Some new static charged spheres
aa r X i v : . [ g r- q c ] F e b Some new static charged spheres
S.D. Maharaj ∗ , S. Thirukkanesh Astrophysics and Cosmology Research Unit, School of Mathematical Sciences,University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa.
Abstract
We present new exact solutions for the Einstein-Maxwell system in static sphericallysymmetric interior spacetimes. For a particular form of the gravitational potentialsand the electric field intensity, it is possible to integrate the system in closed form.For specific parameter values it is possible to find new exact models for the Einstein-Maxwell system in terms of elementary functions. Our model includes a particularcharged solution found previously; this suggests that our generalised solution couldbe used to describe a relativistic compact sphere. A physical analysis indicates thatthe solutions describe realistic matter distributions.
Key words:
Exact solutions; Einstein-Maxwell equations; relativistic astrophysics
Exact solutions of the Einstein-Maxwell system of field equations, for spher-ically symmetric gravitational fields in static manifolds, are necessary to de-scribe charged compact spheres in relativistic astrophysics. The solutions tothe field equations generated have a number of different applications in rel-ativistic stellar systems. It is for this reason that a number of investigationshave been undertaken on the Einstein-Maxwell equations in recent times. Acomprehensive review of exact solutions and criteria for physical admissabilityis provided by Ivanov [1]. A general treatment of nonstatic spherically sym-metric solutions to the Einstein-Maxwell system, in the case of vanishing shearwas, performed by Wafo Soh and Mahomed [2] using symmetry methods. The ∗ Corresponding author.
Email addresses:
[email protected] (S.D. Maharaj), [email protected] (S. Thirukkanesh). Permanent address: Department of Mathematics, Eastern University, Chenkalady,Sri Lanka.
Preprint submitted to Elsevier 4 November 2018 ncharged case was considered by Wafo Soh and Mahomed [3] who show thatall existing solutions arise because of the existence of a Noether point sym-metry; the physical relevance of the solutions was investigated by Feroze et al [4]. The matching of nonstatic charged perfect fluid spheres to the Reissner-Nordstrom exterior metric was pursued by Mahomed et al [5] who highlightedthe role of the Bianchi identities in restricting the number of solutions.In this paper, we seek a new class of solutions to the Einstein-Maxwell sys-tem that satisfies the physical criteria. We attempt to perform a systematicseries analysis to the coupled Einstein-Maxwell equations by choosing a ratio-nal form for one of the gravitational potentials and a particular form for thecharged matter distribution. This approach produces a number of differenceequations, which we demonstrate can be solved explicitly from first princi-ples. A similar approach was used by Thirukkanesh and Maharaj [6], Maharajand Thirukkanesh [7] and Komathiraj and Maharaj [8]. They obtained ex-act solutions by reducing the condition of pressure isotropy to a recurrencerelation with real and rational coefficients which could be solved by math-ematical induction. In this way new mathematical and physical insights inthe Einstein-Maxwell field equations were generated. An advantage of thisapproach is that we generate new solutions to the Einstein-Maxwell systemwhich contain uncharged solutions found previously: Maharaj and Leach [9],Tikekar [10], Durgapal and Bannerji [11] and John and Maharaj [12], amongstothers.We first express the Einstein-Maxwell system of equations for static sphericallysymmetric line element as an equivalent system using the Durgapal and Ban-nerji [11] transformation in Section 2. In Section 3, we choose specific formsfor one of the gravitational potentials and the electric field intensity, whichreduce the condition of pressure isotropy to a linear second order equation inthe remaining gravitational potential. We integrate this generalised conditionof isotropy equation using the method of Frobenius in Section 4. In generalthe solution will be given in terms of special functions. We demonstrate thatit is possible to find two category of solutions in terms of elementary functionsby placing certain restriction on the parameters. We regain known chargedEinstein-Maxwell models from our general class of models in Section 5. InSection 6, we discuss the physical features of the solutions found, plot thematter variables, and show that our models are physically reasonable.
