A vector-metric theory of gravity
by Ronald Ward Hellings
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY in Physics
Montana State University
© Copyright by Ronald Ward Hellings (1973)
Abstract:
A new theory of gravity is presented in which gravity is produced by a massless vector field in addition
to the usual metric field. It-is found that the theory is compatible with present solar system experiments
and cosmological observations. The theory also predicts the existence of black holes and of the most
general types of weak plane waves. A VECTOR-METRIC THEORY OF GRAVITY
by
RONALD WARD HEELINGS
A t h e s i s su b m it te d to the Graduate Fa c ulty in p a r t i a l
f u l f i l l m e n t of t h e re quire m e nt s f o r th e degree
Of
DOCTOR OF PHILOSOPHY
in
Physics
Approved:
Chairman, Examining Committee^
GraduatefDean
MONTANA STATE UNIVERSITY
Bozeman, Montana
A u g u s t , ' 1973
in
ACKNOWLEDGMENTS
Primary acknowledgment must go to Ken Nordtvedt f o r t h e i n i t i a t i o n
and d i r e c t i o n o f my r e s e a r c h on t h e problem and f o r t h e p a t i e n t e x p l a ­
n a t i o n s and r e - e x p l a n a t i o n s o f t h e p r i n c i p l e s o f r e l a t i v i t y physics
which I needed to un d e rs ta n d in o r d e r to do t h i s work.
Several o t h e r s
have had a hand in t h e p r o d u c t i o n o f t h e t h e s i s via h e l p f u l d i s c u s s i o n s .
Among t h e s e a r e B i l l K i n n e r s l e y , Wei-Tou Ni, Roger S t e t t n e r , C l i f f W il l,
Bob Dicke, Kip Thorne, David Lee, Alan Lightman, and Bob Rashkin.
S pe ci al thanks a l s o to Peggy Humphrey f o r he r remarkable calm and
e f f i c i e n t p r e p a r a t i o n o f th e f i n a l copy in t h e f a c e o f a d i f f i c u l t manu­
s c r i p t and s e v e r a l even more d i f f i c u l t d e a d l i n e s .
F i n a l l y , I want to thank Dee, my w i f e , w i t h o u t whose l o v e , s u p p o r t ,
and s a c r i f i c e t h i s r e a l l y would not have been p o s s i b l e .
iv
TABLE OF,CONTENTS
Page
VITA. ................................................
ACKNOWLEDGMENTS......................................................................
ABSTRACT............................................
CHAPTER I : INTRODUCTION. . . . . . . . . . . . . . . .
ii
iii
Vi
.......................
I
A. Metr ic T h e o r i e s ........................................
B. Machian E f f e c t s
...................... ; . . . . . .
C. .Summary o f R e s u l t s ............................................................■ ........................
3
4
6
CHAPTER I I :
A.
B.
C.
D.
THE VECTOR-METRIC THEORY OF GRAVITY. . . . . . . . .
N o t a t i o n .................................................... .... . . ,
...............................
The La g ra n g ia n ...................... ........................................................ • ■ . . . .
The F i e l d Equations ..........................................................................
The S o l u t i o n s
........................... . . . . . . . . . . . . . . .
1 . Weak G r a v i t y . . . ....................................................................................
2. Event H o r i z o n ..................,.........................................................................
3. R a d i a t i o n ....................................................................................................
4. C o s m o l o g y ..................................
CHAPTER I I I :
CHAPTER IV:
9
9
IO
12
13
13
13
14
15
THE PPN METRIC............................................................................... ' 16
THE EVENT HORIZON............................................ .... ......................
24
CHAPTER V : . RADIATION.......................................................................' ...................
30
A.
B.
C.
L i n e a r i z e d G r a v i t a t i o n a l Waves..................................................
R e s u lt s ..................................................................................................................
C a u s a l i t y .............................................................................................................
CHAPTER VI:
CHAPTER VII:
A.
1.
2.
3.
4.
5.
6.
COSMOLOGY........................................ .... . . . ; ...........................
CONCLUSIONS............................... : .
..............................
30
32
37
38
46
Comparison o f Theory and Experiment . ...................... 46
PPN Parameter y ............................................................................................
46
PPN Parameter e ................................................................................
PPN Parameter Ct1- . ........................................................................................
48
PPN Parameter ^ ..........................................................................
Black Holes . . :..........................................................................
49
Speed o f G ra v it y Waves............................... ' ........................................
50
47
49
V
TABLE OF CONTENTS ( c ' o n t ' d . )
Page
7.
8.
9.
Modes o f G ra v it y Waves...........................................................................
Ren orm al iz a tio n o f G.................................................................................
Time V a r i a t i o n o f G .........................................................
B-. Fu tu r e E xp e rim en ts ..................................................... - .................................
1 . n T5 0 . . . .
.............................................................................................
2. n = 0 ............................
51
52!
53
54
54
56
APPENDICES.......................................................................................................................
57
Appendix
Appendix
Appendix
Appendix
Appendix
A.......................................................................................
58
B...................................................................................................................
70
C.......................................................................................
87
D........................................................................................................................102
E.............................................................................................................. . 105
BIBLIOGRAPHY........................................................................................................................112
vi
ABSTRACT
A new th e o r y o f gravity i s p r e s e n t e d in which gravity i s produced
by a ma ss le ss v e c t o r f i e l d in a d d i t i o n to t h e usual m e t r i c f i e l d .
Iti s found, t h a t the th e o r y i s c omp at ibl e with p r e s e n t s o l a r system
ex periments and cosmological o b s e r v a t i o n s . The t h e o ry a l s o p r e d i c t s
t h e e x i s t e n c e o f b la ck hole s and o f t h e most gener al ty pe s o f weak
pla ne waves.
CHAPTER I
.
INTRODUCTION
Whenever a new t h e o r y i s propos ed , i t should be made c l e a r why t h e
t h e o r y i s needed.
Usua lly t h i s in v o lv e s showing how e x i s t i n g t h e o r i e s
f a i l to adequately- e x p l a i n c e r t a i n exper imen ta l o r o b s e r v a t i o n a l r e s u l t s .
Yet t h i s need not be t h e only j u s t i f i c a t i o n f o r a new t h e o r y , and i t is
n o t t h e j u s t i f i c a t i o n f o r t h e new t h e o r y o f g r a v i t y to be p r e s e n t e d h e re ,
As a m a t t e r o f f a c t , t h e r e i s p r e s e n t l y nothing o b s e r v a t i o n a l Iy
i
wrong w ith E i n s t e i n ' s g en eral r e l a t i v i t y , t h e most widely acce pte d
t h e o r y o f g r a v i t y to d a y .
As time goes on and t h e e vid enc e mounts in
f a v o r o f g en eral r e l a t i v i t y , i t i s s t r a n g e to remember t h a t a few y e a r s
ag o, as r e c e n t l y as 1960, only two o f g en eral r e l a t i v i t y ' s p r e d i c t i o n s
which d i f f e r e d from Newton's th e o r y had a c t u a l l y been obs e r v e d ,
2
and
based on only t h e s e two t h e th e o r y was almo st u n i v e r s a l l y a c c e p te d .
One
t h i n g which probably gave impetus to t h e growth o f ex perimental g r a v i t y
d u ri n g t h e s i x t i e s and c e r t a i n l y was t h e m o t i v a t i o n behind t h e r e c e n t
l i g h t d e f l e c t i o n and ti m e - d e l a y e x p e r im e n t s , was th e appeara nce in 1961
o f a new r e s p e c t a b l e , c o v a r i a n t t h e o r y o f g r a v i t y —t h e Brans-Dicke t h e o r y .
^See a summary o f experimental r e s u l t s in C. M. W i l l , Le ctures in
B. B e r t o t t i , e d . , Pro ceeding s of Course 56 o f th e I n t e r n a t i o n a l School
o f Physics "Enrico F e rm i" , Academic Pr ess (1973).
p
These were t h e d e f l e c t i o n o f l i g h t and th e p e r i h e l i o n p r e c e s s i o n
o f mercury. The t h i r d " c l a s s i c a l t e s t " o f r e l a t i v i t y , t h e g r a v i t a t i o n a l
red s h i f t , was o r i g i n a l Iy co nfirmed, but subse quen t o b s e r v a t i o n s showed
t h e r e s u l t s to be i n c o n c l u s i v e . I t was not u n t i l th e Pound-Rebka e x p e r ­
iment in 1960 t h a t t h e red s h i f t was v e r i f i e d . See E. F in l a y - F r e u n d li c h ,
P h i l o s . Mag. 4_5, 303 (1954); and R.V. Pound and G.A. Rebka, Phys. Rev.
L e t t e r s 4, 337 ( I 9 6 0 ).
2
In f a c t , what had been l a r g e l y m is si ng during t h e p e r io d 1916-1960
was t h e usual i n t e r p l a y between th e o r y and ex per im ent .
Most t h e o r i s t s
were s a t i s f i e d w it h g e ner al r e l a t i v i t y and were r e l u c t a n t t o c o n s i d e r
any o t h e r p o s s i b i l i t y .
Most ex p er im en t er s were r e l u c t a n t to t a c k l e
any o f t h e g r e a t d i f f i c u l t i e s in volve d in measuring t h e e xtre m el y weak
post-Newtonian e f f e c t s o f g r a v i t y , e s p e c i a l l y when t h e expected outcome
o f t h e experiment would merely be a f u r t h e r c o n f i r m a t i o n o f what e v e r y ­
one a l r e a d y "knew".
There i s an a d d i t i o n a l d i f f i c u l t y f o r t h e e x p e r im e n t e r t h a t a r i s e s
when t h e r e i s only one th e o r y p r e s e n t , and t h a t i s t h a t i t i s not c l e a r
which ex periments a r e s i g n i f i c a n t .
This i s e s p e c i a l l y t r u e f o r gener al
r e l a t i v i t y which p r e d i c t s a val ue o f z er o f o r th e r e s u l t o f many e x p e r ­
iments.
A nu ll ex per im ent al val ue i s r e a l l y only meaningful a g a i n s t a
background o f o t h e r t h e o r i e s p r e d i c t i n g non-nu ll r e s u l t s f o r the same
ex p er im en t .
For example, gener al r e l a t i v i t y p r e d i c t s t h a t a l l n e u t r a l
t e s t bodies f a l l a t t h e same r a t e in a g r a v i t a t i o n a l f i e l d (weak e q u i v a ­
le n c e p r i n c i p l e ) , and t h a t t h e r e s u l t of any lo c al ex per im ent i s in d e ­
pendent o f t h e v e l o c i t y o f t h e a p p a r a t u s ( p a r t of the s t r o n g e quiv a- ,
le n c e p r i n c i p l e ) .
I f t h e second p r e d i c t i o n must f o ll o w from the f i r s t
in any r e a s o n a b l e t h e o r y o f g r a v i t y , then an experimental r e s u l t
v e r i f y i n g t h e second p r e d i c t i o n does n o t i n c r e a s e our co nfid e nce in
gener al r e l a t i v i t y .
However, i f a th e o r y appears which, p r e d i c t s t h e
f i r s t but not t h e second (as t h e V e ct or- M e tri c Theory d o e s ) , then a
3
v e r i f i c a t i o n o f t h e second p r e d i c t i o n s e r v e s to e l i m i n a t e th e i n t e r l o p e r
and i n c r e a s e our c o n f id e n c e in g en eral r e l a t i v i t y .
The re aso n t h a t th e
p r e s e n t th e o r y i s needed, as a r e o t h e r t h e o r i e s , i s to p o i n t t h e way to
f u t u r e s i g n i f i c a n t e x p e r i m e n t s , and to make " n o n - n u l l '' some o f the l a t e n t
nu ll p r e d i c t i o n s o f gener al r e l a t i v i t y .
A.
M etric T h e o r i e s .
An a n a l y s i s by R. H. Dicke
3
has shown t h a t the
high p r e c i s i o n nu ll e x p e r im e n t s -- E o tv o s e x p e r i m e n t s , Hughes-Drevor
e x p e r im e n t s , and e t h e r d r i f t e x p e r i m e n t s - - r u l e o u t t h e e x i s t e n c e o f
v e c t o r o r a d d i t i o n a l t e n s o r f i e l d s ( b e s i d e s t h e m e t r i c f i e l d ) which
coup le d i r e c t l y to m a t t e r .
As p o i n t e d o u t by Will and N o r d t v e d t , 1
however, v e c t o r and t e n s o r f i e l d s may e x i s t along w ith t h e m e t r i c f i e l d
as long as th e e q u a ti o n o f motion f o r m a t t e r does not i n c l u d e them
explicitly.
These f i e l d s may couple to t h e m e t r i c f i e l d an d.b e invol ve d
in d e te r m in in g th e f u n c t i o n a l form o f t h e m e t r i c , but once th e m e tr ic
f u n c t i o n s a r e found
t h e m a t t e r e q u a t i o n s o f motion w i l l depend only
on them and on t h e m a t t e r v a r i a b l e s ( p o s i t i o n , v e l o c i t y , , e t c . ) .
5
which have t h i s p r o p e r t y a r e c a l l e d " m e tr ic t h e o r i e s " . 3*
Th e ori e s
3R. H. Dicke, The T h e o r e t i c a l S i g n i f i c a n c e o f Experimental Rela­
t i v i t y , Gorden and Breach (1965).
^C. Will and K. N ord tv e dt , J r . , Ap. J . 177, 757 (1972).
3K. Thorne and C, W i l l , Ap. J . 16 3 , 595 (1970).
4
G r a v i t a t i o n a l ex per im ent al r e s u l t s depend on t h e m e t r i c f i e l d s and
can be expecte d to i n d i r e c t l y d e t e c t t h e e x i s t e n c e o f t h e v e c t o r or
t e n s o r f i e l d s which go i n t o de te r m in in g t h e m e t r i c .
In n o n - g r a v i t a t i o n a l
e x p e r i m e n t s , however, t h e . m e t r i c only s e r v e s as a background.
Coor­
d i n a t e s a r e always chosen so t h a t t h e m e t r i c i s l o c a l l y L o r e n t z , and
t h e e f f e c t o f whatever went i n t o d e te r m in in g th e m e t r i c i s washed o u t.
The E o t v b s , Hughes-Drevor, and e t h e r experiments a r e a l l n o n - g r a v i t a t i o n a l
in t h i s s e n s e .
They a r e based on t h e fundamental idea o f a m e t r i c and
measure no n-m e tri c p e r t u r b a t i o n s to g e odes ic beh avi or .
B.
Machian E f f e c t s .
Will and Nordtvedt
6
have s t u d i e d e x t e n s i v e l y the
7
e f f e c t s o f g r a v i t y which have been v a r i o u s l y c a l l e d "Machian",
" p r e f e r r e d - f r a m e " , and " e t h e r " .
Of t h e s e l a b e l s th e l a s t two a r e f a r
i n f e r i o r , implying t h e e x i s t e n c e o f a b s o l u t e space in l o g i c a l c o n t r a ­
d i c t i o n to th e s p i r i t o f a l l r e l a t i v i t y .
In f a c t , one need not p o s t u ­
l a t e a b s o l u t e space in o r d e r to f i n d a frame which i s in some sense
preferred.
Since g r a v i t y i s a l o n g - r a n g e u n i v e r s a l i n t e r a c t i o n , one
might e x p e c t t h e globa l d i s t r i b u t i o n o f m a t t e r to a f f e c t l o c a l g r a v i t a ­
t i o n a l ph y s ic s in a Machian way and to e s t a b l i s h a p r e f e r r e d frame as
t h e mean r e s t frame o f t h e u n i v e r s e .
^Will and No rdv ed t, op c i t .
7
The term ."Machian" has been l o o s e l y a p p l i e d to r e f e r to th e d e t e r ­
m in a ti on o f t h e p r o p e r t i e s o f space by t h e global d i s t r i b u t i o n of m a t t e r .
See E r n s t Mach, The Scie nce o f Mechanics, Open Court P u b l i s h i n g (1902),
Chapter I I .
I
5
As Will and Nordvedt p o i n t o u t t h e mystery i s n o t , "How can we have
a p r e f e r r e d r e s t frame in s p a c e ? " , but "How can we ever avoid having
one, r e l a t e d to t h e u n i v e r s e r e s t - f r a m e " .
Theories which have only a
m e t r i c f i e l d , o r only a m e t r i c and a s c a l a r f i e l d , avoid t h e s e e f f e c t s
in t h e f o ll o w i n g way.
1.
• ‘
A th e o r y which c o n t a i n s only a m e t r i c f i e l d y i e l d s lo c al g r a v i ­
t a t i o n a l ph y s ic s which i s i d e n t i c a l in a l l frames which a r e a s y m p t o t i ­
c a l l y Minkowskiian.
This foll ow s from t h e i n v a r i a n c e under boosts of
n^v ( t h e a s y m p t o ti c form o f gy v ) , t h e only f i e l d c o up lin g t h e lo c a l s y s ­
tem a s y m p t o t i c a l l y to t h e u n i v e r s e ; and from general c o v a r i a n c e , which
allo w s us to f i n d a c o o r d i n a t e system in which t h e m e t r i c ta kes t h i s
Minkowski form a t t h e boundary between t h e u n i v e r s e and t h e l o c a l s y s ­
tem .
2.
A th e o r y which c o n t a i n s a m e t r i c f i e l d and a s c a l a r f i e l d
y i e l d s ph y s ic s which i s i d e n t i c a l in a l l asy mp to tic Minkowski frames.
This f o ll o w s from i n v a r i a n c e o f n
yv
and t h e s c a l a r f i e l d under boosts
( t h e s c a l a r f i e l d , of c o u r s e , i s g e n e r a l l y i n v a r i a n t ) .
In t h e p r e s e n t t h e o r y , we in c l u d e with th e m e t r i c a v e c t o r f i e l d ,
K^, whose components depend on t h e c h o ic e o f Lorentz frame..
In some
Lorentz frame, t h e v e c t o r f i e l d a t a p o i n t w i l l have a z e r o t h component
g
on ly . This i s th e only frame in which space is l o c a l l y i s o t r o p i c and
O
'
’
Any non-zero space p a r t of t h e background v e c t o r f i e l d must p o i n t
in some p r e f e r r e d d i r e c t i o n , d e s t r o y i n g t h e lo c al i s o t r o p y o f space.
6
we d e f i n e t h i s frame to be th e p r e f e r r e d frame in which, f o r s i m p l i c i t y ,
we w i l l do much o f t h e work t o f o l l o w .
The a p p a r e n t i s o t r o p y o f the
m a t t e r d i s t r i b u t i o n seen through th e t e l e s c o p e le a d s us to b e l i e v e t h a t
t h i s p r e f e r r e d frame i s a l s o the mean r e s t frame o f t h e u n i v e r s e , and
t h a t t h e p r e f e r r e d frame i s determined in a "Machian" way.
C.
Summary o f R e s u l t s .
In t h e c h a p t e r s to f o l l o w , we compute the p r e ­
d i c t i o n s o f t h e v e c t o r - m e t r i c th e o r y f o r :
I ) s o l a r system e x p e r i m e n t s ,
2) t h e e x i s t e n c e o f bla ck hole s o r e v e n t h o r i z o n s , 3) g r a v i t a t i o n a l
r a d i a t i o n , and 4) cosmology.
