •

/ 0...

\-J11ere

e CR jU- J hen

'.L

an indicator - +1 if

- - 1 if

(( I

•

lS p:::.st

("..

R. c..

•

directec.

. 1u-cure C.irecteC: ~

·\{a.: b] =0.

Tl1us V:"" is a cone;ruence of ge odesic irrotational timo-lilcr-; v ect; ors .

.i3y condition ea) ,

-~her· efor e

the

con~ ruence

R°"b va... v h > o

•

sin~t1l<.tr

must have a

point on e <::Lc'-"

seodesic ( by equation 1) either in the future or in the

~ast .

£urther , by the homogeneity, the distance al ong euch 3e oiesic 1·ro1n l1 J to the singular poi11t must be the same for each ,-;eo«ce::.:ic . ~nus

if tne surfaces of

transitivi~y

r emain s·Jace - like, 2 must degener E·t e into , at the most , a 2 - surfaco c 111hich be uniquely ciefined •

Let M be the subset of the

of the matt er which intersect empty subset of

group transitive on

c'2- .

intersected by

1f , /. .

Let

fY .

flo\·1-li~;..es

be 1..11e 21on-

Since there is a

must be [--/ 3 itself .

Thus all t L.• e

flow-lines t trough

must intersect the 2 - surfuce

·bhe denoity •:: ill be infini"'ce tl1ere r>hJJical sinGularity .

CL"1..:.

Alternatively if

c2 .

Thus

-C;h0re i::ill be a ~he

surfaces of

transi ti vi ty do not remain space-like, there must be <::.t least one .::>urface \vhich is null - call this

P..~ J

s3 . }~t s3, F -

• ( if~.,0.. is zero, vte can talce any other sca.J....:..r

~) olynorr:iul

in the curvature tensor and its covari.::nt d.e1· ivatives .

·.!. hey cunnot all be zero if s9ace - time is not station:.:::-y) .

..e

..,.

introduce a geodesic irrotational null coigruence ons? t:ith t an00nG

vec~or

Le-.

111here ['

ti1e.L·e will be a singular

~ . : ()...

poin~c

"hen by e ci ua+i·on v

;, /~),

-·

of each nul L geodesic in ;:_;.

vii thi11 i'i11i tG t..ffine distance si ther in ille future or in t:~e past . ~he 2- surface of these sincular points will be uni~1;ely defined .

·The sa1ne arr;umen·c used before cho;...,rs tha·t tl1e dc:r1si ty

becomes ini'ini te and there is a physical singulaI·i ty . In fact US

83 is a surface of

sin~ul~:c

homo~eneity,

the whole of

will bE

and it is not 1neaningfu1.· ~o cal.!. it null or to

distin~uish t~is ccse f r om the case ~here the surfaces of tr~nsitivity

remain space-like.

:J;he conditinns (a), (b) , (c) may be v1eakened in ti·.Jo ·,·1a:.rs . Condition (b) tnat there is a group of motions throu~hout space - Lime m~y be replaced by (b/) and (d) .

•

3 . Thei"B is a space - liice hyper surf ace !1 ir1

(b )

~re three independent vector fields X.~

ix.°' 9

~ 01

j 1Z

xl;

-·-

hcd. e f?3 one homoGeneous space section . bC..

(d)

"".:1ere

such thlit .

. ·ihat is , therP-

'I'here exist equations of stat;o such that the 1.Jaucl1y

develoument of

Then succeeding are

, .

~·1i.1icr1

ho.ao~eneous

is determinate . so~ce - like

surfaces of constant

and. much the same proof can be siven t::ict vu.ere I

are no non-sin~ul~r models satisiying (a) , (b) , (c) , (d) . Th0 only property of perf cct fluids that has been

u~ed

i11 the a ·o ove nroof is ·that they have well defined f loi:1-li1:cs intersection of v-1hich implies a physical .=ingulcL.Ci ty . howe v er ,

·t~is

proper·ty will be possessed by a much ::ore

clcss of fluids .

