On the Generation of a Class of Einstein-Maxwell Solutions 1

Transcription

1 General R elativity and Gravitation, VoL 15,No. 12, 1983 On the Generation of a Class of Einstein-Maxwell Solutions 1 M. HALILSOY King Abdul Aziz University, Department of Nuclear Engineering, P.O. Box 9027, Jeddah, Saudi Arabia Received March 3, 1983 Abstract We make an alternative approach to Bonnorification by making use of effective Lagrangians for the gravitational fields. Using this technique we obtain a new Einstein-MaxweU solution. Some time ago Bonnor [1] gave an example of a solution for the static axially symmetric Einstein-Maxwell fields, generated from the rotating Kerr metric. The same method was elaborated on by various authors [2-4] in the generation of similar static Einstein-Maxwell solutions associated with any stationary axially symmetric vacuum solutions. Included among these solutions are the ones corresponding to TS solutions as well as to Taub-NUT solution. The aim in these derivations as initiated by Bonnor is to exploit the degree of freedom brought in by the rotation and to define an electromagnetic potential from this new dimension. Furthermore the method is not restricted to rotating solutions but rather to any line element which includes a nontrivial cross-term representing rotation, twist, or cross-polarization. For each of these solutions, however, the electromagnetic potential cannot be given in closed form and the physical interpretations of the resulting solutions become even more vague for more complicated vacuum solutions. It is the purpose of the present paper to show, using Lagrangian methods, how this method works in general: An explicit example is also given for the timedependent fields related with cosmological models. In the first section we make 1 This research was supported by the Engineering College of King Abdul Aziz University under Project No / J Plenum Publishing Corporation

2 1116 HALILSOY a review of the stationary axially symmetric fields, while in the second section we give a new solution as an example. (1) The stationary axially symmetric gravitational fields are described in the cylindrical coordinates p, z, by the Weyl-Papapetrou line element ds z = e2~ (dt - cod~) z - e -2~ [e27(dp 2 + dz 2) + X2 d~ 2] (1) As far as field equations are concerned, besides some physical questions the same field equations of this line dement can equivalently be obtained by the variational principle for the Lagrangian density [5] : Ll = 2(TpXp +')'zxz)- 4X ~ + t~2z - -~~ Note that the coordinate condition X = t9 is to be used after variation. Among the various representations of Bonnor's original transformation we adopt the one given by Yamazaki [4], which states that if (1) is a vacuum solution then ds z = e4~ dt 2 - e -4~ [est (dp = + dz 2) + p2 cl~2] (3) is an electrovacuum solution with the electromagnetic field tensor Ft z = _p-1 e4to cop sin a, F~o = cop cos a Fro = P-1 e 4~ COz sin a, F~oz = coz cos a (4) a = const (arbitrary) This form of Bonnor's transformation has the virtue that the complex unit i does not exist. By the choice of the constant a, the field can be either pure electric or pure magnetic. Note also that the choice with hyperbolic angles is also a solution but this case does not possess both pure electric and pure magnetic limits. The proof of Bonnor's above-stated theorem can be described in terms of isometries of the Lagrangian (2). An alternative description lies within the formulation of harmonic maps employed between manifolds [8, 10]. Our aim is not to give detailed information about harmonic maps-since detailed exposition of this subject exists in the literature-but we will just provide the basic elements. Let M and M' be two Riemannian manifolds with respective dimensionalities n and n' and the line elements, ds 2 = gab dxadx b ; ds,2 = gab dxa dxb. If we consider the maps, fa :M-+ M', then an energy functional is defined from such maps by [12] ~fm ~fa ~fbgabgl/2 dx n (5) E(f) = gab ~xa OX b The mapsf A for which the variational principle, 6E(f) = 0, i's satisfied are called harmonic maps. Einstein's field equations of general relativity with two

