Rotating and counterrotating relativistic thin disks as sources of stationary electrovacuum spacetimes
aa r X i v : . [ g r- q c ] O c t Rotating and counterrotating relativistic thin disksas sources of stationary electrovacuum spacetimes
Gonzalo Garc´ıa-Reyes ∗ Universidad Tecnol´ogica de Pereira, Departamento de F´ısicaA. A. 97, Pereira, Colombia
Guillermo A. Gonz´alez † Escuela de F´ısica, Universidad Industrial de SantanderA.A. 678, Bucaramanga, Colombia
Abstract
A detailed study is presented of the counterrotating model (CRM) for electrovacuum sta-tionary axially symmetric relativistic thin disks of infinite extension without radial stress, inthe case when the eigenvalues of the energy-momentum tensor of the disk are real quantities,so that there is not heat flow. We find a general constraint over the counterrotating tangentialvelocities needed to cast the surface energy-momentum tensor of the disk as the superpositionof two counterrotating charged dust fluids. We then show that, in some cases, this constraintcan be satisfied if we take the two counterrotating tangential velocities as equal and opposite orby taking the two counterrotating streams as circulating along electro-geodesics. However, weshow that, in general, it is not possible to take the two counterrotating fluids as circulating alongelectro-geodesics nor take the two counterrotating tangential velocities as equal and opposite. Asimple family of models of counterrotating charged disks based on the Kerr-Newman solution areconsidered where we obtain some disks with a CRM well behaved. We also show that the disksconstructed from the Kerr-Newman solution can be interpreted, for all the values of parameters,as a matter distribution with currents and purely azimuthal pressure without heat flow. Themodels are constructed using the well-known “displace, cut and reflect” method extended tosolutions of vacuum Einstein-Maxwell equations. We obtain, in all the cases, counterrotatingKerr-Newman disks that are in agreement with all the energy conditions.
Key words: general relativity, thin disks, exact solutions, Einstein-Maxwell equations. ∗ e-mail: [email protected] † e-mail: [email protected] Introduction
Several methods are known to exactly solve the Einstein and Einstein-Maxwell field equations, orto generate new exact solutions from simple known solutions [1]. However, the above mentionedmethods in general lead to solutions without a clear physical interpretation or to solutions thatdepend on many parameters without a clear physical meaning. Accordingly, it is of importance tohave some appropriate procedures to obtain physical interpretations of these exact solutions. So,in the past years such procedures have been developed for static and stationary axially symmetricsolutions in terms of thin and, more recently, thick disk models.Stationary or static axially symmetric exact solutions of Einstein equations describing relativisticthin disks are of great astrophysical importance since they can be used as models of certain stars,galaxies and accretion disks. These were first studied by Bonnor and Sackfield [2], obtaining pres-sureless static disks, and then by Morgan and Morgan, obtaining static disks with and without radialpressure [3, 4]. In connection with gravitational collapse, disks were first studied by Chamorro, Gre-gory, and Stewart. Also thin disks with radial tension were considered [6]. Several classes of exactsolutions of the Einstein field equations corresponding to static thin disks with or without radialpressure have been obtained by different authors [7 - 15].Rotating thin disks that can be considered as a source of a Kerr metric were presented by Bi˘c´akand Ledvinka [16], while rotating disks with heat flow were studied by Gonz´alez and Letelier [17].The nonlinear superposition of a disk and a black hole was first obtained by Lemos and Letelier [10].Perfect fluid disks with halos were studied by Vogt and Letelier [18]. The stability of some generalrelativistic thin disks models using a first order perturbation of the energy-momentum tensor wasinvestigated by Ujevic and Letelier [19].Gonz´alez and Letelier [20] constructed models of static relativistic thick disks in various coordinatesystems. Although the disks have constant thickness, the matter density decreases rapidly with radiusand the z coordinate, and in principle they also can be used to represent both the disk part and thecentral bulges of galaxies. Also Vogt and Letelier [21] considered more realistic three-dimensionalmodels for the gravitational field of Galaxies in the General Relativistic context. Essentially theyformulate the General Relativistic versions in isotropic coordinates of the potential-density pairsdeduced by Miyamoto and Nagai [22, 23] and Satoh [24].Disk sources for stationary axially symmetric spacetimes with magnetic fields are also of as-trophysical importance mainly in the study of neutron stars, white dwarfs and galaxy formation.Although disks with electric fields do not have clear astrophysical importance, their study may be ofinterest in the context