I.
The r e s u l t s a r e as fo l l o w s :
S o l a r system e x p e r i m e n t s .
I t i s found t h a t with r e s t r i c t i o n s
on t h e s t r e n g t h o f some c ou plin g c o n s t a n t s
t h e th e o r y i s co mpatible
w it h a l l e x i s t i n g s o l a r system e x p e r im e n t s , and t h a t f o r a p a r t i c u l a r
c h o i c e o f th e c ou plin g c o n s t a n t s
as does gener al r e l a t i v i t y .
t h e t h e o r y makes t h e same p r e d i c t i o n s
This l a s t p o i n t i s o f p a r t i c u l a r importance
s i n c e i t means t h a t no s o l a r system experiment p r e s e n t l y en v is i o n ed can
d i f f e r e n t i a t e between t h i s th e o r y and gener al r e l a t i v i t y .
I t is evident
t h a t experiments which a r e to choose between t h e s e t h e o r i e s must in v o lv e
hi g h e r o r d e r e f f e c t s such as occu r in a) g r a v i t a t i o n a l r a d i a t i o n , b)
cosmology, o r c) e xtr em el y p r e c i s e s o l a r system e x p e r im e n t s.
I t i s not
c l e a r which ty pe o f exp eriment o f f e r s t h e b e s t p o s s i b i l i t y , but t h e o r i e s
such as t h e s e should s t a n d as a c h a l l e n g e to the g r a v i t a t i o n a l e x p e r i ­
menter to d e v i s e new and b e t t e r ways to measure th e e xtr em el y small
e f f e c t s o f post-Newtonian and p o s t- pos t- N e w to nia n g r a v i t y .
7
2.
Black h o l e s .
I t i s found t h a t an e ven t hori zo n e x i s t s in a
s t a t i c s p h e r i c a l l y symmetric c o n f i g u r a t i o n .
Moreover, t h i s s o l u t i o n
has t h e same m e t r i c be h a v io r c l o s e to t h e horizo n as t h e Swarzschild
s o l u t i o n in gener al r e l a t i v i t y .
. 3.
G r a v i t a t i o n a l r a d i a t i o n -.' We have found t h a t t h e r e a r e two
c l a s s e s o f r a d i a t i o n in t h e t h e o r y , depending on th e ty pe o f coupling
between t h e v e c t o r and m e t r i c f i e l d s .
I f t h e co upling i s " s c a l a r - t y p e "
( t h a t i s , K a ppea rs only as K Ky in t h e i n t e r a c t i o n L a g r a n g i a n ) , then ■
v
u
we f i n d waves o f c l a s s N^ 1^ t h e c l a s s t y p i c a l o f s c a l a r - m e t r i c t h e o r i e s .
In t h e most g en eral c o u p l i n g , t h e c l a s s i s H g , r e f l e c t i n g t h e v e c t o r
nature of the i n t e r a c t i o n .
Also, t h e s e l a s t waves a r e seen to prop ag at e
a t speeds e i t h e r g r e a t e r o r l e s s than t h e speed o f l i g h t , depending on
t h e v a lu e s of t h e c ou plin g c o n s t a n t s .
4.
Cosmology.
This i s d i s c u s s e d in Chapter V.
The v e c t o r - m e t r i c t h e o ry i s found to posse ss an
a c c e p t a b l e c l o s e d cosmology, th e p r e c i s e be h a v io r depending on the
c o u p l i n g c o n s t a n t s as in Brans-Dicke t h e o r y .
I t is found t h a t the lo cal
c o n s t a n t o f g r a v i t y i s dependent on t h e cosmological s t r e n g t h of the
vector f i e l d .
The observed g r a v i t a t i o n a l c o n s t a n t i s o b t a i n e d i f one
p o s t u l a t e s an energy d e n s i t y in t h e u n i v e r s e which i s 1 , 000' times the
observed s t e l l a r m a t t e r d e n s i t y .
This re q u ir e m e n t sounds l e s s outrageous
9For a d e s c r i p t i o n o f t h i s c l a s s i f i c a t i o n , see D. Eardle y , e t a I . ,
Phys. Rev. L e t t e r s 30, 884 (1973). There i s a l s o a summary o f the
scheme in Chapter V o f t h i s work.
When we r e a l i z e t h a t g e ner al r e l a t i v i t y r e q u i r e s a . f a c t o r o f TOO more
energy in th e u n i v e r s e than t h e s t e l l a r m a t t e r p r e s e n t l y observed in
o r d e r to produce a c l o s e d u n i v e r s e .
CHAPTER I I
THE VECTOR-METRIC THEORY OF GRAVITY
A . ' Notation.
The n o t a t i o n used in t h e remainder o f t h i s work follows
t h a t o f A d le r , B azi n, and S c h i f f e r t S I n t r o d u c t i o n to General R e l a t i v i t y . 41
1.
Greek i n d i c e s t a k e on v a lu es from 0 to 3.
from I to 3.
La tin i n d i c e s run
Repeated i n d i c e s a r e summed over t h e i r range
of values.
2.
The m e t r i c t e n s o r i s denoted
n
yv
and has s i g n a t u r e (+ - - - ) •
i s t h e Minkowski m e t r i c t e n s o r with t h e same s i g n a t u r e ,
Occasionally g
w i l l be used to mean g ^ , g2 2 , or g3 3 , and
t h e r e w i l l be no sum over s.
3.
A comma ( , ) i n d i c a t e s o r d i n a r y p a r t i a l d i f f e r e n t i a t i o n .
A
v e r t i c a l bar ( I ) i n d i c a t e s c o v a r i a n t d i f f e r e n t i a t i o n :
A
= A
- r aA
ylv
y,v
yv a
4.
Sign c onven ti ons in t h e d i f f e r e n t i a l geometry
r
a r e as f o ll o w s :
Ct
yv
Ra
R yBv
+ gv6 , p - w
> '
= Fa
- Fa
r- Fa F^ ' - Fa F^
By,v
y v ,6
Xv By
XB yv
R = R"1
yctv
yv
1R. A dl e r, M. Bazin, and M. S c h i f f e r , I n t r o d u c t i o n to General R el a t i v i t y , McGraw-Hill (1965).
10
5.
The box o p e r a t o r i s d e f i n e d
irF( ) - na ~( ) , a g
and we d e f i n e
a cov a r i a n t box o p e r a t o r Q2 ( ) = ga g ( ) | a g*
6.
B.
We work in u n i t s whefe c = I .
The Lagra nqi a n .
E i n s t e i n ' s t h e o r y o f g r a v i t y can be d e ri v e d from
v a ria tio n of the action i n t e g r a l :
J’
•g[l B7rGL
Rld4 X
where g i s the d e te r m in a n t o f g ,
■ .
yv
G i s Newton's g r a v i t a t i o n a l c o n s t a n t ,
Lm = Lm(g; ^ , m a t t e r v a r i a b l e s ) i s t h e m a t t e r Lagrahgian which
is a function of q
and i t s d e r i v a t i v e s and o f whatever v a r i yv
a b l e s a r e used to d e s c r i b e th e s t a t e o f m a t t e r in t h e system
(position, velocity, e tc .) .
The re q u ir e m e n t t h a t th e a c t i o n be i n v a r i a n t under an i n f i n i t e s i m a l
variatio n of g
produces th e f i e l d e q u a t i o n s of gener al r e l a t i v i t y .
V a r i a t i o n o f t h e m a t t e r v a r i a b l e s produces t h e e q u a t i o n s o f motion of
matter.
In 1961, Brans and Dicke
p
proposed a th e or y in which g r a v i t a t i o n
was produced by two f i e l d s - - a m e t r i c t e n s o r f i e l d and an a u x i l i a r y
scalar field .
Their action in teg ral is w ritten
2 C. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961 ).
Tl
A = y / I g[167TLm + .<t>R +
vJ d4x S
where Q i s t h e s c a l a r f i e l d and w i s a dim e ns io nle s s p a r a m e te r .
a g a i n , Lm i s a f u n c t i o n o f
enter.
and m a t t e r v a r i a b l e s o n ly ;
Once
does not
The Brans-Dicke f i e l d e q u a t i o n s come from ind ep en den t v a r i a ­
t i o n o f g ^ and <j>.
The
e q u a t i o n s o f motion o f m a t t e r come from v a r i a ­
t i o n o f t h e m a t t e r v a r i a b l e s in Lm.
Since <f> does not ap pea r in
Lm,
t h e e q u a ti o n o f motion o f m a t t e r w i l l i n v o lv e only th e m e t r i c and w il l
produce a m e t r i c th e o r y o f g r a v i t y .
Our re a so n f o r review ing t h e s e t h e o r i e s i s t h a t t h e new th e o ry to
be p r e s e n t e d here adds onto general r e l a t i v i t y in a. way s i m i l a r to
t h a t o f t h e Brans-Dicke
theory.
We propose a th e o ry o f g r a v i t y in
which a m a ss le ss v e c t o r f i e l d appears in a d d i t i o n to t h e , m e t r i c f i e l d .
Committed to t h e s p i r i t as well as to t h e law o f general c o v a r ia n c e in
p h y s i c s , we i n t r o d u c e no a p r i o r i f i e l d s o r r e f e r e n c e frames i n t o th e
theory.
We r e q u i r e a Lagrangian s u b j e c t to th e f o ll o w i n g c o n d i t i o n s :
1.
The Lagrangian d e n s i t y i s a f o u r - s c a l a r d e n s i t y .
2.
I t g e n e r a t e s p o s i t i v e d e f i n i t e f r e e f i e l d e n e r g i e s f o r both
t h e m e t r i c and th e v e c t o r f i e l d s .
3.
I t produces a " m et ric t h e o r y " .
4.
I t g e n e r a t e s f i e l d e qu a ti o n s c o n t a i n i n g no h i g h e r than second
d e riv a tiv e s of the f i e l d s .
Such a Lagrangian is
12
A = / / - g [ l 6irG L + R - F
Fpv + coK KyR + nKyKvR ] d 4x
J
o m•
yv
y
yv
( I I . I)
where Lm = l m(g^^, m a t t e r v a r i a b l e s ) as b e f o r e ,
^yv = V i v ~ ^viy in analogy with e l e c t r o d y n a m i c s ,
K is the vector f i e l d ,
y
to and n a r e d im e n s io n le s s p a r a m e t e r s , and
Gq i s an a p r i o r i o r "bare" g r a v i t a t i o n a l c o n s t a n t .
C.
The F i e l d E q u a t i o n s .
The f i e l d e q u a t i o n s a r e c a l c u l a t e d by requir ing
t h a t t h e a c t i o n , e q u a ti o n I I . I , be s t a t i o n a r y under ind ep end en t v a r i a t i o n
of the f i e l d s .
D e t a i l s o f t h e d e r i v a t i o n a r e given in Appendix E.
Varia­
t i o n o f g ^ giv e s the e q u a ti o n
B„v - I=MVr + " i KMKvR + KRmv - I=MVkr + K, mv - =MvD2 k )
+ TlKa KfV g pctRxl6 + S v f i R m - i t i p A g ) - I " [ = M V ( K a K S ) l a 6
( I I . 2)
+ I t ( K pKv ) - (Kp Ka ) l v o - (KvKa ) lpct]
+ A cT v + I=MVpCieFaS ' - 8 ”G0T„v
where K e K Kaand T
a ~
7^
/ - g 6g
( / ^ 1 6 v G py .
The c o n t r a c t i o n of equa-r
t i o n I I .2 i s
+ n (Ka Ke ) lag = 8TTGogyvT^v E St G0T.
R * (3to + Tjti
|(
( I I . 3)
V a r i a t i o n o f K in e q u a ti o n I I . I gives
OjRKy + nRyvK + 2Fyv
V
IV
= 0.
( I I . 4)
13
D.
The S o l u t i o n .
In t h e c h a p t e r s t o f o l l o w ^ th e above e q u a t i o n s w il l
be s o lv e d in f o u r d i f f e r e n t c o n t e x t s .
To each of t n e c a s e s , t h e r e
c or re sp on ds some app rox ima tio n or sy mm e tri z ati on o f t h e m e t r i c which
can be used t o s i m p l i f y t h e f i e l d e q u a t i o n s .
The f o u r c as e s a r e as
follow s:
1.
Weak G r a v i t y .
I t i s assumed t h a t t h e r e e x i s t s a re g io n o f
s pac e where g r a v i t y i s weak and t h e motion of so ur c es i s slow.
By weak
g r a v i t y i t i s meant t h a t , f o r each massive so ur c e o f t h e m e t r i c ,
<< I
(where r i s t h e d i s t a n c e from th e s ourc e o f mass M). Slow motion o f th e
2
s o u rc es n a t u r a l l y means v « 1 .
These c o n d i t i o n s a r e s a t i s f i e d in th e
s o l a r system and in most l o c a l r e g io n s o f th e u n i v e r s e .
When th e con­
d i t i o n s a r e s a t i s f i e d , one can make a gener al expansion o f t h e m e tr ic
in powers o f both — and v*\
Keeping terms second o r d e r in t h e combina--
2 2 2
2
t i o n ( t h a t i s , terms l i k e G M / r and GMv / r ) produces t h e P a rame ter ize d
3
Post-Newtonian (PPN) m e t r i c o f Nordtvedt and W il l.
In Chapter I I I , we
o b t a i n t h e PPN m e t r i c f o r t h e v e c t o r - m e t r i c t h e o r y .
2.
Event H ori zo n.
Vishveshwara^ has shown t h a t , f o r a s t a t i c
m e t r i c , a s u r f a c e o f i n f i n i t e r e d s h i f t ( i . e . , a s u r f a c e on. which g^g
3C. M. Will and K. N o r d t v e d t , J r . , Ap. J . 1 7 7 , 7 5 7 (1972).
4C. V. Vishveshwara, J . , Math. Ph ys. 9, 1319 (1968).
0 )43
i s a l s o an e v e n t h oriz on o r a one-way membrane.
Therefore, the question
o f whether o r not t h e v e c t o r - m e t r i c th e o r y p r e d i c t s th e e x i s t e n c e of
b l a c k ho le s as does g e ner al r e l a t i v i t y can be answered by a sea rc h f o r
a s u r f a c e o f i n f i n i t e r e d s h i f t in a s t a t i c c o n f i g u r a t i o n , which f o r
s i m p l i c i t y we a l s o t a k e to be s p h e r i c a l l y symmetric.
I d e a l l y one would
l i k e to have an e x a c t s o l u t i o n f o r t h i s c o n f i g u r a t i o n , but t h i s t u r n s
o u t to be an e xtr em el y d i f f i c u l t problem whose s o l u t i o n has n o t y e t
been found.
R a t h e r , we have expanded t h e f i e l d s in a power s e r i e s about
t h e e ve nt h o r i z o n , keeping t h e le a d i n g terms only.
3.
Radiation.
Weak plane: g r a v i t a t i o n a l waves a r e d e s c r i b e d by
w r i t i n g t h e m e t r i c and v e c t o r f i e l d s as t h e sum o f a c o n s t a n t background
p a r t and a wave d i s t u r b a n c e p e r t u r b a t i o n :
^iav
riUV +
and r e q u i r i n g t h a t h
<< n
and t h a t Ay << <py .
3
yv
pv
•
Here ■
n
i s t h e Minkowski m e t r i c t e n s o r ,
' pv
.
<j>y i s t h e cosmological v e c t o r background,
h
pv
i s t h e m e t r i c wave,
Ay i s th e v e c t o r wave;
h
and Ay a r e assumed p r o p o r t i o n a l to e x p (i k xy ), with k
c o n s t a n t p r o p a g a ti o n v e c t o r .
as some
S o l u t i o n s g iv in g r e l a t i o n s h i p s between
15
‘f y ’ ilUVs Ay and kU ^o r weak r a d i a t i o n a r e found in Chapter V.
4.
Cosmology.
The Robertson-Walker cosmological m e t r i c with a
n e u t r a l d u s t model f o r m a t t e r i s used to e s t a b l i s h a cosmology f o r th e
v ecto r-m etric theory.
The m e t r i c i s assumed t o be o f t h e form
Soo = 1
gIr-Trff^1
V
In Chapter VI5 t h e f i e l d e q u a ti o n s a r e s olv ed giv in g S 5 t h e " s i z e " of
t h e u n i v e r s e , as. a f u n c t i o n o f ti m e .
CHAPTER I I I
THE PPN METRIC
.1
Will and Nordvedt
have a r r i v e d a t a general form f o r t h e f i r s t
post-Newtonian m e t r i c v a l i d in any i n e r t i a l c o o r d i n a t e fr am e .
CM.
jOO
1 "
2
I" iT
. GM. 9
GM.v.‘
+
^ 1 + “ 3 + c I 1?1 “
GM. •
- 2 O - 28 + 5 )I — • I
i ri m
GM.
— +
rij
GM
+ “ pi --- + (Cl, - Q p - OgXE y.
■ i r /
1
1
^
GM..
— ^Cv. ^ rr 1. y
i r_.
1 1
GM
V '
d i
ri
GM.
(a 1 "Z(Xg)I " ( W - V i )
I
GM.
K
9Ok = 2 (4 y + 3 + 01I ' a 2 + c I ^ T : vi
1 1
-I
+ 2 (1 + a 2 “
GM.
T
T ^i
GMGM
- a g ) ! TTW^ + «2%
i ri
'i r-
X C vt-ftix.k +
GM,
3Jim
where x-
i/
£m
+ 2 ^ l ” ft)
I ri
a r e c a r t e s i a n components o f t h e i
vector,
"fch
source-to-field-point
k
"fch
Vi ' a r e c a r t e s i a n v e l o c i t y components o f t h e i
source
T-ar>>
k
w a r e c a r t e s i a n components o f t h e v e l o c i t y o f t h e i n e r t i a l
................... c o o r d i n a t e system r e l a t i v e t o t h e u n i v e r s e r e s t frame.
1C. M. Will and K. N o r d t v e d t 5 J r . , Ap. J . 177, 757 (1972).
M. i s t h e g r a v i t a t i o n a l mass, o f t h e i
f !i
body, and
G i s th e e f f e c t i v e g r a v i t a t i o n a l c o n s t a n t .
I t should be a p p r e c i a t e d t h a t t h i s form i s based on very few assumpt i o n s (s e e N ordtvedt ).
The para mete rs B, y , ct-j, a ^ , Ct3 ,
in
t h e m e t r i c a re th e o r y - d e p e n d e n t and may depend on cosmological f a c t o r s
through th e i n f l u e n c e o f cosmological f i e l d s .
In general r e l a t i v i t y
t h e PPN pa rame te rs have t h e value
Y = S = I
01I " a 2 = a 3 = ^l =: ^2' =
and in t h e Brans-Dicke s c a l a r - m e t r i c t h e o r y
_ I + to
Y " TTU
a I = a 2 =' a 3 = c I-
■
z Z =2
2 K. N ord tv e dt , J r . , Phys.. Rev. 169, 1017 (1968).
18
I n s p e c t i o n o f t h e PPN m e t r i c shows t h a t one can o b t a i n a l l the
PPN para mete rs by c a l c u l a t i n g in t h e u n i v e r s e r e s t f r a m e , and t h a t
one can o b t a i n a l l bu t
by c o n s i d e r i n g a s i n g l e .s o u rc e .
