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)

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 junction 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 12rKsin 2 2 d 2 2 KtFt d d d 2 a source of the spacetime. Due to the rotation, the boundary 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 equation 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 EDEF  Daa  FF4 , equations in [2] that are required here, to the spherical aa4 11 (11) polar coordinates and time rt,,,  with HBH =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 spatial metric tensor  ab which is given by 12 12 3 4 uF4 uFw. (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-  = Fr32 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 r22 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 rrr 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  22r  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  22wwwr 2 (16)   rrsin   TM E (8) 2Fw Fw  where  22Fwrr 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  

1  2  cot   Re2   r      22nn  2  rr 22 bCrAarP  21nn 21 cos  2  r rr n1 (18)  222  FF2cot Fw ar 0  π 2 + 22 2 4 2   rF rF rsin   0 r,   (27)  ar  = π 2 3*e 2Fw  RF4  22w 2FFrrw 2 (19)  2sinrr  22nn   bCrAarP ()  21nn 21cos   n1 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 2sinrr 1 Fwcot F 2  Fwrr 222 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 Vr0 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  22Cr where  and  are harmonic functions. With a star  2 (32)  222 nn  denoting complex conjugation, the metric functions F bCrAarP   21nn 21 and   are then given by n1

1   *22 K A F exp  . (24)  2221n n   wba 2  F n1 41n  (33) If we denote the real and imaginary parts of 1 by 2nC r r12nn mr 2  P P  and  , then 22nn 2   AbCrFw3  () 22 22. (25)    A bnCaarm2212 21n 2  nn (34) We now choose the functions  and  as follows: n1 41n  m PP  bC r C  r 1 22nn 2 r (26) A bF C r b. (35) ar 0  π 4  

It is a little difficult to solve the two equations in (28) will be required. We express rr, and   as to find w in (33). It is even more difficult to solve the  bm b m bY two Equations (29)-(30) to find A3 in (34) and com- rr=22 = bX  . (42) plete details of the calculation are not given. Whenever rr sin there are two signs in a term, the upper sign gives the  The components of G are therefore calculated using expression in 0  π 2 and the lower sign the ex- the exterior functions (32)-(35) with Equations (16)-(22) pression in π 2  π, as in Equation (32). and, whenever necessary, bearing in mind the first of The function B defined by Equation (39). The calculations give the following non- Ca 0 < π 2 zero components:  12  1 B   0  = π 2 (36) GG12  8π E 1  22 22 Ca π 2< π 2 (43)    mbmX2  22 bY Fb 42X 22 has Legendre polynomial expansion of the form rr rsin   GrG12 28π  E 1 BAP  2121nn 21  2 n1 2 (44) 1 (37) 2  mY 41n  2bF2 X A21nn  BP21 xd x r sin 2 1 3 2  where An22n 0 1,2, It therefore follows from 2bm F Y GE338π  (45) (27), that a,  0. The function B in (36) satisfies 44r 42sin  the conditions for such an expansion [6] and we have for GE448π the odd coefficients A21n 33 22 n1  2 mbmX2  1412nn  2!n 2 F wb 2 X A  Ca,  42 21n 21n 2  rr 2!n (38)   2 22 n=1, 2, bFmY2  bF wY (46)  With a,0, by Equations (24) and (26), the rr222sin  + 32 metric function F at ra becomes 2  bF w mY F  a,12221bCa . It will be shown in Sec-    42  tion 4, that Fr ,1  everywhere in the interior. In r sin    order to satisfy the junction condition at ra therefore, we must have Fa ,1 bCa22 1. It is easily 43+4   GG43  8π E 4 seen that, as r , Fr,11 b22 , which is   22 2 a constant. If we take this to be equal to 1, we obtain  mbmX2  22 F  42 (47) b 1 1 and collecting these relationships toge-  rr ther we have 22 bFmwY2  22 bY   22 21 bX 22  42 . bb11 2 rrsin sin Ca  (39) 2   Here, E are the nonzero components of the elec- 2 2mm  . tromagnetic energy tensor. The components of  E aa  were obtained from (10) the third of (3) for F , the The third of Equation (39) is the result of substituting exterior electromagnetic potentials in (34) and (35). the second of these equations into the first, bearing in Equation (22) gives R  0 in V  and so by (6),   mind the second of Equation (26) for ra . GR   whether  is equal to  or not. Another For the calculations that follow the functions X and consequence of the result R  0 , is that the matter en-  Y defined by ergy tensor M  will be null as should be the case in the  electrovac V  . X  An2 arP221nn (40)  21nn21 The sourceless Maxwell equations in the first of (5) n1   give FF41,  42,r  0 and FF23,r 31, 0 . By the third    A 22nn 21n of (3) and with A3 and A4 given by (34) and (35), Ynn 22 1arPP22nn 2 (41)   n1 41n  these become AA3,rr 3, 0 , AA4,rr 4, 0 which