of exact solutions. Thin disks have been discussed as sources for Kerr-Newmanfields [25], magnetostatic axisymmetric fields [26], conformastationary metrics [27], while models ofelectrovacuum static counterrotating dust disks were presented in [28]. Charged perfect fluid diskswere also studied by Vogt and Letelier [29], and charged perfect fluid disks as sources of static andTaub-NUT-type spacetimes by Garc´ıa-Reyes and Gonz´alez [30, 31].In all the above cases, the disks are obtained by an “inverse problem” approach, called by Syngethe “ g-method ” [32]. The method works as follows: a solution of the vacuum Einstein equations istaken, such that there is a discontinuity in the derivatives of the metric tensor on the plane of the disk,2nd the energy-momentum tensor is obtained from the Einstein equations. The physical propertiesof the matter distribution are then studied by an analysis of the surface energy-momentum tensorso obtained. Another approach to generate disks is by solving the Einstein equations given a source(energy-momentum tensor). Essentially, they are obtained by solving a Riemann-Hilbert problemand are highly nontrivial [33 - 39]. A review of this kind of disks solutions to the Einstein-Maxwellequations was presented by Klein in [40].Now, when the inverse problem approach is used for static electrovacuum spacetimes, the energy-momentum tensor is diagonal and its analysis is direct and, except for the dust disks, all the obtaineddisks have anisotropic sources with azimuthal stress different from the radial stress. On the otherhand, when the considered spacetime is stationary, the obtained energy-momentum tensor is non-diagonal and the analysis of its physical content is more involved and, in general, the obtained sourceis not only anisotropic but with nonzero heat flow. Due to this fact, there are very few works aboutof stationary electrovacuum disks and they are limited to disks obtained with solutions that lead todisks without heat flow [25, 31].The necessary condition to obtain a thin disk without heat flow is that the eigenvalues of theenergy-momentum tensor must be real quantities, which can be only for very few known electrovac-uum solutions. In [31] we consider a Taub-NUT type solution such that the energy-momentumtensor can be written as an upper right triangular matrix, so that the diagonalization is trivial andthe eigenvalues are real quantities. However, the obtained disks are not really rotating disks since thespatial components of their velocity vectors are zero with respect to the coordinates and so the disksare “locally statics”. The first true rotating electrovacuum thin disks were obtained by Ledvinka,Bi˘c´ak, and ˘Zofka [25] by applying the “displace, cut and reflect” method to the Kerr-Newman solu-tion. The so obtained disks have no radial pressure and no heat flow. However, the authors do notshow if the eigenvalues of the momentum-energy tensor are real quantities for all the values of theparameters, is that, if these disks can always be interpreted as a matter distribution with currentsand purely azimuthal pressure, or if there are some case where can exist nonzero heat flow (complexeigenvalues).The above disks can also be interpreted as made of two counterrotating streams of moving chargedparticles, as was also indicated in [25]. Now, in order to do this interpretation, the counterrotatingtangential velocities of the two streams must to satisfy a constraint, which in general is not satisfiedfor disks obtained from generic stationary electrovacuum solutions. In [25] the authors take the twocounterrotating streams as circulating along electro-geodesics, but they do not show if such decom-position can be done. In addition, as we will show in this paper, in general the electro-geodesicsmotion do not agree with the above mentioned constraint and so it is necessary to consider anotherpossibility for the complete determination of the counterrotating tangential velocities. Another pos-sibility, also commonly assumed, is to take the two counterrotating velocities as equal and oppositesbut, as we will show, in general the counterrotating velocities are not completely determined by theconstraint, so that the corresponding interpretation as two counterrotating streams is not possible.The above interpretation is obtained by means of the Counterrotating Model (CRM) in whichthe energy-momentum tensor of the source is expressed as the superposition of two counterrotatingperfect fluids. Now, even though this interpretation can be seen as merely theoretical, there are3bservational evidence of disks made of streams of rotating and counterrotating matter (see, forinstance, [41, 42, 43]). These disks are made of stars and gas so that they are disks with pressure.Nevertheless, as is suggested in [42], the preexisting galaxies have a component originally constitutedmainly by a gas free stellar disk, i.e., collisionless matter or dust. A detailed study of the CRM forgeneric relativistic static thin disks was presented in [15] for the vacuum case, whereas the extensionfor static electrovacuum