For most o f
ou r work we t h e r e f o r e use
2.2
♦ 26^
g00 = I -
sOk
- ( 2t + I + »3 +
' ? (4'r + 3 + O1 - O2 + C1J^Vk + 1(1 + O2 - C1Il yi r • Vjxk^ _ _
where i t i s noted t h a t a pos t- p o s t- N e w t o n ia n term i s added to g
£m
In a d d i t i o n t o t h e PPN m e t r i c e xpan s io n, we w i l l need a s i m i l a r
expansion f o r K ,
4[i
»3^
V)2 + f ^ ( r
r
t)]
( H I . 2)
= ygid^v*' + d ' ^ C f .-v)x&]
where 4, i s a c o n s t a n t , as a re t h e pa ra m et e rs a . , b , f , d , d ' .
These
ex pansions a r e s u b s t i t u t e d i n t o the f i e l d e q u a ti o n s ( I I . 2, I I . 3, and
I I . 4 ) , t h e d e t a i l s o f t h e s o l u t i o n be ing given in Appendix A.
The
l i n e a r i z e d e q u a t i o n s a r e f i r s t s ol ved f o r a s t a t i c s o u r c e , giv in g f o r y
7 -
I +
2m - n - 2
1 - tocp(4w - i~y
19
L ig ht d e f l e c t i o n and r e t a r d a t i o n ex periments
we s p e c i a l i z e to t h a t c a s e .
be r e a l i z e d .
<j> «
3
show t h a t y = I , so
There a r e t h r e e ways t h i s c o n d i t i o n can
The f i r s t i s to r e q u i r e t h a t
I.
This weak cosmological f i e l d c o n d i t i o n reduces a l l r e s u l t s a r b i t r a r i l y
c l o s e to gener al r e l a t i v i t y .
esting.
For t h a t r e a s o n , i t i s t h e l e a s t i n t e r - •
There a re two o t h e r c o n d i t i o n s which give y = I .
w = Jf1 +
"I
These are
(Case I )
and
( I I I . 3)
w=0
(Case I I )
Proceeding with t h e s o l u t i o n s f o r each c a s e , i t i s found, t h a t
Case I
I
I
‘
Case II
= I
-I
'
i =¥
and t h e g r a v i t a t i o n a l c o n s t a n t in each case i s re no rm a li z e d
Case I
Case II
( I I I . 4)
I +
I +' o<t> + I n 2*
3 John D. Anderson, et^ al_., Proceedings o f the Conference on
Experimental T e s ts o f G r a v i t a t i o n a l T h e o r i e s , ( NASA-JPL Technical
Memorandum 33-499, 1970).
20
This msans t h a t G, t h e e f f e c t i v e g r a v i t a t i o n a l c o n s t a n t which e n t e r s
t h e m e t r i c and deter mine s t h e s t r e n g t h o f g r a v i t y in t h e s o l a r system,
depends on background s t r e n g t h o f K f a r from th e s o l a r system.
In
Chapter VI i t i s seen from cosmological c o n s i d e r a t i o n s t h a t t h i s pro ­
duces a time development o f G coupled t o t h e e v o l u t i o n o f t h e u n i v e r s e ;
and t h a t th e extreme weakness of G in t h e s o l a r system can be viewed
as being due t o a r e n o r m a l i z a t i o n of Gq (.-1 ) by a l a r g e cosmological r ^ ,
The pa rame te rs g and 6
e q u a t i o n s to second o r d e r .
come from t h e s t a t i c s o l u t i o n o f the f i e l d
For t h e two cases we ge t
(Case I)
6
=
1
which ag re es with t h e value in gener al r e l a t i v i t y and
6 = 1+
2<j)
---------+ 2a) ( i - 4 (d)
which does n o t , general r e l a t i v i t y having 6 = 1 .
( I I I . 5)
Unfortunately, th is
pa rame te r does not a f f e c t e x i s t i n g experiments to a measurable degree.
In t h e o t h e r c a s e , g and 6 a r e found to be
(Case I I )
2pi(n + 2 ) (n + 4)(f)
4 + n^Cn + 4)
i
(n + •3) - In^Cn + 4)
I - ?n^(n + 4 ) ------ -
( I I I .6)
21
The ex per im ent al r e s u l t g ~
I r e q u i r e s Ti z O1 - 2 , o r -4 (none of which '
p ro v id e s t h e small cy r e q u i r e d below).
The u n o b s e r v a b i l i t y o f 6 e f f e c t s
in p r e s e n t s o l a r system exp erimen ts l i m i t s <j> ~ I or l e s s . , .
So lv in g t h e t o t a l dynamic l i n e a r i z e d f i e l d e q u a t i o n s grves th e
a d d i t i o n a l PPN pa ra m e te rs :
(Case I)
"3 = 0
( I I I . 7)
-I
2*1
(Case I I )
4tj>
4n
+ 4 + 6n "i" n
?
Cl = C3 = 0
_
iI
2n&(3 + fi)
I + ni) +
I 2
( I I I . 8)
^
I
1_ 3n<j>(n + 2)
“ 2 “ 2*1 “ 2 ~ . ~ ~ 2
^ 2 + 4n<j) + n^<|>
An examination o f t h e c o n f i g u r a t i o n o f a p o i n t source i n s i d e a massive
s p h e r i c a l s h e l l y i e l d s t h e second o r d e r -two-mass PPN p a r a m e te r ;
f o r both c a s e s .
The % para mete rs measure 4-momentum n o n - c o n s e rv a ti o n and are
e xpe ct ed t o be zero in t h e o r i e s d e r i v e d from Lagrangian a c t i o n p r i n - ■
ciples.^
The a para mete rs measure t h e e x i s t e n c e of liMachian" e f f e c t s
^Will and N o r d t v e d t 1 op c i t .
22
in g r a v i t a t i o n .
Except f o r t h e case n. = 0 , both c as e s p r e d i c t such p r e ­
f e r r e d frame e f f e c t s .
Will and Nordtv edt^ have an aly ze d v a ri o u s geo­
ph y s ic a l and p l a n e t a r y o r b i t a l e f f e c t s t o a r r i v e a t upper l i m i t s on
t h e a p a r a m e te r s .
The r e s t r i c t i o n s are®
V-'
ci 2 < *I
in Case' I , t h e r e a r e two ways t h e s e r e s t r i c t i o n s can, be met:
n > 34
( I I I . 9)
n
<
•I
Of s p e c i a l i n t e r e s t i s t h e ca s e where n = 0 (w = I ) in eq uat io n
I I I . 7; the n
and a 2 a r e s t r i c t l y z e r o .
In t h i s ' c a s e , r e n o r m a l i z a t i o n
o f Gq is l o s t and t h e t o t a l s e t o f PPN pa rameters i s i d e n t i c a l to th o s e
o f general r e l a t i v i t y .
6 s t i l l d i f f e r s , however, as can be seen by
s e t t i n g w = I in e q u a t i o n . I I I . 5.
6
I + I
3d)
IT 3d> - I
In Case I I , t h e - a r e s t r i c t i o n s r e q u i r e 65
5C. M. Will and K. No rd tv e dt, J r . , Ap1 J ,
177, 775 (1972).
6TIie o r i g i n a l va lu e o f a? < ,03 given by Nordtvedt and Will
probably r e p r e s e n t s an o v e r l y ^ o p t i m i s t i c e v a l u a t i o n o f gra v im e te r
results.
See K. No rdt ve dt, J r . , Science 178, 1157 (1972).. ■
23
Case I I a l r e a d y had w - 0, so t h i s a d d i t i o n a l r e s t r i c t i o n or, th e magni­
t u d e o f n "shows t h i s c a s e to r e p r e s e n t weak coup lin g t o t h e m e t r i c .
To sum up, exp erimental r e s u l t s l i m i t t h e th e o r y to a form f u l f i l l i n g
one o f s e v e r a l c o n d i t i o n s .
The f i r s t c o n d i t i o n i s t h a t o f weak cosmo­
logical vector f i e l d ,
«
I:
which i s no t i n t e r e s t i n g because one can always p o s t u l a t e any kind
of
f i e l d one wants as long as i t i s too weak to a f f e c t exper imen ta l r e s u l t s .
A second c o n d i t i o n i s Case I I where
0) << I
n
<< I
which i s a l s o no t i n t e r e s t i n g , in v o l v i n g as i t does an extremely weak,
c o u p li n g o f t h e v e c t o r f i e l d to t h e m e t r i c .
The l a s t p o s s i b l e way of
s a t i s f y i n g exp erimental r e s t r i c t i o n s i s t h a t o f Case I , which is
U = TjTl + I
n % 34
or
,
.
n s .1.
In t h e i n t e r e s t o f g e n e r a l i t y , su bse que nt s e c t i o n s w i l l n o t r e q u i r e
t h e s e c o n d i t i o n s a p r i o r i bu t w i l l r e f e r to them from t i m e - t o - t i m e .
CHAPTER IV
THE EVENT HORIZON
We f i r s t produce t h e e x a c t f i e l d e q u a t i o n s f o r a so u rc e which i s
S p h e r i c a l l y symmetric, s t a t i c , and a t r e s t in th e u n i v e r s e .
Then an
approximate s o l u t i o n i s found which i s v a l i d n e a r t h e e ven t h o r i z o n .
Choosing S w a r z s c h i l d - ty pe v a r i a b l e s , t h e f i e l d s a r e w r i t t e n
9OO
9rr = -e
Gee = - r
2
Q,, = - r s i n e
<p<t>
.
2
K0 =
K. «= 0
a
Where y, v , and x a r e f u n c t i o n s o f r o n l y .
The s p h e r i c a l symmetry and
time independence a r e m a n i f e s t in t h e form o f t h e m e t r i c .
The absence
Of motion through t h e u n i v e r s e i s r e f l e c t e d in t h e va nis hin g o f K
.
I f a s p h e r i c a l l y symmetric b la c k ho le were moving with r e s p e c t to th e
f i s t frame o f t h e u n i v e r s e , an asymmetry could be communicated to th e
m e t f i c v i a a non-zero space p a r t o f t h e v e c t o r f i e l d , producing an
asymmetric e v e n t h o r i z o n .
This p o s s i b i l i t y i s worth i n v e s t i g a t i n g f o r
i t s a s t r o p h y s i c a l i m p l i c a t i o n s , though t h a t w il l not be done here.
25
Continu ing w it h t h e s o l u t i o n , we w r i t e t h e a c t i o n i n t e g r a l in
terms o f t h e Sw a rz sc hi ld v a r i a b l e s .
R = ^ " ( 2 e A - 2 + 2r%' +
R00 = eV a C ^ 1V1 -
K0 Ir = ^ljl -
V
First,
- r2v" - 2rv‘ - J r A ' 2)
- ^ v 1 ~ ^ v '2)
' )4e^ •
Krl0 = " r V *e1J
with a l l o t h e r c o v a r i a n t d e r i v a t i v e s o f Ky being z e r o .
d i f f e r e n t i a t i o n w it h r e s p e c t to r . )
e>
(Prime denotes
Equation I I . I can then be w r i t t e n
-v+X
A - J d V e 2 [1 6 „ G 0 Lm + R + oi<|>Re2 y - v + n *R 0 0 e 2 ‘, ' 2 v + 2 t p ' 2 e 2 ll"v " x j
I n v a r i a n c e o f t h e a c t i o n under v a r i a t i o n o f ^ gives t h e e quat io n
4-gp + (Pu + n )“j~ - 4F - 4o) ^e i=( eA - l + r x , ) = 0.
(IV.I)
V a r i a t i o n o f v gives
( 2w + n )-gp - ( 2a) + n
- e 2 ( e A - I + r x 1) + uxjie'’ f e A - I + r x 1)
+ F = -S ttGqTq •
( I V . 2)
V a r i a t i o n o f x gives
y-X
e 2 ( e A “ I -- r y ! ) + a)ct>ee ( e x - I + r y ' - 4 r y 1) - F = O.
( I V . 3)
26
In all of the above
1 1
£ = 2y - gV - 2^
P = <j>r"eS'
Q - <pr2e ?v ‘
F = (2w +
+ ^r^e^y'
These t h r e e f i e l d e q u a t i o n s can be made s i m p l e r b o t h . f o r e xact
s o l u t i o n and f o r t h e p r e s e n t approximate s o l u t i o n , by some recombina­
tion.
Adding t o g e t h e r IV .I and IV.2 give s
v-A
(2w + n)^r(P - Q) - e 2 ( rv 1 + rx') + 2a)(f,ec (ex - I - r? 1) = 0
( I V . 4)
where t h e s o l u t i o n i s to be taken e x t e r n a l to th e so u rc e (Tq0 = 0 ).
One -fo urt h o f eq ua ti on IV. I s u b t r a c t e d from eq uat io n IV .3 gives
y-A
2w + n ) ^ " ™ 2o)(j)e^(ex - I - r v 1) - e
(ex - I - r ; ' ) = 0 .
( I V . 5)
The f i e l d e q u a ti o n s f o r a. s t a t i c so urc e ( I V . I to I V . 5) have been
w r i t t e n in a form which, i t i s hoped, should be amenable to e xact
s o l u t i o n , though we have no t made p r o g r e s s in t h i s d i r e c t i o n .
Here we
w i l l d e s c r i b e an approximate s o l u t i o n , v a l i d ne ar an e ven t h o r i z o n , •
27
SbtSnfieci' by a method which should be usef ul in o t h e r com pli cat ed t h e o r i e s
o f g r a v i t y as w e l l .
The e v e n t hor iz on around a s t a t i c s p h e r i c a l l y symmetric source i s
a s u r f a c e r - r = c o n s t on which gnn = 0. Close to t h e horizon
r - f.
0
ou
- < I and we can expand t h e f i e l d s in a power s e r i e s .
Soo B ^
r.-.r
Oxn‘ (1 T ^ ) 5 Ca + Zan C - V - ^ ) " ]
n=l "
o
o
. r - r
S r r ** "S
Kg % ^
+
o
^
O
oo
r - r
Oxn
i b t—
)"]
n=l
o
(IV.6 )
.r - r
Ovn
^ C^C
)"]■
n=l1
Requ irin g t h a t s > O g u a ra n te e s t h a t r = r 0 i s indeed an e ven t ho ri z o n .
S u f f i c i e n t l y c l o s e to t h e h o r i z o n , t h e m e t r i c and v e c t o r f i e l d s may
be approximated by t h e i r le a d i n g terms a l o n e .
When t h i s approximation
i i S u b s t i t u t e d i n t o e q u a t i o n s IV .5, IV.4 , and IV.3, th e y become,
respectively
X(u - S)[ (w = D x w ^ + 2xw i*] -
x l ( s
+ t ) + 2ujxw( b x u - I
c
=O
‘
( I V . 7)
(U + i x s ) [ ( w - i ) x w“ 2 + Zxw^l J - \ x p (bxt - I - | )
- Zwxw(Pxt - I ~
- O
(IV.8 )
28
Axw-2
'
(u - -gs) +
Xw - 2 U 2 - - ^ r x P Cbxt -
I
-
(V.9)
+ o)XWCbxt - I " I - 4^ ) = 0
where we have de fi n e d
.r -. r
x =
o.
and
w = 2u - -^s -
¥
A = 2ii> + n.
The d e t a i l s o f t h e s o l u t i o n f o r s , t , and u a r e given in Appendix
B.
The r e s u l t i s
r - r
9oo “ a - T
'rr
— -j r ,
l-[I" + T4 Ta - 'C2u
- + n + 4' ') J■'r
r - r,
wit h a , c , and r Q s t i l l
arbitrary.
The a r b i t r a r i n e s s o f a stems from
t h e freedom to r e d e f i n e time anyway we choose,
c j u s t r e p r e s e n t s th e
freedom in t h e cosmological va lu e o f t h e v e c t o r f i e l d .
Si nc e th e m e t r i c has t h e same dependence on r - Tq as does the
Sw a rz sc h il d m e t r i c , t h e b e h a v io r n e a r t h e horizo n i s s i m i l a r to t h a t
o f gener al r e l a t i v i t y .
In p a r t i c u l a r , t h e s i n g u l a r i t y in gr f a t r =
i s on ly a c o o r d i n a t e s i n g u l a r i t y , t h e p hys ic a l components o f th e
Riemann t e n s o r remaining r e g u l a r a c r o s s t h e boundary.
CHAPTER V '
RADIATION
A . ' L i n e a r i z e d G r a v i t a t i o n a l Waves.
We now c o n s i d e r t h e pro pag at io n o f
weak g r a v i t a t i o n a l waves through a re gi on o f empty s pa c e .
The f i e l d s
are w ritten
uv
+ A
where nyv and ^
f i e l d s and h
a r e t h e c o n s t a n t cosmological background v a lu es o f t h e
and Ay a r e t h e l o c a l wave p e r t u r b a t i o n s .
By "weak"
wav es, i t i s meant t h a t , o ve r t h e re g io n o f space to be c o n s i d e r e d ,
h" «
uv
n
UV
and Ay << <f>y .
Minkowski m e t r i c .
Coordina tes a r e chosen so t h a t n
In a l l t h a t fo ll ows
.
UV
i s the
we c o n s i d e r pla ne waves, w r i t t e n
ik xy
h
= e e y
uv
uv
i k xy
Ay = aye
y
.
k^ i s a c o n s t a n t p ro p a g a ti o n f o u r - v e c t o r which w ill be taken t o r e p r e ­
s e n t waves moving in t h e z - d i r e c t i on, i . e . , k^ = ( w , 0 , 0 , - k ) .
The speed
o f pr op a g a ti o n o f t h e r a d i a t i o n is
and may o r may no t be equal to th e speed o f l i g h t (I in t h e u n i t s we
have c h o s e n ) .
31
S u b s t i t u t i n g t h e s e e x p r e s s i o n s f o r t h e f i e l d s i n t o th e f i e l d equa­
t i o n s . and d i s c a r d i n g terms second o r d e r o r h i g h e r ( i . e . , terms l i k e
2
2
A 5 Ah, h )
produces t h e f o ll o w i n g l i n e a r i z e d e q u a t i o n s ( d e t a i l s o f
t h e d e r i v a t i o n a r e given in Appendix C).
The g ( c o n t r a c t e d ) e q u a ti o n (e q u a t i o n I I . 3) i s
R+
Y ag
+ 3C?P - 2n#a F-6 .
a jg
where P = (2U'+ n)(<i>a Aa ease.
= 0
ha g ) has been d e fi n e d for- r o t a t i o n a l
The K e q u a ti o n ( e q u a ti o n I I . 4) i s
yUKfr R + Tllfra R
p •
The g
pv
+ 2F a
pa
p ia
e q u a ti o n ( I I . 2) i s s i m p l i f i e d by adding combinations o f o t h e r
equations to i t ;
RPV O +
+ V
CDY
lfr
.
)
“ Fc,6 ,H
+
pavg.
TllfrctIfr^R
T Y
.
- uxfrYpT
ifrvR + i^g
+ pC
V
,
pvc f P
,
+ P ,p v
+ Z ) ( \ V , = -" V y a ,=.1
= 0.
