Volume 12.k number 4,5 PHYSICS LETTERS 23 July 1979

SOMEEXACTCOSMOLOGICALMODELSWITHGItAVITATIONALWAVES

J. WAINWRIGHT and B.J. MARSHMAN

Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3GI

Received 17 May 1979

Three classes of exact solutions of the Einstein field equations with perfect fluid source are presented. These solutions can be interpreted as spatially inhomogeneous cosmological models, in which the inhomogeneity is due to the presence of gravitational waves.

In this note we show that certain spatially homo- geneous [I] or spatially self-similar [2] fluid-filled cosmological models can be modified so as to be

strictly spatially inhomogeneous. The inhomogeneity manifests itself by the appearance in the line-element of an arbitrary function, which is constant on null hypersurfaces. The presence of this arbitrary function enables one to use these solutions to construct grav- itational wave pulses which propagate in a spatially

homogeneous or spatially self-similar cosmology [3] . This interpretation, together with the fact that the pressure and density of the fluid are required to satis- fy the equation of state for stiff matter, namely p = 1-1, suggests that the solutions may be of possible interest in connection with the early universe [4].

We now present three classes of solutions of the Einstein field equations, with perfect fluid source. The line-element is of the form

ds2 = e2k(- dt2 t dx2) tgQpdxaY@, (1)

where k and the gap, a,/3 = 2,3 are functions oft and x. The local coordinates t,x,y =x2, z =x3 are co- moving, so that the fluid velocity is

u=e-k a/at. (2)

In order to present the different cases within the same framework, it is convenient to write

gap dxa tip = r[f(Ady +&)2

tf-l (Cdy tDdz)2],, (3)

where r, f, A, B, C, D are functions oft andx, and AD - BC = 1, although there is clearly redundancy

in this representation.

Each solution contains two parameters q,m which satisfy

q#O, m > - 3116. (4)

Only the sign of 4 is essential, however, and 4 may be set, without loss of generality, to equal t 1 or - 1. The solutions also contain two functions w = w (t - x) and

n = n (t - x), which are related according to

2qn’ = (w’)2 . (5)

The coordinates assume the values - 00 <x, y, z <=withO<t<=,or-=<t<Odependingon whetherq>Oorq<O.

CaseI.A=D=l,B=w(t-x),C=O,with

r=qt, f = (qt)-112

and

ek = (qt)m en(t- x).

The density and pressure are

16n~ = 16ap = q2(2m t g) t-2e-2k.

CaseII, A=D=cosw(t-x), B=--C = sin w(t -x), with

r = e2qx sinh 2qt, f = (tanh qt)l/2

275

Volume 72A, number 4,s PHYSICS LETTERS 23 July 1979

and

ek = (sinh 2qt)m exp [n( t - x) + (1 - 202) 4x1 .

The density and pressure are

167rn = 16np = 4q2 (2m t i) (sinh 2@)-2 e-2k. (6)

CaseHI. A=D=coshw(t-x),B=C = sinh w( t - x) , with

r = eW2qx sinh 29 t, f= [tanh qt] U2,

and

ek=(sinh2qt)m exp[n(t-x)-(1 -2m)qx].

The density and pressure are given by eq. (6) (but note that the expression for ek is different).

The line-elements admit an abelian G2 of local iso- metries generated by the Killing vector fields

+JJ = alay, q2) = a/az,

but, unless w( t - x) is restricted, do not admit a G3 or a similarity group H3. If w is given by

w=sq(t-x), (7)

where s is an arbitrary constant, each solution is spa- tially self-similar, and ifs is suitably restricted, spa- tially homogeneous. The similarity group H3 is gen-

erated by E(J), l(2) and

$3) = alax tq(-; ~2yts~)alay - ;qs22 alaz,

= a/axtq(-Mytsz)a/ay-q(Mztsy)a/az,

= a/axtq(Mytsz)a/aytq(Mztsy)a/az,

in cases I, II and III, respectively, with M = f (4m + s2) in case II and M = 3 (4m - s2) in case III. $3) is a Killing vector iff s = 0 (case I) or M = 1 (cases II and III). Ifs # 0 (case I) and M # 0,l (cases II and III),

the similarity group is of class D in Eardley’s classifi- cation [2]. Referring to table 1 in ref. [2], the

Bianchi group types are: case I: type ,IV, case II: type fVIIh (or type ,V ifs = 0),

case IIl: type fVIh (or type ,III if M2 = ~2, or type fVifs=O).

It is interesting that all Bianchi types in class

D (except type ,111, which does not admit an abelian G2 as a subgroup) are obtained as special cases, when eq. (7) is assumed. The preceding discussion es- tablishes that our new solutions generalize certain classes of spatially homogeneous and spatially self- similar solutions in the sense that linear expressions for w (t-x) and n (t-x) [see eqs. (7) and (5)] are re-

placed by arbitrary functions, subject to eq. (5). Whether this type of procedure can be used to gener-

alize other exact solutions is at present not known. The properties of the general solutions will be dis-

cussed in detail elsewhere [3,5]. We note that case I has been given previously [6]. Cases II and III are new,

although certain spatially homogeneous limits of case II have been given previously. For example, the case II solution in which (7) holds and M = 1, has been given

by Lukash [7] when m = - 3/16 (vacuum) and by Barrow [4], without this restriction. We finally men- tion that the solutions were derived and checked using a library of programs [8] written in the algebraic com-

puting language CAMAL [9].

References

[l] G.F.R. Ellis and M.A.H. MacCallum, Commun. Math. Phys. 12 (1969) 108.

[2] D.M. Eardley, Comm. Math. Phys. 37 (1974) 287. (3 ] J. Wainwright, A gravitational wave pulse in a spatially

homogeneous universe, preprint, Univ. of Waterloo (1979). [4] J.D. Barrow, Nature 272 (1977) 211. [S] J. Wainwright, Non-rotating inhomogeneous cosmologies:

exact solutions, in preparation. [6] J. Wainwright, W.C.W. Ince and B.J. Marshman, Gen. Rel.

Grav. 10 (1979) 259. [7] V.N. Lukash, Nuovo Cimento 35B (1976) 268. [8] J. Wainwright, CAMAL programs for GRT: A user’s

guide, unpublished (1978), available from the Depart- ment of Applied Mathematics, Univ. of Waterloo.

[9] J.P. Fitch, CAMAL Manual, Univ. of Cambridge (1976).

276