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A Global Solution of the Einstein-Maxwell Field Equations for Rotating Charged Matter

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A Global Solution of the Einstein-Maxwell Field Equations for Rotating Charged Matter Journal of Modern Physics, 2012, 3, 1301-1310 http://dx.doi.org/10.4236/jmp.2012.329168 Published Online September 2012 (http://www.SciRP.org/journal/jmp) A Global Solution of the Einstein-Maxwell Field Equations for Rotating Charged Matter Andreas Georgiou School of Physics Astronomy and Mathematics, University of Hertfordshire, Hatfield, UK Email: [email protected] Received June 27, 2012; revised July 31, 2012; accepted August 9, 2012 ABSTRACT A stationary axially symmetric exterior electrovacuum solution of the Einstein-Maxwell field equations was obtained. An interior solution for rotating charged dust with vanishing Lorentz force was also obtained. The two spacetimes are separated by a boundary which is a surface layer with surface stress-energy tensor and surface electric 4-current. The layer is the spherical surface bounding the charged matter. It was further shown, that all the exterior physical quantities vanished at the asymptotic spatial infinity where spacetime was shown to be flat. There are two different sets of junc- tion conditions: the electromagnetic junction conditions, which were expressed in the traditional 3-dimensional form of classical electromagnetic theory; and the considerably more complicated gravitational junction conditions. It was shown that both—the electromagnetic and gravitational junction conditions—were satisfied. The mass, charge and angular momentum were determined from the metric. Exact analytical formulae for the dipole moment and gyromagnetic ratio were also derived. The conditions, under which the latter formulae gave Blackett’s empirical result for rotating stars, were investigated. Keywords: Gravitation; Exact Solutions; Einstein-Maxwell Equations; Rotation; Charged Dust 1. Introduction 2. The Einstein-Maxwell Field Equations There are difficulties in finding exact solutions of the Consider electrically charged pressure-free matter (char- Einstein or of the Einstein-Maxwell field equations for a ged dust) bounded by the hypersurface r = a and rotating volume distribution of rotating bounded matter [1]. Such with constant angular velocity about the polar axis solutions should consist of an interior filled with matter 0 under zero Lorentz force. It is assumed that the and an asymptotically flat vacuum or electrovacuum ex- current is carried by the dust. The transformed expression terior, these being separated by a surface on which ap- (2.1) in [2] for the Weyl-Lewis-Papapetrou metric for a propriate boundary conditions should be satisfied. The stationary axially symmetric spacetime V is main aim of this work is to obtain an exterior and mat- 2222 ching interior solution of the Einstein-Maxwell field dddserr (1) equations with finite bounded rotating charged matter as F 12rKsin 2 2 d 2 2 KtFt d d d 2 a source of the spacetime. Due to the rotation, the boun- dary will actually be an oblate spheroid, but it is assumed where we have taken the signature of the spacetime met- that it is a spherical surface with equation r = a. The ric tensor g to be 2. It is implicit in the form (1) main objective and emphasis after all, is to see how far of the metric that we have assumed, without loss of gene- the attempt at finding a solution can be taken—a solu- rality, that LF K222 r sin and so the component tion with finite bounded rotating matter as a source of the g33 of g is g33 L. We shall use units c = G = 1 spacetime. The additional complication of spheroidal co- where c is the vacuum speed of light and G the Newto- or-dinates is avoided, in a problem which is already enor- nian gravitational constant. Unless otherwise specified, mously complicated. we shall adopt the convention in which Roman indices Most of the equations and expressions for the various take the values 1, 2, 3 for the space coordinates r,, physical quantities are difficult to derive and they require which are spherical polar coordinates co-moving with the involved and lengthy analysis. It is not therefore possible dust, and Greek indices take the values 1, 2, 3, 4 for or desirable to include these calculations in the paper, but the spacetime coordinates rt,,, . Semicolons and directions in which to proceed are indicated. commas indicate covariant and partial derivatives respec- Copyright © 2012 SciRes. JMP 1302 A. GEORGIOU