There a r e a t o t a l o f f o u r t e e n f u n c t i o n s which a p p ea r in t h e s e equa­
t i o n s , te n components o f hyv and f o u r o f K .
s c a l a r P, h
However, e xce pt f o r t h e
appea rs only in th e components o f t h e Riemann t e n s o r
PV
which only s i x a r e in d e p e n d e n t, and A appea rs only in F
UV
5Ct
of
which only
32
has t h r e e ind ep en den t components.
T h e r e f o r e , a c a r e f u l check o f any
s o l u t i o n i s n e c e s s a r y t o be s u r e t h a t i t i s c o n s i s t e n t with a l l f o u r ­
t e e n ind ep en den t e q u a t i o n s .
As was mentioned, c o o r d i n a t e s were chosen so t h a t t h e background
m e t r i c i s l o c a l l y Minkowskiian and th e z - a x i s i s along t h e d i r e c t i o n o f
p r o p a g a ti o n o f t h e wave.
There s t i l l
remains t h e freedom o f i n e r t i a l
frame which we choose t o be th e r e s t frame o f th e u n i v e r s e ( i ^ e . , t h a t
frame in which ^
components o f R
= 0).
yavg
The v a n is h i n g o r non-va nis hin g o f t h e va ri ou s
and A w i l l o f c o u r s e depend on t h i s c h o ic e of
v
Lorentz frame, b u t , once t h e l i n e a r i z e d s o l u t i o n s have been worked out
i n t h i s frame, they may be found in any frame by Lorentz t r a n s f o r m a t i o n s
s i n c e we a r e working in a f l a t background m e t r i c .
B.
Results.
The s o l u t i o n s a r e worked o u t in d e t a i l in Appendix C.
Here we simply p r e s e n t t h e r e s u l t s w i t h i n a c l a s s i f i c a t i o n scheme worked
o u t by Douglas E a r d l ey, e t aj_.
In t h i s scheme, th e s i x independent
components o f t h e Riemann t e n s o r a r e combined i n t o f o u r Newman-Penrose
f u n c t i o n s , two of which a r e complex.
The reaso n f o r t h i s reco mbination
i s t h a t each o f t h e s e f u n c t i o n s a f f e c t s a g r a v i t a t i o n a l wave antenna
in a c h a r a c t e r i s t i c way.
The r e a d e r i s r e f e r r e d to th e work of
^D. E a r d l ey, et_ al_., Phys. Rev. L e t t e r s ,30, 884 (1973).
33
E a r d l ey, ert
for d e ta ils.
The f o u r f u n c t i o n s a r e (where t h e 3-
a x i s i s ta ken al ong th e d i r e c t i o n o f p ro p a g a ti o n )
^2 = " ^R0303
^3 = t v/-R0103 + 1R0203^
*4 = R0101 ' R0202 + 2 i r OI02 '
^22 “ ~ (R0101 + R0202^
Besides a f f e c t i n g an a n te n n a , t h e pass ag e o f r a d i a t i o n could a l s o
I
■ .
a f f e c t t h e g r a v i t a t i o n a l c o n s t a n t in PPN-type e x p e r i m e n t s - ( s e e Chapter
III).
The e f f e c t i s p r o p o r t i o n a l to
K //"
= (4% +
+ h" ") ..
In t h e u n i v e r s e r e s t frame t h i s becomes .
W
uv - V
+ 2»dAo - * o \ o
-.
.
where t h e f a c t t h a t h00 = -h„„ has been used.
oo
We d e s c r i b e t h e s e p a r a t e modes in Tables I and TI..
34
TABLE I .
n = 0; f i v e ind ep end en t modes.
Independent
Mode
Speed ( c 2 =
)
Other F i e l d s
C o r r e c ti o n to
Ku ff-
1
None
Reifi^
I
None
None
Imifi4
I
None
None
I
None
None
I
None
None
*22
A1
A2
Reifl3 = Imiflg = Iflg = O
At t h i s p o i n t i t i s r e c a l l e d t h a t t h e s o l u t i o n has been found only
in a p a r t i c u l a r Lorentz frame.
E a r d l e y 5 e t al_., have shown, however,
t h a t i f ifig and ifg a r e nu ll waves (c = I ) and i f Ifi3 = Ifig = O in any
frame, then t h e y a r e zero in a l l f r a m e s .
35
In t h e most gener al case (n f 0) i t i s found t h a t a l l f o u r o f the
N e wa n-P enr ose f u n c t i o n s a r e p r e s e n t , though they t r a v e l a t d i f f e r e n t
s p ee ds .
TABLE I I . n
^ 0 ; f i v e indepen de nt modes.
Independent
Mode
Re*,
Speed (c
I -
=
)
______ Tiy
Other F ie ld s
C orr e c ti o n
to Ku Ktj
9
None
None
n*
2
2
I + co* + n*
None
None
'2
'
I + co* •■+ n*
Im*,
I -
Re*,
n2*2
I + I
4 I + CO*2 + n*2
e^ aI =
None
Im*,
n 2*2
I + I
C^ a 2 =
4 II +
x CO
A2
* X
+ n*A2
None
I + I n*2 + n 2*2
3 I + co*2 + n*2
pi + CO* + Tl*
2co + n
9
*2 = r - l !*22
1 -c
X ( - c 2k2 ) * „
a =1 ^ 2 2
*F0g a i s used i n s t e a d o f Ag o r A3 to avoid p o s s i b l e non-ph ys ic al
" c o o r d i n a t e r i p p l e " waves.
See Appendix D.
9
.36
E a r d l e y 5 e t al_., have c l a s s i f i e d t h e o r i e s o f g r a v i t y acc ord in g to
which o f t h e metric-wave f u n c t i o n s v a n i s h , as f o ll o w s :
Class H g .
*2 Z 0.
All o b s e r v e r s in such Lorentz frames measure
a . n o n - z e r o am plit ud e in t h e ^
a m p li tu d e .
mode
and a g r e e on t h e va lu e o f t h i s
(But th e y w i l l g e n e r a l l y d i s a g r e e about t h e pre s e n c e or
absence and a m plit ud e o f a l l o t h e r modes.)
Class I I I 5 .
- O^
1^2 and th e pre s e n c e o f
p r e s e n c e o r absence o f ^
Class Mg.
All o b s e r v e r s a gre e on t h e absence of
(But th e y g e n e r a l l y d i s a g r e e about th e ■
and
)
E 0 - ^3 ’ ^4 ^ 0 ^ ^22’
'observers ag re e about
t h e pr e s e n c e o r absence o f a l l modes.
Class N2 -
^2 = O H
Class O1 ?
Ip0 = 0 =
^ O E ^ 2-
All o b s e r v e r s a g r e e .
^ e 0 ^ $99.
"3' :4
All o b s e r v e r s a g re e .
The v e r s i o n o f th e th e o r y with n = 0 i s t h e r e f o r e o f c l a s s N g 5
w hil e t h e g en eral n / 0 v e r s i o n i s o f c l a s s I I g .
In a d d i t i o n th e
n Tt 0 ca s e has g r a v i t a t i o n a l waves t r a v e l i n g a t speeds g r e a t e r or l e s s
than t h e speed o f l i g h t , depending on t h e va lu es o f w and nspeed i s i n t e r e s t i n g - but not sh ocking.
additional discussion.
The slower
The f a s t e r speed r e q u i r e s some
.
37
C.
C ausality.
The usual argument a g a i n s t th e e x i s t e n c e o f r e a l f a s t e r -
t h a n - ' l i g h t s i g n a l s i s t h a t such s i g n a l s would v i o l a t e c a u s a l i t y .
I f by
c a u s a l i t y i t i s meant t h a t causes must precede t h e i r e f f e c t s , as seen
by a l l o b s e r v e r s , then c a u s a l i t y i s c e r t a i n l y v i o l a t e d , bu t t h a t i s an
a r b i t r a r y and u nnece ssa ry co ncept o f c a u s a l i t y .
I f by c a u s a l i t y i t is
simply meant t h a t l o g i c a l paradoxes ( i . e . , A both sends and does not
send a s ig n a l to B) may not o c c u r , then c a u s a l i t y in t h i s sense is not
v i o l a t e d by t h e f a s t e r - t h a n - 1i g h t s i g n a l s o f t h i s t h e o r y .
In th e r e s t frame o f th e u n i v e r s e , t h e r e a r e t h r e e " g r a v i t y cones"
in a d d i t i o n to t h e l i g h t cone, c o rr e sp ondin g to th e t h r e e d i f f e r e n t
speeds o f p r o p a g a ti o n o f th e I I ^ waves.
l i g h t cone
t y p i c a l g r a v i t y cone
As long as t h e r e e x i s t s a maximum speed o f p r o p a g a t i o n , i t is p h y s i c a l Iy
i m p o s si b le to produce c lo s e d causal loops in s p a c e - t i m e , so i t is
im po s si ble f o r any e x p e r im e n t e r to i n f l u e n c e his p a s t and produce
causal p a r a d o x e s .
CHAPTER VI
COSMOLOGY
To i n v e s t i g a t e cosmological s o l u t i o n s o f th e t h e o r y , we use a
homogeneous d u s t model f o r m a t t e r and a Robertson-Walker cosmological
m etric.
s OO “ 1
S(t)‘
9Ij
( l + f r 2)2 y
from which we can c a l c u l a t e
R00 " 3S
R = - U s s + S2 + k ) ,
S
t h e do t d e not in g time d i f f e r e n t i a t i o n .
Kq E
In t h e u n i v e r s e r e s t frame
0
kI =
and we can w r i t e down t h e
(KotKe )
- <}> + 3 ^ +
•*
*>j
n f (J) = (f, + 3 (J)g-
*100 = *
derivatives.
+ 6 <j) ^
39
£0 (KqKq) - "(#) + 3t}> g- - 6<j)— 2
;2
^ “ ^0 ^ io = *
* 3*
3
S u b s t i t u t i n g t h e s e i n t o e q u a ti o n I I . 4, give s
V
Q2 4.
(2w + n ) | + Zixi 9 K' = 0.
( V I . I)
And in e q u a ti o n I I . 2, t h e r e r e s u l t s
VS2 +
^^
(I - UHj))-—
K
.S2
- (2m + n ) ^ + (2m + n)X<j>§ “
^
0
^
T .
O OO
Using V I . I to e l i m i n a t e second d e r i v a t i v e s o f S reduces t h i s to
'2
+ K
.S2
(I + mj))— -y - + n<i>-^2 + (2m + n)^i>|- = ^ G qTc
The
( V I . 2)
and gy^ e q u a t i o n s v a nis h i d e n t i c a l l y , and the g ^ eq u at i o n is
c o n s i s t e n t with V I . I and V I . 2.
Equation VI. I i n t e g r a t e s to give
S2 + K = (cp-)P
where S p i s a c o n s t a n t o f i n t e g r a t i o n ,
2m
p E ------ » and we d e f i n e
m + gH
q
e
—
r
m + Tpl
f o r f u t u r e use. ■
( V I . 3)
40
Equation V I. 3 i n t e g r a t e s a g a i n , y i e l d i n g
b
'
t - ■ 5 ds
I / S p - S p
( V I . 4)
.0
The ti m e development o f S depends only on t h e s c a l e c o n s t a n t Sq and on
t h e pa ra m et e rs w arid n-
I t i s indepen de nt o f T ^ .
Equation VI.2 i s made f i r s t o r d e r in
•
I
V I . 3 to e l i m i n a t e S and d i v i d i n g by to +
■
+ 4
+
S
S
where we have d e fi n e d x = ■=—.
as f o l l o w s .
Using eq uat io n
eq u at i o n VI.2 becomes
- T t L i r 6 OT0= - £ P
to + -^n''
S Xr
2
M u lt ip ly in g by S
)
and d e f i n i n g
0+1
2G0M . 8 tgOt OO5'
35O
produces t h e f i r s t o r d e r eq u at i o n
3.
)
tqiT
1V
w it h s o l u t i o n
I
-
KXP
- % dx +
iq+1
(to + •^n)x^
( V I . 5)
■Jq+l
(I - KX^
41
where x i s a c o n s t a n t o f i n t e g r a t i o n .
The s p e c i a l c a s e n = 0 , m = I i s o f p a r t i c u l a r i n t e r e s t .
I t i s the
c a s e Which e x a c t l y re pro duces t h e PPM pa rame te rs o f g en eral v^ e i a t i v i t y .
I t w i l l here be shown to have a cosmological s o l u t i o n d i f f e r e n t from
t h e cosmology o f g en eral r e l a t i v i t y .
Also t h i s ca s e r e p r e s e n t s an
approximate s o l u t i o n f o r one o f t h e c a s e s allowed by e q u a ti o n I I I . 9,
namely n < . 1 , w = I .
Choosing th e k = I o f a c lo s e d cosmology and
not in g p = 2, q = 0, e q u a ti o n VI.4 i n t e g r a t e s to give
S = /t(2S
- t) ,
o
and e q u a ti o n V I . 5 y i e l d s
T + * = X"
X --
I f we demand 4 ->■ 0 as t
Dicke^) , then f o r <j> »
Assuming Sq »
;
0 ( t h e i n i t i t a l cosmological s i n g u l a r i t y , see
I ( e q u i v a l e n t to Gq >> G),
S 5 t h e s e become
S ~ /2S0t
(VI.6)
1C. Brans and R. H. Dick e5 Phys. Rev. 124, 425 (1961 ).
V I r ’«
42
The r e l a t i o n s h i p between Hubble time (T^) and the p r e s e n t age of
the universe t^ is
M
M
-
S
u
Writing
H 5 ^v
= I m s 0Th ) 3/ 2 .
some s u b s t i t u t i o n s giv e 4— in terms o f o b s e r v a b l e s ,
bo
.
(J) _ 4 tt T 2
( T " F 131H *
Using t h i s in e q u a ti o n I I I . 4 gi ve s
■g- -
pnT|_j^ ** IO4
(e.g.s units)
based on p = I 0" 31 g/cm3 and
= 2 x IO10 y e a r s .
( V I . 7)
The a c t u a l value o f
I i s about IO7 , but t h e r e i s s u b s t a n t i a l u n c e r t a i n t y in t h e t o t a l energy
fa
d e n s i t y (p) o f t h e u n i v e r s e .
From eq u at i o n VI.6 th e e a r l y - t i m e beh avi or
of the universe is
S
as compared with general r e l a t i v i t y ' s
S - t 2/3
(VI.8)
43
A second l i m i t i n g cosmology, c om pat ib le with t h e second case allowed
I
by e q u a ti o n I I I . 9, i s w ~ Tj-n » 1 .
Then p = q = I , and f o r k = I we f i n d
-S- = s i n ~ ^ / x - fx - x 2"
5o
. _ J u - 1) / T ^ r
* " 2
"X
I
(/ r
sin"^ ZxT
)
/X
where we have a g a in assume S -» 0, <j)
0 as t -»■ 0.
The e a r l y time
b e ha vio r i s
t =
3/2
( V I , 9)
( V I . 10)
and
O
IS11S0 2 ■
From e q u a ti o n V I . 9, th e age o f t h e u n i v e r s e i s r e l a t e d to t h e Hubble
age by
Then M can be w r i t t e n
44
M = T p s O1 H *
and ^)/G
becomes
4 _'32 t , 2 ,TH;2/3
% - W 1H t S ^
•
F i n a l l y , using t h i s in e q u a ti o n I I I . 4 giv e s
H
^
if Y S o
t„2£
.i.
(V L ll)
) 2/3 - IO4
.
A f i n a l c h o ic e o f pa rame te rs c om pat ibl e with Chapter I l l ' s case II
i s to = 0.
In t h i s c a s e , e q u a ti o n V I . I re duces to
n f = 0,
having s o l u t i o n
S = vt
where v i s a c o n s t a n t .
Equation VI .2 becomes
S2 + K .
S2
I ;S _ S ttg ,
12
+ ^ 7 2 + ZnijlS " T V o o
which has s o l u t i o n
» t -2 ,
nv
v2 +
nv
nv2
K
45
and i s s i n g u l a r a t t h e o r i g i n .
T h e r e f o r e h e r e , a s in Chapter I I I , case I I i s seen to be much l e s s
i n t e r e s t i n g than ca se I .
The most i n t e r e s t i n g p r e d i c t i o n s in t h e ca se I
s o l u t i o n s a r e t h e e a r l y time be hav io r ( e q u a ti o n s VI. 8 and V I . 10) and th e
f u n c t i o n a l form o f t h e v a r i a t i o n o f G with t h e e v o l u t i o n o f th e u n i v e r s e
( e q u a t i o n s V I . 7 and V I . 11 ).
The former i s i m p o r ta n t in for mati on of
t h e elements duri ng t h e big bang;
2
t h e l a t t e r det ermi ne s t h e h i s t o r y
o f t h e g r a v i t a t i o n a l c o n s t a n t in t h e u n i v e r s e .
^See f o r example G. Gamow, Revs. Modern Rhys. 2J_, 367 (1 949).
CHAPTER VII
CONCLUSIONS
A.
Comparison o f Theory and Experim ent.
In t h i s work we have made
s e v e r a l p r e d i c t i o n s o f o b s e r v a b l e r e s u l t s o f th e v e c t o r - m e t r i c t h e o r y .
■We would l i k e to review t h e s e here and make some comments on the
e m p ir ic a l s t a t u s o f each r e s u l t .
A.good summary o f experimental
g r a v i t y i s found in Kip T h o r n e ' s Le c tu re to th e I n t e r n a t i o n a l Union o f
Pure and Applied Ph ys ic s in 1972J
I.
PPN Parameter y .
The most p r e c i s e t e s t o f y to d a t e has been
t h e time d e la y ex periment usi ng a c t i v e r a d a r to Mariners VI and VII.
2
Anderson, ejt al_., f i n d a r e s u l t o f
1(1
+ Y)
=
1.0 0 + .04,
or
Y
- 1 . 0 0 + .08. .
We chose to c o n s i d e r w = -g- + I so as to p r e d i c t a va lu e of
Y = I1
S. Thorne, A v a i l a b l e as Cal Tech P r e p r i n t 0AP-321 (1 972).
2
John D. Anderson, et^ al_., Proceedings of the Conference on E xpe rimental T e s t s o f G r a v i t a t i o n a l T he or ie s ( NASA-JPL Technical Memo 33-499,
TgToy:
47
in agreement with: ex per im ent .
2.
PPN Parameter g .
The v e c t o r - m e t r i c th e o ry p r e d u c t s a val ue
B = I ..
The b e s t ex perimental d e t e r m i n a t i o n o f B comes from t h e r e l a t i v i s t i c
3
p e r i h e l i o n s h i f t o f Mercury, which i s known to be
6ft = 43 + I s ecs . / c e n t u r y = 4 3 ( 1 .0 0 +_ .02)
When 5ft i s determined t h e o r e t i c a l l y i t i s found t h a t i t i s a c t u a l l y
p r o p o r t i o n a l to B, y , and a l l t h e a - p a r a m e t e r s , but i t can be shown,
by comparison with t h e e a r t h ' s p e r i h e l i o n s h i f t , t h a t t h e a dependence
i s small and t h e d e f l e c t i o n i s b a s i c a l l y j u s t
6ft =
2y + 2 - B) •
The l i m i t i n g f a c t o r on t h e a cc ura cy i s seen to be th e a ccu racy of y .
