S Hawkins Thesis
S Hawkins Thesis
S Hawkins Thesis
DATE
!JV
;·1Cb/ci
\~ 1~~1~·
- -
----
----
-----
-----
. 1
THE BOA Rn I'~ Rf Sf ARCf! STUDIES
Ar" · ·
f(lr?
; Pb. D. ,, . '·II..' '
~ .
< ' ffB 1966
·-------- --__j
01
l l l
• •
ONIVi:Rc
LB l
CAMUKliJ<JE
•
.L te e re o h. ivc .• ·t o~
bri o
om i lica ion und c n~e u c o oft
't l niv r c r ...._,i d. In vh >t r 1 it is o n t t
t i er u.te r ve ifric l i~o for t oyle-
_ l i ·r r .... o 0 Vl.
•
ion . ter e l i th
·t tion of •
n OU i ot o ic
•
u iver • c t l· i ..., c 0
or r t, 0 t .L'O • r ... ur h re
initi l l . he nro •
ion tion of r Vl.-
•
•
l .... o i VC.Jti in i.; roxj atio .
In Jt1 ptcr r· vit tion 1 r . .... . •
"' l. n n r univ r 0
i inc ;t 0 0 o ic •
10 •
t
l' -
t
0 viour n t c s to Lie u e i d.
l Lt l t i o . . urrcnc o • •
in co olo c l
o el . t i ... o n t t i u l. i i it 1 n ovi e
t err. i v r l c n r •
l. •
J..
... 0
-
• 0 i 1 i 0 t
l lt"} ll
it i
t •
•
' -
- '
•
'
- l •
i v
'
-
• -
•
- • - t
t
i
' • 0
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- •
•
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. - I
' -
• .l
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' ...
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•
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..
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k :;: . 0
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-
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• •
• 4
-
- •
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1. •• -
i
•
i - • . ' 1
'
• i
• l 0
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J. u . l . {.
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•
CI-i.A.PT~rl 1
of Gravitation
1. Introduction
- -
,I\ I< y I ...... . ..... I
I- L J A~V
"A"'l /
It \'Jill bG ...;ho.:n that in an exp;.11ding universe t·!1e
advanced fields are infinite, and the retarded fields finite.
This is because, unlike electric charges, all Toasses h[tVA tho
•
same ...:;i.;;;n .
l(( .
£ x. x /)
--
/-3
wl1ere j is the dei:;erminant of 3.j • -=>inc:e t~e double sum
in the action A is symcetrical between all pairs of
pi:irticles a.,b , or1ly thc..t part of 4·(0...b) that is
symmetrical betv.reen c.... and b v1ill contribute to the action
i.e. the action can be writ·ten
A
\vhere ~··)!(-( ~ _ b)
·lt
'rhus G must be the time- symmetric Green function, and can
be i.,rrittien:
()"'
'1 ::
..!..
~ qre-t' -r 1qo..dv v1here
are the r3tarded r~nd adval1ced Green f·L1nctions.
Ey requirin~ that the action be stationary under variations
of the g~ , Hoyle and Narlikar obtain the field - equations:
)i,
/l1. .
; r\
(o.);r (b)J
- m M.,
, >
(o..)
- £ /VL (x) =
..-
D 1 : - "f
'·"'he re is the density.
'
In an infini te staLic unive~se ,
.
wo1Jld be infini~e, since
the source al~ays has the same ~ign . ~he diffic ulty was ~ssol-
ved v1hen it v1as reoliz ed th<~t the uni versP. ·.-1as expa11d.in;;, since
in .:tn exoand ing univer se the retard ed soluti on of the above
eq~ation is finite by a sort of ' red- shift ' ef~ect . ~he
• - .
Since t.1ey are conformalJ.y flat, one c~n caoose coordinates
in \vhich tr.Ley become
2
2 :z i :L J. . Q
cls -
-- - t~_(c) .c7)
/ tl~ I ·t i K ( 't r-f )iJ [I ~ ~ ii ( ~ - ( )~J
__ Q ~)
("'C= JJt
For the steady- state (de Sitter) universe
•
W\Gre the integr ation ~s over the fu t ure light cone . This
~·1 ill normally be infinite in an expandir13 :.ni verse , e ~ 2-: . in
th~ ~ instein -de 0itter universe.
-2 oO
M. o...).; ( '-f,) - -
- '1: I ~ ( fl - vc ) I
cL 1 .2
I
..."
/., I
-
•
In the s't:;eady- state universe
3
- -l 0 ·-I (
/ll\...CL J ./. ( '1;( ) ·-- - I - n. ,/\
l-
1.
- l ,) cL c:.z .
l T. ~
I -r.
I
-
•
where the integration is over the past light cone. This will
normally be finite , e . g . in the ~in~tein-de 0i~cer univers=
-2 f. -Q.
'L . (
t'.2.. - 'L )
T
' ' _ /?_,
I
c{ '"'L1. ::. '
-(L
J.
I I
-- - -I
·rI
IV1.. ( 't, )
-1
n. -I
;~ {;
'"f I ·-cO
~
l;i.
-
- -2..i (L
·-:i.
J
equation
DG(X,X')
.I 0....
&
.rhen
1
c &-J o S' e( X, 1) J '< ( lj ) J K j - 3 cl_ 'X 4J
oc -- J
- I'\
JK
,
hoyle and Narl ikar claim thut if th~ action of the
' C' - fielQ is included al on,~ vii th the action of tl1e ' r~ '-field,
a univerue \1ill be obtained that approximates to th8 3teady-
will ~how that the advanced field is infinite while the re~ar -
- 1-\
.dut J }K
is the rate of creation of matter= ft,. (canst . ) •
in
-- rv V.
