SPACETIME Symmetrles and SOME SOLUTIONS of the FIELD EQUATIONS

SPACETIME SYMMETRlES AND SOME SOLUTIONS OF THE FIELD EQUATIONS

S. D. Maharaj1

1 Department of Mathematics and Applied Mathematics University of Natal King George V Avenue Durban 4001 South Africa

59 Abstract

We investigate the role of symmetries, in particular conformal symmetries, in general relativity. Conformal Killing vectors in the k = 0 Robertson-Walker spacetimes, the pp-wave spacetimes and spherically symmetric spacetimes are generated. Examples of solutions with a conformal symmetry are presented to the Einstein, the Einstein-Klein-Gordon and the Einstein-Maxwell field equa tions. An irreducible Killing tensor is found in static, spherically symmetric spacetimes.

1. Introduction

Exact solutions of the Einstein field equations are important because of wide applications to cosmology and relativistic astrophysics. For a comprehensive list of

1 exact solutions to thspacetime geometry, assuming a special

form for the matter distribution or imposing ~n equation of state. Very often a

solution is found by making ad hoc choices for the geometrical and matter variables. A more recent and systematic approach is to impose a symmetry requirement on the spacetime manifold. This places additional conditions on the gravitational field and may simplify integration of the field equations. A variety of symmetries may be defined on the spacetime manifold by the action of the Lie derivative operator on the metric tensor, the connection coefficients, the Ricci tensor, the curvature tensor and the Weyl tensor. A detailed discussion of the geometric interpretations and interrelationships of these symmetries is given by

2 3 4 5 Katzin and Levine • • • • We are ·particularly interested in conformal symmetries which are associated with constants of the motion along null geodesics for massless particles. The objective is to obtain ne'w so}utions to the Einstein field equations with a conformal symmetry. Solutions with a conformal symmetry applicable in relativistic astrophysics have been investigated by Herrera at af', Herrera and Ponce

7 8 de Leon and Maartens and Maharaj , amongst others. Dyer at a'fl, Maharaj et

60 al10 and Havas 11 have studied spherically symmetric· conformal symmetries in the cosmological context. Our main intention in this paper is to consider conformal symmetries and present some examples of spacetimes admitting conformal Killing vectors. We also briefly

study an example of a symmetry which is a irreducible Killing tensor arising from ~ solution of the Liouville equation. Ideally we should generate physically reasonable solutions to the field equations which are invariant under the action of a symmetry

vector. In §2 we provide background material on spacetime symmetries and confor mal symmetries in particular. Conformal Killing vectors for the Robertson-Walker spacetimes, the pp-wave spacetimes and spherically symmetric spacetimes are found in §3. Solutions to the Einstein field equations are generated in these spacetimes with a conformal symmetry. In §4 we generate an irreducible rank two Killing tensor in the Kimura spacetime by analysing a constant of the motion for a collision-free gas. We briefly discuss our re~ults obtained in §5 and consider some avenues for future work.

2. Spacetime symmetries

The Lie derivative provides a coordinate independent description of a symmetry property in the manifold. A number of symmetries may be defined on the manifold by the action of the the Lie derivative operator £e on geometric quantities defined on the manifold. For example Dugga.112 defines the curvature inheritance symmetry by

£eRi;k1 = 2aRi;k1 where a= a(xi) and R is the curvature tensor. The curvature inheritance symmetry generalises a curvature collineation for which £eRi;kt = O. Curvature inheritance

5 regains many of the other symmetries listed by Katzin and Levine • A conformal Killing v.ector e is defined by the action of the operator le on the

61 metric tensor field g by (1)

where t/J = t/J(xi) is the conformal factor and g is the metric tensor field. If g is specified, then for some spacetimes it possible to explicitly solve (1) to obtain the conformal Killing vector e. The set of all conformal Killing vectors forms a Lie

algebra Gr (r ~ 15) with basis {eI}, .Celii = 2t/Jr9if

re1i eJ] - cK ut/JK

ert/JJ - eJtPI - CK IJtPK

1 13 where C JK are structure constants • The maximal G15 Lie algebra is attained when the spacetime i.s conformally fiat; the fifteen conformal Killing vectors in Minkowski spacetime are given by Choquet-Bruhat et a/13 and those in Robertson-Walker space