T h e re fo re B can be no more a c c u r a t e than y:
B = 1 . 0 0 + .08.
The v e c t o r - m e t r i c th e o r y i s in agreement with the ex perimental b ••
^C. M. W i l l , Le c tu re s in B. B e r t o t t i , e d . , Proceedings, o f Course 56
o f t h e I n t e r n a t i o n a l School o f Physics "Enrico Fermi", Academic Press
(1973), p. 129.
;
:
48
3.
PPN Parameters a ...
The: v e c t o r - m e t r i c t h e o r y p r e d i c t s values
f o r a., o f
^
4 < ( ) ^ + 4 + 6ri + o^
___________ 4n________ _
^
4 < j ) ^ + 4 + 6ri + ri^
«3 =
A combination o f e a r t h t i d e , e a r t h r o t a t i o n r a t e , and p e r i h e l i o n s h i f t
data
4
g i v e exper imen ta l l i m i t s of
Ct-j ^ • I
«2 < • I
The p r e d i c t e d a . para mete r i s well w i t h i n experimental l i m i t s . .
r e q u i r e d sm al ln es s o f t h e a-j and
n <■ - I
pa rame te rs can only be met i f
•'
:y
or
^ I b i d . , p. 146 and K. N o r d t v e d t , J r . , Science 178, 1157 (1972).
The
49
n £ 34
T h e r e f o r e , a v e r s i o n o f t h e th e o r y .in which n s a t i s f i e s t h e s e r e s t r i c ­
t i o n s w i l l p r e d i c t a - p a r a m e t e r s in agreement with ex per im en t.
4.
PPN Parameters y .
zero f o r both ^
a r e co nse rv e d.
and
The v e c t o r - m e t r i c th e o ry p r e d i c t s val ues o f
as do a l l t h e o r i e s in which energy and momentum
The b e s t experimental t e s t o f t h e s e pa rame te rs i s th e
Lunar Laser Ranging e xpe ri me nt , whose r e s u l t s a r e not y e t i n .
5
The
p r e d i c t e d change in l u n a r range i s given by
SS = 20(4B - 3 - y - ot-j +
~
i - g-tg) m e t e r s ,
and th e expected a cc u ra cy o f t h e ex periment should be a bout 0.1 m e t e r s .
A non-zero v a lu e of ^
r e t u r n e d by t h i s experiment would c e r t a i n l y
d is p r o v e t h e v e c t o r - m e t r i c t h e o r y , but i t would a l s o d is p r o v e general
r e l a t i v i t y , Brans-Dicke t h e o r y , arid any o t h e r c o n s e r v a t i v e t h e o ry .
5.
Black H o le s .
The v e c t o r - m e t r i c th e o r y p r e d i c t s t h e e x i s t e n c e
o f b la ck ho le s as do g en eral r e l a t i v i t y and many o t h e r t h e o r i e s .
The
p r e c i s e d e t a i l s o f c o l l a p s e and a n a l y s i s o f the i m p l i c a t i o n s o f p o s s i ­
b l e n o n - s p h e r i c a l e v e n t horizo ns (s e e page 24) might have ob s er v ab le
5
N ord tv e d t, op c i t .
50
a s t r o p h y s i c a l p r o p e r t i e s but t h a t work has not y e t been done.
As f o r
t h e e x i s t e n c e o f bla ck h o l e s , o b s e r v a t i o n a l evidence i s s t i l l very
tentative.
6.
Speed o f G ra v it y Waves.
One form (n = 0) o f t h e v e c t o r - m e t r i c
t h e o r y p r e d i c t s t h a t g r a v i t y waves w i l l t r a v e l a t the speed o f l i g h t .
The o t h e r form (n f 0) p r e d i c t s as many as t h r e e s e p a r a t e speeds which
may be g r e a t e r o r l e s s than t h e speed o f l i g h t , depending on th e v a lu es
I
o f w.and n. I f oi = Tjn + I and i f t h e background s t r e n g t h o f th e v e c t o r
f i e l d i s ta ken to be l a r g e compared to one, th e speeds a r e
2q
Sn + 2
I +
C '
2
1
n
2
2 Sn + 2
2 n + n2
3 Sn + 2 *
No ex periment has been done to measure t h e speed o f pro p a g a ti o n of
gravity.
What i s needed i s a d d i t i o n a l re fi n e m e n t o f g r a v i t y wave
d e t e c t o r s to g iv e us more c o n fi d e n c e in them and to en abl e us to d e t e c t
t h e incoming d i r e c t i o n with more a c c u r a c y .
Then th e time o f a r r i v a l of
g r a v i t y waves and e l e c t r o m a g n e t i c waves c r e a t e d by a supernova or o t h e r
d i s c r e t e e ven t can be compared to de ter mi ne t h e speed.
Eardley5 et a I
^D. E a r d l ey, e t a K , Ph ys.. Rev. L e t t e r s 30, 884 (1 973)..
51
have c o n s id e r e d t h i s typ e o f ex periment and e xpect t h a t i t could have
an a c c u ra c y o f - K f 9 x
7,
Modes o f G ra v it y Waves.
.
The two forms o f t h e v e c t o r - m e t r i c
t h e o r y make d i f f e r e n t p r e d i c t i o n s a bout t h e kinds o f p o l a r i z a t i o n s
present.
$22
Wg . ,
The n = 0 v e r s i o n has
»
present;
Wg not p r e s e n t .
The n Tf 0 v e r s i o n has a l l f o u r f u n c t i o n s (both r e a l and complex p a r t s )
present.
The only d e t e c t i o n o f g r a v i t a t i o n a l r a d i a t i o n to d a t e has
been t h a t o f Weber^ whose antenna i s not s e n s i t i v e t o t h e p o l a r i z a t i o n
o f t h e wave which i s e x c i t i n g t h e a p p a r a t u s .
However,' he has used a
s p e c i a l d e t e c t o r , a d i s k , which should be p r e f e r e n t i a l l y s e n s i t i v e to
$22 type waves and he has seen no r a d i a t i o n in t h i s mode. I t should
O
a l s o be p o in te d o u t t h a t j u s t r e c e n t l y Tyson has p u b li s h e d r e s u l t s
d e s c r i b i n g his o b s e r v a t i o n s with a d e t e c t o r about 10
times more sens!
t i v e than Weber's o r i g i n a l a n t e n n a , and he has seen no r a d i a t i o n in
any mode.
The whole q u e s t i o n . o f d e t e c t i o n o f g r a v i t y waves i s y e t to *7
7
J . Weber in B. B e r t o t t i , e d . , Proceedings of Course 56 o f the
I n t e r n a t i o n a l School o f Physics "Enrico Fermi11, Academic P ress (1973).
^ J . A. Tyson, Phys. Rev. L e t t e r s 31_, 261 (1973).
52
be s e t t l e d .
When i t i s , i t should prove to be a powerful tool f o r
te stin g gravity th eo ries.
8.
Re no rm al iz a tio n o f G.
When n = 0 t h e r e i s no r e n o r m a l i z a t i o n
o f t h e g r a v i t a t i o n a l c o n s t a n t (s e e e q u a ti o n I I I . 4 ) , but n t5 0 allows us
t o c a l c u l a t e G from some b a s i c cosmological d a t a .
When n i s small but
n o t z e r o , t h e g r a v i t a t i o n a l c o n s t a n t i s given by
I
G
2
n.
When n i s l a r g e , i t i s
I _ 32 tt v 2 / TpK 2/3
G " GFp lH
Th i s th e Hubble time which i s c a l c u l a t e d from cosmological o b s e r v a t i o n s
g
o f r e d s h i f t v e rs u s d i s t a n c e . I t nas changed over t h e y e a r s as a s t r o n ­
omers have r e - e v a l u a t e d such t h i n g s as t h e p e r i o d - l u m i n o s i t y r a t i o f o r
Gepheid v a r i a b l e s , but i t appea rs to have s t a b i l i z e d a t ab out 2 x I O1
years.
The d e n s i t y , p , i s a l s o t i e d up with t h e d e t e r m i n a t i o n o f the
Hubble t im e , s i n c e we use H ubble's c o n s t a n t (H =
J-)
to deter mine t h e
1H
d i s t a n c e to g a l a x i e s whose mass we a r e c ountin g in o r d e r to a r r i v e a t
a mean d e n s i t y .
The p r e s e n t e s t i m a t e o f p is- p = 1 0
“ 31
3
.g/cm , based 9
9See f o r example E. L. Schutzman, The S t r u c t u r e o f {t h e U n iv e r s e ,
McGraw-Hill (1968), p. .12.
53
on t h e c o un tin g o f g a l a x i e s .
I f t h e s e v a lu es a r e s u b s t i t u t e d i n t o t h e
■]
expressions fo r
there re su lts
I
G
rn
n «
I
J T h/ S 0,)2/3
n » l .
IO5 x
The a c t u a l va lu e o f
1
7
i s 10
in c . g . s u n i t s .
.
At t h i s p o i n t , however,
i t i s not known how much energy d e n s i t y might be p r e s e n t in th e u n i v e r s e
in terms o f g a s , d u s t , or o t h e r forms o f en ergy.
Therefore, observa­
t i o n a l cosmology i s . n o t in a p o s i t i o n a t p r e s e n t to e i t h e r prove or
disprove the theory.
9.
Time V a r i a t i o n o f G.
I f 9 i s l a r g e , we have t h e r e l a t i o n
G
I .
Tjri(J)
T h e re fo re |j- i s given by
SG = .--^=
- -I=
<p
o
-H = -IQ' 10 y e a r s 1 •
s i n c e in each ca s e * = S.
Dicke has shown10 t h a t such a v a r i a t i o n i s
no t i n c o n s i s t e n t with g e o p h y s i c a l , p l a n e t a r y , and a s t r o p h y s i c a l d a t a .
10R. H. Dicke and P . J . E . P e e b l e s , Space Science Reviews 4_, 419
(1965).
54
B.
Fut ur e Ex p e ri m en ts .
We have d e s c r i b e d nine p r e d i c t i o n s o f the
v e c t o r - m e t r i c t h e o r y , some o f which a r e well w it h in well e s t a b l i s h e d
exper imen ta l r e s u l t s , some o f which may well be proven wrong by f u t u r e
e x p e r im e n t s .
Here we w i l l b r i e f l y d e s c r i b e what we f e e l i s the most
promising way to co m p le te ly e l i m i n a t e t h e v e c t o r - m e t r i c th e o r y using
f u t u r e exp eriment s and o b s e r v a t i o n s .
We t r e a t th e n ^ 0 and t h e n = 0
c ase s s e p a r a t e l y .
I. n
7* 0 .
As was seen in number 3 above, p r e s e n t r e s u l t s f o r ct-j
and Ct2 e f f e c t s a l r e a d y l i m i t n to
n < •I
or
n > 34.
G r e a t e r ac c ura cy in t h e s e kinds o f Machian e f f e c t experiments or new
ty p e s o f more s e n s i t i v e experiments could f u r t h e r r e s t r i c t n .
No
m a t t e r how a c c u r a t e l y t h e s e experiments a r e done, however, th e n
•parameter can always s l i d e away in e i t h e r t h e very small or th e very
l a r g e d i r e c t i o n to pro v id e agreement with ex periment.
What i s needed
a r e two a d d i t i o n a l o b s e r v a t i o n s which n a i l th e th e o r y down--one f o r
each d i r e c t i o n .
55.
The b e s t c a n d i d a t e to s t o p t h e n . . « I s l i d e i s an o b s e r v a t i o n a l
■J
e s t i m a t e o f t h e energy d e n s i t y in t h e u n i v e r s e . ^ i s p r o p o r t i o n a l to n ,
I _ 2^ T 2
G " T -p lH n '
The Hubble time (Ty) i s f a i r l y well known.
I f astonomers can s e t a
maximum o r d e r o f magnitude v a lu e on t h e energy d e n s i t y o f th e u n iv e r s e
(p) then t h e observed g r a v i t a t i o n a l c o n s t a n t
~ !Cr) can provid e a
lo w e st a c c e p t a b l e v a l u e f o r n •
In t h e n ^ 0 r a d i a t i o n s o l u t i o n s (s e e number 6 above) we found t h a t
t h e speed o f p ro p a g a ti o n o f some o f th e waves was p r o p o r t i o n a l to u ,
I
I + ?
2 3n + 2
2
I
When n >> I j t h i s reduces to C = I +
.
I f c u r r e n t work in g r a v i t a ­
t i o n a l r a d i a t i o n d e t e c t i o n c o n t i n u e s , one might e xpec t to observe the
l i g h t and t h e g r a v i t y waves from a supernova o r o t h e r l a r g e d i s c r e t e
e ve nt and compare t h e i r a r r i v a l times to s e t a l i m i t on t h e amount by
which t h e speed o f p ro p a g a ti o n o f t h e g r a v i t y wave d i f f e r s from the
speed o f l i g h t .
This can s e t a maximum va lu e on n-
T h e r e f o r e a combination of Machian e f f e c t experiments and energy
d e n s i t y o b s e r v a t i o n s , and Machian e f f e c t experiments and g r a v i t y wave
p ro p a g a ti o n speed experiments s ta n d s a good chance o f showing the
n Ti 0 v e r s i o n o f t h e v e c t o r - m e t r i c th e o r y to be i n c o r r e c t .
56
2.
n - 0.
The exper iment al d i s p r o o f o f t h i s form o f th e th e o ry
w i l l be much h a r d e r . ' I t s PPN m e t r i c i s e x a c t l y th e same as general
re la tiv ity 's.
I t s g r a v i t y waves t r a v e l a t th e speed o f l i g h t .
i s no G r e n o r m a l i z a t i o n to be t e s t e d .
There
The only r e a l o p p o r t u n i t y to
ne g a te t h i s th e o r y duri ng t h e ne xt decade o r so i s in t h e p o l a r i z a t i o n
of gravitational ra d ia tio n .
General r e l a t i v i t y p r e d i c t s t h a t m a t t e r
g e n e r a t e s and t h a t space pro p a g a te s ^ - p o l a r i z e d waves. . We have seen
t h a t t h e r> = 0 v e c t o r - m e t r i c t h e o r y p r e d i c t s th e p ro p a g a ti o n o f both
and $p2 waves.
I t i s c r u c i a l here t o s o lv e t h e problem o f g e n e r a ­
t i o n o f g r a v i t a t i o n a l r a d i a t i o n in t h e v e c t o r - m e t r i c th e o r y to see i f
these scalar-ty p e
waves a r e c r e a t e d by g r a v i t a t i o n a l even ts or i f
t h e y a r e in some way i n h i b i t e d .
show t h a t
I f t h e c r e a t i o n problem i s solved to
waves should indeed e x i s t , then th e de m o n st ra t io n of
t h e i r n o n - e x i s t e n c e (v ia d i s k - t y p e antenna d e t e c t o r s ) would e l i m i n a t e
t h i s t h e o r y along with s e v e r a l o t h e r s and would g r e a t l y i n c r e a s e our
c o n f id e n c e in general r e l a t i v i t y .
Of c o u r s e , th e d e t e c t i o n o f Ogg
waves would f a v o r t h e v e c t o r - m e t r i c th e o r y over general r e l a t i v i t y
(though s c a l a r - m e t r i c t h e o r i e s a l s o p r e d i c t Gu? waves).
APPENDICES
APPENDIX A
The c a l c u l a t i o n o f t h e PPN m e t r i c para mete rs c o n s i s t s o f t h r e e
parts.
F i r s t , s o l u t i o n o f t h e dynamic (moving sources') l i n e a r i z e d
e q u a t i o n s i s unde rtaken to o b t a i n a l l p o t e n t i a l s f i r s t o r d e r in — .
Second, t h e s t a t i c f i e l d e q u a t i o n s a r e s o lv e d to second o r d e r in
o r d e r to f i n d 3 and 6 .
in
T h i r d , a method i s de vis ed to g e t t h e two-
s ou rc e m e t r i c p o t e n t i a l w it h PPN c o e f f i c i e n t inv olv in g
Using t h e PPN m e t r i c f o r a s i n g l e s ourc e ( e q u a ti o n I I L l ) , and th e
expansion o f
( e q u a ti o n I I I . 2 ) , one can c a l c u l a t e th e fo ll o w i n g d e r i ­
v a t i v e s to l i n e a r o r d e r in
,IrVvT
KJK
OinOIOO =
+
r2
V oiO k = V oik O = ' ^
V o ik k " ^ ^ 1
^
+
1 +
" 2a ^ [ 3l ^ L
‘
xl< -
- /]
- 2 4 ^ f r.l
- 2 Tr<f)GM<5 ( I r ) [ 2 a-| + 2 agV^ + 2 + v ^ ( 2y + I + ag + ^ ) ]
_ vk]
K0Kk l 00
V kiO k
r
r
r
+ d 1 - d ) [ 3 ^ ^ - v2] + A d ' - d ) f . T
. r
r
- 2 Tr<f)GM6 ( r ) [2 + V2 ( 2y + 1 +
+ C-j) ]
59
VkIkO ~
“ d + -^x-j' - a2 +• I + c-j - y ) [ 3 —r - ^ ----- + F-"a]
VkIAA ' V
a IkA
= " ^ d ' - d)E3J^ - x k - vk] - 4irGM4,6(r)dvk.
r
r
Components o f R a r e
yv
R00 ” ^ ( y " 1 + a 2
I
Za I
v2] +
GM
(y
-
I
+
I
- C-J )r »a - 2 ttGM6 ( r ) [2 + v^(l + 2y + ctg + t-j)]
Rpk = ^ ( I
- Y + ^ 1)
x k - v k] - TrGMvkS( r ) (4 y + 3 +
- Ci2 + C1 ).
We w i l l need t h e 6- f u n c t i o n p a r t o f R to s t a t i c o r d e r only
R = 2Rp0 + I GitGMy S ( r ) .
Next we w r i t e t h e f i r s t o r d e r approxima tio ns to t h e f i e l d e q u a t i o n s .
F i r s t , t h e g ^ ^ - e q u a t i o n ( I I .2) i s w r i t t e n
(R00 - 1 r ) ( 1 - „») + (2[o + „ U t R 00 + K0Ko l u ) = - S r f 0T0 0 I
(A .U
n e x t , t h e g - e q u a t i o n ( I I . 3)
R + (6w + 3 n ) K p K ° , p p - (6w +
= SttG0T;
- n K p f K ^ ^ f K^,p^)
(A.2)
60
and t h e Kg-equation ( I I . 4)
ioKgR + n KgRg0 - 2 (K q iAA - Ka i q j i ) .
,
(A .3)
I t i s p r o f i t a b l e to e l i m i n a t e R in t h e above t h r e e e q u a ti o n s by
I
I
adding ^ o f A.2 and - -^Kq times A.3 to A.I , giv in g
r O0 O
+
+ K0 kQIM ( 1 " w + I f1) + V o i b 0 Ow + | n )
" I tikOkAIAO '
(1 + i n)K0KAl0A = - 8 it g O 1 OO +
(A-4)
Keeping onl y t h e 6 - f u n c t i o n p a r t s o f each eq u at i o n and s e t t i n g v to
z e r o , t h e f i r s t o r d e r s t a t i c e q u a t i o n s can be w r i t t e n .