F or t hese ,

\•Te

~e~ ;: · 2.

define the fl ov1 vector as

time - like eige11vect or (assumed unique) of the tenaor .

Obv: Jusly,

1..:1.. :

energy-n~omentu:u

•.:.•hen \-Je can reol ace condition (c) on tl1e nature of

tl1e matter b;y· the mu c h \'lea..."l{er condition ( e). (e) <..-\

If the model is singularity- free, the flow - lines -f Q">"•-:"" -

.L. - -·

s;nooth time -like congruence \-Jith no s in~ulo.r poi11ta -.:iLh

a line t.:1--ough e a ci:l point o f space-time. Conci..i tion ( e) v1ill be satisf:Bd automat ically if conditions (a) and (c) are .

'l'his proof rests strongly on the ;:ssumption of homogeneity 11hich is clearly not satisfied by the physico..l 1

universe

loc ~lly

however it

~Âˇ/Ollld

though it may hold on a large enough acule . seem to indicate that la.r.'ge scale e.f'fects

like roÂˇt;at ion cannot preve11t the singularity. It is of interest to

ex~mine

the nature of the

singul ~rity

i n the nomogeneous anisotropic models since thJ s is more likely to be representative of tae general case than that of ~he iso~ropic

models .

It seems that in general the collapse

.!ill be in one direction., 5

that is, the uni verse \-:ill co::..ls.pse

down to a 2- surface . Near the singularity, the volume will be proportional to the time from the singularity irre spective of the precise

nature of the matter .

It also ap pears that the

nature of the particle horizon is different . a particle horizon in every direction except

There will be th~t

in which

the collapse is taking place.

din~ularities

in Inho!o3eneous

~odels

~ifshitz and Khalatnikov6 claim to have proved that ~ general solution of the field equations \vill not have a sinsularity.

Their method is to contract a solution with a

singul&rity which they claim is representat ive of the general solution with a singularity, and then show that it has one fewer arbitrary function than a fully general solution .

â&#x20AC;˘

CleurLy tt1eir whole proof rests on whether their solutiun is fully representative and of "cl1at ·they :_;ive no proof . Indeed it would seem that it is not representative since it involves collapse in two directions to a 1- surface

\~hereas

in general one woul d expect collapse in one direction to a In fact their claim has been proved false by

2- surface.

Penrose? for the case of a collapsing star using the notion of a ' closed trapped surface ' •

...~

similar method v1ill be

used to prove the occurrence of singularities in 'open ' universe models .

5. ' Onen '

~nd

' Closed ' Models

The method used by Penrose to prove the occurrence of a physical s i ngulari ty depends on the existence of a non- compact Cauchy surface .

A Cauchy surface will be taken

to mean a complete , connected space-like surface that intersects every time- like and null line once and once only . ~ossess

exalliples of those 8 that do not include the plane - i,.1ave metrics , the Godel model ,·9

Not all spaces

and N. U. T . s pace:o significct.i."1ce .

a Cauchy surface :

Howev er none of these have any p'iysic.:l

Indeed it v1ould seem re a sonable to deznand of

•

any physically reali st i c model that it possess a Gauchy surf ace .

If the Cau chy surf ace is compact , tl::1e model is

commonly said to be ' closed ';

if non- compact , it is said to

\ The surfz.ces , t = constant, in the . 1.obertson.

·to be ' open ' • •

ialker solutions for normal matter a.re exanrples of Cauchy

surface:..> .

If K = - 1 , they have negative curv<J:ture and it is

f'requently stated that they are non- compact; . necessarily so : they are compact .

:l 'his is not

topolo~ies

the r e exist possible

for which

Howev er , tt1e f ollo\vin;s statements may be

made about the topology of the surfaces t

constant .

