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Cosmological Solutions of Bi-metric Theories of Gravitation.

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Annalen der Physik. 7. Folge, Band 41, Heft 3, 1984, S. 172-178
J. A. Barth, Leipzig
Cosmological Solutions of Bi-metric Theories of Gravitation
By D.-E. LIEBSCHERand J. MUCKET
Zentralinstitut f i i r Astrophysik der Akademie der Wissenschaften der DDR, Potadam-Babelsberg
Abstract. The evidence in solar system tests of gravitation theories shows that in cosmological
models the propagation velocities of gravitation and light may differ only by action of the local gravitational potential. The condition of approximately equal propagation velocities is shown to be consistent with cosmological models of some bi-metric tetrad theories. The way to analyze the situation
is explained. We find a connection between the linear approximation of the theories with those properties of cosmological models, which are responsible for the local propagation of gravitation.
Kosmologische Lbsungen Bi-metriseher (fravitationstheorien
Inhaltsubersicht. Die Beobachtungen allgemeinrelativistischer Effekte im Sonnensystem zeigen, daB in kosmologischen Modellen sich die Ausbreitungsgeschwindigkeit der Gravitetion von der
dea Lichts nur durch die Wirkung lokalen Gravitationspotentials unterscheiden kann. Es wird gezeigt, da.13 die Bedingung der angeniiherten Gleichheit der Ausbreitungsgeschwindigkeiten mit bimetrischen Gravitationstheorien vereinbar ist. Die Analyse der Theorien wird erliiutert. Es zeigt sich,
da13 es eine enge Verbindung zwischen der linearen Niiherung der Theorien und den Eigenschaften
der kosmologischen Liisungen gibt, die die lokale Ausbreitung der Gravitation bestimmen.
Bi-metric Gravitation Theories
The authors, remembering the long-term discussion of bi-metric theories with
D. D. IVANENKO
and the seminar of the department of theoretical physics a t the MGU,
would like to present this paper in his honour. Bi-metric and tetrad theories have always
been the means to discuss alternative ways to describe gravity, and to study real and
supposed characteristics of EINSTEIN’S
General Relativity [1- 221. They have been
shown to be adequate alternatives t o test relativistic gravitation [20, 23, 241. Mainly
they have been constructed ad hoc, and only later i t was shown that they all fit into a
common scheme, which exhibits the main characteristics of bi-metric theories [25, 261.
The main objective against the most widely discussed bimetric theory of ROSEN[21]
is the behaviour of cosmological solutions [27], which yields wrong PPN-characteristics,
if the effective metric of the Riemann space time does not turn out to be conformal to
the Minkowski background metric. I n fact, if gik is not proportional to q i k in the cosmological solution, the propagation velocity of light (determined by g i k ) differs from that
of gravitation (in general determined by l;lik). The PPN-parameter will show this straightforward [241 :
D .-E. LIEBSCHER
et al., Cosmological Solutions of Bi-metric Theories of Gravitation
1i 3
N. ROSENavoids this problem later by modifying his theory in such a way, that the
cosmological model becomes itself the background metric, and is no more object of the
field equations [28]. We will show that the general problem does not need such a complication in identifying a subclass of theories allowing for proportionality of g i k with qik.
We look for cosmologies of bi-metric theories with respect to the condition
in order to single out viable theories. We discuss the general class of homogeneous bimetric theories presented in [ 2 6 ] . In deriving the equations for Robertson-Walker models we find the desired subclass.
