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General Relativify and Gravitation, Vol. 9, No. 7 (1978), pp. 575-583 Plane Waves in the Bimetric Gravitation Theory ITZHAK GOLDMAN 1 Department of Physics, Technion-lsrael Institute of Technology, Haifa, Israel Received May 24, 19 77 Abstract Field equations for plane waves are set up and some solutions are obtained. Transverse, longitudinal, and mixed waves are possible. In the purely transverse case the energy density is positive definite. In the purely longitudinal case it may be negative. However, in one ex- ample investigated it is positive if the Riemannian metric tensor satisfies a condition corre- sponding to space-time having satisfactory physical properties. w Introduction The purpose of the present paper is to investigate the solutions of the field equations of the bimetric theory of gravitation [1, 2], describing gravitational plane waves in empty space. The field equations in vacuum are given by N, ~ =g(g xvl~)l~' -0 (1.1) where a vertical line denotes a covariant derivative with respect to the flat-space metric tensor 7~v, determined by the choice of coordinate system, and where g~v is the Riemannian metric tensor, to be determined by the equations. From (1.i) one gets 1 Nu u =- gA~,tx - - 0 (1.2) where Ax -- Fgo,~x (1.3) i Present Address: Department of Nuclear Physics, Weizmann Institute of Science, Rehovot, Israel. 575 0001-7701 / 78/ 0700-0575505.00/0 1978 Plenum Publishing Corporation

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Page 1: Plane waves in the bimetric gravitation theory

General Relativify and Gravitation, Vol. 9, No. 7 (1978), pp. 575-583

Plane Waves in the Bimetric Gravitation Theory

ITZHAK GOLDMAN 1

Department o f Physics, Technion-lsrael Institute o f Technology, Haifa, Israel

Received May 24, 19 77

Abs t rac t

Field equations for plane waves are set up and some solutions are obtained. Transverse, longitudinal, and mixed waves are possible. In the purely transverse case the energy density is positive definite. In the purely longitudinal case it may be negative. However, in one ex- ample investigated it is positive if the Riemannian metric tensor satisfies a condition corre- sponding to space-time having satisfactory physical properties.

w In troduct ion

The purpose o f the present paper is to investigate the solutions of the field equations of the bimetric theory of gravitation [1, 2] , describing gravitational plane waves in empty space.

The field equations in vacuum are given by

N , ~ = g ( g xv l~) l~ ' - 0 (1.1)

where a vertical line denotes a covariant derivative with respect to the flat-space metric tensor 7~v, determined by the choice of coordinate system, and where g~v is the Riemannian metric tensor, to be determined by the equations. From (1 . i ) one gets

1 N u u =- gA~, tx - - 0 (1.2)

where

Ax -- F g o , ~ x (1.3)

i Present Address: Department of Nuclear Physics, Weizmann Institute of Science, Rehovot, Israel.

575 0001-7701 / 78 / 0700-0575505.00/0 �9 1978 Plenum Publishing Corporation

Page 2: Plane waves in the bimetric gravitation theory

576 GOLDMAN

and a line under an index indicates raising it by means of @v. From the two metric tensors one can form the scalar

K = (gl,y)l/2

and one sees that

(1.4)

Aa = 2~,x/~ (1.5)

where a comma denotes an ordinary partial derivative. In the bimetric theory one has the gravitational energy-momentum density

tensor t J given by [2] :

32rrtuV= gaOgOrgaotugorlv_ �89 u 1 v at) or _ $6u (g g gaolo~gort~ �89

The component to o is the gravitational energy density.

Let us take a coordinate system in which 7uv reduces to r~uv = diag (1, - 1, -1, -1) , the metric tensor of special relativity. To obtain plane-wave solutions, let us look for solutions depending only on x ~ and x : . It is readily seen that, if one takes guy = g~v( x~ + x l ) , the field equations are satisfied. A detailed knowl- edge of the source is needed if one wishes to establish relations among the com- ponents g~v. In the general case, in contrast to the situation in the general rela- tivity theory, it is not possible to impose coordinate conditions [1, 2] , and one can have longitudinal, as well as transverse, components in the plane wave.