disks was presented in [28, 30]. On the other hand, the CRM for stationarythin disks has not been completely developed, neither for the vacuum case, and only a preliminaryversion of it was presented in [17] for the case of stationary thin disks without heat flow and withpositive radial stress (pressure).The purpose of the present paper is twofold. In first instance, we present a detailed analysis ofthe energy-momentum tensor and the surface current density for electrovacuum stationary axiallysymmetric relativistic thin disks of infinite extension without radial stress, in the case when theenergy-momentum tensor of the disks can be diagonalized, so that there is not heat flow. And, inthe second place, we present the complete study of the Counterrotating Model for these stationarythin disks. The paper is structured as follows. In Sec. II we present a summary of the procedureto obtain models of rotating thin disks with a purely azimuthal pressure and currents, using thewell-known “displace, cut and reflect” method extended to solutions of Einstein-Maxwell equations,in the case when the eigenvalues of the energy-momentum tensor of the disk are real quantities. Inparticular, we obtain expressions for the surface energy-momentum tensor and the surface currentdensity of the disks.In Sec. III the disks are interpreted in terms of the counterrotating model (CRM). We find thegeneral constraint over the counterrotating tangential velocities needed to cast the surface energy-momentum tensor of the disk as the superposition of two counterrotating charged dust fluids. We thenshow that this constraint can be satisfied if we take the two counterrotating tangential velocities asequal and opposite as well as by taking the two counterrotating streams as circulating along electro-geodesics. However, we show that, in general, it is not possible to take the two counterrotatingfluids as circulating along electro-geodesics nor take the two counterrotating tangential velocities asequal and opposite. We also find explicit expressions for the energy densities, current densities andvelocities of the two counterrotating fluids.In the following section, Sec. IV, we consider a family of models of counterrotating charged dustdisks based on the Kerr-Newman metric, perhaps the only simple electrovacuum solution that leadto stationary thin disks without heat flow. We show that for Kerr-Newman fields the eigenvaluesof the energy-momentum tensor of the disks are always real quantities, for all the values of theparameters, and so they do not present heat flow in any case. We also analyze the CRM for thesedisks and study the tangential velocities, energy and electric charge densities of both streams whenthe two fluids move along electrogeodesics and when they move with equal and opposite velocities.Also the stability against radial perturbation is analyzed in both of the cases. Finally, in Sec. V, wesummarize our main results. 4 Electrovacuum rotating relativistic thin disks
A sufficiently general metric for our purposes can be written as the Weyl-Lewis-Papapetrou lineelement [1], ds = − e ( dt + W dϕ ) + e − [ r dϕ + e ( dr + dz )] , (1)where we use for the coordinates the notation ( x , x , x , x ) = ( t, ϕ, r, z ), and Ψ, W , and Λ arefunctions of r and z only. The vacuum Einstein-Maxwell equations, in geometric units in which8 πG = c = µ = ε = 1, are given by G ab = T ab , (2a) F ab ; b = 0 , (2b)with T ab = F ac F cb − g ab F cd F cd , (3a) F ab = A b,a − A a,b , (3b)where A a = ( A t , A ϕ , ,
0) and the electromagnetic potentials A t and A ϕ are also functions of r and z only.For the metric (1), the Einstein-Maxwell equations are equivalent to the system [44] ∇ · [ r − f ( ∇ A ϕ − W ∇ A t ] = 0 , (4a) ∇ · [ f − ∇ A t + r − f W ( ∇ A ϕ − W ∇ A t ] = 0 , (4b) ∇ · [ r − f ∇W − r − f A t ( ∇ A ϕ − W ∇ A t )] = 0 , (4c) f ∇ f = ∇ f · ∇ f − r − f ∇W · ∇W + f ∇ A t · ∇ A t + r − f ( ∇ A ϕ − W∇ A t ) · ( ∇ A ϕ − W∇ A t ) , (4d)Λ ,r = r (Ψ ,r − Ψ ,z ) − r ( W ,r − W ,z ) e − r ( r e − − W e )( A t,r − A t,z )+ 12 r ( A ϕ,r − A ϕ,z ) e − r W ( A ϕ,r A t,r − A ϕ,z A t,z ) e , (4e)Λ ,z = 2 r Ψ ,r Ψ ,z − r W ,r W ,z e − r ( r e − − W e ) A t,r A t,z + 1 r A ϕ,r A ϕ,z e − r W ( A ϕ,r A t,z + A ϕ,z A t,r ) e , (4f)where ∇ is the standard differential operator in cylindrical coordinates and f = e .In order to obtain a solution of (2a) - (2b) representing a thin disk at z = 0, we assume that thecomponents of the metric tensor are continuous across the disk, but with first derivatives discontin-uous on the plane z = 0, with discontinuity functions b ab = g ab,z | z =0+ − g ab,z | z =0 − = 2 g ab,z | z =0+ . (5)5hus, by using the distributional approach [46, 47, 48] or the junction conditions on the extrinsiccurvature of thin shells [49, 50, 51], the Einstein-Maxwell equations yield an energy-momentum tensor T ab = T elm ab + T mat ab , where T mat ab = Q ab δ ( z ), and a current density J a = j a δ ( z ) = − e − Λ) A a,z δ ( z ),where δ ( z ) is the usual Dirac function with support on the disk. T elm ab is the electromagnetic tensordefined in Eq. (3a), j a