Homogeneous Bi-metric Theories
Bi-metric theories are in fact multi-metric. Given two symmetric tensor fields qik,
and gik, we may introduce the root of their quotient by .the anholonomous transformation coefficients:
g i k = qlnaabr*
(3)
If both tensors are metric tensors, we interpret these coefficients as the anholonomous
transformation of a tetrad hp (orthonormal with respect to q i k ) into a tetrad hf (ortho0
normal to g i k ) :
h%
4 -,,k
- hAa41 )
(4)
AhR gik = q A B h i k - q A B ~ f ~ ~ a =
! @q1rna~'r*
Instead of using g i k as dynamical variable we may use ar. In doing this we get access
to the matrix functions of a to construct Lagrangians. Furthermore, every function of
a leads to a metric tensor, formally
Fik
= qlmfi(a)
fk"(a)
-
(5)
Of course these metrics have no physical meaning per se. They acquire such a physical
meaning, if they are used to construct terms of the Lagrangian, and we will use such a
construction to describe different propagation properties of gravitation.
The matter Lagrangian is supposed to be constructed by force of the weak equivalence principle with g i k and the covariant derivative induced by it. We will not try
to change here this concept of General Relativity. Only the Lagrangian for the gravitational field will be generalized. Its building blocks are
- the flat-space derivatives of a in the combinations
li
i
= qij(a )u ak,w(a')t,
y/,vw
- the scalar quotient of the determinants
1:
= det a ,
'
- the metric
G,,
= qii(a")i(an)$, G,,G"" = SV,,
to contract the covariant tensor indices. All the powers of a used in these construction
elements could be substituted by other functions of a. In using only power functions we
Ann. Physik Leipzig 41 (1984)3
174
get homogeneous theories. The Lagrangean considered reads
L = I/-r (det a)m~
U V W ~ X Y Z
[A,GuvGx~GWZ
+ A,,GuxGv~Gwz+ A,Gu~GVxGwz
+ A31GuWGxzGVU
+ A3,GVwGUZGux
A 33 (GUWGUZGXV + GVWGXZGUV)
+ A 41(GUVGZWGYZ + GXUGVWGUZ1
f A (GUVGMWGVX + GxUGUWGuz
)
42
f
+ A5,G*GxWGYZ+ A,2GVzQUwGux
f A,3(GUzG~WGVx
+ GVzGxWGu~
)I.
We use the abbreviations
A,,
= A,,
=
iA5k9
+
+
f
f
=
2A33,
A, =
A, =
2A53,
A, = A,
A,.
The coefficients A are numerical constants, which together with the exponents 1, m, n,
and r define the theory. Special emphasis should be given to the exponent n. If n = 0,
the propagation velocity of gravitation is determined by the flat background, and in
weak and quasistatic fields gravitation is faster than light [29]. For n = 1, gravitation
propagates by the same metric as other fields (Gt, = gik). For n > 1, gravitation is
slower than light in weak quasistatic fields. Terms with indices contracted by cikZrn
would destroy parity, and are not considered here.
The class presented contains
- elementary linear tetrad theories (m = n = 1 = r = 0),
- linear theories for the metric tensor (m = n = r = 0, I = 1) [16],
- the KOHLER
class [8] (m = n = I = 1, r = 0), which itself contains
- EINSTEIN'S
General Relativity (m = n = 1 = 1, r = 0 , A , = -1, A , = A, = I ,
+
6
+
+
-21,
TREDER'S
equations
[l5, 20, 251 (m = n = 1 = 0, r = -l),
- ROSEN'S
bi-metric theory [all (m = n = r = 0, 1 = -1).
In the last two cases A , = -2A, = 1, the other coefficients vanish. The following properties are given without proof.
Two cases are of special importance. First, if only A,, A,,, and A , do not vanish,
then we get wave equations for all components of the tensor a straightforward (Poisson
case). We are forced, otherwise, to define gauge-reduced quantities to exhibit the propagation properties. Secondly, we may postulate - in analogy t o EINSTEIN'S
theory that the linear approximation allows for gauge transformations of the solutions. That
means, together with a solution ui, also ui. Buh with
- the special class of bi-metric tetrad theories developped from
+
Buk =
+ uj.,/tl
(7)
shall solve the field equations [16]. Gauge invariance in the linear approximation is
equivalent to
+
A,
A4 = 0,
A,, f A63 = 0,
-421
+ A61 = 0,
f
+ A4 = 0.