In the following, more general solutions will be considered.

w Diagonal Metric

Let us first take up the simple case in which guy is diagonal. If we write

goo = e : % ) , gll = - e 2~(1), g22 = - e2%), gaa = - e : % ) (2.1)

equation (1.1) gives

~b(~),oo - ~b(a),ll = 0 (2.2)

so that, in the general case,

r = f a ( x~ - x : ) +ga(x ~ +x l) (2.3)

where fa and gx are arbitrary functions of their arguments. As mentioned above, the relations among the components of the metric de-

pend on the nature of the source. On the basis of equation (1.2) we could have, for example,

Ax : 0 (2.4)

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P L A N E WAVES IN BIMETRIC G R A V I T A T I O N T H E O R Y 577

in which case we can write, to within an additive constant

= 0 ( 2 . 5 )

Another possibility is the condition (which is inconsistent with the field equations in the general case, but which can be taken for guy diagonal)

(Kg~V)lv = 0 (2.6)

a covariant condition corresponding to the De Donder condition in general rela- tivity. This gives two relations:

[~(0) - ~ (1 ) - ~ (2 ) - 4 ( 3 ) ] ,0 = 0 (2.7)

[0~1) - r - r - r ,~ = 0 (2.8)

If we disregard static or spatially constant functions, we can take, to within additive constants

~(1) = 0(o), 4(3) = -4(2) (2.9)

There are two special cases of interest. The transverse case is given by

q~(o) = 4~(t) = 0, 4(3) = -~(~) (2.10)

In this case equation (2.4) holds, in addition to (2.6). The longitudinal case is given by

0(1) = ~b(o), ~(2) = 4(3) = 0 (2.1 1)

Let us now look at the energy density to o , as given by (1.6). In the general diagonal case we have

327rto ~ = 2 ~ [0(x),o] 2 + 2 ~ [0(~),1 ] 2 _ ~-~/b(h), ~ ~ 0(o),o

- Z ~b(x), 1 Z0(o ) ,1 (2.12)

In general one can have to o < 0. However, if either condition (2.4) or (2.6) holds then to o >~ O.

w Transverse Plane Waves

Let us take the metric guy corresponding to the line element

ds 2 = (dx~ 2 - (dxl) 2 - cosh Xe 2~ (dx2) 2 - 2 sinh Xe~+Xd3c~dx 3

- cosh )~e2X(dx3) 2

where r x a n d ~,are functions o f x ~ a n d x ~, so that

= e ~+x > 0

(3.1)

(3.2)

Page 4: Plane waves in the bimetric gravitation theory

578 GOLt)MAN

We have here a metric describing transverse plane waves. This can be ssen with the help of the method of Amowitt et al. [3]. One can also see it by noting that Rloxo vanishes while R2o2o and R~oao may be different from zero. It follows that two masses connected by a spring and constrained to move along the line joining them will not be set in motion by the field if this line is along the x 1 axis [4].

The field equations (1.1) now take the form

N22 =r /~[~b ,~ + �89 sinh 2 X ( ~ - X),~],~ = 0 (3.3)

N33 -=-~/~r + �89 sinh2 X(X- ~b),~],t~ = 0 (3.4)

N23 = �89 ~/~r [e ~o -xx,,~ _ sinh X cosh Xe ~ -x( ~b - X),~] ,t~ = 0 (3.5)

N32 = �89 ~/~ [e • X,,~ - sinh X cosh Xe x-~ (• - ~b), ~1 ,~ = 0 (3.6)

The last equation is a consequence of the others. Adding (3.3) and (3.4) gives

r/~r + X),et~ = 0 (3.7)

Subtracting (3.4) from (3.3) gives

,r/sO (cosh 2 Xo,~),# = 0 (3.8)

with

o = @- • (3.9)

while adding (3.8) and (3.10) leads to

~t~ [cosh oX,,~ - sinh Xcosh X (cosh o) ,a] ,# = 0 (3.10)

Combining (3.7) and (3.6) leads to

~?a~ ( X , ~ - sinh Xcosh Xo,~ o,t~)= 0 (3.11)

Let us consider a rotation of the coordinate system about the x 1 axis through an infinitesimal angle �89 Under such a rotation the general form of the line element (3.1) will remain unchanged. If we write

x 2 >x 2 +~x 2, x 3 >x a +6x 3 (3.12)

then

5 x ~ = - e x ~, 6 x ~ = e x ~ (3.13)

If we consider the transformation of the metric tensor

guy - - - - ~ g u v + 6g~u (3.14)

and write

gz2 = - cosh Xe ~r , g~3 = - cosh Xe ~x, gzz = - sinh Xe ~+x (3.15)