If the curvature is negative , K = - 1 , the universal

:s3 .

covering space is non- compact and is di ff eo111orpl1ic to ~ny

other topology can be obtained by

points .

identific~tion

'l'hus any other topol ogy \·.;ill not be simpl;>r connected.

and , if compact , must have elements of infinite order in tJ:1e

fundamental t;roup . l<"u r t her if compact , they can r1ave no . 12 group o f mo ions . t

If tl1e curvature is zeI'O , I\ = 0 , ·the universal coverint;;

space is ..£ 3 •

1 · h teen possi· bl e t opo 1 0 0- ies · 'rhe r e a r e eig . 3

compact they have a G

of motions and Betti numbers, 3

I.L.co

- 3,

B2 = 3 . 12 If the curvature is positive , K

coverin~ space is

= +1 ,

the universal

s3 .

Thus all topologies are compact . 12 Betti numbers are all zero.

·.i'he

Since a singulari ty in the universal covering space implles a singularity in the space covered , Penrose's mGthod is applicable not only to spaces that have a non- compact Cauchy .

l I

surface but also to spaces whose universal covcrins sDace has a non-co:-:-ipact Cauchy surface .

hus it is applic::;.ble to

1 1

.i.

models which , at the present time , are homogeneous and isotropic on a large scale with surfaces of approximate homogeneity which have negative or zero curvuture . 6.

Âˇ1

he Closed

'11rapped

Surf&.ce

Let r3 be a 3- ball of coordinate radius r in a 3-surf ~ce

(t = const . ) in a

Robertson-~Jalker metric with K - O or - 1 .

Let qa be 1Jhe outward directed unit normal to T2 , the boundary of 'i 3 , in H3 and let Va be Âˇthe past directed u!li t normal to 1

H3 .

Consider the outgoi ng family of null ceodesics which

intersect T2 orthogonally . At T~

be:

-, ( vo... l

. .. ~ b

f ,

tl1eir conver~ence will

q()._ h 1( ~ ~ ~ b +I

/; ~ {: b)

\.,rhere fO'- to.. are unit snace - like vectors in H3 orthot;ons.l , to q a and to each other , i;teref ore If JV-. 1 0

f and 1(

...

'1 R

(Jj_. '3

-K

- -r I ;- kt I

Q. -]

or - 1 , by taking r large enough, vre raa.y

make1 negu.tive at T2 â&#x20AC;˘ TheI'efore , in the lan3uage of Penrose, T2 is a closed trapped sur face . .;mother way o.f s e,ei ng this is t o consider the diagram

in which the flow- lines are drawn at their proper distance from an ooserver .

at t

'l'hey all meet in tl1e

sp~tial

singulc::.rit:~i

J . lf the past light cone of the observer is drawn

on ·tais

dia~ram,

it initially diverges from his world- line

It reaches a raaximurn pi"oper radius ( (:> .:. 0

and then converges again to the sine;ulc.ri ty (

f > ()

)

\) .L he

1 1

•

intersection of the convergin3 li 0 ht cone and the surf uce 3 3 gives a closed trapped surface T2 . If the red- shift of ~he ~uasi-stellar

poi11t

3C9 is cosmological then it will be beyond if

\·Te

~he

are living in a Robertson- .ialker type

universe with normal matter .

However, tae assumptions of

homogeneity and isotropy in the large seem to hold out to the

dis·tance of 309 .

.rhus the.re is

our universe does in

fac~

tjOOd

reason eo believe th'-L

contain a closed trapped

sur~ece .

It should be pointed out that the possession of a closed

trapped surface is a large scale property that does not

Qe~end

on the exac-t; local metric . Thus a P.J.od.el thr t had locc..l irre0uJ..ari ties, rotation and shear but \vas similar on a larse scc:.le at the

pi~esent

time to a Robei"tson- 'l"Jallcer model \-.rould have a

closed trapped surface . Following Penrose it will be shown that space - time has a singularity if there is a closed trapped surface and :

I• .