Thus A. I , A.3,
and A.4 become, r e s p e c t i v e l y .
(I - axi))2GMy6(r) + (2co + o ) (a-j + 2)<j>GMS(r) = ZG^T^^
(A.5)
(2a^ + 4coy - 2m - n)<f>0M6(r) = 0
(A.6)
(I + mcj) +
)GM6 (r) + (I - m +
) (a-j + I )(f)GM5 (r)
- (-^n + l)<f>GM6(r) = G0 (2T00 - I )
■(A.7)
Using some r e l a t i o n s v a l i d in Minkowski space.
To o ^ *
=
=
pdt
-^ 4 = 4 = 2 =
(A.8)
61
Td3X =
=
(A.9)
- v2 d3x = M/1 - v2 .
e q u a t i o n s A.5 to A.7 i n t e g r a t e (with v = 0) to
•
G0
2 y (I - uxf)) + ( 2 lu + n)<f>a-] - 2 q—
(A. 10.)
- 2 ( 2 w + n)d
(A, 11)
2y(4co) + 4a^ = 2(2w + n)
G
,
(A,12)
• (2 - 2m + n)tf>a-| - 2 q— = - 2 ( 1 + gfti)) .
These can be so lve d s i m u l t a n e o u s l y to give
_ I - (oj) ( 2(0 + n + I )
^
I + (0& ( I - 4(0)
•
Experimental ev ide nc e i n d i c a t e s y . ~ I , so we proceed with t h e s p e c i a l
case
.
a) = -^n + I .
( A . l 3)
The pr oc ed ure f o r t h e a l t e r n a t i v e p o s s i b i l i t y , cu = 0, i s s i m i l a r and
w i l l not be given h e re .
Using A.13 and A.11 gives
a-] = - I .
p
Keeping t h e v
'
p a r t o f t h e d e l t a - f u n c t i o n terms and using A.13,
A.4 i n t e g r a t e s to
Q
(I + ^T|(j))[2 + v2 (l + 2y + Ug + e-j )] = 2 ^ ( - ^ = ^ = = = "
which gi v e s
V2 )
.
62
(A.14)
I + 1Jrntp
as a r e n o r m a l i z a t i o n o f Gq , and t h e r e l a t i o n s h i p
I + 2y' + Kg +
(A.15) ■
= 3.
Having found t h e 6 - f u n c t i o n s ourc e t e r m s , we ne xt f i n d s o l u t i o n s to
t h e t o t a l dynamic f i e l d e q u a ti o n s o u t s i d e t h e s o u r c e .
The g^Q eq uat io n
(A.I) becomes
(2 m + n ) <j>^[(3-^—-7T----- v2 ) ( 2 ? i + 3y +
+ "r-"a(3y + Og -
- I - 2y - ^
- I - 2y - ?1 - 2 f )] = 0.
- 2ag)
(A.15)
This g iv e s two req u ir e m e n ts
?1 + a 2 - I a 1 - 2a3 = 0.
I
-S-j + o i g " "2^] ~ 2f = 0,
where we have used y = I .
a3 “ f '
Equation A.3 becomes
.
• (A.17)
S u b t r a c t i n g one from th e o t h e r gives
(A.18)
63
(2w +
. v2 )(c
.r
E - Za V + r *a ^0I2 " Z 01I - c I ^
(A.19)
v^)(2 a_ - d + d 1) + r - a ( Z f - d + d 1)]
J
+
r
r
0,
g iv in g
(2co + n ) («2 - "Tja -J') = 2(d - d 1 - 2 a g )
(A.ZO)
(Zto + n ) (“ 2 - "^ct] - ?-]) = 2(d - d 1 - 2 f ).
S u b t r a c t i n g and usi ng A.18 y i e l d s
(2 uj + n + 4 ) ^ | = 0.
This r e q u i r e s t h a t
(A. 2,1)
C1 = O5
u n l e s s 2u + n + 4 = ' 0 . However5 i f t h i s l a s t r e l a t i o n i s combined with
]
to = 2^ + 1 5 one f i n d s unique va lu es f o r co and n which, when used in
e q u a t i o n A.4, giv e
C1 = 0 anyway,
Equation A.18 y i e l d s th e a d d i t i o n a l .
result
a3 = f.
‘ ■
In e q u a ti o n A.16 i t was assumed t h a t 2w + n f 0.
c a s e , 2co + n = O5 e q u a ti o n A.19 w i l l giv e a^ = f .
(A.22)
In th e o t h e r
Using t h i s in A.4
• 64
r e s u l t s a gai n in ^
= 0.
Th e re fo re t h e same PPN pa ra m et e rs a r e o b t a i n e d ,
r e g a r d l e s s o f s p e c i a l r e l a t i o n s h i p s between w and o.
C- J =O and
y
= 1 i n e q u a ti o n A .15 g iv e t h e a d d i t i o n a l r e s u l t
C3 = O .
Using C- J =O and a^ = f ; A .17, A.20, and A.4 become
I
a 2 ~ 2^] = 2f
(n + I ) fa 2~
(I + <f> +
) ~ d 1 - d - 2f
(a? - ^ a ] ) = (I + n ) ( d ' - d) .
Solving t h e s e s i m u l t a n e o u s l y gives
2f = a 2 - ^<x.j = d' - d = 0.
To g e t t h e i n d i v i d u a l valu es o f
t i o n s a r e needed.
Equation I I .4 gives
"ml Jim
or
(A.23)
and a ^ ,
the K£ and g^^ equa­
65
Equation I I . 2 y i e l d s
(I + (u<t> +
n+)R0» +
(2oi + n)K0Kc
IOJi +
™ Krn!Jlm^
°’
or
- v ^ j [ ( l + ox#) + n<|))
r
) - ln<j)(d + d ' ) ] = 0.
r
Solving f o r a-j and usi ng u = -^n + I
"" O^1
_I
4(f)
4n
(A.24)
O m
+ 4 + 6n + n
Also
d = d' , I
4 | ~ 1 + 4 + 6n
4(j)
+ 4 + 6n + n
The onl y PPN pa rame te rs remaining i n v o lv e second o r d e r s t a t i c terms in
the m etric.
The s t a t i c e q u a ti o n s to second o r d e r a r e 11 ,2 ,
Rgod + w* + n*)-^R(T - w*) - (2w + n ^ K g K o ) , ^ ^ " F ' o / o n / ' " = 0%
(A.25)
I I . 3,
R + (6w + 3n)K°K°,og + (6w +
+ 2^ o i A
ioSm
+
+ " rOOk0r0 - °-
(A.26)
66
and I I . 4 ( m u l t i p l i e d by Kq ) ,
» 0 R ♦ U* R 0 0 + (K 0 K0 ) 11/ " 1 - ZK0 u K0
Adding
I
o f A.26 and -
I
- ZK q K i i i ^ g " = 0 . ( A . 2 7 )
o f A.27 to A .25, and using to.=
I
+ I,
R0 0 Cl + | n * + * ) + O n + 3 )K 0 KO | 0 0 - ( 2 n + 2 JK t m Kt l 0 ,
M n - M J K 0 K11l 0 t = O
:
where we have a l s o used t h e f a c t t h a t to f i r s t o r d e r th e only non-zero
d e r i v a t i v e o f K i s Kn i n .
y
x. I U
To second o r d e r ,
R0 0 = ( 2 - 2 6 ) ^
■
2 2
v ,o
_ i/
u
- y r
KoR ioo
rJu o N io
4
I
K0 K1 m t M 2 B - 3 ) ^
'
.
Using t h e s e give s t h e r e l a t i o n
(2 - 2 3 ) 0 + 2^4)) - O3
o r 3 = I as in gener al r e l a t i v i t y .
To g e t 6, which only appears in R,
: .
I
we s o lv e e q u a t i o n s A.25 and A .27 f o r R. M ul ti p ly in g A.27 by to + ^ri
and adding to A.25
67
.JjROl + w*(2w + n + D - F0/ 0[y
m - (2a» + n ) K g K ^ , ^ g ^ = 0
where we have used t h e f a c t t h a t Rqq = 0 when 3 = I .
Solving and using
n = ZtO - 2 a
(■4»-1)^5Id „
2
r ,
i - to^(4o) - ry
Now,
R = ( - 4 6 - 4 6 + 8 ) ^ - = ( 4 - 4 5 ) —- r - ,
/
so t h e 6 term can a l s o be found:
6 = 1 + — _-r—
-------- .
2(f) + 2co(l - 4m)
The ^2 p a ra m et e r a p pea ri ng in t h e two mass i n t e r a c t i o n term can be
found by means o f a c o n v e n ie n t t r i c k .
The c o n f i g u r a t i o n o f a p o i n t mass
m i n s i d e a s p h e r i c a l s h e l l o f mass M and r a d i u s R >> r i s so lved by two
approaches and t h e r e s u l t s a r e compared.
F i r s t , M i s in c lu d e d as a
s ou rc e o f t h e m e t r i c and we w r i t e
g0Q = I - 2G^ - 2G^ + 43G
- (2 - 43 + 2 ^ ) 6 ^ +
0(m2 ,M2 )
(A.28)
9 SS =
~
"
2y6T
•
\
68
This has t h e a sy m p t o ti c l i m i t (as ^
0 b u t always R >> r )
gOO = 1 ‘ 2Gf
gss =
(A.29)
' 2yt^ *
Second, we no te t h a t t h e post-Newtonian l i m i t i s v a l i d i n s i d e th e
s h e l l , so one must be a b l e to w r i t e to l i n e a r o r d e r .
(A.30)
9 ss
-I
where i t i s re c o g n iz e d t h a t t h e pre s en c e o f th e mass s h e l l may a f f e c t
G, and t h a t t h e c o o r d i n a t e s w i l l be d i f f e r e n t to allow a' Minkowskiian
a sy m pto tic l i m i t .
Comparison o f A.29 and A.30 shows t h a t t h e c o o r d i n a t e
t r a n s f o r m a t i o n must be
ax0.
" I + GM
R
GM
ar _ i
ar'
' - yF - •
Applying t h i s to A.28 give s
gOO = I - 2G^ + (SB - 6 - 2?2)G ^
But r = (I
y G ^ r ' so
.
9 qq = I - 2(^ T 'I- (83 - 6 - 2y -
^
Comparison with A.30 shows the e ff e c t of M on G,
G* - G[1 - (43 - 3 - Y - ■^2
'
However, the e f f e c t of M on G is well known from previous analysis
Equation A.14 gave
- G^Cl + l n ( 1 + 2 ^ * ( 1 - {1)(1 - ^ ] - T
I t -Tjjng
Kq Kq
G* = -----V - = G.
I + -TjTKf)
Therefore 43 - 3 0.
y
- ?2 = 0 ’ anc^ since 3 = y = I ,■ this implies
APPENDIX
B
The s u b s t i t u t i o n o f t h e le a d i n g terms of IV.6,
3OO ' axS
S r r = " bxt
K0 = cxu ,
i n t o t h e f i e l d e q u a t i o n s has produced t h e fo ll o w i n g e q u a t i o n s (e q u a ti o n s
IV .7, IV.8, and IV. 9):
X ( u - s ) [ ( w - l ) x W™2 + Bxw' 1] - ^ x p' 1 ( s + t ) + 2a)XW(bxt - I - ^ ) = 0
(B. I )
( u 4 x s ) C( W - I ) X w" 2 + Bxw" 1] - 4 xP( b x t - l - | - ) - BuxW(bxt - l 4 ) = 0
4
C4
x
x
( B. B)
XXw" 2 is ( u 4 5 ) + Xw- 2U2 - W ( b X ^ - l 4 ) + wXW( b X ^ - l 4 - 4 ) = 0.
Z
Z
X
x x
(g 2 )
These a r e polynomials in x whose powers depend on t h e v a lu es o f s , t ,
and u.
For v a r i o u s ranges o f va lu es o f t h e s e p a r a m e t e r s , c e r t a i n terms
in th e polynomials w i l l be dominant n e a r t h e s i n g u l a r i t y .
The method o f
s o l u t i o n w i l l be to assume some range o f ex ponen ts , s , t , and u, keep
only lo w e st o r d e r terms in x, and examine th e r e s u l t i n g s i m p l i f i e d
e q u a t i o n s t o se e i f a c o n s i s t e n t s o l u t i o n can be found.
The c r i t e r i a
f o r i n c o n s i s t e n c y a r e as fo l l o w s :
I.
a , b, o r c = 0.
This i s i n c o n s i s t e n t with t h e assumption
t h a t axs , f o r example, i s t h e le a d i n g term in t h e power
series.
71
2.
s _< 0.
This i s i n c o n s i s t e n t w it h t h e assumption t h a t r = r
i s an e ven t h o r iz o n .
[I.
w = 1]
F i r s t assume w - I .
Then B.l to B.3 become ,
2X(u-s) - 5^-X*3"^ ( s + t ) + 2ti)(bxt + ^ - l ) = 0
(B.4)
C
2(u+^Xs) -
- 2(i)(bx^ ^ - I ) = 0
2^ s { u - ^ s ) +
^
+ o)(bx^ ^ - s - 4 u ) - 0.
(B.5)
(B,S)
I f t > - I , t h e le a d i n g p a r t s become
[la.
t > -1]
2A ( u - s ) - -TfX^ ^ ( s + t ) - 2w = 0
( B. 7)
C
2 ( u+^Xs) + —
s + 2co = 0
• | ^ s ( U--^s) + ~u^ + ^ x *3
= 0.
I f P < I , t h e second e q u a ti o n reduces to
[1a(i).
4 =
p < 1]
o.
(B.8)
(B.9)
72
So e i t h e r a - 0 o r s -- 0 and we have a c o n t r a d i c t i o n .
Now assuming
t h a t p > I 5. t h e t h r e e e q u a t i o n s become
[T a(ii).
p > 1]
X( U-S) -W= 0
U+^-XS + w = 0
Xs(U-TjS) + 2u
= 0.
The f i r s t e q u a ti o n giv e s
. w
U= 5 + X
which, when s u b s t i t u t e d i n t o t h e second e q u a ti o n give s
These values* s u b s t i t u t e d i n t o t h e t h i r d e q u a t i o n l e a d t o an i neonsis
tency.
T h e r e f o r e , i t can be concluded t h a t p = I .
[I a ( i i i ) .
p = 1]
Now i t should be remembered t h a t
I I
w = 2u—g S - ^ t = I
p = ^s-^t = I
'
73
which can be s u b t r a c t e d to give
u = -gS.
S u b s t i t u t i n g t h i s i n t o e q u a ti o n B.9 w it h p = I , gives
which i s a gai n i n c o n s i s t e n t .
Thus no val ue o f p giv es a s o l u t i o n and we
a r e fo r c e d to conclude t h a t t h e assumption le a d i n g to B.7-B.9 i s not
valid.
•
[lb.
Thus t <. - I , and i t i s s t i l l assumed t h a t w = I .
I f t < - I , B.4 and B.5 become
t < -1]
- ^ x p" 1 ( s + t ) + 2o)bxt+1 = 0
c
- ^ x p (bx*) - E w b x ^ = 0.
c
Adding t h e s e and d i v i d i n g by xp gives
^•(s+t) + bxt = 0.
But t < -I implies, b =0 which i s i n c o n s i s t e n t .
remaining val ue o f t i s t = - I ..
In t h i s ca s e e q u a ti o n s B.4 to B.6 become
T h e r e f o r e , t h e only
74
[lc.
t = -1]
2 x ( u - s ) - ■^'(s+tjx^ ^ + 2 u ( b - 1 ) = O
c
2(u+^- s ) - ^ - ( b - s ) x ^ ^ - 2w(b-l) = O
(u-^s) +
- ^"X*3 ^ (b-s.) = 0,
I I
and w = Zu-gS-gt = I im p li e s s = 4 u - l .
Now i f p < I , t h e f i r s t e quat io n
becomes
[lc(i).
P < 1]
—^ • ( s - l ) = 0
c
or s = I.
This however, along with t = - I , means p = ■
7)5—Jpt - I , which
contradicts p < I.
p = I is also quickly ruled out.
In t h a t c a s e ,
t = - l , s = I , u = - g , and t h e t h i r d e q u a ti o n i s
pc(ii).
p = 1]
¥-°
which i s i n c o n s i s t e n t .
The only remaining p o s s i b i l i t y i s p > I .
t h i $ c a s e t h e t h r e e e q u a t i o n s a r e ( us in g s = 4 u - l ):
In
75
[Ic(Iii).
p > 1]
X(I-Su) - a ) ( b - l ) = O
I
U-XU+^X -
) = O
X( 4 u - l ) (-u+2") + 2U - 0.
Adding t h e f i r s t two and s o l v i n g f o r u gives
U
4(4X -1) '
'
This r e s u l t , used in t h e t h i r d e q u a t i o n , le a d s to an inc ons is te nc y=
All work th u s f a r has been to show t h a t w = I i s not c o n s i s t e n t
w it h t h e f i e l d e q u a t i o n s .
I t may now be assumed t h a t w j M , and th e
same ty pe o f a n a l y s i s w i l l be performed, t h i s time l e a d i n g to a so lu tio n,.
I f w f I , the n Xw " 1 may be dropped w ith r e s p e c t to xw~2 , and eq uati ons
B.l th ro ug h B.3 become
WM ]
X ( u - s ) ( w - 1 ) - ^ " X ^ ( s +1 ) + Z u b x ^ = 0
C
(B.10)
(u+^Xs) (w-1) - —2-x C*(bx^-'~) - Zubx^ ^ = O
(B,T1)
x-gs(u-^s) + u2 - ^ x c*(bx*— ) + cjbxt+2 = 0,
C
( B . l 2)
where we have d e fi n e d
i
p-w+2 = s+2-2u :: q •
and where terms l i k e 2cowx have been dropped in f a v o r o f t h e c o n s t a n t .
f i r s t terms.
Assuming t h a t t > - I , equations' ES. 10 through B.12 become
[2 a .
t > -1 ]
x(u-s)(w-1) - —
c
1 ( s +t ) = 0
■
( u + | x s ) ( w - l ) t- -Tjxq" 1s - 0 .
c
• | s ( u - ^ s ) + u2 +
[2a(i).
’
= 0.
q < 1]
The a d d i t i o n a l assumption t h a t q < I le ad s to
= 0 in th e second
c
e q u a t i o n , which i s i n c o n s i s t e n t with t h e b a s ic c o n d i t i o n s .
I f q > I , th e
f i r s t e q u a ti o n i s j u s t
[2a ( i i ).
q > .1]
( u - s ) (w-1) = 0 '
which has s o l u t i o n (w ^ I ) , u = s .
t h i s l e a d s to
s ( I +J*) = 0
I n s e r t e d in t h e second e q u a ti o n ,
77
which i s a g a i n i n c o n s i s t e n t .
effort.
[2a(iii).