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PLANE WAVES IN BIMETRIC GRAVITATION THEORY 579

then we obtain the relations

6(~0 + X) = 0 (3.16)

6X = -2e sinh a (3.17)

8a = 2e tanh )t cosh o (3.18)

From the last two equations we can write

6 (cosh X cosh o) = 0 (3.19)

Hence the quantities invariant under a rotation are ~ + X, which satisfies the wave equation (3.7) and

f = cosh )t cosh a (3.20)

l fwe go over to new field variables

v = tanh a, w = tanh X/cosh o (3.21)

so that

f = (1 - v 2 - w2) -112 (3.22)

then equations (3.8) and (3.10)lead to

r c~),~ = 0 (3.23)

rlc~(f2w, o~),~ = 0 (3.24)

One can get a class of solutions by setting

w = By (B = const) (3.25)

or

tanh ?t = B sinh a (3.26)

One finds

tanh cr = (1 + B2) -1/: tanh h (3.27)

where h(x ~ x ~) is an arbitrary solution of the wave equation. Calculating the energy density to o one finds

= 2 + cosh: X(a,o ~ + 0,12) (3.28) 32rrto ~ X,o: + X,1

Since this is positive definite, it follows that a transverse wave propagating in a definite direction will carry positive energy.

Since in (3.28) only X and a appear (and not ~ + X), one is inclined to regard X and o as describing the true dynamical degrees of freedom.

Page 6: Plane waves in the bimetric gravitation theory

580 GOL DM AN

Finally it should be noted that if, instead of (3.1), one takes as the line element

ds 2 = (dx~ 2 - (dx2) 2 - cosh X e 2 ~ ( d x l ) 2 - 2 sinh XeqJ+Xdxldx 3

- cosh Xe2X(dx3) 2 (3.29)

one gets the same field equations and the same energy density as before. How- ever, one now has longitudinal-transverse, or mixed waves as one can see by examining the curvature tensor.

w Longitudinal Plane Waves

To describe a purely longitudinal plane wave let us take

ds 2 : cos 7te20(dx~ 2 + 2 sin k e ~ d x ~ 1 - cos X e 2 ~ ( d x l ) 2 - (dx2) 2 - (dx3) 2

(4.1)

where ~, if, and X are functions o f x ~ and x 1. One sees that

= e ~+~ > 0 ( 4 . 2 )

It is found that R2o2o and R3o3o vanish, while Rlmo is in general different from zero, as expected.

The field equations (1.1) now take the form

No ~ = r/a#[O,a + �89 sin 2 X(@ - 0),al ,t~ = 0 (4.3)

Nx 1 = r/a#[@,a + �89 sin 2 X(q~ - tP),a] ,# = 0 (4.4)

Nx ~ = �89 [eq)-q~X,a - sin Xcos Xe ~ -r - q~),a] ,t~ = 0 (4.5)

No 1 = - �89 [e ~-0 X,e - sin X cos Xe~-r - ~O),a] ,~ = 0 (4.6)

Equation (4.6) can be derived from the other equations. Adding (4.3) and (4.4) gives the wave equation

na#(~ + if), a# = 0 (4.7)

Subtracting (4.4) from (4.3), one can write

r/a# (cos 2 Xa, a),# = 0 (4.8)

where now

a = ~b - 6 ( 4 . 9 )

Subtracting (4.6) from (4.5) leads to

rt e# [cosh ok ,~ - sin X cos X (cosh a),~] ,t~ = 0 (4.10)

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PLANE WAVES IN BIMETRIC GRAVITATION THEORY 581

and combining it with (4.8) gives

~ ( X , ~ - sin Xcos X a , ~ a , ~ ) = 0 (4.1 1)

In analogy to what was done in the preceding section, let us now carry out an infinitesimal Lorentz transformation on x ~ x 1, so that

8 x ~ = e x 1, 6 x 1 = e x ~ (4.12)

Under this transformation the general form of the line element (4.1) remains unchanged, but the components of the metric tensor are changed. One finds

6 ( r @)= 0 (4.13)

8X = 2e sinh a (4.14)

8 o = 2e tan X cosh a (4.15)

so that

6 (cos X cosh a) = 0 (4.16)

Hence the quantities invariant under a boost are 4~ + @ and

f = cos X cosh a (4.17)