(f)

E); O for any observer with velocity

(g)

there is .a global time orientation

the universal covering space has a non- compact

(h)

Ca~chy

surf ace H3 •

Assume space - time is singularity free .

Let F

be the

3et of points to the past of H3 that cun be joined by a 2 smooth future directed time - like line to T or its interior T3.

Let B3 be the boundary of F 4 •

Local cons:iierations

that ~3 - T3 is null where it is non- 8ingular and

shO\'!

•

t;;enerated by ·the outgoing family of past u.irected. null geodesic sec;ments v1hich have future end- point on T

and past end- point

·.!here or before a singular point of the null geodesic 2 > 0 and , the convergence, T at ~ince . conbruence

si11ce

Ro..b L ~ lb ~ 0

by (f) , the conversence must

become infir1i te vii thin finite affine distance .

hus B3 - r.r3

'..L

will be compact being generated by a compact family of cown&ct seg1ne11ts .

1ience B3 v1il l be compact . 7,

approximate

tnen as follows:

Penrose' s metho·i is

arbitrarily clo~ely by a

smooth space - like surface and project B3 onto normals to

~his

surfuce .

3 by t~e conti~uous

6ives a many- one

0ince B3 is compact , i·ts image B3¥ .

I I

must be compact . ma1ped to a point

Let ci(Q) be the number of points of B3

of I-13 .

ci((\))

will change only at the 7.

intersection of caustics of the normals \vi th H by continuity

•

l\1oreover ,

can only change by an even number .

•

since this ~his

l S

the identi·ty D-i'1.d

is a contradiction, thus the assumption

is non- si11gular mu s t be false .

th~t sp~tce -ti1ne

al ternv..ti ve proceedure

hl1

which avoids t11e sliGt1tly questionable step of approximating

B3 by a space -like surface i s possible if we adopt condition ( e) on the nature of the mu.tter , tl1en B3 may be proj ~c·bed continuously one - to-one onto H3 by the flow- lines .

~his

again l;eads to a contradiction since B3 is compact and H3 is not .

In tbe

abov~

n~c~ssary

proof it bras

to deQand that ~

£' b& a Cauchy surfac8 oth@rwis~·the w~o l e of B' might not have been projeoted onto

li3.

We will define a somi-Jauchy

'iur:faoe (o,G,s) as a cottlplo·t;e connected epace- lika eurfc.ee i;h.gs,t

intersect~

eve:I.'Y tii:t1@-lilce iiind null line at 1t1:os·t

O"i.1i0ie.

~· s . C.s .I? will l/Q a c;'3ucti.y · sur:f'ace for. ;oi:Q.t• i'.tear it, tim~-likQ

is:!:&i5 is, it· will intersect ove;;:-y

tl1rou,;:1 these points •

nOV.' OVOr,

anC. i1ull l:i.nie

further &>1.,,_y lihoro 1Tiay be

?e! iono for whieh it ic not a Gauohy curfaoe .

~et of uoints for which H3

. f it oxis·l; s, l:

oallod "bhO Q!:U;Cl?:Y: liori z.on rel a ti 1re to H3,

I J

•

is a Cauchy su.rf-.i.c" e..nC.. lot

tac boundary of those points.

•"blrfaoo,

Let

CQ>

will

fft\!:18is 1':

b c- the

<Ql -oe 8:

b ~

r.~M

:1/u;p·i;hO.PHl.O:Fw

if condition (f) boJ..de tho Rl:ill

e@@&@di&C

I '

at lo&ot one sxnsul &r thftiet thoir v iitUet

w~..:.n•

·s:i:m~e c::~&ltl:plo

l·~lec

ho ~~

-DOUnGo - Au

poin~

tthien

· one ttirccu -· .... ion. ·

+ ave~G ~

• ln

of s • J, s "ff·ith o. Cauchy l1origo n is a

~pGooe

· ou:efae c of eons·sa nt negati ve our..,la ture ooeple tel5•

\titl1in tac nu±l eone of the oeint in

1 "ia 1n:o·:; S .<:i • space.

r 11

B:ull oonc forao the tJauohy horiso :a. ~£ 88Bdi tioao (o)

and (g) hold, than a model 'iith a

- . s, ~mpaet o. ·G - -EE 3 ----8 must h :\,..e theepo t ±ogy:

4•4

t: .