11.S the point ? is taken further into the future , the volume
'Yta..dV
1 finite , but this has certainly not been demonstrated
and seems unlikely in vievJ of the fc..\ct th1 . . t , in t11e Tio~rle -
Narlikar direct - particle interaction theory of their 'C 1 -field,
which is derived from a very similar action- principle, it can
be shown without assuming a smooth distribution that the
advanced 1 C' field \-.rill be infinite in an expandin3 uni verse
G~~, b) clo. cL b
'
A particle of ne~ ative 1 in a positive 'M '-field •:J(;u ld
'
REFERE1~CES
2. Notation
Space-time is represented as a four- dimensional Riemannian space
with metric tensor gab of signature +2 . Covariant differentiation
in this space is indicated by a semi - colon. Square brackets around
indices indicate antisymmetrisatio n and round brackets symmetrisation.
The conventions for the Riemann and Ricci tensors are: -
a
where Ua is the vel~city of the fluid, Ua U - - 1 :
~ is the density .
ft is the pressure
'
C o.b
cd _
8
u [o.. t b]
tc.-· ..,tJ
\.A
cC'- Ed]
- 4- O(o.. !,)
-
'
')
)
,
E Q...
().
-=-- Ha. o. = o
E. bc;ol b b c Lide \ h b (1 )
1.
n (j.
b hcd
+ ~ Ho.h W - '? o..~c.d U <Y e n = '5" l'A
l
}"-;" J
(2)
'
• •
+ 2. H
d
Q YJ bcc=I. e
Uc ue -- - t(fi ~) () ob 'J
(3)
•
L H c:o..b - h (o. .f ') 'b) c d. e \). c. E -f o~e + Hott. 8 ..- H (o. w b)c.
- T cob'·b -= 0 )
(f.+ ~) Gl +
0
r.; 'p h ba :: 0
(6)
(8)
'
•.
..·'..
~
(9)
'
2. 06
\Vhere 2 W =: W ab W
'
\/Ve also obtain \\rr1at may be regarded as eq_uati ons of constraint.
( 1 0)
) ( 11 )
0 abcd 0 •
•
If \Ve assume an equation of state of the form, ~ e. it"\. ( ~) ,
•
1:Jien by ( 6) , ( 1 0) , :::. o = Uo. ..
This implies that the universe is spatially homogeneous and isotropic
since there is no direction defined in the 3- space orthogonal to Ua.
In this universe we consider small perturbations of the motion
of tl1e fluid and of the '.ifeyl tensore Ne neglect products of small
1
E ci-b j b -
- -s.h
I
0
b
f;b
'
( 13)
)
( 14)
•
r-
- c;a..
b+E 0 L8t-
r,:>
)
( 1 5)
( 1 6)
'
•
e -- .. I
-3 e2. + •
lA.
Q
• Q.
.I
( 17)
-- - -3 'l.
Ll0-b 8 + J ( 1 8)
( 19)
From these we see. that perturbations of rotation or of Eab or Hab do
not produce perturbations of the expansion or the density. Nor do
perturbations of Eab and Hab produce rotational perturbations.
4. The Undisturbed
- Metric
-l"i····--
Since in the unperturbed state the rotation and acceleration
are zero, Ua must be hypersurface orthogonal.
Ua...-s~;o. '
where measure, the proper time along the world lines. As the
surfaces 't = constant are homogeneous and isotropic they must be
3-surfaces of constant curvature. Therefore the metric can be
written,
where ,
1
dy is the l1ne element of a space of
zero or unit positive or negative curvature.
.-.'c define
1
t by,
d. t I
clT :0. TI ,
then •
In this metric, .
'
' ..••
(20)
••
·3 G -:. -
(2
±- (f 1" ; ft) • ( 21 )
If vve know the relation betvveen µ and fi. , 'vve may determine (l_ •
\Ve will consider the tv10 extreme cases, 1'"\. = 0 (dust) and }1 ~ %
(radiation). .Any physical situation should lie between these.
For 'b ;: 0
• I
• 1.. D'2,- =: E E = const.
••
M rt )
(a) For E 0 9
-
n ~
2 'r- (cos h~
-
E: t - 1)
i 1t - - i E(~
-
1
'
.
~ll'l h~r-M
-~ t -· t ) ;
- 2. EM
o~
-
If E )>O
)
3
~ R- - ) M -= E •
[=O ) :>
•
E <O ) b -3
. n.c.:.
-2 /Y\ ~ ,::::
...t
)
- •
F'or .?\
..,.__.....,:/ -=---)J./"S-- •
~2
- t-
n
••
3 (2 -
- )
)l. . '\..
j-J· -::- )
•
• •
3 (2
fY\
-
I
0. ... -E
(a) For I~ > o,
(1 -- -E I
~.inh t ) 'L - -E' ( C.c•S·h t - I) .,. R -- - 0.'l..
(:,
,•
•
)
(b) For E - o,
'.
,..,.. -- -2. I t 2. •
-, t )
L
) <( -<. -- 0
,
]
I
( c) Fo1"' E 0~
-- t -- ~ -1)
F'
S ..i. YI (c...ost
- .) E )
By (6)
•
•
•• w Q..b •
For )
- (&),.,
(2 '2.
For 1ri.-:. 1 )
•
w -- - w et e +--'+ ~)
f'
I
)
-- - -'} wI
e )
... - w --
6. Perturbations of__Densi.!il.
For fl·= 0 we have the equations,
•
fJ. - -t-8
•
fJ -- -TeI 'a I
--r: fA
These involve no spatial derivatives. Thus the behaviour of one
region is unaffected by tl1e behaviour of another. PeI'turbations
'
will consist in some regions having slightly higl1er or lovrrer values
of E than the average. If the untverse as wh "1..e has a value of E
greater than zero, a small perturbation will till have E greater
than ~ero n.nd will continue to expand. 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. However it will only begin
contracting at a time 8 1" earlier than the wl1ole universe begins
contracting, \l\There
--·
..
'(:"
0
is the time at vvl1ich the wl1ole m1iverse begins contracting.
There is only any real instability vvhen E = o. This case is of
measure zero relative to all the possible values E can have .