3 EINSTEIN-MAXWELL SOLUTIONS Killing vectors can be cast into this form, and finding exact solutions becomes equivalent to finding the harmonic maps. Also the density of the energy functional is equivalent to the Lagrangian density of Ernst's reduced formalism [13, 14]. The Lagrangian (2) is nothing but the energy functional (action density) between the following manifolds: M: ds 2 = dp 2 + dz 2 M': ds '2 = 4dTdX- 4X d e4~ 2~ (5') 42-4~-dco:; A nontrivial isometry (or imbedding) of the M' manifold will map a solution to a new solution. Bonnor's trick amounts then to the identification c@ + COz 2 = Bp 2 + Bz 2 ; X2e-SO (A~ +Az:) (6) which is equivalent to an isometry of M' manifold given by dco 2 = db 2 T- X~e -s~ da 2 (7) Here A and B are the components of vector potential along the Killing directions (A v =A6~ + B6~). The minus and plus signs in (6) and (7) correspond to trigonometric and hyperbolic phase angles, respectively. Since both choices constitute exact electrovac solutions, we adopted in (4) the trigonometric form, whereas in the next section we shall choose the latter one. There is no need to point out that starting from a static axially symmetric geometry by means of imbeddings one can generate an axially symmetric electrovac solution. (2) As an example to the above given method let us consider a cosmological metric given by Szekeres which represents also the collision of gravitational waves [6-10]. The Lagrangian of the system is given by # L2 =2(TuXv + TvXu)- X u~v + X-Z-~Our (8) where the null coordinates are defmed by 21/2u = t - z and 21/2o = t + z. In this model instead of rotation we have an additional degree of freedom representing the cross polarization of gravitational waves. In order to obtain the model given by Szekeres one should make the substitutions X = e -U, e -v-v cosh W = e ~ r = e V tanh W, and the resulting line element reads e 2v = e -M-U-v cosh W ds 2 = 2e-U dudv - e-v (e v cosh W dx 2 + e -v cosh W dy 2-2 sinh W dxdy) (9) (10)

4 1118 HALILSOY The fact that W 4:0 gives us a cross-term in the line element and we can use it to define an electromagnetic field. Taking the case of a vector potential, A. = 6~A + 5~B, we then have the field tensor of electromagnetic field given by Fuy = c~ e 2. cosh c, Fux = 6o u sinh c X Foy - ~~ X e 2~ cosh c, Fox = wo sinh c, c = const (11) The compatibility condition and Maxwell equations are satisfied via the Euler- Lagrange equation, 6L2/Sw = 0. The Einstein-MaxweU Lagrangian now e2r ~ L3 = 2(')'uXv + 7vXu) - X ucv - e-2q'auav - -~BuB (12) and the Maxwell equations are given by 8L a/sa (B) = 0, implying (Xe-2~Au)o + (Xe-=~Ao). = 0 (X -1 ezqj Bu)v + (X-* ez~ Bv)u = 0 These equations arise also from the usual methods by considering the line element (13) ds 2 = 2e 8v-2~ dudv - XZe -2~ dx 2 - e 2~ dy = (14) The resulting Einstein-Maxwell solution is therefore completed. The explicit expressions for the impulsive wave solution are given by [8] X = 1 - p2 _ q2 = t 2 ' CO = -i (1 + k)(1 + k) where e~ =t = (1 +k)(1 +k) 1 - kk (15) t e 2,r =--(1 + k)(1 + k) /'w k = ei~pw + ei~qr, p = uo(u), q = vo (v) r 2 = 1 - p2, w 2 = 1 - q2 a,/~ are arbitrary constants. The exact integrals of the vector potential components A and B are rather involved; however, if we consider the incoming regions where the v (or u) dependence vanishes the results are relatively simple. The electromagnetic field

5 EINSTEIN-MAXWELL SOLUTIONS 1119 tensors in these regions are given by the expressions and F,,x = 20(u) sin c~ sinh c(1 - p2) (1 + 2p cos a + p2)2 Fuy = 20(u) sin a sinh c sin/3 sinh c(1 - q2) Fox = 2 0 (v) (1 + 2q cos t3 + q2)2 (16) Fry = 2 0 (v) sin t3 sinh e respectively. These fields can be eliminated in the rotated coordinate frames separately but not simultaneously [11]. The resulting gravitational and electromagnetic field strengths both diverge on the hypersurface t 2 = 0, which represents the characteristic hypersurface of singularity of colliding Einstein-Maxwell fields. A peculiar property of the obtained Einstein-Maxwell solution is that it does not possess separate Einstein and Maxwell limits. Finally, the solution (15) represents the first member of a family of solutions arising from the known solutions of colliding gravitational waves [9]. To find the rest of the family is easily done. References 1. Bonnor, W. B. (1966). Z. Phys., 190, Misra, R. M., Pandey, D. B., Srivastava, D. C., and Tripathi, S. N. (1973).Phys. Rev. D, 7, Wang, M. Y. (1974).Phys. Rev. D, 9, Yamazaki, M. (1978). Phys. Lett., 67A, Ernst, F. (1968).Phys. Rev., 167, Szekeres, P. (1972). J. Math. Phys., 13, Bell, P., and Szekeres, P. (1974). Gen. ReL Gray., 5, Nutku, Y., and Halil, M. (1977).Phys. Rev. Lett., 39, Halil, M. (1979). J. Math. Phys., 20, Halilsoy, M. (1981). Phys. Lett., 84A, Halilsoy, M. (1981).Phys. Lett., 84A, Eels, J., Jr., and Sampson, J. H. (1964).Am. J. Math., 86, Nutku, Y. (1974). Ann. Inst. Henri Poincar~, A21, Matzner, R. A., and Mimer, C. W. (1967). Phys. Rev., 154, 1229.