Copyright © 2012 SciRes. JMP A. GEORGIOU 1305 are trivially satisfied. The source-containing Maxwell given by   Equations (12) and (13) with A3 and A4 given by  3 4  22A21n nn2 Equations (34) and (35) will give J  0 and J  0 K wAb223  ar  41n  and so the 4-current is null in the electrovac V . n1 (52) 2.naC a m P22nn P 2 4. The Interior Solution The functions Z and U defined by

In accordance with the results of [2], the functions    A21n F ,  and A4 are constant which we shall take as Z  2naC a m n1 41n  (53)   F 1  0 A4 1 b (48) 221nn 2na r P22nn P 2   The functions K and A3 satisfy an equation of the  form *2 0 with *2 given by (15). This implies 22nn UAnaCamarP21nn 2  21 (54)   that K for example, is obtained from n1 1 will be required to simplify the components of the Ein- Krsin r sin r cos  (49) stein tensor.   where  is a harmonic function, which therefore satis- The components of G are calculated using the in- fies Laplace’s equation 2 0 with 2 given by terior functions (48) and (52) with Equations (16)-(22) (14). and, whenever necessary, bearing in mind the first of We choose  as Equation (39). The calculations give the following non-  zero components: 2n   Dr21n P2n (50) 22 n1 12  14UZ GG12 8π Eb 1 422  (55) rrsin  where the constants D21n are determined from the   junction condition Ka,  Ka, . We use Equa- 22   tion (49) for K with  given by (50) to find 334UZ GEb338π 422  (56)  rrsin  K  Dn21r22n  cos P  21nn 2 22 n1  UZ . 4444 GMEb4448π  3 422  (57) 21n   rrsin  PPcos  41n  21nn21   22  4444 UZ  We can further show that GMEb3338π  4 422  w(58) rrsin  1cos 2  P 2n 4 12 2  12bUZ cos GrG218π E 22 . (59) Pn22nn22P1n1P2n1 r sin 41n       Here, E and M  are the nonzero components and with this, the above expression for K becomes of the electromagnetic and mass energy tensors respec-  tively. The components of  E were obtained from (10) KD 2conr2n Ps. P   21nn2 2n1 the third of (3) for F , the interior functions in (48) and n 1     (52). The components of M  were obtained from Finally, after a little manipulation, the above expres- Equations (3) and (9) together with the interior functions  sion for K becomes (48) and (52). Equations (55)-(59) state that Einstein’s   Field Equations are satisfied in V . The sourceless  22nn1 2n KD 21nrPP2n2 2n. (51) Maxwell equations in the first of (5) give n1 41n    FF23,r  31,  0 . By the third of (3) and with A3 given The junction condition for the continuity of K implies  in Equation (52), this becomes AA3,rr 3, 0 which  that on ra , Ka,  Ka,  . Using the expres- is trivially satisfied. The source-containing Maxwell sion (33) for w+ and bearing in mind that Fa ,1,      Equations (12) and (13) with A4 and A3 as in (48) and we have Ka,  wa, . It therefore follows from (52) respectively, will give

(33) and (51), that the constants D21n are given by 42 2 34 bU Z 22n JJ0  422. (60) Dn21nn22n1a2bA212naCam. 2π rrsin     This implies that K , but also w and A3 are It is easily seen that