Ho-Yrever this cannot really be used as an arguement to dismiss it
as there might be some reason why the universe should have E = o.
F·or a region with energy - cE , in a universe with E = 0
r"l -
>l - 4 ·~E ( t'l.- t't
12 t ... )
I
T = 1'1. se
ti E )'l.~
(----· )
f- ::: '""t'" '1./~ 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 i~ these
could occt1r. Hovv0ver 9 since an evolutionary universe has a partic1.e
horizon (Rindler(B) 9 Penrosc( 9 )) different parts do not communicate
in the early stages . This makes it ov0n more difficult for
statistical fluctuations to occur over a rc;g: -.n until light had time
to cross th0 region.
--
• I
n _o i_ •
,,,,: ,,.,
...
0 = - .., 0 - )A + V'- ... '
. . . I o.b .. • ~
l-l 0..: 0. -= t,..l <>-i b 1 + u 0.. I.A
to our approximation.
I1 OA.\7 I b :;:/
Ve I o. vb is tl1.e Laplacian in the hypersurface c-r = constant.
·.~re represent the perturbatior1 as a sum of eigenfunctions S(n) of
this operator, v1i1ere, (->) <:.
S ,•cU. -;-.O
- - Q'2.
Yt.7... s {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.
I/ I I
1.1/hen B (11) + B (t1J rf.
' 0
. ..,.,
...
r e {_ .1:3" t
7. Tpe Steady-stat~_!.!D-jverse
To obtain the st0ady - state universe \Ve must add extra terms to
the energy-momentum tensor . Hoyle and Narlikar( 1 0) use,
where,
c 'CA
'
J
c ·~ 0.. .c. -
'
J
-r ·b
Sin.ce 1 c..b' =- o )
( 21 )
•
lJ-
,/,,
Cl.+ I '-)
\- b
b n CL -
h ~,
Q.
c c cl ;d..= o .
b
(22)
(23)
e f1
•
= - -
\.
•
For f' *ft
e= -•
~
I
e -2'Z. I (
f-+ 3 r. +
)
l
.-----·
:~ 9 _,, ~ 3 ( 1:- (1)
These results c onf j_rm those obtained by Hoyle and l'iarlikar ( 1 4). Vie
uni verse by the gro,•rth of small perturb at ions. However• this does not
exclude the possibility tha·c there migl"rt by a self-perpetuating
system of finite perturbations which could produce galaxies .
(Sciama( 1 5), Roxburgh and Saff'man( 16 )).
8.
~- ·-..-----------
Gravitational ·1
raves
... ----~ •
•• -
=
r: ~b
0
17
. (1-1)
where V I
O~>
= 0 }
_ yt~
v (1"1)
Q '2.
Q. b
)
V Cl'-
0
~o
•
Then
••
-
·-
•
Similarly ,
Then by (19)
Substituting in (24)
Al"') [ n ....
3
n''
(2 t
2
D(~}
[ r2 (j-tt f'L) t i 0 (f fi-)]
i
::. 0
1
so the gravitational field Eab decreases as Cl - and the "energy"
t(Eab Eab + Hab Hab) r....
as 1 t..
- 6 • We might expect this as the
a local co- moving Cartes ian coordi nate system .which depend s only on
first deriva tives. Since the ~requency will qe invers ely propo rtiona l
to r2 , the energy measur ed by the pseudo - tensor will be propo rtiona l
to ll
~L
-4 as for other rest mass zero fields .
Since
we have
(25)
b e.
(J
c.. b
;
h Ol.
-:::: 0
(26)
Equations (15) (16) become
~;e
H~ =
(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 e- "'t
~ . Neglecting expansion for the moment,
suppose we have a wave of the form ,
~V"t
E o.b ::::. 0
E- ~be .
'
energy densities should be equal. .As the uni verse expanded they wouJ. ':.
both cool adiabatic ally at the same rate. As we know the temperatu~c
~
2.
::. 0
E'l. '))
-i
2. -2.
But the energy density of the radiation is 4 0
E v
•
e I 2 I
f
•• -:; - 5 8 - Z }J- G - { (f t- 3 }1.)
Gravitational .Liadi&tion In An
Expanding Unive1·se
'
acymptotically flat space , tna.t is, space that ~pproach~s
; A ( i ·- cas: c) (;. 1. )
. ·1 :i I~ ~ l. ).
C(
I
~
'2.
;..
J2 'l. cl t: .
(b)
_\)_ ;. ~ At 2. (I, 2 )
•
0 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.
"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
,
v1nere :
lf'
L,,., nf' (/,r\ Mp 1iil1v.: lf- (Y)i~ -- (lt' /l?rl; [ ,
f- . ·- f.J.
lf (L = l) Mr M ;; -}
c~b
-- - -
uq ? -- 0
( 0
l 0
0
0
0 ( ?__:__!)
•
0 0 0 -1
l o 0 -I 0
pv
L O.b
vie have -- 2p. 2v
-~
)
.
7 ().-
h
..
- l 11
IA f'L +
fl p- L II ·- rr1 ,~ m ·- fA
v __ fV1 fYl
v
(2- j_)
( 2 3)
-
In f'ac·t it is more convenient to ':.:01:l{ in t ·~ rms of 1J-;.,'el 1e
1
/\ -- (
I~ I
·- - -y
2. lt I
- .L n /Al. v~ 1Yl
- {v<.; v I
I 1iVif l v)\
l -
• 'I
- ( )'l I
I
.'.:l
•
'I -- 1 L //.__,.;..;
3~1
fA - v
- 0 - Lr: - fl1
I v 1'l!J
- •
·-....
·-
·-
-
·-
·--
.