We assume that the spacetime is spherically symmetric and static which isconsistent with the study of charged compact objects in relativistic astro-physics. In Schwarzschild coordinates ( t, r, θ, φ ) the generic form of the line2lement is given by ds = − e ν ( r ) dt + e λ ( r ) dr + r ( dθ + sin θdφ ) . (1)The Einstein field equations for the line element (1) can be written as1 r h r (1 − e − λ ) i ′ = ρ + 12 E , (2a) − r (cid:16) − e − λ (cid:17) + 2 ν ′ r e − λ = p − E , (2b) e − λ ν ′′ + ν ′ + ν ′ r − ν ′ λ ′ − λ ′ r ! = p + 12 E , (2c) σ = 1 r e λ (cid:16) r E (cid:17) ′ , (2d)for charged perfect fluids. The energy density ρ and the pressure p are mea-sured relative to the comoving fluid 4-velocity u a = e − ν δ a and primes denotedifferentiation with respect to the radial coordinate r . The quantities associ-ated with the electric field are E , the electric field intensity, and σ , the propercharge density. In the system (2a)-(2d), we are using units where the couplingconstant πGc = 1 and the speed of light c = 1. This system of equationsdetermines the behaviour of the gravitational field for a charged perfect fluidsource. A different, but equivalent form of the field equations, can be found ifwe introduce the transformation x = Cr , Z ( x ) = e − λ ( r ) , A y ( x ) = e ν ( r ) , (3)where the parameters A and C are arbitrary constants. Under the transfor-mation (3) the system (2a)-(2d) has the equivalent form1 − Zx − Z = ρC + E C , (4a)4 Z ˙ yy + Z − x = pC − E C , (4b)4 Zx ¨ y + 2 ˙ Zx ˙ y + ˙ Zx − Z + 1 − E xC ! y = 0 , (4c) σ C = 4 Zx (cid:16) x ˙ E + E (cid:17) , (4d)where dots denote differentiation with respect to x . This system of equationsgoverns the behaviour of the gravitational field for a charged perfect fluidsource. When E = 0 the Einstein-Maxwell equations (4a)-(4d) reduce to theuncharged Einstein equations for a neutral fluid. Equation (4c) is called thegeneralised condition of pressure isotropy and is the fundamental equation3hat needs to be integrated to demonstrate an exact solution to the Einstein-Maxwell system of equations (4a)-(4d). Our objective is to find a new class of solutions to the Einstein-Maxwell systemby making explicit choices for the gravitational potential Z and the electricfield intensity E . We make the choice for Z as Z ( x ) = (1 + ax ) bx , (5)where a and b are real constants. Note that the choice (5) ensures that thegravitational potential e λ is regular and well behaved in the stellar interiorfor a wide range of values of the parameters a and b . In addition, when x = 0then Z = 1 which ensures that there is no singularity at the stellar centre. Aspecial case of (5) was studied by Komathiraj and Maharaj [13]. The choice(5) does produces charged and uncharged solutions which are necessary forconstructing realistic stellar models. On substituting (5) in (4c) we obtain4(1 + ax ) (1 + bx )¨ y + 2(1 + ax )[ b (1 + ax ) − b − a )] ˙ y + " ( a − b ) − E (1 + bx ) Cx y = 0 , (6)which is a second order differential equation.The differential equation (6) may be solved if a particular choice of the electricfield intensity E is made. For our purpose we set E C = αa ( b − a ) x (1 + bx ) , (7)where α is a constant. The electric field intensity specified in (7) vanishes atthe centre of the sphere; it is continuous and bounded in the stellar interior forwide range of values of x . The quantity E has positive values in the interior ofstar for relevant choices of the constants α, a and b . Therefore the form givenin (7) is physically reasonable to study the behaviour of charged spheres. Withthe choice (7) we can express (6) in the form4(1 + ax ) [ b (1 + ax ) − ( b − a )] ¨ y + 2 a (1 + ax ) [ b (1 + ax ) − b − a )] ˙ y + a ( b − a )( b − a − αa ) y = 0 . (8)4n (8) we assume that a = 0 and a = b so that the electric field intensity ispresent. When α = 0 there is no charge. To find the solution of the Einstein-Maxwell system we need to integratethe master equation (8). We consider two cases on the integration process: α = ba − α = ba − α = ba − ax ) [ b (1 + ax ) − ( b − a )] ¨ y + a [ b (1 + ax ) − b − a )] ˙ y = 0 . (9)Equation (9) is easily integrable and the solution can be written as y ( x ) = c s a (1 + bx ) b − a − arctan s a (1 + bx ) b − a + c , (10)where c and c are constants of integration. Therefore, the solution of theEinstein-Maxwell system (4a)-(4d) becomes e λ = 1 + bx (1 + ax ) , (11a) e ν = A c s a (1 + bx ) b − a − arctan s a (1 + bx ) b − a + c , (11b) ρC = ( b − a )(6 + bx )2(1 + bx ) − a x (11 + 6 bx )2(1 + bx ) , (11c) pC = (2 a − b )(2 + bx )2(1 + bx ) + a x (3 + 2 bx )2(1 + bx ) + 2 ac (1 + ax ) q a (1+ bx ) b − a c (1 + bx ) (cid:18)q a (1+ bx ) b − a − arctan q a (1+ bx ) b − a (cid:19) + c , (11d) E C = ( b − a ) x (1 + bx ) . (11e)Observe that because of the restrictions α = ba − b = a the chargedsolution (11) does not have an uncharged limit. Therefore this solution models5 sphere that is always charged and cannot attain a neutral state. Note thatthe solution (11) is expressed in a simple form in terms of elementary functionswhich facilitates a physical analysis of the matter and gravitational variables. α = ba − α = ba −
1, equation (8) is difficult to solve. Consequently we introducethe transformation y = (1 + ax ) d U (1 + ax ) , (12)where U is a function of (1 + ax ) and d is constant. With the help of (12), thedifferential equation (8) can be written as4(1 + ax ) [ b (1 + ax ) − ( b − a )] ¨ U +2(1 + ax ) [ b (4 d + 1)(1 + ax ) − d + 1)( b − a )] ˙ U + " bd (2 d − ax ) − ( b − a ) ba − − α − d ! U = 0 . (13)Note that there is substantial simplification if we take ba − − α = 4 d . Then (13) becomes2(1 + ax ) " (1 + ax ) − ( b − a ) b ¨ U + " (4 d + 1)(1 + ax ) − d + 1) ( b − a ) b ˙ U + d (2 d − U = 0 , (14)where b = 0. We observe that the point 1 + ax = b − ab is a regular singularpoint of the differential equation (14). Therefore, the solution of the differentialequation (14) can be written in the form of an infinite series by the methodof Frobenius: U = ∞ X i =0 c i " (1 + ax ) − ( b − a ) b i + r , c = 0 , (15)where c i are the coefficients of the series and r is the constant. To complete thesolution we need to find the coefficients c i as well as the parameter r explicitly.The indicial equation determines the value of r from c r (2 r −
3) = 0 . c = 0 we must have r = 0 or r = 3 /
2. We can express the structure forthe general coefficient c i in terms of the leading coefficient c as c i = ba − b ! i i Y p =1 [( p + r − p + 2 r + 4 d −
3) + d (2 d − p + r )(2 p + 2 r − c , (16)where the conventional symbol Q denotes multiplication. We can verify theresult (16) using mathematical induction. We can now generate two linearlyindependent solutions to (14) with the help of (15) and (16). For the parametervalue r = 0, we obtain the first solution U = c ∞ X i =1 ba − b ! i i Y p =1 [( p − p + 4 d −
3) + d (2 d − p (2 p − × " (1 + ax ) − ( b − a ) b i . For the parameter value r = 3 /
2, we obtain the second solution U = c " (1 + ax ) − ( b − a ) b / [1+ ∞ X i =1 ba − b ! i i Y p =1 [(2 p + 1)( p + 2 d ) + d (2 d − p (2 p + 3) " (1 + ax ) − ( b − a ) b i . Since the functions U and U are linearly independent we have found thegeneral solution to (14). Therefore, the solutions to the differential equation(8) are y ( x ) = c (1 + ax ) d ∞ X i =1 ba − b ! i × i Y p =1 [( p − p + 4 d −