is the current density on the plane z = 0, and Q ab = 12 { b az δ zb − b zz δ ab + g az b zb − g zz b ab + b cc ( g zz δ ab − g az δ zb ) } is the distributional energy-momentum tensor. The “true” surface energy-momentum tensor (SEMT)of the disk, S ab , and the “true” surface current density, j a , can be obtained through the relations S ab = Z T mat ab ds n = e Λ − Ψ Q ab , (6a) j a = Z J a ds n = e Λ − Ψ j a , (6b)where ds n = √ g zz dz is the “physical measure” of length in the direction normal to the disk.For the metric (1), the nonzero components of S ba are S = e Ψ − Λ r (cid:2) r (Λ , z − , z ) − e WW , z (cid:3) , (7a) S = − e Ψ − Λ r (cid:2) r W Ψ , z + ( r + W e ) W , z (cid:3) , (7b) S = e Ψ − Λ r (cid:2) e W , z (cid:3) , (7c) S = e Ψ − Λ r (cid:2) r Λ , z + e WW , z (cid:3) , (7d)and the nonzero components of the surface current density j a are j t = − e Ψ − Λ A t,z , (8a) j ϕ = − e Ψ − Λ A ϕ,z , (8b)where all the quantities are evaluated at z = 0 + .These disks are essentially of infinite extension. Finite disks can be obtained introducing oblatespheroidal coordinates, which are naturally adapted to a disk source, and imposing appropriateboundary conditions. These solutions, in the vacuum and static case, correspond to the Morganand Morgan solutions [3]. A more general class of solutions representating finite thin disks can beconstructed using a method based on the use of conformal transformations and solving a boundary-value problem [4, 5, 6, 15, 30, 31].Now, in order to analyze the matter content of the disks is necessary to compute the eigenvaluesand eigenvectors of the energy-momentum tensor. The eigenvalue problem for the SEMT (7a) - (7d) S ab ξ b = λ ξ a , (9)6as the solutions λ ± = 12 (cid:16) T ± √ D (cid:17) , (10)where T = S + S , D = ( S − S ) + 4 S S , (11)and λ r = λ z = 0. For the metric (1) D = 4 e − Λ) r (4 r Ψ ,z − W ,z e ) = A − B , (12a) T = 4 e Ψ − Λ (Λ ,z − Ψ ,z ) , (12b)where A = 4Ψ ,z e Ψ − Λ , B = 2 r W ,z e − Λ . (13)The corresponding eigenvectors are ξ a ± = ( ξ ± , ξ ± , , ,X a = e U − Λ (0 , , , ,Y a = e U − Λ (0 , , , , (14)with g ( ξ ± , ξ ± ) = 2 N ± e (cid:18) ξ ± S − S ± √ D (cid:19) , (15)where N ± = √ D ( −√ D ± A ) . (16)We only consider the case when D ≥
0, so that the two eigenvalues λ ± are real and differentand the two eigenvectors are orthogonal, in such a way that one of them is timelike and the other isspacelike. Since | A | ≥ √ D , from (16) follows that when A >
A < ,z determines the sign of the norm.Let V a be the timelike eigenvector, V a V a = −
1, and W a the spacelike eigenvector, W a W a = 1.In terms of the orthonormal tetrad or comoving observer e ˆ ab = { V b , W b , X b , Y b } , the SEMT and thesurface electric current density may be decomposed as S ab = ǫV a V b + p ϕ W a W b , (17a) j a = j ˆ0 V a + j ˆ1 W a , (17b)where ǫ = − λ ± , p ϕ = λ ∓ , (18)7re, respectively, the surface energy density, the azimuthal pressure, and j ˆ0 = − V a j a , j ˆ1 = W a j a , (19)are the surface electric charge density and the azimuthal current density of the disk measured bythis observer. In (18) the sign is chosen according to which is the timelike eigenvector and which isthe spacelike eigenvector. However, in order to satisfy the strong energy condition ̺ = ǫ + p ϕ ≥ ̺ is the effective Newtonian density, we must choose ξ − as the timelike eigenvector and ξ + as the spacelike eigenvector. These condition characterizes a disk made of matter with the usualgravitational attractive property. Consequently Ψ ,z must be taken positive. So we have ǫ = − λ − , p ϕ = λ + , (20)and V = νe − Ψ √− N − ( S − S − √ D ) , (21a) V = 2 νe − Ψ √− N − S , (21b)where ν = ± so that the sign is chosen according to the causal character of the timelike eigenvector(observer’s four-velocity), W = 2 √ M S , (22a) W = 1 √ M ( S − S + √ D ) , (22b)where M = √ D n g √ D + 2 r W B + ( r e − + W e ) A o . (23) We now consider, based on Refs. [15] and [30], the possibility that the SEMT S ab and the currentdensity j a can be written as the superposition of two counterrotating charged fluids that circulate inopposite directions; that is, we assume S ab = S ab + + S ab − , (24a) j a = j a + + j a − , (24b)where the quantities on the right-hand side are, respectively, the SEMT and the current density ofthe prograde and retrograde counterrotating fluids.8et U a ± = ( U ± , U ± , ,
0) = U ± (1 , ω ± , ,
