The post-Newtonian limit of Rosen's theory of gravity is evaluated and is shown to be identical to that of general relativity, except for the PPN parameter alpha sub 2, which is related to the difference in propagation speeds for gravitational and electromagnetic waves. Both the value of alpha sub 2 and the value of the Newtonian gravitational constant depend on the present cosmological structure of the Universe. If the cosmological structure has a specific but presumably special form, the Newtonian gravitational constant assumes its current value, alpha sub 2 is zero, the post-Newtonian limit of Rosen's theory is identical to that of general relativity--and standard solar system experiments cannot distinguish between the two theories

Caves, C. M.

Lee, D. L.

English

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https://ntrs.nasa.gov/search.jsp?R=19760003862 2020-03-22T19:16:52+00:00Z
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THEORETICAL FRAMEWORKS FOR TESTING RELATIVISTIC GRAVITY.
V. POST-NEWTONIAN LIMIT OF ROSEN'S TIIEORY^
DAVID L. LEE and CARLTON M. CAVES 
California Institute of Technology, Pasadena, California 91125
WEI -TOU NI
Montana State University and National Tsing Hua University$
and
CLIFFCkD M. WI%0
University of Chicago and Stanford University#
THEORETICAL FRAMFWGFKS FOR TESTING
RELATIVISTIC GRaVITY. 5: POSY-NFWTONIAN
LIMIT OF ROSEN'S THEORY (California Inst. of
Tech.)
	 11 p HC $3.25 
	 CSC' 03B
N76-10950
Uncla.9
G3/90 39446
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Supported in part by the National Aeronautics and Space Administration
[NCR-01-002-25F, at Cal*_ech; NCR-27-001 -03J at Montana Sta-el and by
the NaLional Science Foundation 'MPS75-01398 at Caltech; GP-34721X at
Chicago; MP:1	 521• at Stanford].
t
NSF predoctoral Fellow.
$ Present address.
§Alfred P. Sloar. Foundation Research Fellow.
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C92P sue_
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ABSTRACT
The post-Newtonian limit of Rosen's theory of gravity is
evaluated and is shown to be identical to that of general
relativity, except for the PPN parameter a Z (which is related
to the difference in propagation speeds :or gravitational and
electromagnetic waves). Both the value of 02 and the value of
the Newtonian gravitational constant depend on the present
cosmological structure of the Universe. If the cosmological
structure has a specific (but r:esumably special) form, the
Newtonian gravitational constant assumes its current value,
or2 is zero, the post-Newtonian limit of Rosen's theory is
identical to that of general relativity--and standard solar
system experiments cannot distinguish between the two theories.
-i-
T	
1-7
I . I NT RODUCT ION
Nathan Rosen (1973, 1974) has recently devised a new theory of
gravity. It is a bimetric theory in the sense that it possesses two
f^
metric fields--a flat metric Tj ij with "light cones" along which weak
gravitational waves propagate, and a curved metric g ij with "light
cones" along which light propagates. It is also a "metric theory"
In the sense that the Einstein equivalence principle holds in the
I
local Lorentz frames of the "physical metric" g ij (cf. § 39.2 of
Misner, Thorne. and Wheeler 1973--cited henceforth as nl'6,). The field
equations of the theory are derivable from a variational principle. In
forthcoming papers Rosen and his colleagues will use the theory to
analyze cosmology and neutron stars.
In this paper we evaluate the post-Newtonian limit of Rosen's
theory by considering an isolated system with weak internal gravity
(such as the solar system). We express our results in the language
of 'he Nordtvedt-Will Parametrized Post-Newtonian (PPN) Formalism
(see Chapter 39 of MTW; also the review by Will 1974). At any particular
epoch in the evolution of the Universe, the boundary conditions in the
asymptotically flat region far outside the system are determined by the
cosmological structure of the Universe c.t that epoch. We assume
boundary condition-3 which are appropriate for a homogeneous, isotropic
cosmological model. We find that the Newtonian gravitational constant
and the value of the PPN parameter or2 depend on the cosmological
boundary values. For a particular choice of cosmological boundary
values, the Newtonian gravitational constant assumes its present value,
a2 is zero, and the post-Newtonian limit is identical to that of
general relativity.
The notation and format of this paper will be the same as in
Ni's (1972) compendium of metric theories of gravity.
II. EVALUATION OF POST-NEWTONIAN LIMIT
a. Principal References: Rosen (1973, 1974)
b. Gravitational Fields:	 g ,r)
A semicolon ";" and a slash "I" denote covariant derivatives with
respect to g and tl, respectively.
c. Fiel d Equations:
lklgi l j k l _
	