-
--
3 . Coordi.:1atcs
---
.Li lee i:·l e \fJrnan a nd Penrose , \J e introduce a nu ll coo1:-dir1ate
u..( =xj)
lL ·v
I
~ o <~ . i)
vte take L~ • Thus Lr ':Till be
3eodes i c and irrotati onal . This implies
v
(l -- 0
-
( ~ f-
cc- ::: - £
vie take
:L + ~
~o be parallelly transporu~d
(1. b)
In these
{ ~ f)
(
l =
The lield ~~u~tions
the rela·tions
a.._b c..-.:.L o-. bc: (L
{.-{ ~ 't ·-
- ,-
•
")
"~) c, J'
.._L b
p () ( J. 10)
j) G' ~ I!)
- ···-
]) 'f ( ·s.. i2)
/)d. (1.1.3)
..,_ -
j)~ ( 1 I~)
-
Pr
y ,\
( :s . /~)
- - --
(s. I b)
Pr (~·- tJ)
vv {l . IY)
--
~· f ~ ~, c;- -- ( {? r ;_ ) l -t ( .~ - J C\) b' ·- f, r po ' ('5 . 'l c'/)
-
~j_ - r~ -' f" f ->. G ·- J<X ~ 7 O(~ ,- ~? - y; ·t- /\ r ~" 6 '2::_ 1)
-
J~ - f 3 f1
r;/t,' , ( oi. .,. fe )f- ..,. (;} - ~ ~) ). - T J
- (
(? fA - J f )" f .,-
1J b ~ J l. (S
.
i.·1here
D L {" Vr " d (! <.f)
-- ~ '/,..
?_
LJ - rit'
- vr -- v t;.) ~r- ()
·-
-;- )\ i J .
J (.A. Jxl (3- ·29)
.' ~ J_
J -- .5 JXL (~ 3o)
--
•
--
t ·--
•
g -
~~I -- ·--
1,o- ·-I
J. 1] (s. s3)
·-- -J_ R ·--
-
cf, ·1 2 13 1~I ( 3 . ·5 "!)
--
1>01. ·-- - J. R
J ':G
-- c/> ;J.o (.s 3 s-)
--.
--
f l l
..- -1 R
.< J.~ = 1J.~
(1 . 10)
~
/\ --
l~
<~. 3 f)
2 3 4 5
~ -c
) ·2 I 3
--
(~. y /)
: - c
•
'
Expres s i11.~ the rotation coefficients in terms of th8 metric,
we huve : I
• •
l L -;t.
·- .
"L g £., ,.
, - I
g lf'o - Y f, .,_]) cj;c:rl - $' foo"' <r "'- 'fo - ~ f 'f, -( ')_:; ~!]) ~oo
i· '2f cf 0 1 .,.- 16' 1,o . (3. S 1)
6 'fa - ~·y <t D~o~ - ~101:: (·4~ - ~·}'-('o - 2 {2-r_ -t- .~) lf7
-t- S6f;. - °Jo/ ?oJ. (l.S~)
00
- 2P101 .,_ ~b1,,<>1-(
·-
3(5115; - oy,),. 2(J>1. -s ~,o) $' <P", - Li ~00 :: 1> t - 9r t ·t
· 1
( 3. <j8-)
]) f,'.<- r p,, - S' <f>oa ~<Pa,-.- 3 ~II =<~ 0- f - 2;;.,) o, • vtf
·- . .
"1- 0
- ).1,o- 0
JJ"' ~ f; ,i, - ~ r1,lo ..,. 6 rJiI + .1 ])Ii = t 2. r- t- -t-.2i -~) rrA - 2~ ~ i) Toi
'" '+'10 I 00 OD
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,,
L). - -
( 4 1)
is a null coordinate
ro co.lculate t', the affine parameter ,
1
\•re note th,_t C
~ /3 s "'t A - i/J'lfJ ~
N O\r.f 0 (s ·- •)
5)_ fE.S-s
s
v1here S' 1. .:. t
'l 'he1·efore if v-1e try to expand /J'" as a series ·i n po\v:r of S"'
t, l1e r esult v1ill b e very messy and will involve terms of
L If\ S
the form .05 *
s n.
cl r
- _f2_
- I A 5_ s !J4r 1-1?~ ~,.
- 252 ~ <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 . \·J e \vill take these as I\ 1'foo
,-\) 1 rti
lO( •
ci> 0 0
I ·r
t 11 ~oo
•
·- •
-
cp, ~ -- t -- b /\ ~()/ Q.
2 I l + -</J.o •- .~Qi
~oo b (\ -.rp0 0
·- ~
.
40~ -·
- cf i o -
·-
o/01
• (er. 10)
cp (.:> 0
~oo -- 'S R
Sl~
1fl_
4 - t1
-
452. 3
cp ~:t -- lE
452
( ~ . 11)
cf Q I
-,, ~01 =: 0
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
-- s· ~
- ·- I -
- G
• - 2 i,,.-
2Al7~e.
4
T . .
•
5L 2.
Jl'
--- (-t
- Jq 'l.( { ..- e. Q..v.)SL- 2t (]_3(1 t- 2. e 1. )--Jt -s+- . • • ....
25l 4- ~ •
J J} 2
'
- J
5. Boundary Conditions
above .
et>Or> and /\ will then h&ve the values
•
61.Ven o.bove plus term3 of' smaller order. To de·termine this
order o.11d the order of
4:>01 and % '
t l1ere · are tv. 0 1
•
ways 1.Il v1hich \'le may proce ed. ••"e may talce the smallest oraers ~
--
5L~
+ () (JJ_ -<i) ( ')_ 2)
-- 01-52-=t)
~, (see next section) (~-. 3)
-- 0(51-=t)
(S-~)
\•/ e also assume 11 unif orm s inoothness" , thic,t •
J.S:
d d • . . .. ' • d ~ -- 0(-51-r)
(....