3) + d (2 d − p (2 p − " (1 + ax ) − ( b − a ) b i (17)and y ( x ) = c (1 + ax ) d " (1 + ax ) − ( b − a ) b / ∞ X i =1 ba − b ! i × i Y p =1 [(2 p + 1)( p + 2 d ) + d (2 d − p (2 p + 3) " (1 + ax ) − ( b − a ) b i . (18)7hus the general solution to the differential equation (6), for the choice of theelectric field (7), is given by y ( x ) = A y ( x ) + A y ( x ) , (19)where A and A are arbitrary constants and d = (cid:16) ba − − α (cid:17) /
4. From (19)and (4a)-(4d), the exact solution of the Einstein-Maxwell system becomes e λ = 1 + bx (1 + ax ) , (20a) e ν = A y , (20b) ρC = (3 + bx )( b − a )(1 + bx ) − a x (5 + 3 bx )(1 + bx ) − αa ( b − a ) x bx ) , (20c) pC = 4 (1 + ax ) (1 + bx ) ˙ yy + a (2 + ax ) − b (1 + bx ) + αa ( b − a ) x bx ) , (20d) E C = αa ( b − a ) x (1 + bx ) . (20e)We believe that this is a new solution to the Einstein-Maxwell system. In gen-eral the models in (20) cannot be expressed in terms of elementary functionsas the series in (17) and (18) do not terminate. Consequently the solutionwill be given in terms of special functions. Terminating series are possible forparticular values of a and b as we show in the next section. It is possible to generate exact solutions in terms of elementary functions fromthe series in (19). This is possible for specific values of the parameters a, b and α so that the series (17) and (18) terminate. Consequently two categories ofsolutions are obtainable in terms of elementary functions by placing restric-tions on the quantity ba − − α . We can express the first category of solution,in terms of the variable r , as 8 ( x ) = A ax ) n × n X i =0 ( − i − bb − a ! i (2 i − i )!(2 n − i + 1)! " (1 + ax ) − ( b − a ) b i + A ax ) n " (1 + ax ) − ( b − a ) b / × n − X i =0 ba − b ! i ( i + 1)(2 i + 3)!(2 n − i − " (1 + ax ) − ( b − a ) b i , (21)where ba − − α = 4 n relates the constants a, b, α and n . The second categoryof solution, in terms of the variable r , is given by y ( x ) = A ax ) n − / × n X i =0 ( − i − bb − a ! (2 i − i )!(2 n − i )! " (1 + ax ) − ( b − a ) b i + A ax ) n − / " (1 + ax ) − ( b − a ) b / × n − X i =0 ba − b ! i ( i + 1)(2 i + 3)!(2 n − i − " (1 + ax ) − ( b − a ) b i , (22)where ba − − α = 4 n ( n −
1) + 1 relates the constants a, b, α and n . Thus wehave extracted two classes of solutions (21) and (22) to the Einstein-Maxwellsystem in terms of elementary functions from the infinite series solution (19).This class of solution can be expressed as combinations of polynomials andalgebraic functions. The simple form of (21) and (22) helps in the study of thephysical features of the model.From our general classes of solutions (21) and (22), it is possible to generateparticular solutions found for charged stars previously . If we take b = 1 and K = − aa then it is easy to verify that the equation (21) becomes y ( x ) = D (cid:20) KK + 1 + x (cid:21) n n X i =0 ( − i − (2 i − i )!(2 n − i + 1)! (cid:20) xK (cid:21) i + D " KK + 1 + x ) n (cid:20) xK (cid:21) / × n − X i =0 ( − i ( i + 1)(2 i + 3)!(2 n − i − (cid:20) xK (cid:21) i , (23)where K − α = 4 n , D = A (1 − a ) n and D = A (1 − a ) n − / . Also, equation (22)9ecomes y ( x ) = D (cid:20) KK + 1 + x (cid:21) n − / n X i =0 ( − i − (2 i − i )!(2 n − i )! (cid:20) xK (cid:21) i + D (cid:20) KK + 1 + x (cid:21) n − / (cid:20) xK (cid:21) / × n − X i =0 ( − i ( i + 1)(2 i + 3)!(2 n − i − (cid:20) xK (cid:21) i , (24)where K − α = 4 n ( n −