UZ22 bm b3 42 s4   wa,, Y a bZa , 8π Mb4 8π 2 422 . 222  rrsin  4πaa2π sin  It follows from this and Equation (60) that   2  The electromagnetic junction conditions are or, in dimensional units,  2 G . Dnˆ 4π s4 If N is any function in V, we write 3 bm 2b   +,,,wa  Ya  bZa  NNa :, Na, aa222sin   Na,:li mNa ,  (61) 2b3  0 Hsn = 4π ˆ Ya,,  bZa θˆ Na , :limNa , sin  0 where nˆ is the unit normal to the sphere and θˆ is the where the second and third of Equation (61), represent unit vector in the  direction. In these equations, the   the values of N on the V and V sides of . contravariant component D1 of D and the covariant It follows from Equations (32)-(35), (48) and (52) that component H2 of H from the second and third of  g  0 and  A   0. The functions g and A Equation (11) were used.  are therefore continuous across , but one degree of The Equations (16)-(22) for R and R, will give smoothness is lost because the first order partial r-deri- rise to terms with factors of delta-functions and first or- vatives of these functions are discontinuous on . It der partial r-derivatives which are discontinuous on  . follows that the ordinary junction conditions requiring Denoting these terms by Gothic symbols, the Einstein  the continuity of the directional derivatives of these func- tensor G and the associated matter stress-energy ten-  tions normal to , cannot be applied. The discontinue- sor M are connected through the field equations, and ties of these normal derivatives will generate a surface so on  we have layer on  with surface stress-energy tensor and sur- 1 G:=R-  R G=-M  8  (64) face 4-current and a more complicated set of junction  2   conditions will apply. The Equations (12) and (13) for  J 3 and J 4 will give rise to expressions with factors of Bearing in mind that exp F and that in V , exp  F 1, we display below the components delta-functions and first order partial r-derivatives which   R1 and R 2 as examples: are discontinuous on . We shall denote these terms by 1 2 1 Gothic symbols, and we find from (12) and (13) that RR12raF . these are 122 r  22 3 4πresin  J The surface stress-energy tensor S is expressed in (62) terms of the limits as   0 of the integrals of KAFA A ra   2 43rr3r M e with respect to r from ra to ra    and with M given in Equations (64). The junction 4πre22sin   J 4 conditions on  are [2,7] (63)  2GRkkk1(3)ba 2 LA43rr  K A  A3r r  a 1 ab 1 b   Gn21 k 2;b k ;2 where A34, A and A3 , are given in (34), (35) and (52) (65) 4 bbb respectively. To obtain the surface 4-current s and s , 8πSkaaa  k we form the integrals of J 3 and J 4 with respect to STnb 1 0. proper distance measured perpendicularly through  2;b  2 1 from ra to ra and then find the limits as Here, kab is the extrinsic curvature tensor of  de-

  0. There are no sign indicators with the metric fined by knab  a; b , where the covariant differentiation functions K, F and L in (62) and (63) because their is connected with the metric of V. Since   0 on    values in both, V and V , are required in these inte- this gives kgab ab,1 2 . grations, where the only nonzero contributions will arise The hypersurface scalar curvature invariant of  is 3 4 3 33  ab 3 from the delta-function parts J and J of J and RRg: ab where the Ricci tensor Rab is given J 4 in Equations (62) and (63). With φˆ the unit vector by in the  direction, this gives 3 ddnddn Rab  ad,, b ab d ad bn ab dn , 3 3 b d being the Christoffel symbols of the second kind s sYabZaφφˆˆ,, ab 2πa22sin  based on the metric of  . With these, all the elements in

Copyright © 2012 SciRes. JMP A. GEORGIOU 1307 the junction conditions (65) may be calculated and these With these, the magnetic dipole moment is therefore, 32 2 conditions may be shown to be valid. PbAa=-1 ()1 l and on using (69), this gives