~1100}'

there ''ere a rc5ion '/ V l.j throug h which taerc were

·8uppo~e

no flow lines iatcrs eoting S 3 , then ~I l the bounda ry of

be a t:hme- liko surfao e t;onera ted

•if",Y§t

inters ect H3 ,

t/li __ ~- -

1)y floi·:= lines ri·hioh '~ e

Procee ding •long thoso !lo .i:-linoi ;i in 1

·1;;1;;i,,e

d ireoti on of their intors oction with H.?, ¥JO :Q:Uat reac a 6.jji ., 3 ~4d-point of tho gon@r ator sine~ V does not inters ect l I•.

&:it..

the -0xiste nco of

~plios

thi~

onO.-p oint contra dicts (o)

a singu larity of tho i'lo·. x-.lin8 cone;ru once,

•

~inco ii;

'?hue

1/ ,, ~ •

:t'lus b be em1> b;y and ewer;, point has a fle;1 line ti'lrous a i-V interc oeting :a3, 0

,..,

Thus wo

1 :.,/ , C. I -. huy assign • • ~-e ing

hd.'l9 ,

a hoi:aoo morphi se of the space

every

poin~

of iho s ·J ae e i ·t .a

6:S:o=6anoe along the flow-l ino from 53 and tho point of

s·eetio n of li:1c flo\11- line •11ith H3.

f •

intG~-

It oan also be shoi... n ·that

I ..1.x>. =rx

s e""'"ne +.hl:. -~~vn--

r--1t

.._ '00 a ffiUSv

,,v&Uehy

"6t1ere '!'fore a Cauohy horizo n ·

eacb !lo•·r= line a.t n1ost once . •

GUPfao e,

(((-1

M'or

this c&n

•

~herefore

H? .

~'urtj;j.er,

.w:ust \i.nter~Qct .cQ 3 .

Tnus

~ 3 is h.omeo~o:r1)hic

O.irecti c-.1* 6.way from H3 e~ch

"by ( e)

If condit ion (f) h.olde every null

of ~·3 fi&s· at least one end point .

~l1ere

intor~ocv

L±OX00 -

every :floli!= l :i.ne -:o H.3 and. is

g~od~sic

3Bno:» .to:F-

sine~ in tho direct ion to\tJard s

aenera tor auo=6 be unboun dsd.

oin~ularities

....

Tnio must be in the

. a ire,"}o ssiole l:O r iniso· n·eve

m, .

uuu •OcJ.1€

there is a

FBorpl:iistfl. of ~:l in"t;o ·

€01ripa ot.

nun

•

. . ses univer ' Closed in ' .

is a singu larity in every model which satisf ies

(a) , (g) and (i) . •1ihere exists a compa ct Cauchy surfac e H3 \·1l1ose

(i) normal

l l l l l t:

3 has positi ve expans ion everyw here on h •

PROOF the proof it is necess ary to establ ish a

~or

cou ~ le

11.Ssume that space- time is sin6u lari ty-fre e .

le111mas.

follo;,. 1ing result is quoted v1ithou t proof , it

'I'he

m'~-..l :a:.~ac-~]:y

l)e

derive d from lemmas proved in refere nce 11 . if

and

ci,.

are conjug ate points along a geode sic (

and 'X is a point on point in I

f} ·.

not in fCj... then X' must have a conjugc-.;.te

iln

e lon3

i~mediate

corrollary is that if q is

conjugace t o p

conju5ate points in

and y •

•

J.n

J.S

~he firs~ ~ oint

then

•

a.G..3

Also since the result that x

h o.s a conjugate point in pq ca11 only depe11d on the values of

in pq, any irrotational 5eodesic congruence includin5

C\. M f\

the geodesic

must have a sin.t;ular point on

.rhus if q is a point on

f"i3

in pq.