1) + 1, D = A (1 − a ) n − / and D = A (1 − a ) n − . Thus we haveregained the Komathiraj and Maharaj [13] charged model; our solutions allowfor a wider range of models for charged relativistic spheres. We illustrate thisfeature with an example involving a specific value for n . For example, supposethat n = 1 then b = (5 + α ) a and we get y = a (7 + α + 3(5 + α ) ax ) + a (1 + (5 + α ) ax ) / ax (25)from (21) where a and a are new arbitrary constants. From (25) and (4a)-(4d) the solution to the Einstein-Maxwell system becomes e λ = 1 + (5 + α ) ax (1 + ax ) , (26a) e ν = A a (7 + α + 3(5 + α ) ax ) + a (1 + (5 + α ) ax ) ax , (26b) ρC = a (3 + α − ax )1 + (5 + α ) ax + 2 a (1 + ax ) [3 + α − (5 + α ) ax ][1 + (5 + α ) ax ] − αa (4 + α ) x α ) ax ] , (26c) pC = 2 a (1 + ax )[1 + (5 + α ) ax ] × h a (4 + α ) + a (1 + (5 + α ) ax ) (13 + 3 α + (5 + α ) ax ) ih a (7 + α + 3(5 + α ) ax ) + a (1 + (5 + α ) ax ) i − a (3 + α − ax )1 + (5 + α ) ax + αa (4 + α ) x α ) ax ] , (26d) E C = αa (4 + α ) x [1 + (5 + α ) ax ] . (26e)Note that the solution of the form (26) cannot be regained from Komathirajand Maharaj [13] charged models except for the value of a = α ) . This indi-10ates that our model is the generalisation of Komathiraj and Maharaj chargedmodels with more general behaviour in the gravitational and electromagneticfields. In this section, we briefly consider the physical features of the models generatedin this paper. For the pressure to vanish at the boundary r = R in the solution(20) we require p ( R ) = 0 which gives the condition4(1 + aCR ) " ˙ yy r = R + a (2 + aCR ) − b + αa ( b − a ) CR bCR ) = 0 , (27)where y is given by (17)-(19). This will constrain the values of a, b and α . Thesolution of the Einstein-Maxwell system for r > R is given by the Reissner-Nordstrom metric as ds = − − mr + q r ! dt + − mr + q r ! − dr + r (cid:16) dθ + sin θdφ (cid:17) , (28)where m and q are the total mass and the charge of the star. To match thepotentials in (20) to (28) generates the relationships between the constants A , A , a, b and R as follows − mR + q R ! = A [ A y ( R ) + A y ( R )] , (29a) − mR + q R ! − = 1 + bCR (1 + aCR ) . (29b)The matching conditions (27) and (29) place restrictions on the metric coef-ficients; however there are sufficient free parameters to satisfy the necessaryconditions that arise for the model under study. Since these conditions aresatisfied by the constants in the solution a relativistic star of radius R isrealisable.From (20a) and (20b) we easily observe that the gravitational potentials e λ and e ν are continuous and well behaved for wide range of the parameters a and b . From (20c), the variable x can be expressed solely in terms of theenergy density ρ as 11 .2 0.4 0.6 0.8 1.0 r Ρ Fig. 1. Energy density. x = 12 b (3 a C + bρ ) (cid:20) b C − bρ − a C − abC ± q ( a − b ) C × q [ − a bC + a C (25 + 6 bα ) − b ( bC + 8 ρ ) + ab (3 bC + 8 ρ + 2 bαρ )] (cid:21) Hence, from (20d) the isotropic pressure p can be written as a function of en-ergy density ρ only. Therefore the solutions generated in this paper satisfy thebarotropic equation of state p = p ( ρ ). Many of the solutions found previouslydo not satisfy this desirable feature. We illustrate the graphical behaviour ofmatter variables in the stellar interior for the particular solution (26). Weassume that a = − . , a = C = 1 and a = α = 1 / ≤ r ≤
1. To generate the plots for ρ, p, E , dp/dρ and p vs ρ , we utilised the software package Mathematica. The behaviour ofthe energy density is plotted in Fig. 1. It is positive and monotonically de-creasing towards the boundary of the stellar object. In Fig. 2, we have plottedthe behaviour of matter pressure p , which is regular, monotonically decreasingand becomes zero at the vacuum boundary of the stellar object. In Fig. 3, wedescribe the behaviour of the electric field intensity. It is well behaved anda continuous function. In Fig. 4, we have plotted the speed of sound dp/dρ .We observe that 0 ≤ dp/dρ ≤ p verses thedensity ρ and we find that this approximates a linear function. This behaviouris to be expected as the gradients of p and ρ have similar profiles in the stel-lar interior. Thus we have demonstrated that the particular solution satisfiesthe requirements for a physically reasonable stellar interior in the context ofgeneral relativity. 12 .2 0.4 0.6 0.8 1.0 r Fig. 2. Matter pressure. r E Fig. 3. Electric field intensity. r (cid:144) d Ρ Fig. 4. Speed of sound dpdρ . Ρ Fig. 5. Pressure vs Density