The assumption t h a t q = I r e q u i r e s more
In t h i s c a s e t h e t h r e e e q u a t i o n s reduce to
q = 1]
A(S-I)(W-I) + ■^|-(s+t) = 0
C
(2s +2+As)(w-l) + ^ | s = 0
C
Xs + (s+1)^ + —
= 0
C
where we have used t h e f a c t t h a t q = s+2-2u = I im p li e s 2u = s + 1 .
Solving t h e f i r s t two e q u a t i o n s f o r s and t y i e l d s
S = -I - h
c
~ H
c
which i s used in t h e t h i r d e q u a t i o n , le a d i n g to an i n c o n s i s t e n c y .
There
f o r e one must have t <_ - I .
[2b.
t = -1 ]
The assumption t h a t t = -I in e q u a ti o n s B.10, B . l l , and B.12, gives
X( u - s ) (w-1) -
^(s-1) = 0
C
(u+^-Xs) (w~l) - ^gX^
(b-s) = 0
■|s(u-^s) + u2 - —g - x ^ ( b - s ) = 0.
78
[ 2 b ( i Jl
q > I.]
q > I reduces t h e f i r s t e q u a ti o n to
u -s = 0
which can be used i n t h e second e q u a ti o n t o give
(l+^-Xjs = 0,
an i n c o n s i s t e n c y .
[2b(ii)o
q < U
q < I im p l i e s a s o l u t i o n when
s = l
b = I,
and in o r d e r to have q < I , i t i s n e c e s s a r y t h a t x+2-2u < I , o r
u > I.
This a pp ear s to be a v a l i d s o l u t i o n .
However, s u b s t i t u t i n g t h e s e r e s u l t s
i n t o t h e y e q u a t i o n (e q u a t i o n IV. I ) g i v e s ^
The re a so n f o r doing t h i s i s t h a t duri ng t h e a l g e b r a i c s i m p l i f i ­
c a t i o n which le d to e q u a t i o n s IV. 5 and IV .6 (B.l and B . 2 ) , we in tr oduc e d
i n t o t h e u e q u a t i o n elements which Were lower o r d e r in x tha n th e o r i g ­
i n a l . Thus, t h e s a t i s f y i n g o f t h i s lo w e s t o r d e r p a r t g iv e s th e i l l u s ­
t r a t i o n o f s a t i s f y i n g a l l t h r e e e q u a t i o n s ; but i t i s t h e lo w e st o r d e r
p a r t o f t h e o r i g i n a l t h r e e e q u a ti o n s which must in f a c t be s a t i s f i e d .
|(u+^-A)[(2u-l )x
+ 2x ^] - to - ~ - ( u - l ) + - ^ ^ x ^ 0 = 0,
whose l e a d i n g terms a r e
(u+^-A) ( 2 u - l ) - 2"(u—^-) +
= 0,
wit h s o l u t i o n
u = 0
;
or
u = I.
This is. i n c o n s i s t e n t with th e r e q u ir e m e n t o f u > I , so t h i s i s in f a c t
no s o l u t i o n .
[ 2 b ( i i i ).
q = 1]
A s o l u t i o n i s found when q = s+2+2u = I .
Then 2u = s + 1 , and the
e q u a t i o n s B.10, B . l l , and B.12 become
X( s - 1 ) (w-1) + —^-(s-l) = 0
C
( 2s+2+Xs) (w-1) - —2"(b0s) = 0
c
Xs + (s+1
- —Tj-(b-s) = 0.
c .
The f i r s t e q u a ti o n i s s o lv e d by s = I .
and t h i r d e q u a ti o n s give s
Using t h i s in both t h e second
80
b = I +
.
Here, t h e n , i s a s o l u t i o n
r-r
900 =
o
■
I c'
9r r = - n + 4 i “ ^ +4^ r - r .
r-r
kO = c~ r ~
S u b s t i t u t i o n o f t h i s s o l u t i o n i n t o th e o r i g i n a l t h r e e e q u a t i o n s gives
c o n s i s t e n c y , with a , c , and r 0 s t i l l a r b i t r a r y .
F i n a l l y i t i s n e c e s s a r y to make t h e assumption o f t < -I in equa­
t i o n s B.10, B . l l , and B.12.
F i r s t , we t r y t < - 2 .
e q u a t i o n s become
[2c.
t < -2]
-
( s + t ) + 2u)bx^+^ = 0
c
- ^ 7XqBxt - 2tobxt+2 = 0
C
■^2bq+^ + Oibx^+2 “ 0.
Adding t h e second and t h i r d gives
Then t h e t h r e e
81
-mb = O
which i s i n c o n s i s t e n t with t h e b a s i c c o n d i t i o n s .
[2d.
Therefore t >
t = -2 ]
The assumption t h a t t = -2 produces t h e t h r e e e q u a t i o n s
x ( u - s ) (w-1) - ~ 2"X^ ^ ( s - 2 ) + 2mb = 0
c
(B.13)
(u+^XS) (w-1 ) -
(B.14)
|s (u -^ s )
[ 2 d ( i }.
- 2mb - 0
+ U2 - ^ | x q ™2 +
mb = 0 .
q < 2]
I f q < 2 t h e second e q u a ti o n i s
7
“ °
which i s i n c o n s i s t e n t .
[ 2 d ( i i ).
q > 2]
I f q > 2, t h e t h r e e e q u a ti o n s a r e
x ( u - s ) ( w - 1 ) + 2mb = 0
(u+^Xs) (w-1) - 2mb = 0
( B. 1-5)
82
(u—IjS) +
+ tub = 0.
Adding th e f i r s t and second e q u a t i o n s t o g e t h e r and remembering t h a t
w.?M gi ve s
,, 3Xs
u
4n+ iy Using t h i s , th e second and t h i r d e q u a t i o n s reduce to
Xs2^ ± l l % 2 )
(4A+4)^
- 2™b = 0
2 .( ^ - 4 ) ( 4 A - 2 ) + gwb = 0
As
(4A+4)d
which a r e i n c o n s i s t e n t w it h s > 0 .
Now i f q = 2, s = 2u and e quat io ns
B.13, B.14, and B.15 become
[ 2 d ( i i i ).
q = 2]
o
-Au^ + 2cob = 0
U2 + TfAu
2
U2 -
- -H- - 2(ob = 0
c2
+ cob = 0.
C
S u b t r a c t i n g t h e t h i r d from t h e second and adding o n e - h a l f o f the f i r s t
yields
-2wb = 0
83
which i s i n c o n s i s t e n t .
We have now e xhau st ed a l l p o s s i b i l i t i e s with
t = - 2 , and have l i m i t e d a p o s s i b l e second s o l u t i o n to th e range
-2 < t < - I .
In t h i s c a s e , e q u a t i o n s B.10, B . l l , and B.12 reduc e to
£2e.
-2 < t < - I J
X( u - s ) (w-1) - —
c
(s +t ) = 0
(u+^-Xs) (w-1) - ^gX^ t = 0
c
^ s ( u - ^ s ) + u2 - ^ x c,+t = 0.
[ 2 e ( i ).
q < 1}
I f q <. I , then t < -I im p li e s t h a t q+t < 0, and t h e second equa­
tion is ju s t
which i s i n c o n s i s t e n t .
f i r s t e q u a ti o n to
R e(ii).
A > Tl
X( u - s ) (w-1) = 0
o r, since w ^ I ,
The remaining p o s s i b i l i t y ( q . > I ) reduces the
.84
u = s.
P u t t i n g t h i s i n t o th e second and t h i r d e q u a ti o n s give s
(l-flx)s(w -l) -
^
= 0
c
(B.16)
( I t l x ) S 2 - SjjxOt t = 0
C
which can only be s olv ed i f
S = W-I
-
Zu-vjS—T jt- I
or
t = s-2.
I f t h e s e r e s u l t s a r e s u b s t i t u t e d i n t o e q u a ti o n B . l , t h e r e r e s u l t s
x ( s - s ) [ s x s ^ t 2xs ] - -Tf(Zs-Z) + 2to[bx^s ^ - x
^ - (s+I )x^J = 0
C
(B.17)
Now t h e l i m i t s on t (- 2 < t < - I ) g iv e l i m i t s on s (0 < s < I ), so i t
i s p o s s i b l e to compare t h e o r d e r s o f some o f the terms in B.17.
s-1 < Zs-I < s
s-1 < 0 < s
The lo w e s t o r d e r term (xs ~^) v a n i s h e s .
However, i t must be remembered
t h a t t h e b a s i c expansion o f t h e f i e l d s was a power s e r i e s , and i f one
85
goes beyond t h e le a d i n g term in. th e power s e r i e s , t h e n e x t o r d e r c o n t r i ­
b u t i o n w i l l be a t l e a s t one power h i g h e r , o r xs .
T h e r e f o r e , t h e second
lo w e st o r d e r terms in ES.17,
-
- •^2"(2s - 2) + 2ti)bx^S ^ ,
c
(B.18)
must a l s o v a n i s h , s i n c e any terms a r i s i n g from c o n t i n u i n g t h e power
s e r i e s expansion must be o f d i f f e r e n t ( g r e a t e r ) o r d e r than B.18.
I t is
th e re fo re necessary t h a t
- - | ( s - l ) + 2(obx2s " 1 = 0 .
C
I
If s f j ,
t h i s reduces to
s = l
-
b = 0
which i s i n c o n s i s t e n t .
I
I f s = ^ , then
'
2wc2
and
Now, i f t h e s e va lu es a r e used in e q u a t i o n B.2, t h e r e r e s u l t s
(B.19)
'8 6
- ^-JX ^ ^
- 2o)b - 0.
I /O
The x™ z term v a n is h e s because o f e q u a t i o n B.16, and an argument l i k e
t h e one l e a d i n g to t h e v a n is h i n g o f B.18, l e a d s to
—rf - 2wb - 0
c
which i s i n c o n s i s t e n t with B.19.
There i s , t h e r e f o r e , only one s o l u t i o n to th e t h r e e e q u a t i o n s ,
B. l , B.2, and B.3, which i s c o n s i s t e n t with th e assumptions o f an even t
h o r i z o n ; and t h a t is
9r r
a
K Zl
0
wit h a , c, and r 0 a r b i t r a r y .
o
APPENDIX C
We f i r s t d e r i v e t h e l i n e a r i z e d form o f th e f i e l d e q u a t i o n s .
The
f ie ld s are w ritten
9 uv
K
u
where h
hwv
V
+ AX
wv
«
n
wv
, A «
w
<j> , and n
and <j) a r e c o n s t a n t cosmological
w
wv
w
v a lu e s o f t h e f i e l d s ( c o o r d i n a t e s a r e chosen so t h a t nyv i s th e Minkowski
metric te n so r).
^yavB
Then t h e c u r v a t u r e t e n s o r s a r e
2 ^ w v ,aB
^aB »wv
r XV = I fcj2hPV
WV + h , p v -
R = D h - h ctB
where h = nyvh^v .
^wB iOtv
^ a v ,y'B^
2h“ p , J
»aB
The d e r i v a t i v e s a p pe a ri ng in th e f i e l d eq uati ons a r e
(remember K e K^K^g^^)
K,pv = 2* X , p v - ^
6hCS-PV
D 2 K = 2 * t ] 2 A^ - ♦ “ 't'f S 2 Ila 6
(Ka K6) la6 ■ Z A 6 i6a ♦ A 6Ra6 - A 6D 2Ha6
D 2 IK K W
lP 2Av + ^ A
-
- \
O11A d X cP
H6ct,Bv - h V 5Bct
- h ^ ^ )
.
88
^ ( V 0 ) 111 0- * \ , v a
+ tX
lll^ V
a i =V +
V ai=P
+ 2tat8R p«v6 - t a t 6 h =3,pv
- ^ t X h8O1V6 " h6V1=B + c 2 h VC1
- V aI fh8O1P6 ‘ h8P1O6 t c 2 h PC1
in which c a s e , one can a l s o w r i t e t h e combination
.
= 4> F a
+((,Fa
- 2<#>(V 'F
D 2 ( V ) " (KpKa ) ]va - (Kv Ka ) Iya
y v ,a
v y ,a
'yav3
+ t a t 8 h O6 l PV - t a X 1VO +
Thus th e
A
v,ya
).
e q u a ti o n ( I I . 2) can be w r i t t e n ,
R, „ ! 1 + , i t Z ) + s f l o t Pt V
" K v
" K 2Spv1 t 2 r o ta fX 1PV " S p K '
- ( « + f i ) A 8 ( h a e , ; V - S 110 2 Kc t 6 ) + T i f l t llRv a + ♦ / „ „ )
+ P A X R v a v 6 - S v v X s ) - P S u v A X o t I n t a f X 1V = t l X 1P = 1
- I n f t PpVa i = t X p a i O1 = 0
where < T =
R+
<f>a .
■
(C.l)
The g e q u a ti o n ( I I . 3) i s
+ a fp -
= O
where we have d e fi n e d P e (2w+n)(^A^ The Ky e q u a ti o n ( I I . 4) i s
(C.2)
01^ h a g ) f o r r o t a t i o n a l ea s e .
89
w* R + n<f)a R
y
ya
+ 2F a
y ,a
= 0.
(C.3)
Based on C.3 a l o n e , t h e fo ll o w i n g e q u a t i o n can a l s o be w r i t t e n .
M u l t i p l y i n g e q u a ti o n C.2 by
■K
vr
I.
yields
+ r> 9yV<fra / Rae + f g yv^ p
-
= 0.
D ott ing e q u a t i o n with *p and m u l t i p l y i n g by ^g^ y gives
1“V
t2fi + ^ 9Pvta t6 fi a S +' 9Pvt a p C6 1B =
Adding t h e s e l a s t t h r e e e q u a t i o n s to C.l r e s u l t s in a s i m p l i f i e d v e r s i o n
o f th e gyv e q u a t i o n .
Rpv O * M 2 ) + Tl^qi6Rllav6 -
" I 9 t a IpPp1V + pCv1P1 -
+ I s illP 2P + Pjllv t- V t V
I? 9 + fiX t PpVa i C + t VpP0 1C1 = ° :
A s o l u t i o n i s now assumed in which t h e p e r t u r b a t i o n s (h
a r e pl a ne waves moving in th e z d i r e c t i o n .
h
yv
= e
yv
e
iK
y
A" - aye ll<l|X .
yv
1B
(C' 4 )
and A )
y
90
i s a p ro p a g a ti o n 4 - v e c t o r whose components a r e given by
kp = (Ck9Os O5 - k)
( C . 5)
where c i s no t n e c e s s a r i l y th e speed o f l i g h t (which i s equal to one in
o u r u n i t s ) , bu t i s t h e speed o f p r o p a g a ti o n o f g r a v i t y waves.
There a r e f i f t e e n non-zero components o f the Riernann t e n s o r , not
c ou n ti n g p e rm u ta ti o n s o f t h e s u b s c r i p t s , b u t o f t h e s e , only s i x a re .
in d e p e n d e n t.
R0101
These s i x can be ta ken to be
' R0102
’
R0202
’
The remaining nine a r e given by
R1313 = 1T r OIOI
c
D
K1 323
= I-' P
T T l 02
_ I■
R2323 ~ T * 0 2 0 2
c
R0131 = " Cr OIOI
R0132 = R0231
I
= ' c R01 02
R0232 = " 0*0202
]
R0313 = 0*0103
R0323 = c*0203.
R0103
’
R0203
R0303.
91
The components o f
a r e a l s o l i n e a r combinations o f t h e f i r s t s i x .
R00 " " R0101 " R0202 “ R0303
R01 “ " 3 * 0 1 0 3
R02
“ 3*0203
*03 “ c ^R0101 + R0202^
rH = 0
- 1 ^ r 0IOI
c
R12 = (1 " 1T )R0102
R22 " (I " ^2)* 0202
R13 " R0103
R23 = R0203
33
2^R0101 + R0202^ + R0303 ’
m d R is
R ? "2(1 - 1 2 ^ R0101 + R0202^ " 2R0303"
■c
We have a l r e a d y chosen c o o r d i n a t e s so t h a t th e background m e tr ic is
Minkpwsiian, and o r i e n t e d t h e t h r e e space c o o r d i n a t e s so t h a t t h e z - a x i s
i s along t h e d i r e c t i o n o f p ro p a g a ti o n o f th e wave; bu t t h e r e s t i l l e x i t s
92
th6 freedom o f c h o ic e o f i n e r t i a l frame, s i n c e any b o o s t w i l l leave n
unchanged.
To s i m p l i f y c a l c u l a t i o n s , we choose to work in th e r e s t
frame o f t h e u n i v e r s e .
In t h i s frame
M0
The v a n is h i n g o r non -v a n is h in g o f v a r i o u s components o f Rviavg w i l l depend
oh t h i s c h o ic e o f r e f e r e n c e frame, b u t once t h e l i n e a r i z e d components
a r e found in One frame , the y may be determined in any frame by the
a p p l i c a t i o n o f Lorentz t r a n s f o r m a t i o n s .
In t h e r e s t frame o f t h e u n i v e r s e , t h e
(We assume for th e time being that n J6 0),
e q u a ti o n (C.3) i s
W hen£=.l, th ere re s u lts
(C .6)
and & = 2 gives
(C. 7)
Setting & = 3 yields
(C .8)
93
The Kq e q u a t i o n i s
* " * R00 + 2Fo " , o
= 0,
which, usi ng t h e e x p r e s s i o n s f o r R and Rqq and th e r e s u l t 'C.8 , gives
(2w + n)*RQ3Q3 = 2FQ *^ + (2w + n However Fq01 a and Fq0 q a r e not ind ep end en t.
n U.
„ r* 3
F0 , a
~ F0 ,3 " “ F03,3 “ c 0 3 , 0 " " c 3 0 , 0 ~ " Cf S ,cr
— n
_ Ir-
_
Ir
_
Ir-
CL
So Rq303 i s given by
R
- 2 0 - c2)r
•0303
n<i> ' 0 , a ’
(C.9)
This i s a l l th e i n f o r m a t i o n a v a i l a b l e from th e f o u r K
There remain
equations.
ten g ^ e q u a t i o n s which w i l l determine Rqiq1 - p-q202,R0303’
A , and c ,
These e q u a t i o n s a r e ( e q u a ti o n C.4)
(II)
^g(l
+ w<j) ) + n*
(11)
R11O + V ) + I t 2R0101 - ^cf P- ^ o a jll = 0
m)
R12O + «*z ) * ^ 2R0202 -
(13)
R^jO + «♦ ) + n4i RgiQ3 -
(33)
R23U + w# ) + n# Rg2Q3 - Zn^oZiS ~ 0
2
2
'
Rq 102 = O
»0
i3 = 0 ,
'
.