kl g mng
mi I kgn j I I 	 Go l7 1 i j - 1 91 
J gk1Tk1 ^	 (1)
Riem( t0 = U	 (^)
The field equations are written in units with the speed of light
c - 1. G0 is a coupling constant with the dimensions of the Newtonian
gravitational constant. The field equations are invariant under the
resealing given by " i
 -^	 1 , j 	 A ^?i''
	9 i' a	 gi	 gi •, andJ	 J	 J	 J	 J	 J
G 00 
= AG 0 , where A is a constant; therefore, G 0 can be chosen toGo -^ 
have any convenient value. We choose Go to be the Newtonian gravitational
constant in the solar system today. In his original presentation Rosen
chose units such that G = 1.
G
d. The Post-Newtonian Solution:
Consider an isolated system with weak internal gravity. As in
general relativity (MTW, Chapter 19), so also in Roser's J ,.eory, there
-2-
exist coordinate systems ix i) in which g ij is asymptotically Minkouskiian:
11g ij 11 - diag(1, -1, -1, -1)	 far from source.	 (3)
However, the cosmological structure of the Universe at the epoch of
interest will typically force Tl ij to have a non-Minkowskiian form in
these coordinates. To avoid this complication, we shall compute in
coordinates txi } with
diag(1, -1, -1, -1)
	
(4a)
(B)
11gi,j,ll -► 	 jjg i , j ,11 0 diag(1, -1, -1, -1)	 far from source.	 (4b)
By definition the cosmological background metric (eq. [4b]) changes only
on cosmological time soles. Since cosmological time scales are so much
longer than the dynamical time scales of a system such as the solar
system, we can ignore the variation of the cosmological background metric
in calculating the post-Newtonian limit.
We can simplify the cosmological background metric, without
destroying the form (eq. [4aj) of r i , j
 „ by performing an appropriate
Lorentz transformation. More specifically, by an appropriate boost we
can set g0 I CY, = 0 , and by a subsequent rotation we can diagonalire
i	
^B)
	
g( , B 	 thereby obtaining
(B)
^^ fi i - 
j 11
	
diag(co, -c l , -c z , -c 3 ) .	 (5)
In general the four constants c  will be different. However, for
the simple case of a homogeneous, isotropic cosmological model, a further
simplification of the cosmological background metric is possible (Rosen
1975, Caves 1975). In the universal rest frame of such a cosmological
-3-
model (frame comoving with the cosmological fluid; frame in which the
black-body background radiation is isotropic), equation (5) holds with
C  = c2 = c3 . He:ceforth, we will restrict ourselves to this case:
	
8 ( B ) ^11	 diag(c , -c , -c , -c )	 (6)1 i
	
o	 1	 1	 1
The simple form of equation (6) will be maintained under boosts if and
only if co = c  . Therefore, if co ¢ cl , assuming equation (6) is
equivalent to assuming that our calculation of the post-Newtonian limit
is carried out in the universal rest frame.
We first solve the field eauations (1) to Newtonian order, The
solution +.s
90,101= c o[1 - 2c l (c oc l ) 1/2coU'] ,	 (7a)
g
o 1a
l = 0 ,	 (7b)
ga ,,` ,
_ - c 1 6op	 (70
where U' is the Newtonian potential:
	
U , (x,.t,)	 =	 P(^,t^)	 dF'	 (d)
where P is the rest mass density in therp oper rest frame of the material
[frame with	 ij^^ = diag(1, -1, -1, -1) and u l = 50 , where u i is
the 4-velocity of the material].
In order to compare with the correct Newtonian limit, we introduce
coordinates ^x i ) in which g ib is asymptotically Minkowskiian:
-4-
Tr
xO ^ C	 xO	 r0
a	 art
x = c l
 x
In these new "unprimed" coordinates the metric (eqs. 771) becomes
Soo	 0= 1 - 2	 (c c 1 )i G0U
9 o = o ,
(9a)
(9b)
(1C..)
(lob)
%P 
s 
-bap '
where we have rescaled the Net, potential appropriately, i.e.,
(10c)
(11)U(x,t) - clU'(x',t') =	 P	 t)	 d 
Ix -^
It follows from equations (10) that, at any given epoch in the evolution
of the Universe. the Newtonian gravitational constant is given by
t
C = (c0 c l ) G0	 (12)
For the remainder of the calculation we choose units such that, at the
particular ep.)ch of interest, G = 1.
We now use standard procedures to solve the field equations (1) to
post-Newtonian order. In the standard post-Newtonian gauge the solution
is
90'0' = c o[ 1 - 2c 1U' + 2ci0 1 ) 2 - 0'j ,	 (13a)
-5-
C2	
1 
C2g 0 ,^, , . ci [(4 - 1 i ) V^ + z cl Wa ] ^	 (13b)
go, , P , _ -c 1 6^[1 + 2c 1 U 1 ] 1	 (130
where the potentials V' , , Wa are as on page 1085 of MTW, where
P(K'.t')4''('.t')
t') -	 d, 	 (14a)
2
cl ( v • ) 2 + c 2 U' + ?c 1 n +	 cl P	 (14b)
0
and where p and TI are pressure and specific internal energy in the
proper rest trame of the material and v' is the coordinate velocity
of the material. v' = dx'/dx 0 .
In order to compare with the PPN formalism, we now transform the
metric (eqs. [131) to the "unprimed" coordinates (eqs. [91). After
rescaling the various potentials accordingly, e.g.,
i
	