0) be the velocity vectors of the two counterrotating fluids,where ω ± = U ± /U ± are the angular velocities of each stream. In order to do the decomposition (24a)and (24b) we project the velocity vectors onto the tetrad e ˆ ab , using the relations [52] U ˆ a ± = e ˆ ab U b ± , U a ± = e ˆ ba U ˆ b ± . (25)In terms of the tetrad (14) we can write U a ± = V a + v ± W a p − v ± , (26)so that V a = p − v − v + U a − − p − v v − U a + v + − v − , (27a) W a = p − v U a + − p − v − U a − v + − v − , (27b)where v ± = U ˆ1 ± /U ˆ0 ± are the tangential velocities of the streams with respect to the tetrad.Another quantity related with the counterrotating motion is the specific angular momentum ofa particle rotating at a radius r , defined as h ± = g ϕa U a ± . This quantity can be used to analyze thestability of circular orbits of test particles against radial perturbations. The condition of stability, d ( h ) dr > , (28)is an extension of Rayleigh criteria of stability of a fluid in rest in a gravitational field [53]. For ananalysis of the stability of a rotating fluid taking into account the collective behavior of the particlessee for example Refs. [54, 19].Substituting (27a) and (27b) in (17a) we obtain S ab = F ( v − , v − )(1 − v ) U a + U b + ( v + − v − ) + F ( v + , v + )(1 − v − ) U a − U b − ( v + − v − ) − F ( v + , v − )(1 − v ) (1 − v − ) ( U a + U b − + U a − U b + )( v + − v − ) where F ( v , v ) = ǫv v + p ϕ . (29)Clearly, in order to cast the SEMT in the form (24a), the mixed term must be absent and thereforethe counterrotating tangential velocities must satisfy the following constraint F ( v + , v − ) = ǫv + v − + p ϕ = 0 , (30)9here we assume that | v ± | 6 = 1.Then, assuming a given choice for the tangential velocities in agreement with the above relation,we can write the SEMT as (24a) with S ab ± = ǫ ± U a ± U b ± , (31)so that we have two counterrotating dust fluids with surface energy densities, measured in thecoordinates frames, given by ǫ ± = (cid:20) − v ± v ∓ − v ± (cid:21) ǫv ∓ , (32)Thus the SEMT S ab can be written as the superposition of two counterrotating dust fluids if, andonly if, the constraint (30) admits a solution such that v + = v − .Similarly, substituting (27a) and (27b) in (17b) we can write the current density as (24b) with j a ± = σ ± U a ± (33)where σ ± are the surface electric charge densities, measured in the coordinates frames, σ ± = " p − v ± v ± − v ∓ ( j ˆ1 − j ˆ0 v ∓ ) . (34)Thus, we have a disk makes of two counterrotating charged dust fluids with surface energy densitiesgiven by (32), and surface electric charge densities given by (34).As we can see from Eqs. (26), (32) and (34), all the main physical quantities associated with theCRM depend on the counterrotating tangential velocities v ± . However, the constraint (30) does notdetermine v ± uniquely so that we need to impose some additional requirement in order to obtain acomplete determination of the tangential velocities leading to a well defined CRM.A possibility, commonly assumed [25, 38], is to take the two counterrotating streams as circulat-ing along electrogeodesics. Now, if the electrogeodesic equation admits solutions corresponding tocircular orbits, we can write this equation as12 ǫ ± g ab,r U a ± U b ± = − σ ± F ra U a ± . (35)In terms of ω ± we obtain12 ǫ ± ( U ± ) ( g ,r ω ± + 2 g ,r ω ± + g ,r ) = − σ ± U ± ( A t,r + A ϕ,r ω ± ) . (36)From (24a), (24b), (31), and (33) we have σ ± U ± = j − ω ∓ j ω ± − ω ∓ , (37a) ǫ ± ( U ± ) = S − ω ∓ S ω ± − ω ∓ , (37b) ω ∓ = S − ω ± S S − ω ± S , (37c)10nd substituting (37a) and (37b) in (36) we find12 ( S − ω ∓ S )( g ,r ω ± + 2 g ,r ω ± + g ,r ) = − ( j − ω ∓ j )( A t,r + A ϕ,r ω ± ) , (38)and using (37c) we obtain12 [( S ) − S S ]( g ,r ω ± +2 g ,r ω ± + g ,r ) = − [ S j − S j + ω ± ( S j − S j )]( A t,r + A ϕ,r ω ± ) . (39)Therefore we conclude that ω ± = − T ± p T − T T T (40)with T = g ,r + 2 A ϕ,r j S − j S S S − S S , (41a) T = g ,r + A t,r j S − j S S S − S S + A ϕ,r j S − j S S S − S S , (41b) T = g ,r + 2 A t,r j S − j S S S − S S . (41c)On the other hand, in terms of ω ± we get v ± = − (cid:20) W + W ω ± V + V ω ± (cid:21) , (42)and so, by using (40), we have that v + v − = T W − T W W + T W T V − T V V + T V , (43)so that, using (17a), we get F ( v + , v − ) = 32 e − Λ) Λ ,z ( r Λ ,z √ D + 4 r Λ ,z Ψ ,z e Ψ − Λ − W ,z e − Λ ) r ( A + √ D ) p ϕ ( S S − S S )( T V − T V V + T V ) × (cid:20) Λ ,z − r Ψ ,r Ψ ,z + 12 r W ,r W ,z e + 1 r ( r e − − W e ) A t,r A t,z − r A ϕ,r A ϕ,z e + 1 r W ( A ϕ,r A t,z + A ϕ,z A t,r ) e (cid:21) . (44)Finally, using the Einstein-Maxwell equation (4f) follows immediately that F ( v + , v − ) vanishes andtherefore the electrogeodesic velocities satisfy the constraint (30) and so, if the electrogeodesic equa-tion admits solutions corresponding to circular orbits, we have a well defined CRM.11nother possibility is to take the two counterrotating fluids not circulating along electrogeodesicsbut with equal and opposite tangential velocities, v ± = ± v = ± q p ϕ /ǫ. (45)This choice, that imply the existence of additional interactions between the two streams (e.g. col-lisions), leads to a complete determination of the velocity vectors. However, this can be made onlywhen 0 ≤ | p ϕ /ǫ | ≤