~x ~XJ- dx (.,
--
etc ••• I
'
6 Intee;~ation
j) er :: ?re- f()
where 3 fl
~00 -... 411 r ·t
-
•
Let
-
•
-- <.) 0
- -
('
Oo
then ·op ~ ? ··2 ., ~ 0·. I)
let ·p -- -(» 'I ) '-/ ·-1 (6.2J
then D1. y ~ -CR! (LJJ
s ince S, . Cf clr ..t{. cD
•
Let
--
Then using:
'l) : J - .) ,_
(\ -; (
I t
n ·-~
A.) J_ -
A ~.f).·).
- et A?.f)_ 1.. ..) )
-
Jt ~ J. JJl
fB ·r0~1-')) .lfJ- -r O(i)_)dJ;_
Integrating ,
3 ~ Jl-; 0(1) a...,.
1
Jl,.- 0(1)
-
.i!'Or 1-L /
::. l .52- ~
\vhere --
h, :;..
(.J 0
Unl ike tf e111man and Unti, we c a nnot m:· .ke \. zero by
1 0
the transf or.:natio11 'r :::. 't" - P , since this would
alter the boundary co11di tion
I A
.I\ - (; 52 :!
- i' K
<\ -=r
.) J_
Cont i nuing the above p rocess \ve de .:·ive:
f ~ - 2Si-2_ A 51 -;_
0
(t.1l)
•
-
•• (A )( _, 7 R ) J T b
rI~n)_~ ')'jlr 1 )
JL J. A 51~A Sl. i(\ 52. ~. e3 52.")..
I ) )_, ) .:.) )
:e~ )3 nic.) l-).J2 JIc~vJ ~
I
(6'. }_O)
By equat ions 3 s J) S, 13) "l. r4 I 3. ~ s I 3. l.f y
A- ~ ')
·- .J 0 fo A f:, A- o o o o G i 5.ft 0
00 0 000000 0 0
0 0
0 0
0 0 £·. ] 0
0 0
0
•
0 •
0
CJ 0 0 f) 0Qe) 0 0 0 a
0 C) .._, -I oo0 0 0 -2 •
0
CJ ~1 0 0 -I o0 0 0 a -2
•
>.Jlnc e '-[ ;;
-oL t- fS
-- CJ (51..- <)_)
\
Usin0 this vie i11tegrate eouatior1 ('3 . /·i) by the same
method as above . ~e obtain .
3
't 0 ( i.A. }<' l) 51- 1 ~ (j ( .5l . )
J
geoa.esic -- -
-
--
•
'
using this in eq_uation ~. 5 i)
Yi= 0 0--6)
by equation
·:;. 0 ( _))_ - 1-)
by equation
by equation (g l( "-1)
.
•
uniformly smooth.
Ad,lin,_; equations a nd: ~. b O : -
--
By equ t..ti ons 3. /G. 3. I
11-,.r G. 2.8'.
- )_
0 0 0
0 0 -I
and o(s'L-2)
are
'.i'heref ore
·v.1- :: o (n·- ~)
~ ·:: . 0 ( 52 - ~)
- CJCJl. -1)
and are uniformly smoctb
From (t, 2 q) and (·1 _It) \-le may shov1
~ •:- - 0
JS:~ -{-f() J2- ST ~ (3 !9(-? o_ f orsO~ .~o6~ )n-<tcy 0 (Jl-S"~,3<;
2 0
~- 31-)
'
•
·rherefore
&(59)
-
By equation (~ · Jo)
v ·= v -- 0
±y ; Jl -!( 1' 0 ( J2 - 5)
By the orthor:ormali ty relc:tions { 2 .1) .
J iJ.4 ; x -(5ic:J ;-
L ~ LW)
.: x(, 0 i- 0 ( ..)'l
-
- 4J
- (S .. S ~ g· _sJ) 1
--
.
By making the coordinate transformation
Ll ( --
II -- I
' '
XL)
I
c~
( (; {,
x ·-- x T ( u,
waere
_ x· 30( I -r x ~o (
'
c3
..J I
-- c_;3
3 ) _ 3
) y-
/
l,.o
X :. 0
·: le still have the coordinate freedom
Xi..::. j)L(xj)
1.ve may use this to reduce the leading term of 3~J, (~,)'; 3.; <-) to
a conformally flat metric (c.f. rTewman and Unti), that is: J
~i ~ - ~PP r1 (·si-~.- 2 Rn-<;),, d.51 -(,) (t lt J
I
9
•
v1here JO •
..._
-
7. Non-r~d ial
- Ecuat ions
By comparing coefficients of the various •uo :Jer.s o.f 1
JL
in the non-radi a l equations of ·{ 3 , relations between
the integ r ation constants of the radial equations may be
obtained:
In equat ion 3. 23 the term in Jl-' is
- l4 A ( 0 0
•
0
-
0
) - J_ A
4
therefore by
0 (5J.-I)
u ·-:.
In equation (3 , Su) , the constant term is
\/o
therefore
p - p . ~ (( 0 ·- 0 )
) I
n~:i
By the t er:m
p - ~i - ..... 0
ther.'efore
- -0 ~
- -l. I
(:;.. · ~).
~ 0 -- 0
if
/vl 0
·:;. - fl ~ - c:; 1.. v v lo s e,Q,U.
I l/ 2 J
·:: -A <)_ [. i + e. Q ~ J
Lt - -
By (·~ · '5 f ) ~ . 60) (s. G t)
2 Oj
~ 0 cn- 1)
00
i.-1nere
•
3
- '-I '
By tne Jl. term in (J . 2..b)
2 ' C}.U..
f~ -
o ~ o 1- A~ i l) 2 2u... :: 0
tnerefore p J ' - -s- f + b - lo ,~ ~
7- Li..
'.
C(x~ e, 3
-
-.,,, I
-t-
(z.1o)
on the hypersurface
.JY tue Jl - Y term in (?. 2 5)
A 0 = 3 ( (3· ~I - 6' 0 ) (T- I I)
By the S2_-<.f term in Q.22)
~ -~ -- ( 6 ~' - cS 0
) e '-- V ~ - ~ e.'-'- SV(o~ ."'-6: "') (t-~
By the JL-2 term in Q. S-o)
0
../. \.A.)
o .- (....J
~ -- Llf-c
':2...