is the geodesic normal to

and (

M3 tl1roug h q , then a point conjugate to q alons ·

canno·t occur

3 until after a point conjugate .to M • If 113 is a complete connected s1)ace-like surf o.ce \·1i1icli .

inters e cts every time-like and null line from a i)oint p, '::e may define a function over M3 as the square of the geodesic

distance from p which i s taken as positive if the seodesic is time-like and negative if the geodesic is space-like. this the world function CY \vith respect to p. set of v<:,lues

:?O

v1here

11." l I

A time-lil{e geo ... c sic

from p will be said to be critical

to a value of <5 for \vhich , € f'J..

Ji'or ~he c l osed

wiJ.l be a continuous ( in

general mult1 i-v<:1.lued) function over

~e call

u,, ,AA ..

C• /.A t,

if it corresponds I i. L..

:. .•, ~ 3>) 1

are three independent vectors in !'13.

Cle~· rly

...

(

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. cannot be maximal for a smooth M3 if there is a

A geodesic

point

where q Let

and

conjugate to M3 but no point conjugate to q is the intersection

of !

"'f

in qp ,

and M3 . ~

f and g be the Jacobi fields along m m respectively. They may be written

which vanish at X

A(s) f/q ,

•

g = B(s)g/q • m mn

h1 (

Then n

any h

e~ x;

<{,

°r

~ J ff¥ I ~

c/..s

must be positive for

since if it were negative for any h

by taking a = ... b h h mb mn beyond q , it would be possible to have a point y on beyond q

conjugate to

before a point conjugate to

would be conjugate to

)(

does .

Since

o£ i

•

If it were zero

This shows that the surface at q

of constant geodesic distance from direction than the surface

lies nearer to

in every

of constant geodesic distance from

X is conjugate to M3 the surface at

constant geodesic distance from

direction than M3 does .

Hence

lies closer

f to Jiil

is not maximal .

in some

,u :;.t 'be

o ":Q 4

•it i

"'~e

G.O

...,(].

.. g

A• A

.!JA 1\

. . bO £ l:Ill: .J..

~ M

'-:>

(.,(

But

*·

a•-0

~E f

g f &' <

cannot be positive definite .;:.t a. fcare there

m1Jst be somq

(v\

dj recti on

1\.

&.nd

~'\

fi 10

1/ .,. Q 1\\: •, rv

~""

\,zbi

c:b

I\.

P.*t

for

i.2 A •

I\.

2 i~ th~ unit tangent vector of tbc congrueLoa

of ooncbant JCodccic distanco from p lies aloser to p thaB:

~he ourfaee M3 6occ,

~horoforo

is not m~xim~l.

If r·1./ is compact; or if the intersection of all

and null limes with M3 is

comp<.~ct,

value, ·c hus there must be a •

a rna:x:irau:..

~eodesic normal to M3 through

'i-ie use this to prove another lera:tiG.. •

.Leinma 2 ..:> If p lies to the future (past) on a time-like ·:·eoO.esic

(

throue;h

q, beyond a point z conjugate to q ,and

exists a compact Cauchy surface l

6' a1ust have

, ... . ."C ime_ll<e

7. 7 II

t~ere

through q, then there

1nust be o..nother time-like geociesic from p to q longer tl1.a1-_ Let y be the last point conjugate to q on

t before p .

Let x be the nearest point to p conjugate to p in pq.

r be a poin·t in yx.

•

Let

Let K3 be the set of •uoints which have

a future (past) directed geodesic of lensth rq from q.

-l

~hen

K3 will be a soace - like

hypersurf~ce

be the set of points •..vhich have at

- t .Le

throuGh r .

l~ast

-~4

i!'

one fu\,ure

directed Geodesic from q of length creater than rq .