14 times have been found by Maarteris and Maharaj • There are four subcases associated with the symmetry (1):

• # • (i) t/J ··= 0 : eis a Killing vector,

(ii)t/J,i = 0 ~ ,,P : eis a homothetic Killing vector,

(iii) .tP;i; = 0 ~ tP,i : eis a special conformal Killing vector and

(iv) tP;&; ~ 0 : e is a nonspecial conformal Killing vector.

These vectors are of physical importance as they help to produce first integrals of the motion. Killing vectors generate constants of the motion along geodesics. A homothetic Killing vector scales distances by the same constant factor and preserves null geodesic affine parameters. Conformal Killing vectors generate constants of the motion along null geodesics for massless particles.

62 The Einstein field equations take the form

(2)

in appropriate units. The energy density µ, the pressure p, the heat flow vector qi

and the stress tensor 'Trij are meas?red relative to the fluid four-velocity vector u. We seek solutions to the field equations (2) which admit a conformal Killing vector ethat satisfies (1). There do not exist many many solutons, with a conformal symmetry, for a perfect fluid energy-momentum tensor as pointed out Castejon-Amenedo and

Coley15 • Exact solutions for a perfect fluid are of importance in cosmology.

3. Solutions with conformal symmetries

We consider three examples of spacetimes for which the conformal Killing equa tions may be explicitly found. The relationship between the conformal vectors and special solutions of the Einstein field equations is briefly investigated . The three spacetimes of cosmological significance that we have chosen are the Robertson

Walker spacetimes, the pp-wave spacetimes and the spherically symmetric spacetimes.

(A) Robertson-Walker spacetimes

For the k = 0 Robertson-Walker model the line element may be expressed in the conformally flat form

(3)

where T = f dt/ R(t). We can integrate (1) for the line element (3) to generate the

14 maximal G15 Lie algebra of conformal vectors in Robertson Walker spacetimes • The Lie algebra is spanned by the vectors

63 M .. a a I] - Xi {)xi - Xj {)xi

Ki - 2xiH - (x;xi)Pi . a H - I x 7Px•

where (xi)= (r,x,y,z) and (xi )= (-r,x,y,z). In addition to the well-known six Killing vectors we have generated an additional nine proper conformal Killing vectors

for the case k = 0. The conformal Killing vectors for the k = ±1 Robertson-Walker

14 spacetimes have also been found by Maartens and Maharaj • If we wish to reduce some of the conformal Killing vectors to homothetic vectors or Killing vectors then the scale factor R(t) in {3) is restricted. For example to regain

a homothetic vector (.1/J,i = 0 =F ,P) we have

R(t) ex t(l/J-t)N

With this form of the scale factor the field equations (2) may be easily solved to obtain the thermodynamical quantities measured relative to a four-velocity u.

(B) pp-wave spacetimes

In coordinates (xi)= (u,v,y,z)·= (u,v,xA)(A = 2,3) the pp-wave spacetime is given by (4) where u, v a.re retarded/advanced time coordinates. The pp-wave spacetimes have the null Killing vector

which is tangent to the null rays { u, xA = constant}, and covariantly constant. The Riemann, Weyland Ricci tensors may be written in the compact form

R;jkl 4k1;xB,i]k[k"xD,1]H,BD

Cijkl - 4k[iXB,j]k[kXD,l]'YBDEFH,EF

64 where

The conformal Killing equation (1) has been completely integrated by Maartens and