,I I '3
~
'-A
vr
S 17 o
"T
I
~ w
o
·- e t.L SV- ~ o J o r7 r
:.:. - CJ v ,) e .....
Then by
Using this in
B~r
v (5 ' s / ). 6. ~ 1) f}.60) d -b t)
. J 101 ·-- () (.FL - :r)
J L-._
d (S2 -9)
d~
1b0 -- 0
_J_ /\ _,
-~
o (SL -:r)
c) v--
l .? . I~)
d --- 0 ( 5L -:;)
J~
Thercf ore if the boundary conditions { S"". i- Y) hold on
one null hypersur!ace , they will hold on succeedinG hypersur-
faces .
By
0 (JL-2_)
~'he "peelin~ off" beho..viour is therefore:
-
- CJ ( t -1)
-- (j ( i -1)
lf 1 -1
0 (r
f~ --
]_.
f. 21)
•
-- o(r - i)
'f L
·2
.
- O {r - 2 )
'f o
As mentioned before , this asymptotic behaviour is inde~en~e~t
'l'hen :
- 1.
.,.,._ [. fl v( 5 - 1 Q;... J. <..l.i..) 5·0~-cl n - b
' _1r ~ --o e - I e - v J '-
• 0 <6' :2 ..
.L [ n ~( 3 ·7- . o0 iu
. o ~Ll (} \.( ,~ 0
u.)· - '-f
+ 3 . 11 - '? er e, -r 0 Q., r-t rl. e., 0 0
-+ ?. 8)
-
•
i _I (~ .o - ·- . ~
1. '-J JI 0° )Jl- - /}6)~
- ,._ - -
2.
v ~ -j (6~-6°)e~PS-
-r 0 ( J1 -5)
7- ]0)
•
e ~ s 5L - -~ (} e ~S' 52 -")T i [ f) 2 e_ 5 (.f
VL r ~ s ~)
- e {,_$ 0- 0
_/12 -<; "r (') (52 ->) (f- i c:)
.
{n,-z_ t(Cf c 7 ~'io)SL-2.,,o(JL-S) (1- 55)
1
CJ ·- () .
(.) . <l - S" 0 , -G \\
S2_ ~· J c
T ""(
i--
OJ . __; G ·f- {
15 - O;
I I
-
. {.?.?rr)
b -::o => r:;:.o I o. n-2 -c_-, -o
~
-- - 6 -r- -=..
2..
u ,/i
-r -- -7. o )ii
.._)c. ._
I
·- C,_ ·3 6~
- - - 2_ "f JI
T _j_
2
6
) fI
o 52-sT
·-
[J'"J..(<3~ ~
lf
a=o- )I
S&
f" )f I
o .\
/
-
-f- /)"- ( - (f 0-r- ]_ 6 6
- ~ 6--:: 0
--
I 2 )/ /II
'f i
x ( 6~~ - 6 °) 2- v(s (~, - 0
6")1;_)
- JI ;T I I
_.. ,
<f lie
-
2 ( <iVv·LI- 6-
&'
v--s-,_
- S'V (<Y c:_ ,;, 6 c: J
)} ~ )I
0
c-
0
)
~ -
. ).
0 --:r
I
:: t J
J1 ·-r
•
(,.0 J
Q, 'r ~ • - •
.
~
l
-- ~
lA..
...
sv +~: + /5 oo) .
'+ -
s ol
I
l --
I
+y bi
· 1fri
1 I)/ ;::
·s,+0,l
•
.... .
·11he mevric has the form:
I '2..
~ li ·--
i$
-... ~
i'f
- CJ ) 3 - I
'J
-- J1?.. r- 0 (52 -<-)
'2.4
~ ~[
-- 0 ( ~ -y)
CJ.'
-- ·- 2 f ~). g J Jl - r ·t-
I / ( ;
3 G} t-
2 R ?'2. S(JA-~ c;(~?_- 6)
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 oo -- ~ R. I CX • •
J I
(~. y)
-
rA ( ( o ~ D r. o< n ~)f.
Q
0 'fo I ::;. '?._ [_ f.- i o<. k I Ji I\ So( 1l JJ I f(J ~ .i o( f( / ( r S-)
To obt~in the asymptotic zroup we deuand
~
- - - -
:( =
J'.J'~:: f3 :S
·-
_.
'95
I 't
.:::. 0
f :s3 .: 2.2
0(51_-<-)
..._ I
~ (..
(j (52,-4)
~3 --
' '
--
S'3 ~J 0 (yt_,-b)
~. &'
-
~) 11 ......... 0 (5L-7)
-
{-5G-1)
r ~OU - - 0
-
o (SL-,
~~ 01
--
dy
/( '2 ·f- (( ( y q
) 2.
) ; .J
,, k'.::. 0
JJ
•
I< I:: 1\0/ (l(~)
(U)
)J.. j J
c\ I k "3 :::- f( OS (v.., 'X- l.) -
( ?. 11)
l '
1. 2) I
w
0-
0 . I ) '""->)
(~ 10)
re. t.2)
. I
0
1"3) imply that K. '- is an
and&. anal:-ftic fu11ction of
)( '3 -t- t,_ 'X.1 . ·rhis is a consec;uence if t t1e f a~t that 1t1e h;.,~"1e
.,
v'
recluceu. the leadi11g term of ~ '.J to a co~for~~llv
., fl~t
'
/,
form . ·~huu the only allowed transfor~ations of )( ·'r""v
c;.. +-'~ e
utJ...
--
c ( x 3 + L xlf-)"I cl
{).cl - be -- i
du s -c 1:1ill be :
-
.. Jl~i.,..
v). --
v ·~
--
v --
4'
(9~)
L
~
, the projection of the ;,..:ave v 0ctor in t;he
f\"'\
observer's rest - space ( the anparent direction of the wave ·
1:1ill be :
2
q, .