1. In the general case, the two counterrotating streams circulate with differentvelocities and we can write (30) as v + v − = − p ϕ ǫ . (46)However, this relation does not determine completely the tangential velocities, and therefore theCRM is undetermined.In summary, the counterrotating tangential velocities can be explicitly determined only if weassume some additional relationship between them, like the equal and opposite condition or theelectro-geodesic condition. Now, can happen that the obtained solutions do not satisfy any of thesetwo conditions. That is, the counterrotating velocities are, in general, not completely determinedby the constraint (30). Thus, the CRM is in general undetermined since the counterrotaing energydensities and pressures can not be explicitly written without a knowledge of the counterrotatingtangential velocities. As an example of the above presented formalism, we consider the thin disk models obtained by meansof the “displace, cut and reflect” method applied to the well known Kerr-Newman solution, whichcan be written as Ψ = 12 ln (cid:20) a x + b y − c ( ax + c ) + b y (cid:21) , (47a)Λ = 12 ln (cid:20) a x + b y − c a ( x − y ) (cid:21) , (47b) W = c kb (1 − y )(2 ax + 1 + c ) a ( a x + b y − c ) , (47c) A t = c p c − ax + c )( ax + c ) + b y , (47d) A ϕ = − k ba (1 − y ) A t , (47e)where a + b = c ≥
1, with a = km (1 − qq ∗ ) , b = Lm (1 − qq ∗ ) , c = 1 √ − qq ∗ , k = √ m − L − e , | q | = em , (48)12here m , L and e are the mass, angular momentum and electric charge parameters of the Kerr-Newman black hole, respectively. Note that c is the parameter that controls the electromagneticfield. The prolate spheroidal coordinates, x and y , are related with the Weyl coordinates by r = k ( x − − y ) , z + z = kxy, (49)where 1 ≤ x ≤ ∞ , 0 ≤ y ≤
1, and k is an arbitrary constant. Note that we have displaced theorigin of the z axis in z . This solution can be generated, in these coordinates, using the well-knowncomplex potential formalism proposed by Ernst [44] from the Kerr vacuum solution [1]. When c = 1this solution reduces to the Kerr vacuum solution.Let be ˜ D = k D , ˜ T = kT , and ˜ j t = k j t , therefore˜ T = 4 c a ¯ y { x (1 − ¯ y )(¯ x + 2 a ¯ x + c ) − (¯ x − ¯ y )[ a (¯ x + 1) + ¯ x (1 + c )] } (¯ x − ¯ y ) / [( a ¯ x + c ) + b ¯ y ] / , (50a)˜ D = 16 c a ¯ y { [ a (¯ x + 1) + ¯ x (1 + c )] − b ˜ r } (¯ x − ¯ y )[( a ¯ x + c ) + b ¯ y ] , (50b)˜ j t = 2 c p c − a ¯ y {− b ¯ y (3 a ¯ x + 2¯ xc − a ) + ( a ¯ x + c )( a ¯ x + ac ¯ x (¯ x − ¯ y ) / [( a ¯ x + c ) + b ¯ y ] / − a ¯ x + 2 b ¯ x − ac ) } , (50c) j ϕ = − c p c − b ¯ y (1 − ¯ y ) {− ab ¯ y (¯ x −
1) + ( a ¯ x + c )(3 a ¯ x (¯ x − ¯ y ) / [( a ¯ x + c ) + b ¯ y ] / +5 ac ¯ x − a ¯ x + 2 b ¯ x + 2 c ¯ x − ac ) } . (50d)In the above expressions ¯ x and ¯ y are given by2¯ x = p ˜ r + ( α + 1) + p ˜ r + ( α − , (51a)2¯ y = p ˜ r + ( α + 1) − p ˜ r + ( α − , (51b)where ˜ r = r/k and α = z /k , with α > D , is enough to consider the expression˜ D = [ a (¯ x + 1) + ¯ x (1 + c )] − b ˜ r , (52)that can be written as˜ D = a (1 + c ) R + [ α ( α −
1) + 2 + ˜ r ] + a (1 + c ) R − [ α ( α + 1) + 2 + ˜ r ]+ R + R − [( c + 1) + a (˜ r + α + 3)] + [( c + 1) + 5 a ]+ ˜ r [ c + 1 + a (˜ r + 2 α + 6)] + α [( c + 1) + a ( α + 2)] , (53)13here R ± = p ˜ r + ( α ± . Since α ( α ∓
1) + 2 > α , from (53) follows that D always isa positive quantity for Kerr-Newman fields and therefore the eigenvalues of the energy-momentumtensor are always real quantities. So we conclude that these disks can be interpreted, for all the valuesof parameters, as a matter distribution with currents and purely azimuthal pressure and without heatflow.We can see also that for the vacuum case, when c = 1, D is everywhere positive. These disks,obtained from the Kerr vacum solution, were previously considered by Gonz´alez and Letelier in thereference [17]. In this previous work, due to a mistake in the computation of the expressions for ˜ T and ˜ D , was concluded that the energy-momentum tensor could present complex eigenvalues for somevalues of the parameters. As we can see from the expressions presented here, this is not correct and,in all the cases, we have a matter distributions with purely azimuthal pressure and without heat flowfor all the values of parameters.In order to study the behavior of the main physical quantities associated with the disks, weperform a graphical analysis of them for disks with α = 2, b = 0 . c = 1 .
0, 1 .
5, 2 .
0, 2 .
5, and3 .
5, as functions of ˜ r . For these values of the parameters we find that Ψ ,z is a positive quantity inagreement with the strong energy condition. Therefore, ǫ = − λ − and p ϕ = λ + . However, one alsofinds values of the parameter for which Ψ ,z takes negative values. Furthermore, we take ν = − V a be a future-oriented timelike vector.In Fig. 1 we show the surface energy density ˜ ǫ and the azimuthal pressure ˜ p ϕ . We see that theenergy density presents a maximum at ˜ r = 0 and then decreases rapidly with ˜ r , being always apositive quantity in agreement with the weak energy condition. We also see that the presence ofelectromagnetic field decreases the energy density at the central region of the disk and later increasesit. We can observe that the pressure increases rapidly as one moves away from the disk center, reachesa maximum and later decreases rapidly. We also observe that the electromagnetic field decreases thepressure everywhere on the disk.The electric charge density ˜ j t and the azimuthal current density j ϕ , measured in the coordinatesframe, are represented in Fig. 2, whereas the electric charge density ˜ j ˆ0 and the