If we now take as variables

v = tanh a, w = tan ~,/cosh a (4.18)

so that

f = (1 + w 2 - 02) -'/2 (4.19)

equations (4.8) and (4.10) lead to

r l ~ ( f 2 v , ~ ) , ~ = 0 (4.20)

~ ' ~ ( f = w ~ ) , ~ = 0 (4.21)

Let us look for a solution for which

w = B y (B = const) (4.22)

One finds, depending on the value of B,

f h, B 2 = 1

tanh a tanh h, < 1 (4.23) = ( l - B 2 ) -112 B 2

(B = - I) -1/2 tan h, B 2 > 1

where h ( x ~ x 1) is an arbitrary solution of the wave equation subject to the re- striction [ tanh a[ < 1.

Page 8: Plane waves in the bimetric gravitation theory

582 GOLDMAN

In the present case the gravitational energy density is given by to o , where

32rrto ~ = cos 2 X(a,o 2 + o, 12) - X,o 2 - X,12 (4.24)

Here again the expression suggests that o and 3, be regarded as describing the dynamical degrees of freedom of the wave. However, in the present case the energy density is not positive definite, in contrast to (3.28). For the solution satisfying (4.22) one finds

327rto ~ = (1 - B 2) (1 + B 2 sinh 2 17)-2(O,0 2 + O, 12) (4.25)

If one wants to o not to be negative, one must take B 2 ~< 1. Moreover, there ap- pears to be a connection between the sign of to o and the geometry of space- time, and this will be investigated in the following section.

w Positiveness o f Energy Density

For the solution satisfying equation (4.22), one sees that with the line element as given by (4.1) one has the relation

2g01 = B(goo + g n ) (5.1)

Hence to o will be non-negative if

Igoo + g u [ >t 2lgoll (5.2)

Squaring both sides, we can write this

Igoo - gn l z /> 4A (5.3)

where

- A = goo g n - go12 (5.4)

Let us consider the eigenvectors of guy, A " , defined by

glav A v = brluvA v (5.5)

The possible values of b are given by the roots of the secular equation

Igtav - b71uvl = 0 (5.6)

For the metric of (4.1), two of the roots are b = 1. The other two are given by

goo - b go1 = 0 (5.7) got gl l + b

so that

b = �89 - g11) +- ~1 [goo + g11) 2 - 4gol 2] 1/2 (5.8)

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PLANE WAVES IN BIMETRIC GRAVITATION THEORY 583

We see that equations (5.2) or (5.3) are the conditions that the roots of (5.7) or (5.8) should be real and therefore that the eigenvectors of (5.5) should be real.

From (5.5) we see that

g u v A U A v = b ~ u v A U A v (5.9)

Hence, one can go further and require not only that the eigenvalues, b, should be real, but that they should be positive. This ensures that three eigenvectors will be spacelike with respect to bo thguv and 3'~v and one will be timelike with respect to both of them.

For the solution under consideration one finds that for both roots of (5.8) to be positive one must have cos ~. > 0.

If one considers the general case of a physical system of finite extent, so that at infinity guy goes over into ~uv [2], the requirement that the roots of (5.6) should be real and positive is necessary for space-time to have the properties that we expect. I f the roots were real and negative in some region, the field would have singularities. I f the roots were complex, then, among other things, there would be difficulties in setting up a Cauchy problem for the gravitational field and for the nongravitational fields present.

Finally it should be noted that all the considerations of the previous section, as well as this one, hold in the ease of the line element

ds 2 = cos ~e2C~(dx~ 2 + 2 sin ~ e ~ + ~ d x ~ 2 - cos ~e2r 2 - (dxl) 2 - (dx3) z

(5.10)

However, this describes a mixed, or longitudinal-transverse, wave since in general both Riot0 and R2o20 are different from zero.

A c k n o w l e d g e m e n t

The author thanks Professor Nathan Rosen for interesting and useful discussions.

R e f e r e n c e s

1. Rosen, N. (1973). Gen. Rel. Gray., 4, 435. 2. Rosen, N. (1974).Ann. Phys. (N.Y.), 84, 455. 3. Arnowitt, R., Deser, S., and Misner, C. W. (1962). Gravitation: an introduction to cur-

rent research, Witten, L., ed. (Wiley & Sons Inc., New York). 4. Weber, J. (1960).Phys. Rev., 117, 306.