tively, and the suffixes r and θ denote partial differentia- Instead of expressing the electromagnetic field equa- tion with respect to r and θ. All the functions are as- tions in 4-dimensional form as in Equations (5), we shall sumed to depend on r and θ only, or they are constant. use the Maxwell form (Maxwell’s equations), because The results to be used in this work may be found in a we can make direct comparisons with the results from number of different publications [2,3] but we shall use [2] classical electromagnetic theory. The electric and mag- where all the necessary equations have been collected netic intensities and corresponding inductions in 3-vector together and written in terms of the cylindrical polar co- form, are [4,5] ordinates and time zt,,, . We shall transform those EDEF Daa FF4 , equations in [2] that are required here, to the spherical aa4 11 (11) polar coordinates and time rt,,, with HBH =Fe Fkp B a e akp F 12 a akp kp rz 22, tan z , , zr cos , 22 akp r sin and tt . where eakp akp , e akp are the com- pletely antisymmetric permutation tensors, The contravariant and covariant forms u and u of the 4-velocity are g F , being the determinant of the spa- tial metric tensor ab which is given by 12 12 3 4 uF4 uFw. (2) ab g ab a b with aa g 4 F , and akp is the Levi-Civita symbol. It is easy to show that The electric 4-current J , the electromagnetic 4-po- = Fr32 2sin . tential A and the Faraday tensor F , are The transformed equations (2.14) and (2.13) of [2] Ju may be written as AA 34+ A (3) 4πreJKAFA22sin 3 *2 *2 34 43 (12) FAA . 22 ,, KrrArKAFArFA4433 rr where is the electric charge density. The Einstein- 4πreJLAKA22sin 4 *2 *2 Maxwell field equations for charged dust are 43 (13) LA r22 LA KA r K A rr4433 rr GT 8π (4) where the operators 2 and *2 are defined by FF,, F, 0 22 2 12cot 1 (5) 14) g FJ4π . rr222 rrr 2 g , 22 *2 1cot Here, G is the Einstein tensor . (15) rr222 r 2 1 GR: R (6) 2 Equations (12) and (13) are the detailed form of the source-containing Maxwell equations given in the second where R is the Ricci tensor of the spacetime defined of (5). by its fully covariant form as The non-zero components of the Ricci tensor obtained R :,, (7) from the transformed Equations (2.16)-(2.21) of [2] are: with the Christoffel symbols of the second kind 2 2 3411 F 2F RR e F based on the metric of V in Equation (1), RgR is 34 22r 2 F F r the spacetime scalar curvature invariant and g is the de- 2 w2 terminant of g . The total stress-energy tensor T is F *2 2 22wwwr 2 (16) rrsin TM E (8) 2Fw Fw where 22Fwrr 2 rrsin M uu (9) 1 1 cot 11 Re EF FFF (10) 1 rr 22 2 rr 4π 4 (17) FFFw2222 are, respectively, the matter and electromagnetic stress- rr r + 222 energy tensors and is the mass density. FrrF sin Copyright © 2012 SciRes. JMP A. GEORGIOU 1303 1 2 cot Re2 r 22nn 2 rr 22 bCrAarP 21nn 21 cos 2 r rr n1 (18) 222 FF2cot Fw ar 0 π 2 + 22 2 4 2 rF rF rsin 0 r, (27) ar = π 2 3*e 2Fw RF4 22w 2FFrrw 2 (19) 2sinrr 22nn bCrAarP () 21nn 21cos n1 3 rK22sin 23R 4 2KR 4 ar π 2< π R 3 (20) 3 F F 2 where bm, and are constants whose significance 1 2 R2 e FFr will emerge later. From now on we shall omit writing the R1 rr222 F2 argument cos of the Legendre polynomials and we shall write, for example, P21n instead of P21n cos . 1 F sin Fr cos (21) We note the significant fact that at = π 2 , P21n 0; Frsin this enables us to set r,0 at = π 2 ar 2 as in (27). 1 sin F wwr +cos r 22 The function wKF: and the electromagnetic 4- sin r r sin 34 potential AA 34 A in the exterior are obtained 2 from r 1 2 F Re rr 22 Fr 2 rFr 2 r wr 2 sin (28) (22) 2 2 2 wr 2sinrr 1 Fwcot F 2 Fwrr 222 1 rF r 2srrin Aw3rr F sin (29) The entire Riemannian spacetime V, will be separated 1 into the following 4-dimensional manifolds: the hyper- 2 Aw3 Fr rsin (30) surface with equation ra separates V into the interior Vr0 a and exterior Var Ab4 (31) spacetimes. We shall use the + and – signs to denote where an arbitrary constant in A was set equal to b quantities in V and V whenever it is necessary to 4 in order to satisfy the continuity condition of A4 . Note do so. Quantities without the + or – indicators, may be that the full expression for wr in the first of (28) is associated either with V or with V . wr 2sin , but by (26), 0 . From Equations (24) and (28)-(31), we obtain the fol- 3. The Exterior Solution 1 lowing expressions for F , exp , w , A In accordance with the formalism in [2], we first form the 3 complex function and A4 : 1 i (23) Fb exp 22Cr where and are harmonic functions.
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