azimuthal currentdensity ˜ j ˆ1 , measured by the comoving observer, are represented in Fig. 3. We observe that theelectric charge density has a similar behavior to the energy and that the current density have asimilar behavior to the pressure which is consistent with the fact that the mass is more concentratedin the disks center. We also computed this functions for other values of the parameters and, in allthe cases, we found the same behavior.We now consider the CRM for the same values of the parameters. We first consider the twocounterrotating streams circulating along electrogeodesics. In Fig. 4 we plot the tangential velocitycurves, v + and v − . We see that these velocities are always less than the light velocity. We also seethat the inclusion of the electromagnetic field make less relativistic these disks. In Fig. 5 we havedrawn the specific angular momenta h and h − for the same values of the parameters. We see thatthe presence of electromagnetic field can make unstable these orbits against radial perturbations.Thus the CRM cannot apply for c = 6 (bottom curve). In Fig. 6 we have plotted the surface energydensities ˜ ǫ + and ˜ ǫ − . We see that these quantities have a similar behavior to the energy density ˜ ǫ .14n Fig. 7, we plotted the surface electric charge densities ˜ σ + and ˜ σ − . We find that these quantitieshave also a similar behavior to ˜ ǫ ± .The Figs. 4 - 7 show that the two fluids are continuous in r which implies to have two particles incounterrotating movement in the same point in spacetime. So this model could be possible when thedistance between streams (or between the counterrotating particles) were very small in comparingwith the length r so that we can consider, in principle, the fluids continuous like is the case ofcounterrotating gas disks present in disk galaxies.Finally, in the case when the two fluids move with equal and opposite tangential velocities (non-electrogeodesic motion) we find that the physical quantities have a similar behavior to the previousone. We presented a detailed analysis of the energy-momentum tensor and the surface current densityfor electrovacuum stationary axially symmetric relativistic thin disks of infinite extension withoutradial stress, in the case when the energy-momentum tensor of the disks can be diagonalized, so thatthere is not heat flow. The surface energy-momentum tensor and the surface current density wereexpressed in terms of the comoving tetrad and explicit expressions were obtained for the kinematicaland dynamical variables that characterize the disks. That is, we obtained expressions for the velocityvector of the disks, as well for the energy density, azimuthal pressure, electric charge density andazimuthal current density.We also presented in this paper the stationary generalization of the Counterrotating Model (CRM)for electrovacuum thin disks previously analyzed for the static case in [28, 30]. Thus then, we wereable to obtain explicit expressions for all the quantities involved in the CRM that are fulfilled whendo not exists heat flow and when we do not have radial pressure. We considered both counter rotationwith equal and opposite velocities and counter rotation along electrogeodesics and, in both of thecases, we found the necessary conditions for the existence of a well defined CRM.A general constraint over the counterrotating tangential velocities was obtained, needed to cast thesurface energy-momentum tensor of the disk in such a way that can be interpreted as the superpositionof two counterrotating dust fluids. The constraint obtained is the generalization of the obtainedfor the vacuum case in [17], for disks without radial pressure or heat flow, where we only considercounterrotating fluids circulating along geodesics. We also found that, in general, there is not possibleto take the two counterrotating tangential velocities as equal and opposite neither take the twocounterrotating fluids as circulating along geodesics.A simple family of models of counterrotating charged disks based on the Kerr-Newman solutionwere considered where we obtain some disks with a CRM well behaved. We also find that the disksconstructed from the Kerr-Newman solution can be interpreted, for all the values of parameters, as amatter distribution with currents and purely azimuthal pressure and without heat flow. We obtain,for all the values of parameters, counterrotating Kerr-Newman disks that are in agreement withall the energy conditions. Finally, the generalization of these models to the case of electrovacuumstationary axially symmetric solutions where the energy-momentum tensor of the disk can to present15omplex eigenvalues for some values of the parameters, the stability of counterrotating fluids takinginto account the collective behavior of the particles, and a thermodynamic analysis of the disks, willbe considered in future works.
Acknowledgments
The authors want to thank the financial support from COLCIENCIAS, Colombia.
References [1] D. Kramer, H. Stephani, E. Herlt, and M. McCallum,