- s - vv /l"'I I
M
M
-2
.,. o(J!--~)
\
- J1.
•
' c~
O(JL-~)
-- --
I I
T
)..
t;2..
0 ( Sl. -- ~)
c:
L --
.3
(}
--
l-
lf
o 4) Cu~ - cI - - 0 ( ._,)<]_ ·- ~)
<:; --
.
0 { ___,'2_ - Q.)
I
~ - O{J2 -z) t --
::i..
'.l
{ .I
.,
f; - - l
- J5.
~
•
5'J .
3 ff
·-I
- t - L'
3 -- .j 2
·-
\./ u Ji (~z. _5)
:e v:ri te
(J..
e.""' --
E -- ,.,. . 0
,,,,,.
-
0 0 ( J1_ -'r) -;- C> 0 0 o(-52-4)
0 r CJ
a I
0 0
•
0 -J I 0
- -
,.... - -;
•
0 (Jl-5)
~
-1
C) I l O(J2-~).- 0 t
..
•
I 0 0 I 0 0
.l-(
0
' 0 0 -
·-r I
•
0
-
0 O{Jl-6)
•
0 --.L
l.. 0 I i;_ \ ,\
0 0 -- ).
I
( _1. r J
'"hi
J. ' ......1ould be cor.ipared to the behaviour for asymptotically
.;.;
"-t-
. -CJ [ l eJ (r-2) . 0 I -1 o(r -1)_1- 1 ~-~{
() v ?"
-~)
-r C>
I 0 0 l 0 0
0
i
-~i 0"'
l 0 0 ·- l
- 0
- o-i
0
!q:;)
\ t l •
•
4. • .L . e 1....
• t1 . yo . ~ 91 1~ 2
d • •J . ti
• • 0 i ver""i J 1
are satisfied , a Robertson- VJalker :::nodel can ' bounce ' or avoid
a singuli:i.ri ty only if the pressure is less than :::iinus one-
third tne density . This is clearly not a property ~ossessed
by normal matter though it might be possessed by a field of
.
ncGative energy density like the 'C ' field . nO\vever there lS
of Jhapter 2 : ~ . v ct ·v b {t )
eI°' V°' =
I
CAb
A uoint;
~
will be said to be a bingular point on a Geodesic
J l\ <>.. -- C)
,
"-
'
J) KC\.
-- v b'
(/.I
kb {!1)
f
JJs I
u~on··
(J
'V
Cl.
v1itn eu
0..
- vo. vie have
fl1
~
oL 'l J\ - Q_
t'v
I\ (4)
M
n..
ol. s CJ,
(If\
v.1~1ere I'\..
(\.. 0....
e
b
R vc vd.
0.. -... e,
M C\., Q, bcl
fVV
-
1 .. solution of (!.t) i'Iill be called a Jacobi field . ·.rhere :ire
clet...rly eight independent solutions. Since Va.. and l{:.,;o.. a::·e
solutions, ~c11c other six indeuena.ent solutions of (f/J m<--y ·oe
ta~en ortho~onal to • i1hen
1
'f... is conjugate to p alon~
a 3eodcsic 't if , and only if, there is a Jacooi fielQ alon§ {
1.-JJ..lich vanishes at p and ~ . :.i1his UJ.ay be bho111n as f ollo'.lS:
•
·-- v
-- d f{
cl~ f
y
.. nere
~
cL A A
M I'\ -
~
cl g
., ear p , A lh ('\
{s) will be oositive
.. definite . ·Ihe:.:·e \:ill "'.:>e
in Chapter
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 dobertson - .~alker models are b~atial~y no;~o;eneous
f
symmetry ib,y considering models tl1;;;t arc spati::.-lly ~!omoger: eot1s
but; anicotrouic (that '.is , t11ey hc....ve a ·thr·ee parameter grou_p
of mol,ions transitive on a space-like hypers11rfc~ce) tnen
tne mbt~er flow may have rotE1tion , acceleration ~n~
..:>ee section 5
le~st one space- like surface but space - time is not s·tationary,
(c) the energy- momentum tensor is that of a perfect fluid,
-- •
flow - lines ~nd is uniquely de fined as the time-like eigen-
vector of the ki cci tensor.
•
R, tl1e curvature scalar 111ust be co11st<:l.nt On
... ,.,
o. --1~"e
bl_ c...tv . -
like sur i'ace of tra11si tivi ty ff 3 of the s roup . ·.::here fore R·. / 0...
\-J11ere
.
e CR jU- J an indicator - +1 if •
lS (( I
J
(".. lS p:::.st directec.
.
- - 1 if R. c.. J
•
lS 1u-cure C.irecteC:
~
1
'.L hen ·\{a.: b] =0.
Tl1us V:"" is a cone;ruence of ge odesic irrotational timo-lilcr-;
v ect; ors . .i3y condition ea) , R°"b va... v h > o •
-~her· efor e the con~ ruence must have a sin~t1l<.tr point on e <::Lc'-"
seodesic ( by equation 1) either in the future or in the ~ast .
P..~
J
10 ( if~.,0.. is zero, vte can talce any other sca.J....:..r
•
. -·
.?
ti1e.L·e will be a singular poin~c 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 homo~eneity, the whole of s3 will bE
•
sin~ul~:c 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
"".:1ere
~re three independent vector fields X.~ such thlit
ix.°' 9 bC..
~ 01
hcd. e
' j 1Z
xl; -·- on
.
. ·ihat is , therP-
develoument of
- H1 is determinate .
i11 the a ·o ove nroof is ·that they have well defined f loi:1-li1:cs
tenaor . •.:.•hen \-Je can reol ace condition (c) on tl1e nature of
(e) If the model is singularity- free, the flow - lines -f Q">"•-:"" - .L. - -·
I
'l'his proof rests strongly on the ;:ssumption of
homogeneity 11hich is clearly not satisfied by the physico..l
1
•
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
.