Vci (x,t) = cl(cl/co);,Vo,(x',t')	 ,	 (15)
the transformed solution is
900 = 1 - 211 + 2U 2 - 0	 (16 a)
	
C	 c
9 = ( 4 - 2 c
o ) Va + 2 c0 Wa	 (16b)OU	 1	 1
ga13 _ -6 aR (1 + 2U)	 (16c)
where the potentials U, Va, W01 , ^ are all as on page 1085 of MTW. Ry
-6-
i	 I
comparing ,  equations (16) with the standard 11 1 1N metric (see, e.g.,
M'rw ,:39,8 and e4. [4] of hill 1973), we obtain the following values
f or the PPN parameters of Rosen's theory:
Y = (l	 l;	 bl ' ; ' b 3 	 e4	 bw 	 otl = a3 - 0
a2 = (co /c 1 ) - 1
	
(17)
The PPN parameters of Rosen's theory are identical to those of
general relativity, except for the preferred frame parameter or which
	
is nonzero whenever c # c, . The value of o'2 call 	 related ro theo	 ,.
relative propagation s;>e-ds of electromagnetic and weak gravitational
waves in Rosen's theory, In the "unprimed" coordinates the physical
metric has the asymptotic formgij^^ -^ diag(1, -1, -1, -1), while
	
the flat metric 't)	 has the form	 ^^ ^ i j^^ s diag(col,-ell, -C
Thus the vacuum, linearized equations for 
gij 
(wave equations for
weak gravitational waves) take the form
	
(co/cl)8i
J+ oo	 gij	 0
	
(18)
	
with s peed v
	
1/1whose solution is a .ave propagating,
 P	 g = (c 1 /c o )	 In
the "unprimed" coordinates, electromagnetic waves propagate with speed
unity. Thus the PPN parameter 
o2 
measures the relative difference
in ;peeds (as measured by an observer at rest in the universal rest
frame) between electromagnetic and gravitational waves in Rosen's
theory, and is given by
-7-
-7
02 	 ( 1 /v g) - 1	 (19)
(see Will 1971 for another example of a the-'.y with this property).
Earth-tide measurements place an upper limit on 
a2 given by (Will 1971)
	
2	 T	 (1 , 'vg) - l I < 7 x 10"?0 	 .	 (20)
Equations (12) ind (17) show that, in Rosen's theory, the values
of both the Newtonian g.-ev tational constant and the PPN parameter 02
depend on the cosmological boundary values. In general, both will
change as the Universe evolves. In a separate paper, Caves (1975)
will analyze the homogeneous, isotropic cosmologies of Rosen's theory.
Ile will show that there exists a special epoch in such cosmologies
at which co s cl = 1. At that epoch the Newtonian gravitational constant
has its current value. and the PIN parameters are precisely the same
as those of general relativity.
Notice that the constants co and c  can he written in terms of
scalars constructed from rl and 5 .
a (3
Co + 3cl
	11
	
gap	 (21a)
I + c - =	 0(3 g 0^i	 (21b)
	
0	 1
where the expressions on the right are to be evaluated in the asymptotically
flat region outside the solar system.
-8-
REFERLNCFS
Caves. C. M. 1975, paper in preparation.
Misner, C. W., Thorne, K. S., and Wheeler, J. A. 1973, Gravi
	 Lion
(San Francisco: W. N. Freeman & Co.).
Ni. W.-T.	 1972, Ap. J. , 176, 769.
Rosen. N. 1973. J. Gen. Rel. and Gray., 4 , 435.
Rosen, N. 1974. Ann. Phys. (N. Y.) , 84, 455.
Rosen, N. 1975, paper in preparation.
Will, C. M. 1971, Ap. J. . 169, 141.
Will, C. M. 1973, Ap. J. . 185, 31.
Will, C. M. 1974, in Experimental Gravitation: Proceedings of Course
56 of the International School of Physics "Enrico Fermi" ed.
B. Hertotti (New York: .Academic Press), p. 1.
1
-9-


Theoretical frameworks for testing relativistic gravity. 5: Post-Newtonian limit of Rosen's theory

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