Maharaj16• The solution is given by

tP - µv + ~~(u)xA + b(u) eu - ~µ6.ABXAXB + aA(u)xA + a(u) ev - µv2 + [a~(u)xA + 2b(u) - a'(u)Jv + F(u,xB) eA - (µxA + aA(u)]v + /ABQDa~(u)xCXD +b(u)xA + c(u)fAnx8 + cA(u)

subject to a. condition on H given by

and the function F satisfies

F,u =· 2(b- a')H - (!µ8ABXAXB + aAXA + a)H,u

-(/ABCDa~xCxD + bxA + CfABXB + CA)JJ,A

This conformal symmetry generalises previous results on Killing and special confor mal Killing vectors of pp-wave spacetimes. TP,e Ein:stein-Klein...:...Gordon equations with a massless scalar field n for the space time ( 4) reduce to

_ n n n;i . _ 0 R ij - H,jH,j, H jl -

A solution with a conformal symmetry is given by

1 2 A B H - 2h0 8ABX x

n - ../2hou

65 The Einstein-Maxwell equations reduce for a pp-wave metric to

R IJ.. - F. t· kFk J!. I'lt)jD[ . . k) ---0 - pii jJ.

This admits the following solution

1 2 A B H - 2ho8ABX X

Fi; - 2kuxA,iJfA(u)

2 fAfA - 2h 0

with a conformal symmetry. Thus we have demonstrated that their exist solutions to the Einstein-Klein-Gordon equations and the Einstein-Maxwell equations with a nonspecial conformal Killing vector e (1/J;ii I= 0). These solutions correspond to a conformally flat pp-wave for which the metric function H = th~8ABXAxB.

( C) Spherically symmetric spacetimes We utilise coordinates (xi) = (t, r, 0,

(e-v, O, 0, 0) so that we can write the sph~rically symmetric line element as

2 2 2 2 2 2 2 ds = - exp(2v(t, r)]dt + exp[2..\(t, r)]dr + Y (t, r )(d0 + sin Od

A class of solutions to the field equations (2) for this spacetime was obtained by

7 Maharaj et a/1 • This class of metrics are characterised by the fact that they are accelerating, expanding and shearing. It is convenient to distinguish the three cases k = 0, k < 0, k > 0 that arise. The appropriate line elements are given by

1 k = 0: ds 2 - -a2 r 2dt2 -) dr2 + (-br2 + (5)

1 k = -n2 < 0 : ds 2 - -a2 r 2dt2 + ( ) dr2 + -n2 + br2

66 2 2 2 r ( csin(2ant) + d cos(2ant) - 2 ~ 2 ) (d0 + sin Od

k = n 2 > 0: ds 2 . 1 2 2 2 2 r (cexp[2ant] + dexp[-2ant] + n ) (d0 + sin 0d

Even though this class of solutions is simple and are expressible in terms of elemen

tary functions it contains as special cases many of the solutions of other authors17• The energy-density and the pressure for these models are related by the equation

p = µ + 6b

which is a generalisation of the the stiff equation of state p = µ. This may be viewed as a thermodynamical characterisation of the spacetimes (5)-(7). We can solve (1), for a special case, and generate a conformal symmetry for the spherically symmetric spacetimes (5)-(7). The re;;ulting conformal Killing vector e

18 19 = (0, e, 0, 0) is very specialised with only one nonvanishing component • • The component e is given by

Co if k = 0

and the conformal factor has the form

if k = 0

if k = n 2

67 where C0 , C+ and C_ are constants. Thus for this conformal Killing vector e both e1, 1/J are static. This conformal vector is proper (1/J f:. 0) and cannot reduce to a Killing vector of the spherically symmetric spacetimes. By finding this conformal vector ewe have characterised our shearing class of spacetimes (5)-(7) geometrically. This analysis suggests the possibility of finding other solutions, with a conformal Killing vector, to the Einstein field equations with nonzero shear. Solutions with nonvanishing shear are rare.