I
l
I
surface but also to spaces whose universal covcrins sDace
has a non-co:-:-ipact Cauchy surface . 1 1
.i. hus it is applic::;.ble to
models which , at the present time , are homogeneous and iso-
l tropic on a large scale with surfaces of approximate homo-
geneity which have negative or zero curvuture .
6. 1
·1 he Closed '11rapped Surf&.ce
Let r3 be a 3- ball of coordinate radius r in a 3-surf ~ce
a3 (t = const . ) in a Robertson-~Jalker metric with K - O or - 1 .
Let qa be 1Jhe outward directed unit normal to T2 , the boundary
of 'i 3 , in H3 and let Va be ·the past directed u!li t normal to
1
I
/; ~ {: b)
..
in which the flow- lines are drawn at their proper sp~tial
I•
.
(h) the universal covering space has a non- compact Ca~chy
surf ace H3 •
L~
Assume space - time is singularity free . Let F be the
I 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 shO\'!
.
0ince B3 is compact , i·ts image B3¥
. '
I
I
must be compact . Let ci(Q) be the number of points of B3
I .
ma1ped to a point q of I-13 . ci((\)) will change only at the
7.
7
intersection of caustics of the normals \vi th H • l\1oreover ,
by continuity can only change by an even number .
•
.
since this l S the identi·ty D-i'1.d
~his is a contradiction, thus the assumption th~t sp~tce -ti1ne
is:!:&i5 is, it· will intersect ove;;:-y tim~-likQ anC. i1ull l:i.nie
oallod "bhO Q!:U;Cl?:Y: liori z.on rel a ti 1re to H3, CQ> fft\!:18is 1': 8: r.~M
.
•"blrfaoo, :1/u;p·i;hO.PHl.O:Fw if condition (f) boJ..de tho Rl:ill e@@&@di&C
I
J
•
I '
•if",Y§t be a t:hme- liko surfao e t;onera ted 1)y floi·:= lines ri·hioh '~ e
d ireoti on of their intors oction with H.?, ¥JO :Q:Uat reac a 6.jji
-
•
.,
3
~4d-point of tho gon@r ator sine~ V does not inters ect l I•. I
•
&:it.. the -0xiste nco of thi~ onO.-p oint contra dicts (o) ~inco ii;
s·eetio n of li:1c flo\11- line •11ith H3. It oan also be shoi... n ·that
•
I
€01ripa ot. If condit ion (f) h.olde every null g~od~sic 3Bno:» .to:F-
« nun me •
~ 3
normal V has positi ve expans ion everyw here on h •
I
PROOF
~or the proof it is necess ary to establ ish a cou ~ le of
point in f} ·.
I
J
iln i~mediate corrollary is that if q is ~he firs~ ~ oint
\
l
which is critical will be said to be maximal if it corresponds to a
local maximum of •
Lemma 1.
n
g = B(s)g/q •
m mn
Then h1 ( e~
<{, J ~ J ff¥ I ~
n x; °r c/..s i r n
must be positive for
any h since if it were negative for any h
by taking a = ... b h h
mb mn
beyond q , it would be possible to have a point y on beyond q 6
conjugate to before a point conjugate to p • If it were zero
X would be conjugate to f . This shows that the surface at q
of constant geodesic distance from p lies nearer to f' in every
direction than the surface o£ i of constant geodesic distance from
1
,u :;.t 'be o •it i "'~e
. . bO
£ l:Ill:.J..
t But a•-0
p ":Q
4 ,. G.O
t.
'-:>
(.,(
i
*· ~E
f I
...,(].
A• A
.. g
.!JA 1\
j ~
M
s
7\
-' MO
g f &' <
lb A
0
.Leinma 2
I
If p lies to the future (past) on a time-like ·:·eoO.esic
..:>
H3 .
the future (p•-st) directed normals to converging a.1~e
•.
.
l
•
But if there is no point conjugate to q along ~in qp , then
cLJnnot be maximal by the first lem!;1a . 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. ·. :. hus
.
is not the seoa.esic of maximum leng"t;h from H3 to p . ·- nis is
a contradiction which shows tha~ ~he ori~inal assu~ption t~et
'i. l)I'OVided t tiat tl1e expansion of its normo..ls was bounded av:ay
l from zero and provided that the intersection of the Cauchy
surf ace \vi th all the time- like and null l ·ines from o. point
v¥as compact .
l
•
1• o. Hecl<:1nan I . A. U. uympo~iun 1961
2. ~~ . J{ayc haudhuri ~hys . ~ev . 98 1 ;23 1965
••
:; . ....\. . Kornar Phys . 1{ev . 1 QL~ 5L1.4 19 56
4. l.i . C . t.ihepley Proc . Nat . Acad . 6ci . 52 1403 1964
t::
/ • c. Benr Z. Ascrophys . 60 266 1965
6. ~.M Lifshitz and Adv . in Phys . 12 185 1963
I . M. ~(halatnil<:ov
'l . l{. Penrose Phys . Rev • .Lett . 1L~ 57 1965
b. rt . Penrose Rev . : .od . Phys . '!!. 215 17 65
9. I( . Godel Rev . Mod . ?hys . ?:d 44-7 l'ftr f
10 . C. •.1 . l'1isner J . Math . Phys . ~ qz4 r~
11 . J . i·:ilnor ' Morse Theory ' No . 51 Annals of
of Maths . ~tudies . 2rincetown
,
Univ. Press .
i~umber·s 1'To . 32
1
12. K. Yo.no and 'Curvature and Bet·!Ji
.::> . Bochner ~nnals of Maths . ~tudies .
Princetown Univ . Press .
13 . a . Heckman and l~rticle in 'Gravit;ation ' ed . .;;it·cer-1
I L~
\ (.A.c•10;,
-
J
\ '
l
\
I
Ph.D.
51:;7