94
(0 1 )
Rq 1 ( I
+ <i)<j>2 ] - T pvP F q -J . q
(02)
Rggtl + wfj)2) - ^ni(J)Fgg
(63)
RggCl + W+2 ) + P , g g - 2 F g » ^ = 0
(00)
RggCl + OJfj,2 ) - OXjj2 R +
(33)
R3 3 (I + ojcj)2 ) + HcjJ2 R0303 " ] r f p + Pj33 + (n - l)*Fo*',a
q
- C^n. + 2 ) 4 'F 1 ^ ^
= 0
- (-gn + 2)<j)Fg^^ = 0
■
+ Pj00 - ( n + 3)4,Fg*^ = 0
= 0
We now r e w r i t e t h e s e te n e q u a t i o n s , i n c o r p o r a t i n g t h e e x p r e s s io n s
for R
yv
and th e in f o r m a t io n gleaned from the K e q u a t i o n s ,
U -
(12)
Rgi Qgttl - -tg-) (I + (jj^) + n'cj)2] = O
c
(11)
Rq101[(I
-^g)(I + ojcj)2 )
(22) Rg202[(I - -tg)(l + ojcj)2)
(0 .1 0 )
+ n<j>2]' - ]jE?P ~ CjjFga ja = O
+ n*2] ~
- CjjFgctsa = 6.
S u b t r a c t i n g t h e s e l a s t two e q u a t i o n s gives
(Rg101 - Rq2 0 2 ^ t(l ” "^2")(I + wcj,2 ) + n.cj) I - 0.
( C. l l )
Adding (11) and (22) giv e s
(Rg101 + R0202^t(l ~ ^ ) (I + wcj,2 ) + rvp2] = .D2P + 2^Fgctja '
(0- 12)
95
S u b s t i t u t i o n o f (%8 f o r Rqiqi + Rq2q2 g i v e s ,
D2P = ~ — 7~'~0 + w*2 + a f 2 )2FQ“
n_(j)
( C .12)
•
u ,a
C ont in u in g , usi ng e q u a t i o n s C.6 and C.7
(13)
[^ (1
(23) '. [ ^ O
(C.13)
+ n.<j)2) - F ^ f i5 Y lirIa ,« = 0
(C.14)
+ w*2 + nci>2 ) - I ^ T T Y l r Ea i Ci = 0 ■’
where we have a l s o used t h e f a c t t h a t
• F02 ,3 = " ! fQE,O = ! f E0 1O = ^
1^
,a
The same type o f c onver sio n is
and a s i m i l a r e x p r e s s i o n f o r Fq1 q .
used in t h e (01) and (02) e q u a t i o n s t o g iv e
(01)
F f - O + OKf,2 ) - 2 + I - ^ f l i - I F 1a
= 0
n<p
^ c - I
’a
(02)
[4 (1
+ Mtj)2 ) - 2 + ~
— ] F 2a
(C.1.5)
■
= 0.
(C.16)
In w r i t i n g t h e (03). e q u a t i o n , Rqq = 1 ( Rq1 q1 + Rq2q2 ) is- e l i m i n a t e d
v i a e q u a ti o n C.8 and t h e f a c t t h a t Fq01
•J ja
(03)
[% 1
o<i>
+ w*2 ) + 2c]F %
u >a
= -cFna
U ,a
■•
i s a l s o used.
- - 4 -----a ? = 0,
2
or
D2P =
ntf)
1 (1 + w*2 + n / ) 2 V
u ,a
(0.17)
96
in agreement with C.12.
Before w r i t i n g down t h e (00) and (.33) equa­
t i o n s , we not e t h a t , based on th e K e q u a t i o n s a l o n e .
R00 = - R0303 " ( R0101 + R0202^
I
R33 ~ R0303
2 vxOI01
( R0101 +' "0202
R0202^
2 p a
0 ,a
■■2Cr a
n<i) 0 ,a
R = ™^R0303 ~ 2 ^1 " " 7 ^ R0101 + R0202^ =
Then, usi ng C.17 f o r □ P, t h e (00) and (33) e quat io ns a r e both
%r[3(c2 - 1)(1 +
+ n4>2) - n<f>2(n + 1 ) ] F^
= 0.
U ,CZ
( C. I 8 )
These a r e a l l o f t h e g ^ e q u a t i o n s , and we proceed to s o l v e them.
Equation C.10 i s s a t i s i f e d i f
n
c
.
.
.2
= I -
(C.19)
I +
+ n<i>2
So Rq-] Q2 must obey
ik xu
u
R0102
eI 6
wit h k r e p r e s e n t i n g a wave with speed given by 0.19.
u
is also s a t i s f i e d i f
(
0 . 20 )
Equation 0.11
1kZ jl
R0101 ~ R0202
e26
(C.21)
w it h t h e same k
Equations C.13, €..14, € . 1 5 , and €.16 a re a l l s a t i s f i e d , i f
2
( 0 . 22 )
I -r cjtj)
+ n<()
which means t h a t ( us in g €.6 and C.7)
i k 1Xy
R0103
CD
R0203
m
Il
I k nXy
. ,
i k i-vy
.. ^
i k '.1xj-i
Fi a , .
Y .. ' ^ 4 '
where
v
i s t h e p r o p a g a ti o n v e c t o r f o r a wave with speed given by
0.22.
F i n a l l y 0 . 1 2 , 0 . 1 7 , and 0.18 a r e s a t i s f i e d i f
2
T +
I
I
n
2 2 .
2
+ n<k
3 i . ,2 , .2
I f ouj) + n<()
and
ik"xy
v
R0101 + R0202
eSe
( 0 .2 3 )
98
‘0303 =
Q
2 -eBe
... '
ik " x
■a . e y
0 ,a
2c2 5
d ?
Ik
J- - 1l CI + H 2 + V k 5G v
= 'c2
where k" i s t h e p ro p a g a ti o n v e c t o r f o r a wave moving a t speed given by
C.23.
To sum up, a l l o f t h e f o u r t e e n e q u a t i o n s a r e s a t i s f i e d by
Rr
'0102
i k xu
u
eIe
i k xy
y
R0101 " R0202
£2e
i k ‘xy
y
R0103
e 3e
i k 1xy
y
R0203 " e4e
ik"xy
y
R0101 + R0202
R
*0303
eB6
c „2
Be
i k 1xy
r
a
hI
= iii_ e
,a
e
y
■
2 c 1 3e
i k ‘xy
c
a
= J Iire
e
y
2c1 4
.
.
Ii v y
I k nX
Bi__e p y
Foa ,«
where
'h<l>
T + r|(j>
~~Z
I + axj)
.2 . 2 . .
I + 4
K3
I/"
,u2
n <f>
,
,2
.
I + co<j)
x
2
+ n<#>
2
=I + I
k3
2 2
tn *
I + coif) + ri(J)
*
In a l l t h a t has been done u n t i l now, i t has been assumed t h a t
n Ti 0.
The o t h e r case n = 0 must be taken s e p a r a t e l y .
The K equaXj
t i o n s (C.3) a re then
(C.24)
For £ =.1 and &= 2, t h i s becomes
D 2 A-, = O 2 Ag = O
For £ = 3, one has
O
F “
' 3 ,a
(C.25)
Using th e l a s t eq u at i o n in t h e Kq e q u a ti o n gives
LO(J)R = -2Fna
= O
. U ,a ■
(C.26)
100
Then t h e g e q u a ti o n (C.2) giv e s t h e a d d i t i o n a l r e s u l t
Scfq = -R = 0
where Q =
(C.27)
otCjs^h ^ ) .
• The (01) e q u a ti o n i s
™X l 0 3 (1 + ^ 2 ) = 2^ l 01ia = 0
where we have used C.24.
Therefore,
R0103 = °'5
which a l s o s a t i s f i e s t h e (13) e q u a t i o n .
S i m i l a r l y th e (02) and (23) e q u a t i o n s a re s a t i s i f e d , b y
R0203 = 0
Fpa
= 0
£ ,a
'
The remaining e q u a t i o n s a r e ( u s i n g C.25 and 0.27)
(12)
Rgiggfl + w(f>‘_)(l
- -^g-) = 0
(11)
Rq i q -] (I +-OXj)^) (I
- Tg-) = 0
(22)
^QgQgfl + w$^)(l
- ^g-) = 0
c
and
(C.28)
101
(o s)
(,(R0101
(OO)
“ ^0 3 0 3 ^ + .
(33)
Rq303CI + (OCj) ) - - y ( R 0101 + R0 2 0 2 ' ^ +
c
+
Rq 2 0 2 ^ 1
+
)
+
Q }0 3
P
^ ~ ( Rqi oi
^0202^ ^
0
O
+ uit^ ^ + ^ OO ~ ^
(C.29)
+ 0,33 = 0
I f c / I then Rq101 and Rq203 must va nish due t o e q u a t i o n s C.28,
and t h e only s o l u t i o n i s t h e t r i v i a l one o f no r a d i a t i o n .
If C = T 5
th e n t h e t h r e e C.28 e q u a t i o n s a re s a t i s f i e d i d e n t i c a l l y and t h e t h r e e
C.29 e q u a t i o n s reduce t o
R
+ R
= Q,00K0101
k0202
, ^
2
I + (Dcj)
R0303 = °*
The s a t i s f y i n g o f C.28 im pl ie s t h a t Rq101 - R0202 and Rq102 are
u n s p e c i f i e d by t h e f i e l d e q u a t i o n s .
Also
means t h a t t h e r e i s no phy s ic a l l o n g i t u d i n a l waveJ
F106
I
= 0
but
and
id
do allow non-zero A1 , Ag, which produce physica l t r a n s v e r s e waves. 1
1See Appendix D.
APPENDIX D
I t i s he re shown t h a t Fq
F0
a “ 0 does not r e p r e s e n t p h y s i c a l waves
■ 0
i s s olv ed i f
Thus, i t would a p p ea r t h a t Fqcx a allows t h e pro p a g a ti o n o f a non-zero
v e c t o r wave with t h e above r e l a t i o n s h i p between i t s components.
However,
we now show t h a t t h i s i s not a ph y s ic a l wave a t a l l , b u t a " c o o r d in a t e
r i p p l e " which can be c r e a t e d by a c o o r d i n a t e t r a n s f o r m a t i o n , and t h a t
.
’
'
. . .
t h i s wave has no o b s e r v a b l e e f f e c t s .
Consider a c o o r d i n a t e system (xy ) in which t h e r e i s no wave d i s ­
tu r b a n c e to t h e v e c t o r f i e l d .
Kp = U 0 , 0 , 0 , 0 ) , .
and e f f e c t a c o o r d i n a t e t r a n s f o r m a t i o n xy ^ x y where
X0 = x° + - ^ - s i n ( a j x ° - Kx^)
UMj,o
In t h e new c o o r d i n a t e system
103
o
or
K0 = <|>0 + S q C O S ( wx° - kx3 ) = <j>0 + Aq
Ko = - - a cos(a)X° - kx3 ) = - -^A
O
0) O
CU
This i s j u s t t h e type o f wave r e q u i r e d f o r Fq” a = 0.
There a r e two ways t h i s wave could be obs erved .
I t could couple
t o t h e m e t r i c v i a t h e f i e l d e q u a ti o n s o r i t could a f f e c t t h e g r a v i t a t i o n
a 'I c o n s t a n t as in Chapter I I I . . In t h e f i e l d e qu a ti o n s i t only appears
in t h r e e e x p r e s s i o n s
'
Y ,*
' and *°(AQ -
The f i r s t two a r e o b v io u s ly z e r o ; th e t h i r d can be shown to be zer o.
The c o o r d i n a t e t r a n s f o r m a t i o n t h a t induced t h e v e c t o r wave
induce a m e t r i c wave in an o t h e r w i s e f l a t background
g
^
.J ib is i5
9xm ax" r
9OO '
12V
f? V
9 X 9X
" (1
+ V +oX1 + V * o )r'oo
9OO ' 1 + 2aO^O 5 1 + hOO
Then
(A0 " ^ 0fl0O} = 0
will also
104
and t h e wave c a n , have no o b s e r v a b l e e f f e c t on th e m e t r i c .
F i n a l l y , t h e c o o r d i n a t e r i p p l e could a f f e c t K
; Which, as was
seen in Chap ter I I I , determines, t h e e f f e c t i v e g r a v i t a t i o n a l coupling
constant.
However,
'
kZ
.
.
" = U p + Ai i H t v + Av H n v v + Kp v )
- f / . + Z t0A0" 00 + t 02h°0
V
Noting t h a t h00 = -P qq = - Z h J , i t i s seen t h a t
*
i s unchanged
(as a s c a l a r must be ).
T h e r e f o r e t h e wave
A0 - A0COSkiiXv
a
A3 = F co sk Px*1
i s an unphysical c o o r d i n a t e wave, and may n o t be counted as a mode o f
propagation.
APPENDIX E
The a c t i o n i n t e g r a l i s
A = jd^x[L.| + L2 + L3 +'
+ L5] ,
wi t h
.
= /-gl6TrG^Lm(gy v , m a t t e r v a r i a b l e s ) ,
L2
F
V ^ g R 1
L_ = w/=gK K^R,
0
y
■L4 = T1Z ^ K 11Kv Rii v ,
L5 *
wi th
.
- K
'vly'
= K
u Iv
We r e q u i r e t h a t t h e a c t i o n be i n v a r i a n t under v a r i a t i o n o f
Vari -
a t i o n o f L-| d e f i n e s t h e s t r e s s - e n e r g y t e n s o r f o r m a t t e r . ,
— —
A g Sgvv
5
’
S ttG T
0 vv
(E -I )
L2 a l s o occurs in general . r e l a t i v i t y , from which t h e v a r i a t i o n i s known
106
t o be
I
. .SL2 '
(Ev2)
yv
/Cg
looks much l i k e a Brans-Dicke Lagrangian term', and
5 Lg
__
ax
— '— = a)/-gK' K R + toK K ------ ( / - g R ) .
gg%v
P v
a gguv
^
^
The l a s t term i s t h e same as t h e v a r i a t i o n Brans and Dicke performed
K' Ky e K co rre sp ond s to th e s c a l a r q>,
^
2
and we can use t h e i r r e s u l t t o w r i t e
to get t h e i r f i e l d eq u atio n s.
/ - I Sgvv
io(Ky KvR + KRy v - 29 y v KR + KIyv - SpvP 2 W
(E.3)
Lg i s analogous t o t h e f r e e f i e l d energy o f th e Maxwell f i e l d of e l e c t f o dynamics, whose s t r e s s - e n e r g y t e n s o r has been found to be
I
6L3
Sg^
2Fy a r v +
3
•ctB
(E.4)
i
See , f o r example, James L. Anderson, P r i n c i p l e s o f R e l a t i v i t y
P h y s i c s , Academic Press (196 7) , p. 345.
^C. Brans and R. H. D ic ke , Phys. Rev. 124, 925 (1961 ).
3
’’
See, f o r example, A d le r , Bazin , and S c h i f f e r , I n t r o d u c t i o n to
General R e l a t i v i t y , McGraw-Hill (196 5) , p. 326. ,
107
In L^5 v a r i a t i o n o f gyv gives
+ AgKVGR
(E.5)
ag
The l a s t t e r m ' s c o n t r i b u t i o n to t h e a c t i o n is
■
SI = Jd4Xv^gKa K3SRag
From t h e d e f i n i t i o n o f R n , one can w r i t e
Ctg
5R=6 = t 6 ^ h 6 where i t shou ld be noted t h a t Sr ^ i s a t e n s o r even though r ^ i s n ' t /
ctg
3
ag
T h e re fo re
s i == / d 4 x ^ [ K V ( 5r a = ) , v - . K l t s (Ora J )
]
/ d ^ A g C f ^ K ^ r ^ - Ka K6S arg J Iy) + (Ka K6 ) liiS r J - ( K V ) 1Iyi SvaJ g'
r]
Iy
ag
W rit in g t h e e x p r e s s i o n in p a r e n t h e s e s as a s i n g l e v e c t o r ,
y = KctKij5r 3 _ Ka K3SF J 5 •
• ag
ag
v i e c o l l e c t t o g e t h e r a p e r f e c t d iv e rg e nce ,
I b i d . , p . '307.
whose c o n t r i b u t i o n t o th e i n t e g r a l va n is h e s when th e boundary' o f the
re g io n o f i n t e g r a t i o n i s ta ken a t i n f i n i t y .
The i n t e g r a l then reduces
to
«1 = / d V g a A i b ^ r "
- (KciKb ) | 66 r 1‘i|] i
Now
/= 9"«% '
6 C-c Sl
S
" VI s 6-c S
(/=9)
^ 6 /-c
C- K s s vV6Sliv) , a + J t z=A) . . V g vv
- K ( S vV6Sv v ) , .
The second t e r m ' s c o n t r i b u t i o n to t h e i n t e g r a l i s then
SI2 = + K
x -c
S ( kA
I n t e g r a t i n g by p a r t s giv e s
s Cs v V6 S i j v )
109
I i' j 4
-,Pv
lag g”
pv 4s
(E.6)
where ( / ^ ? 01) a = / ^ , C a , a has again been used.
The remaining, term i s
fill = J d4XZ^(KljKv ) ^ s r i".
I t can be shown t h a t f o r any t e n s o r Mvv ■
J dttXZ^gMljv s r 01 = %rd4xZ-g[M
+ M
+ M
-M
a
uv
4;
a uy v
avy
yav
. yva
+ M ' ' - M ' • ] Qa e Sgy v ,
vay
v y a 1B3
Applying t h i s t o (KyKv ) 1 give s
Ia
I f ,4
. 61I = 2 / d
) , v :+ W
i ; - '
ag yv
( kZ v ) 1J 16S 6S
J l 1 = l / d 4x/^ g[ (K a K )
+ (K°K )
- n 2 (K K I l i g lrI •
u Iva
v Iya
y v
(E.7)
P u t t i n g E.6 and E.7 i n t o E.5 and g a t h e r i n g E.l through E.5 y i e l d s t h e
g
yv
f i e l d equation:
R
yv
+
- W
I
yv
I
R + io(K KR + KR
- -^g KR + K
- g OD K)
y v
yv
<- y v
Iyv
yv
+ V
p.
( Kv Ka)
IV. - (K/
“ ' .,P„ «J
"X
V
" X
pW
+ 2Fu « F“ v + k v Fa 6F“ e
ci6 +
e
< W '
' 8 t 6Ot UV
no
The K e q u a t i o n is. s t r a i g h t fo rw ard.
Ky d o e s . n o t ap pea r in e i t h e r
L-J o r L2 ; in Lg and L^ t h e v a r i a t i o n i s simply
I
^3
-----------— 2u)K R
vCg 6 KM
—
—
v q f 6K*
' P
= 2n Kv R
,
.
^
Lr can be w r i t t e n
o
4
- - c SCSl l V '
- K3 ,a>9V“=V6.
Then
6 L 5 - - ^ , y K ^ M g ' - V 6 - gv 6g " )
I n t e g r a t i o n o f J d x L5 by p a r t s y i e l d s
SL5 = 4[/^gKa j 3 (gyV 3 - 9 ^ 9 ^ ) ]
6L5 ' ^ < ^ 8
,V
K
11,
, y
- K6 , „ ) s " V 6] , v«Kv .
m
So t h e c o n t r i b u t i o n to t h e f i e l d e q u a ti o n i s
.J-----!L= 4p
gX
The Kjj f i e l d e q u a ti o n i s t h e r e f o r e
" V
+ " kVrpx, + 2Fp“ m
0
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-' "
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D378
H368
co p . 2
S e l l i n g s , Ronald W
A v e c to r -m e tr ic
th eo ry o f g r a v ity
MAMVAND AODMftB
HmFr
APR I a
S E 1
Z D 3 7 8
/ / 3 i >
S
(to p . S i