4. Other symmetries

It is sometime possible to generate synimetries on the spacetime manifold by considering the action of vector fields on the mass shell in the tangent bundle. These symmetries need not be of the type considered earlier. As an example we generate

20 an irreducible Killing tensor in Kimura spacetime • A collision-free gas of particles satisfies Liouville's equation

(8)

where f = f(xi,p"') is the distribution function for particles of mass m (gi;Pipi =

2 -m ). The general solution of the Liouville equation is a fonction of six linearly independent constants of the motion. Equation (8) can be solved in general for the static, spherically symmetric line element20

ds" = -exp[v(r)]dt2 + exp[.\(r))dr2 + r 2 (dtJ2 + sin2 Od2) in terms of the constants of motion. One of the constants is given by

where the quantities E and J are given by

68 For the Kimura spacetime

the constant C has the form

2 2 1 . . . C = (aE - J t J

where the quantity Ki; is given by

Thus we have demonstrated that the constant of motion C is generated by I(; which is an irreducible Killing tensor of rank two. This is a generic new quantity on the manifold and cannot be derived from the metric tensor field and the Killing vectors of spacetime. We have demonstrated that symmetries, other than conformal symmetries, may be generated by considering mathematical structures such as the tangent bundle defined over spacetime.

5. Discussion

Our objective in this paper was to investigate the relationship between symmetries on the manifold and the Einstein field equations. We generated conformal Killing vectors for three spacetimes of cosmological signifance and illustrated how these con formal symmetries may generate exact solutions. Also an irreducible rank two Killing tensor was obtained in static spacetimes by a consideration of the Liouville equation. Clearly the study of symmetries is a worthwhile endeavour in general relativity. It provides us with a deeper insight into the structure of the spacetime manifold. Also we may obtain new solutions to the field equations which are invariant under the action of the corresponding symmetry vector. Such solutions to the field equations are rare and this area should prove to be a fertile area of research. In particular the symmetry approach may help to gener~te solutions to the Einstein equations in

69 areas which have yielded few results in the past using other approaches. An example of a class of line elements with few solutions are spacetimes with nonvanishing shear.

70 References

[1] D. Kramer, H. Stephani, M. A. H. MacCallum and E. Herlt, Exact Solutions of Einstein's Field Equations (Cambridge University Press, Cambridge, 1980).

[2] G. H. Katzin and J. Levine, Tensor NS 21, 51, (1970).

[3] G. fl. Katzin and J. Levine, Tensor NS 21, 319, (1970).

(4] G. H. Katzin and J. Levine, J. Math. Phys. 11, 1578, (1970).

[5] G. H. Katzin and J. Levine, Colloquium Mathematicum XXVI, 21, (1972).

[6] L. Herrera, J. Jimenez, L. Leal, J. Ponce de Leon, M. Esculpi and V. Galina, J. Math. Phys. 25, 3274 (1984).

[7] L. Herrera and J. Ponce de Leon, J. Math. Phys. 26, 2302 (1985).

[8] R. Maartens and M. S. Maharaj, J. Math. Phys. 31, 1'51 (1990).

f9] C. C. Dyer, G. C. McVittie and L. M. Oates, Gen. Rel. Grav. 19 887 (1987).

(10] S. D. Maharaj, P. G. L. Leach and R. Maartens, Gen Rel. Grav. 23, 261 (1991).

(11] P. Havas, Gen. Ret Grav. 24, 599 (1992).

[12] K. L. Duggal, J. Math. Phys. 33, 2989 (1992).

(13] Y. Choquet-Bruhat, C. Dewitte-Morette and M. Dillard Bleick, Analysis, Man- aifolds and Physics (North-Holland, Amsterdam, 1977).

(14] R. Maartens and S. D. Maharaj, Class. Quantum Grav. 3, 1005 (1986).

[15] J. Castejon-Amenedo and A. A. Coley, Class. Quantum Grav. 9, 2203 (1992).

[16] R. Maartens and S. D. Maharaj, Class. Quantum Grav. 8, 503 (1991).

71' [17] S. D. Maharaj, R. Maartens and M. S. Maharaj, fl Nuovo Cimento 108B, 75 (1993).

(18] S. D. Maharaj and M; .S. Maharaj, II Nuovo Cimento (submitted) (1994).

[19] M. S. Maharaj, Ph .D. Dissertation (University of Natal, Durban) (1993).

[20] S. D. Maharaj and R. Maartens, J. Math. Phys. 27, 2514 (1986).