Acknowledgements
ST thanks the National Research Foundation and the University of KwaZulu-Natal for financial support, and is grateful to Eastern University, Sri Lankafor study leave. SDM acknowledges that this work is based upon researchsupported by the South African Research Chair Initiative of the Departmentof Science and Technology and the National Research Foundation.13 eferences [1] B.V. Ivanov, Static charged perfect fluid spheres in general relativity, Phys. Rev.D 65 (2002) 104001.[2] C. Wafo Soh, F.M. Mahomed, Non-static shear-free spherically symmetriccharged perfect fluid distributions: a symmetry approach, Class. Quantum Grav.17 (2000) 3063-3072.[3] C. Wafo Soh, F.M. Mahomed, Noether symmetries of y ′′ = f ( x ) y n withapplications to non-static spherically symmetric perfect fluid solutions, Class.Quantum Grav. 16 (1999) 3553-3566.[4] T. Feroze, F.M. Mahomed, A. Qadir, Non-static spherically symmetric shear-freeperfect fluid solutions of Einstein’s field equations, Nuovo Cimento B 118 (2003)895-902.[5] F.M. Mahomed, A. Qadir, C. Wafo Soh, Charged spheres in general relativityrevisited, Nuovo Cimento B 118 (2003) 373-381.[6] S. Thirukkanesh, S.D. Maharaj, Exact models for isotropic matter, Class.Quantum Grav. 23 (2006) 2697-2709.[7] S.D. Maharaj, S. Thirukkanesh, Generating potentials via difference equations,Math. Meth. Appl. Sci. 29 (2006) 1943-1952.[8] K. Komathiraj, S.D. Maharaj, Tikekar superdense stars in electric fields, J. Math.Phys. 48 (2007) 042501.[9] S.D. Maharaj, P.G.L. Leach, Exact solutions for the Tikekar superdense star, J.Math. Phys. 37 (1996) 430-437.[10] R. Tikekar, Exact model for a relativistic star, J. Math. Phys. 31 (1990) 2454-2458.[11] M.C. Durgapal, R. Bannerji, New analytical stellar model in general relativity,Phys. Rev. D 27 (1983) 328-331.[12] A.J. John, S.D. Maharaj, An exact isotropic solution, Nuovo Cimento B 121(2006) 27-33.[13] K. Komathiraj, S.D. Maharaj, Classes of exact Einstein-Maxwell solutions, Gen.Relativ. Gravit. 39 (2007) 2079-2093.withapplications to non-static spherically symmetric perfect fluid solutions, Class.Quantum Grav. 16 (1999) 3553-3566.[4] T. Feroze, F.M. Mahomed, A. Qadir, Non-static spherically symmetric shear-freeperfect fluid solutions of Einstein’s field equations, Nuovo Cimento B 118 (2003)895-902.[5] F.M. Mahomed, A. Qadir, C. Wafo Soh, Charged spheres in general relativityrevisited, Nuovo Cimento B 118 (2003) 373-381.[6] S. Thirukkanesh, S.D. Maharaj, Exact models for isotropic matter, Class.Quantum Grav. 23 (2006) 2697-2709.[7] S.D. Maharaj, S. Thirukkanesh, Generating potentials via difference equations,Math. Meth. Appl. Sci. 29 (2006) 1943-1952.[8] K. Komathiraj, S.D. Maharaj, Tikekar superdense stars in electric fields, J. Math.Phys. 48 (2007) 042501.[9] S.D. Maharaj, P.G.L. Leach, Exact solutions for the Tikekar superdense star, J.Math. Phys. 37 (1996) 430-437.[10] R. Tikekar, Exact model for a relativistic star, J. Math. Phys. 31 (1990) 2454-2458.[11] M.C. Durgapal, R. Bannerji, New analytical stellar model in general relativity,Phys. Rev. D 27 (1983) 328-331.[12] A.J. John, S.D. Maharaj, An exact isotropic solution, Nuovo Cimento B 121(2006) 27-33.[13] K. Komathiraj, S.D. Maharaj, Classes of exact Einstein-Maxwell solutions, Gen.Relativ. Gravit. 39 (2007) 2079-2093.