3 32 2 5. Mass, Charge, Angular Momentum and PbAa=-1 ()1 l (73) the Magnetic Dipole Moment 2 From (71) and (73), we deduce that the gyromagnetic The mass, charge and angular momentum are defined by ratio is their imprints on the spacetime geometry far from the P source. To obtain the gravitational mass and electric =-b()1.l2 (74) charge therefore, we expand the exterior metric function J  22 F up to the term 3.mr Bearing in mind the first In physical units Equations (71), (73), (74) and the of (39), we then obtain from (32) third of (39), become 2 2  1mm23m æö3 3 2 c F 2211 . (66) ç ÷ Cr rrr Jba= l ç ÷ (75)   2 èøç G ÷ We may transform F + in (66) to the F of the RN 3 c2 Reissner-Nordstrom solution, by the transformation Pba=-22ll()1 2 (76) 2mq2 2 G rr=-m giving F =-1 + with qm= RN r 2 r PG2 22 =1b()-l (77) 2Gm G m J c [8], or in physical units, FRN =-1 242+ with cr cr 2 Gmæö Gm qGm= . This expression therefore implies that the l2 =+2 ç ÷ (78) 22ç ÷ gravitational mass is m and the electric charge is q acèø ac and these are connected by [8] It may be shown that the units of J and P are 21- 12 52- 1 qGm= (67) [J ]= MLT and [PMLT]= respectively, which are the units of angular momentum and magnetic If we now expand K + to O1r we have () dipole moment. We also find from the second of (26) and 2sinbAa222q K + = 1 (68) the second of (39) that r Gm 1 Ca=+1 b = (79) where A is obtained from (38) by setting n =1, () 2 1 ac Ca() which will then give, bearing in mind (39) 3lCa() 3l We stress the fact that all the above formulae are for A1 ==. (69) an electrically charged sphere whose mass m and charge 22b q are related by Equation (67). We note from (75) and If J is the total angular momentum, we have [9] (76) that the angular momentum J and dipole moment P 2sinJ 2 q depend on a2 but also in a somewhat more subtle way, K + = (70) r on the mass to radius ratio through the quantities l and 22 b . The analytical Formula (77) may be applied to a From (68) and (70), we then obtain J = baA1 and on using (69), this gives number of different objects. We note that there exists a formula for the gyromagnetic ratio of stars known as 3 J = bal 2 . (71) Blackett’s empirical Formulas [10-12], which reads 2 PG The dipole field is the part of the magnetic field Η = b (80) whose physical components Hr and Hq contain the J c -3 -3 factors r cosq and r sin q respectively. Since where b is a constant of the order of unity so that -3 only the r power is required, we only need the n =1 (80) becomes mode of the third of the expressions in (11) for Η . We PGæö÷ find that these components are = ç ÷. (81) ç ÷ 32 2 J èøç c ÷ 21bAa1 ()-lqcos H = r r3 Blackett suggested that an explanation of this relation . (72) “must be sought in a new fundamental property of matter bAa321s-lq2in 1 () not contained within the structure of present day physical Hq = r3 theory.” We note in this connection that the factor

Gc , occurs in both our analytical Formula (77) and Formula (81). These reductions however, are only possi- in Blackett’s empirical Formula (80). The explanation for ble in the cases where, b()11-=l2 . Thus, if we con- the presence of this factor in the analytical Formula (77) sider a typical neutron star as a fourth case we have however is implicit in its derivation. Furthermore, the mM==´1.4 2.7846 1033 g a = 106 cm coefficient of Gc in this formula is b 1-l2 , and s () 2 in Blackett’s Formula (80), it is a constant equal to 1, or ll==0.4562170327 0.6754384003 (85) 2 approximately equal to 1. The quantity b()1-l with PG b == 0.959011213 0.521493962 l and b given by (78) and (79) respectively, is ex- J c pected to vary from star to star, but b in Blackett’s where M is the mass of the Sun. Formula (80) is a constant equal to 1 for all stars, an as- S It is seen that PJ¹ Gc and this is because sertion that seems improbable. In the context of our solu- b 1-=l2 0.521493962 ¹ 1. In the context of our equation, it is difficult to see why different objects which can () tions, we found the precise condition under which our be as diverse as the Earth and the Sun, will conform to analytical Formula (77) will give Blackett’s empirical such a requirement as implied by Blackett’s empirical Formula (81). Again, in the context of our equations, this Formula (81). Although the “new physics” idea was sub- provides a full explanation why Blackett’s formula is sequently abandoned, it is nevertheless of interest to in- sometimes valid and why this occurs only for a range of vestigate further under what circumstances, if any, our objects. Our formula for the gyromagnetic ratio PJ is exact analytical Formula (77) reduces to Blackett’s em- not empirical, but an exact analytical formula which is a pirical Formula (81). consequence of the equations derived from the exact In order to gain an insight into the relation between the global solution of the Einstein-Maxwell field equations analytical Formula (77) and Blackett’s empirical Formu- found here. It does not require any new fundamental la (81), we shall consider three cases with different nu- properties of matter or any new physics and it is valid for merical values for the radius a and gravitational mass all values of the ratio ma. m of the sphere. We shall then proceed to calculate the We note that Wilson [12,13] observed that in the case 2 corresponding quantities in l , l , b and PJ in of the Earth and the Sun, the Formula (80) can be ac- (78), (79) and (77): counted for, if we assume that a rotating mass m has ma=´1.989 1033 g = 6.9599´ 1010 cm the same effect as a rotating electrical charge q where 26--3m and q are connected by Equation (67). It is a little l=´4.243406362 10 l=´ 2.059953 10 (82) puzzling that our electrically charged spheres charged as PGthey are in accordance with Equation (67), seem to echo b ==0.999997878 0.999993635 Jcthe above observation by Wilson. In our case however, m 33 11 and q are connected by Equation (67) in reality. ma=´=4.33602 10 g 1.4337394´ 10 cm The quantity of charge required is quite small. As noted ll26=´4.493 10- = 2.119669786493´ 10-3 (83) by Bonnor [8], if the mass m and charge q are related by Equation (67), then if in a sphere of neutral hydrogen PG b ==0.999997755 0.999727877 one atom in 1018 had lost its electron, this would be suf- Jc ficient. ma=´5.976 1027 g = 6.3675 ´ 108 cm 6. Discussion and Conclusions l29=´=1.393554681 10- l3.733034531´ 10-5 (84) Exact exterior and interior solutions of the Einstein- PG b ==0.999999999 0.999999998 . Maxwell field equations for rigidly rotating pressure-free Jc matter were obtained. The exterior and interior space- The above masses and radii were deliberately chosen times are separated by a boundary which is a surface to be numerically equal to those of the Sun, 78 Virginis layer with surface stress-energy tensor and surface elec- and the Earth. These correspond to the three astronomical tric 4-current. objects that are quoted in the literature by later authors in Perhaps one of the most important aspects of this work connection with Blackett’s empirical Formula (80) [10]. is that the source of spacetime, is rotating charged matter It is seen from the numerical results in (82)-(84), that in bounded by a closed surface. As far as we know, a global the case of our electrically charged spheres, the coeffi- solution with a volume distribution of finite bounded cient of Gc is very nearly equal to1 in every case. rotating matter as a source of the spacetime, does not We must conclude that in situations where the ratio ma exist in the literature, although flat disk solutions do in- is such that b()1-l2 is approximately equal to 1, our deed exist [1]. Another important outcome of this work is analytical Formula (77) will give Blackett’s empirical the derivation of analytical formulae for the angular mo-