-462
4
3
(8)
D.-E. LIEBSCHER
et al., Cosmological Solutions of Bi-metric Theories of Gravitation
17.5
TWO combinations of coefficients determine the behaviour of linear effects, C, =
A , A,, and C , = A ,
A , A,. I n the gauge case just considered both C,, and C,
vanish. The coefficient A , is chosen t o be 1, otherwise we would have to redefine the
coupling constant x . The light deflection has the EINSTEIN
value, if
+
+ +
+
( A , - A,) c, = (4 A,) c,.
(9)
I n the Poisson case we get
2A1 A , = 0 .
I n the case of C, = 0 we find
A, = A , = 0.
The only nonlinear effect of the post-Newtonian approximation, the perihelion shift,
is determined by the exponents of a in the Lagrangian. We get the EINSTEIN
value, if
m+n- 1 -r = 1.
(10)
+
We will show in the following section that the desired subclass is given by
4C,
+ C, = 0,
and
2n = m
+ 1.
(11)
Cosmological Models
We want to inspect cosmological models i.e. the question how the Robertson-Walker
metrics
+
+
ds2 = H2(t)dt2 - A ( t ) ( d 9 d$
dz2)
(12)
fit into the theory. We do so only for the simplest case, in which t , x, y, and z are' Cartesian coordinates of the back-ground metric, i.e. qik shall take the usual numerical
values. One can see by simple inspection that the field equations are reduced to two
equations for H , and A , if we substitute into the field equations the form
a!JH, a l - a 22 -- a 3 3- - A , a i = 0 , if i# k.
(13)
Therefore, we may substitute it already in the Lagrangian, and we may derive the field
equations by variation with respect t o H and A . The action is reduced t o
s = $ dt L + smatter,
(14)
with k = 2n - 1 - r ,
+ A,,
01
= 3A1
B
= 3c1,
y=c,+c,.
6L
and -contains no second derivatives.
6H
For a n ideal fluid we get the Euler-Lagrange equations
I n the gauge case, we find?!, = y
= 0,
(1.5)
Ann. Physik Leipzig 41 (1984)3
176
I n addition, by the construction of Smatter
we get the energy conservation equation in
the form
This relation is independent of the gravitational part of the Lagrangian. The structure
of the gravitational part leads to another conservation law. L does not depend on t
explicitely. Therefore,
aL
H-
aL
+A - L + %HA3@= const,
aa
aA
which may be reduced by the homogeneity of L in A and H to
+
L xHA3e = M .
This constant M means something like the energy content of the universe, including
gravitation. We shall see, that a necessary condition to get the models we want to find
is M = 0.
The Euler-Lagrange equations read explicitely
2yD(H) + 2,9D(A)= H2n--nzfk-1A-3m
( ( m - 2n) L - %HA3@),
(18)
2,9D(H) 2aD(A) = H2n-mA-3mfk-'(3mL 3xHA3p) .
(19)
with
H
(m - 2n) - + 3m H
The first question to put is the question about the possibility of A = H . We have two
equations for one variable A , if A = H . They are identical, if
(a ,9) ((m - 2n) L - xHA3e) = (,9 y ) (3mL 3xHA3p).
(21)
Taking into account, that L = M - .HA", we bave to compare the coefficients of M ,
e, and p. Therefore, we find the following cases.
1. I n a radiation universe, e = 32, = NA-4, and eq. (21) is just a condition M =
M ( N ) for the two integration constants M , and N .
2 . Any other equation of state p = p ( e ) requires ,6 y = 0 in order to allow for
H = A.
In t*hissecond case, we still have to ensure that
(OL +j3) (m - 2 n + 1) = 0 .
Again, we have to restrict the parameters by
m - 2n
1 = 0.
(22)
The sum OL ,9 cannot be zero, or the equation of state is restricted again to radiation.