·1·hen the

Since p is in FLJ- o..11d since every l

past (future) direc ted t i me- like and null line from p intersects 4 H3 which is not in F , they must also intersect J 3 •

Bince ri3 is

be the int•ersection of J3 and these lines .

8onsider the function Cf'

compact , L3 must be compact . respect top over K3 .

L' .

\·1ith

Its maximum must lie in the compact

region

T.et _, ,3 ~

But , by the previous lemma (

is not m~ximal ,

1uoreover, local considerations show that a singular ooint in the surf ace J3 cannot b e a max imum of C> ..

n1aximum value of

t:~onal to L3 .

cr-

must

o c~ur

•

'fhus tl1e

for a ..:.:eode sic from p ort:.J.o-

Thi s must als o be a goedesic from p to q of

length cre~1:ter than (

•

. 0111ce Us in.~ tt1ese two lemmas the theore1a may be urovea . . H3 converging to the future (p•-st) directed normals eve.._"Y'·lhere on H3 , there must be a point COnJU(?;O.te to H3 a ~

a.1~e

•

finite distance a l ong eacn future (past) directed geodesic nor1nal . (

Let ~

be the maximum of these distances .

Let .u

be a point on a future (past) directed geodesic normal at a distance

~reater

than

•

Consider the function G

v1i·ch •

•.

. l

Let~ be l-he

respect to p over the compact surface :13 .

~eodesic from p normal to H3 at the point q , where~has its maximum . •

l;here must be a point conjugate to li3 along A in en .

But if there is no point conjugate to q along cLJnnot be maximal by the first lem!;1a .

~in

qp , then

If however there

is a point conjuGate to q ~long~ in qp, then there must be a

lon~er scode~ic

from q to p by the second lemi.a.

is not the seoa.esic of maximum leng"t;h from H3 to p . a contradiction which shows

·. :. hus .

·- nis is

tha~ ~he ori~inal assu~ption t~et

the spc..ce \vas non-singular must be false . Tt1is proof could also be used of a singularity in a model

vJi th

show the occurrence

·non- compact Cauchy

sui~f ~:·.ce

'i.

l)I'OVided t tiat tl1e expansion of its normo..ls was bounded av:ay

from zero and provided that the intersection of the Cauchy surf ace \vi th all the time- like and null l ·ines from o. point v¥as compact .

•

1• 2. •• :;

o. ~~

....\.

I . A. U.

Hecl<:1nan J{ayc haudhuri

~hys . ~ev .

. Kornar

l.i . C . t.ihepley

t::

/ •

~.M

'l .

l{. Penrose

uympo~iun

1961

;23 1965

Phys . 1{ev . 1 QL~ 5L1.4 19 56 Proc . Nat . Acad . 6ci . 52 1403 1964 Z. Ascrophys . 60 266 1965

Benr

Adv . in Phys . 12 185 1963

Lifshitz and I . M. ~(halatnil<:ov

rt . Penrose 9. I( . Godel 10 . C. •.1 . l'1isner 11 . J . i·:ilnor b.

Phys . Rev • .Lett .

1L~

57 1965

Rev . : .od . Phys .

'!!.

215

17 65

?:d 44-7 l'ftr f Rev . Mod . ?hys . J . Math . Phys . ~ qz4 r~ ' Morse Theory ' No . 51 Annals of of Maths . ~tudies . 2rincetown ,

Univ. Press .

'Curvature and Bet·!Ji

12. K. Yo.no and

~nnals

.::> . Bochner

~tudies

1'To . 32

Princetown Univ . Press . l~rticle in 'Gravit;ation ' ed . .;;it·cer-1

13 . a . Heckman and ~.

of Maths .

i~umber·s

·riiley, .r;ev,r York .

i::il1ucl\:ing

' Cosmology ' Cambridge Uni v . ::.'ress

14 . H. Bondi

I L~ -

\ (.A.c•10;,

l \

Ph.D.

51:;7