Copyright © 2012 SciRes. JMP A. GEORGIOU 1309 mentum, dipole moment and gyromagnetic ratio of a ro- which is the Papapetrou solution [15] for which Bonnor tating sphere based on general relativistic equations. has found a matching interior solution [8]. The mass, charge, angular momentum and the mag- Referring to the surface layer that occurs in our solu- netic dipole moment were determined in Section 5. In tion, we note the result obtained by Ruffini and Treves in particular, we derived the analytical Formula (77) for the a non-relativistic treatment, in which they had shown that gyromagnetic ratio and discussed special cases to estab- a magnetized rotating object has surface charge and cur- lish the facts regarding the connection between the ana- rent densities; it is also endowed with a net electric lytical Formula (77) and Blackett’s empirical for For- charge [16]. This agrees with our results and in particular, mula (80) the conditions under which the analytical For- it confirms the existence of a surface layer with 4-current mula (77) reduces to Blackett’s empirical formula, were and stress-energy tensor on the boundary ra= . obtained. No new properties of matter and no new phys- The mass, charge, angular momentum and the mag- ics was required. Perhaps the analytical Formula (77) is netic dipole moment were determined in Section 5. In valid for all rotating objects and in particular for stars, particular, we derived the analytical Formula (77) for the but we have no data to demonstrate this, except for the gyromagnetic ratio and discussed special cases to estab- cases of the Sun, 78 Virginis and the Earth. lish the facts regarding the connection between the ana- All the physical quantities of interest in the interior lytical Formula (77) and Blackett’s empirical Formula and exterior were calculated as well as those associated (80). The conditions under which the analytical Formula with the spherical surface layer. In this problem, the or- (77) reduces to Blackett’s empirical formula, were ob- dinary gravitational junction conditions are inappropriate. tained. No new properties of matter and no new physics In fact there are two sets of junction conditions, the elec- were required. Perhaps the analytical Formula (77), is tromagnetic and the gravitational ones. The former were valid for all rotating objects and in particular for stars, expressed in the familiar form of classical electromag- but we have no data to demonstrate this, except for the netic theory. The gravitational junction conditions in this cases of the Sun, 78 Virginis and the Earth. problem are more complicated than the usual ones, because of the surface layer. These were clearly stated, al- REFERENCES though no detailed formulae were displayed. + [1] M. A. H. MacCallum, M. Mars and P. 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