If m - 2n
1 vanishes, the two Euler-Lagrange equations are equal modulo eq. (16).
Their first integral is given by eq. (17), which represents the equivalent for the corresponding Friedmann equation. If we denote the cosmological time by
+
+
+
+
+
+
+
+
+
dz = H (t ) at,
we may write this equation
It coincides with the Friedmann expansion law, if 3m - 2k - 1 = 0.
(23)
D.-E. LXEBSCHER
et al., Cosmological Solutions of Bi-metric Theories of Gravitation
177
After finding the condition for existence of solutions with H = A in ikl = 0, and
1 = 0, we have t o put the question about the behaviour of the other solutions
to the solutions with H = A . We got one equation free of source. If = - y # 0, it
reads
rn - 2 n
+
D(H) = D ( A ) .
The case o f @= y = 0 will be considered later. Setting
H =A(l E),
we get for the first approximation in E the equation
(25)
+
For a n expanding universe
3m
+ 1 > 2k,
c
-> 0
)
we find damped oszillations in E , if
and
The case of @ = y = 0 represents theories, which are gaugeinvariant in the linear
approximation. I n contrast to the case just considered, eq. (18) yields no further condition on H and A. That means that we are free to choose any function H ( t ) , and we get
again eq. (24), with LX
@ = -2. This is an equation independent of H , because we use
the cosmological time, eq. (23). It corresponds to the Friedmann equation for the
Hubble parameter. H is now open t o be gauged t o H = A , and there is no question of
stability of this choice.
+
Conclusion
Bi-metric tetrad theories may well allow for cosmologies, which do not lead to preference in the local system of coordinates by different causal cones of the background
and the effective metric gik. Therefore, bi-metric theories are not excluded by the evidence for the PPN-parameter a, to be zero. Furthermore, the condition of gauge invariance in the linear approximation just leads to these cosmologies. However, we get
in these theories, for universes filled with matter, the equality H = A only in the case
of eq. (22),
2n=mf1.
If we now remind the reason for our effort, the evidence for a, = 0, we have also to
take into account the perihelion condition, eq. (lo),
k f m = n + 1.
Combining both conditions, the Priedmann equation will be
and the radiation universe expands like
2
-
A oc zsn-4,
the dust universe like
A oc
2
-
t81*-5.
(28)
178
Ann. Physik Leipzig 41 (1984) 3
The conclusion a t hand would be, that n = 1 and by force of eqs. (22), and (28) also
m = k = 1 would be strongly favoured by primordial nucleosynthesis data. However,
if n = 1, there is no problem with the parameter 0 1 ~ . Now, the metric Gik of eq. (6),
ruling the propagation of gravitation, does not differ from gik, ruling the propagation
of non-gravitational fields. Therefore, the coefficient 0 1 ~vanishes independently of the
relation between gik, and qin. The subclass n = 1 of hi-metric theories meets the a*condition without restriction on the other parameters k, m, and A .
A c k n o w l e d g e m e n t s . The authors are obliged to H.-J. TREDER,and U. KASPER
for valuable and helpful discussions.
References
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V. A.: Z. Phys. 54 (1929) 798.
[5] IVANENKO,
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[6] WEITZENBOECH,
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[22] LOCUNOV,
[23] NORDTVEDT,
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D.-E.: Astron. Nachr. 295 (1974) 11.
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[%I KASPER,U.; LIEBSCHER,
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[26] LIEBSCHER,
[27] EARDLEY,
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[29] LIEBSCHER,
Bei der Redaktion eingegangen am 24. Mai 1984.
Anschr. d. Verf.: Prof. Dr. sc. nat. DIERCK-EKKEHARD
LIEBSCHER
Dr. rer. nat. JAN
MUCHET
ZI f. Astrophysik d. AdW der DDR
DDR-1502 Potsdam-Babelsberg
Rosa-Luxemburg-Str. 17a
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