Exact Solutions of Einsteins’s FieldEquations (Cambridge University Press, Cambridge, England, 1980).[2] W. A. Bonnor and A. Sackfield, Commun. Math. Phys. , 338 (1968).[3] T. Morgan and L. Morgan, Phys. Rev. , 1097 (1969).[4] L. Morgan and T. Morgan, Phys. Rev. D , 2756 (1970).[5] A. Chamorro, R. Gregory, and J. M. Stewart, Proc. R. Soc. London A413 , 251 (1987).[6] G. A. Gonz´alez and P. S. Letelier, Class. Quantum Grav. , 479 (1999).[7] D. Lynden-Bell and S. Pineault, Mon. Not. R. Astron. Soc. , 679 (1978).[8] P.S. Letelier and S. R. Oliveira, J. Math. Phys. , 165 (1987).[9] J. P. S. Lemos, Class. Quantum Grav. , 1219 (1989).[10] J. P. S. Lemos and P. S. Letelier, Class. Quantum Grav. , L75 (1993).[11] J. Bi˘c´ak, D. Lynden-Bell, and J. Katz, Phys. Rev. D , 4334 (1993).[12] J. Bi˘c´ak, D. Lynden-Bell, and C. Pichon, Mon. Not. R. Astron. Soc. , 126 (1993).[13] J. P. S. Lemos and P. S. Letelier, Phys. Rev D , 5135 (1994).[14] J. P. S. Lemos and P. S. Letelier, Int. J. Mod. Phys. D , 53 (1996).[15] G.A. Gonz´alez and O. A. Espitia, Phys. Rev. D , 104028 (2003).[16] J. Bi˘c´ak and T. Ledvinka, Phys. Rev. Lett. , 1669 (1993).[17] G. A. Gonz´alez and P. S. Letelier, Phys. Rev. D , 064025 (2000).[18] D. Vogt and P. S. Letelier, Phys. Rev. D , 084010 (2003).1619] M. Ujevic and P. S. Letelier, Phys. Rev. D , 084015 (2004).[20] G. A. Gonz´alez and P. S. Letelier, Phys. Rev. D , 044013 (2004).[21] D. Vogt and P. S. Letelier, to be published in MNRAS.[22] M. Miyamoto and R. Nagai, Publ. Astron. Japan , 533 (1975)[23] R. Nagai and M. Miyamoto, Publ. Astron. Japan , 1 (1976)[24] G. Satoh, Publ. Astron. Japan , 41 (1980)[25] T. Ledvinka, J. Bi˘c´ak, and M. ˘Zofka, in Proceeding of 8th Marcel-Grossmann Meeting inGeneral Relativity , edited by T. Piran (World Scientific, Singapore, 1999)[26] P. S. Letelier, Phys. Rev. D , 104042 (1999).[27] J. Katz, J. Bi˘c´ak, and D. Lynden-Bell, Class. Quantum Grav. , 4023 (1999).[28] G. Garc´ıa R. and G. A. Gonz´alez, Phys. Rev. D , 124002 (2004).[29] D. Vogt and P. S. Letelier, Phys. Rev. D , 064003 (2004).[30] G. Garc´ıa-Reyes and G. A. Gonz´alez, Class. Quantum Grav. , 4845 (2004).[31] G. Garc´ıa-Reyes and G. A. Gonz´alez, Phys. Rev. D , 104005 (2004).[32] J. L. Synge, Relativity: The General Theory . (North-Holland, Amsterdam, 1966).[33] G. Neugebauer and R. Meinel, Phys. Rev. Lett. , 3046 (1995).[34] C. Klein, Class. Quantum Grav. , 2267 (1997).[35] C. Klein and O. Richter, Phys. Rev. Lett. , 2884 (1999).[36] C. Klein, Phys. Rev. D , 064033 (2001).[37] J. Frauendiener and C. Klein, Phys. Rev. D , 084025 (2001).[38] C. Klein, Phys. Rev. D , 084029 (2002).[39] C. Klein, Phys. Rev. D , 027501 (2003).[40] C. Klein, Ann. Phys. (Leipzig) , 599 (2003).[41] V. C. Rubin, J. A. Graham and J. D. P Kenney. Ap. J. , L9, (1992).[42] H. Rix, M. Franx, D. Fisher and G. Illingworth. Ap. J. , L5, (1992).1743] F. Bertola et al . Ap. J. , L67 (1996).[44] F.J. Ernst, Phys. Rev. D , 1415 (1968).[45] C. Klein and O. Richter, Ernst Equations and Riemann Surfaces , Lecture Notes in Physics (Springer) (2005)[46] A. Papapetrou and A. Hamouni, Ann. Inst. Henri Poincar´e , 179 (1968)[47] A. Lichnerowicz, C.R. Acad. Sci. , 528 (1971)[48] A. H. Taub, J. Math. Phys. , 1423 (1980)[49] E. Israel, Nuovo Cimento , 1 (1966)[50] E. Israel, Nuovo Cimento , 463 (1967)[51] E. Poisson, A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics . (CambridgeUniversity Press, 2004)[52] S. Chandrasekar,
The Mathematical Theory of Black Holes . (Oxford University Press, 1992).[53] L.D. Landau and E.M. Lifshitz,
Fluid Mechanics (Addison-Wesley, Reading, MA, 1989).[54] F. H. Seguin, Astrophys. J. , 745 (1975).18 ǫ ˜ p ϕ ˜ r ˜ r ( a ) ( b )Figure 1: ( a ) The surface energy density ˜ ǫ and ( b ) the azimuthal pressure ˜ p ϕ for Kerr-Newman diskswith α = 2, b = 0 . c = 1 . .
5, 2 .
0, 2 .
5, 3 .
0, and 3 . r . ˜ j t − j ϕ ˜ r ˜ r ( a ) ( b )Figure 2: ( a ) The surface electric charge density ˜ j t and ( b ) the azimuthal current density j ϕ forKerr-Newman disks with α = 2, b = 0 . c = 1 . r ), 1 . .
0, 2 .
5, 3 .
0, and 3 . r . 19 ˜ j ˆ0 − ˜ j ˆ1 ˜ r ˜ r ( a ) ( b )Figure 3: For Kerr-Newman disks we plot, as function of ˜ r , ( a ) ˜ j ˆ0 with α = 2, b = 0 .
2, and c = 1 . r ), 1 . .
0, 2 .
5, 3 .
0, and 3 . b ) j ˆ1 also with α = 2, b = 0 .
2, and c = 1 . r ), 1 . .
0, 2 .
5, 3 .
0, and 3 . v + − v − ˜ r ˜ r ( a ) ( b )Figure 4: The tangential velocities ( a ) v + and ( b ) v − for electrogeodesic Kerr-Newman disks with α = 2, b = 0 . c = 1 . .
5, 2 .
0, 2 .
5, 3 .
0, and 3 . r . 20 h − ˜ r ˜ r ( a ) ( b )Figure 5: The specific angular momenta ( a ) h and (b) h − for electrogeodesic Kerr-Newman diskswith α = 2, b = 0 . c = 1 . .
5, 2 .
0, 2 .
5, 3 .
0, 3 .
5, and 6 . r . ˜ ǫ + ˜ ǫ − ˜ r ˜ r ( a ) ( b )Figure 6: The surface energy densities ( a ) ˜ ǫ + and ( b ) ˜ ǫ − for electrogeodesic Kerr-Newman disks with α = 2, b = 0 . c = 1 . .
5, 2 .
0, 2 .
5, 3 .
0, and 3 . r . 21 ˜ σ + − ˜ σ − ˜ r ˜ r ( a ) ( b )Figure 7: The surface electric charge densities ( a ) ˜ σ + and ( b ) ˜ σ − for electrogeodesic Kerr-Newmandisks with α = 2, b = 0 . c = 1 . r ), 1 .
5, 2 .
0, 2 .
5, 3 .
0, and 3 . rr