Non-regular Potentials and Sources for Static Axisymmetric Electrovacs
aa r X i v : . [ g r- q c ] J un Non-regular Potentials and Sources for StaticAxisymmetric Electrovacs
L. Fern´andez-JambrinaDepartamento de F´ısica Te´orica II,Facultad de Ciencias F´ısicas,Universidad ComplutenseE-28040-Madrid, Spain
Abstract
In this lecture a new formalism for constructing electromagnetic sur-face sources for static axisymmetric electrovacs is presented. The electro-static and magnetostatic sources are derived from the discontinuities ofthe scalar potentials. This formalism allows the inclusion of two kinds ofdipole sources: Sheets of dipoles and the dipole moment of a distributionof monopoles. It is a generalization of a previous formalism in order tocope with asymptotically monopolar electric fields.
The problem of finding compact physically reasonable material sources thatcould be matched to an appropriate asymptotically flat vacuum spacetime isa task of great relevance in general relativity. Unfortunately, very few exactsolutions are known for the stationary axisymmetric case (cfr. [1] for a recentreview).Instead of considering volume sources this lecture will be devoted to thecalculation of electromagnetic sources for static axisymmetric asymptoticallyflat electrovacs. The electric charge distribution is easily obtained from theintegration of the Maxwell equations [2], but electric and magnetic momentdistributions cannot be calculated in that way. In order to achieve that goal ageneralization of the approach followed in [3], [4] will be attempted. In thesereferences magnetic and electric sources for static electrovacs were constructedthanks to the introduction of an asymptotically cartesian coordinate Z , providedthat the electric field was not monopolar. A way to circumvent that difficultywill be shown here. This will also allow to calculate the contribution of thecharge density to the electric dipole. 1n the next section the Green identity will be used to rederive the classicalexpression for the dipole surface density that arises from a discontinuous scalarpotential. This will be helpful to understand its generalization to curved space-times in section 3. An example of an application of this formalism is presentedin section 4. The results will be discussed at the end. Let us consider a non-relativistic physical vector field, E , (electric or magnetic)which can be obtained by differentiation of a scalar potential, V , ( E = − dV )that fulfils the flat-spacetime Laplace equation. From the classical theory ofpotential [5] it is known that, if the field is discontinuous across a surface S ,(and therefore the normal derivative of V is discontinuous) then on crossing S a layer of monopole charge is encountered and the surface density thereof, σ ,is given by: σ = 14 π [ E · n ] = − π (cid:20) dVdn (cid:21) (1)where n is the outer unitary normal to S and a square bracket denotes thedifference ([ a ] = a + − a − ) between the values of a quantity on the outer ( a + )and inner ( a − ) sides of S .If, besides, not only the field but also the potential is discontinuous on S ,then the dipole density on S can be constructed in the following way: Sinceboth the cartesian coordinate z and the scalar potential V satisfy the Laplaceequation out of S , then the Green identity is valid on R ∪ R − , that is, theeuclidean space outside and inside the surface S :0 = Z R ∪ R − d x ( V ∆ z − z ∆ V ) == Z S ( ∞ ) ∪ S − dS ( V dzdn − z dVdn ) − Z S + dS ( V dzdn − z dVdn ) (2)since the boundary of R consists of the sphere at infinity and S and theboundary of R − is just S .Taking into account that the asymptotic behaviour of V is known from itsexpansion in Legendre polynomials and inverse powers of the spherical radius,the integral at infinity can be performed and the other terms can be identifiedas the dipole surface density σ on S : σ = 14 π (cid:26) n · u z [ V ] − z (cid:20) dVdn (cid:21)(cid:27) (3)The first term in this expression arises as the contribution of a sheet ofdipoles [5] whilst the second one is the moment density of the σ distribution.2 Relativistic thin shells
In this section a way of generalizing the expression for the dipole density toMaxwell fields in curved spacetimes will be introduced. The metric for thestatic axially symmetric electromagnetic-gravitational system can be written inWeyl coordinates: ds = − e U dt + e − U { e k ( dρ + dz ) + ρ dφ } (4)where U , k are functions of ρ and z .The scalar potential V for either the electric or the magnetic field satisfies thefollowing equation, that can be derived from Maxwell’s vacuum equations only,even if the electromagnetic stress tensor is not the source of the gravitationalfield: 1 √ g ∂ µ (cid:8) √ g e − U g µν ∂ ν V (cid:9) = 0 (5)where the metric g is the one induced by (4) on each of the hypersurfaces t = const. Hence the results will be valid also for Maxwell fields in the geometry definedby (4).Our aim will be to cope with compact sources, therefore only asymptoticallyflat metrics will be considered: ds ≃ − (1 − mr ) dt + (1 + 2 mr ) { dr + ( r + α r )( dθ + sin θdφ ) } (6)in terms of the total mass m and a constant α in a coordinate patch describedby { t, φ, r, θ } . On the other hand, the scalar potential will be required to havethe following expansion, where Q is the total monopole charge, d is the totaldipole moment and β is another constant: V = Qr + d cos θr + βr + O ( r − ) (7)If the source for the field is located on a surface S , then equation (5) can beintegrated on the regions V +3 and V − in which the hypersurfaces t = const. aresplit by S . Since the integrand is a total derivative, the integral can be reducedto a surface integral on S and another on the sphere at infinity and therefore ityields the expression for the monopole charge density: σ = − π e − U (cid:20) dVdn (cid:21) (8)which is the formula that was obtained by Israel in [2], but written in terms ofthe potential instead of the field. The only difference with the non-relativisticsituation is the appearance of a metric factor.3fter the fashion of [3], [6], [7], an asymptotically cartesian function Z willbe introduced and will be taken to satisfy the same differential equation as V :1 √ g ∂ µ (cid:8) √ g e − U g µν ∂ ν Z (cid:9) = 0 (9)Hence a Green identity can be used to reduce the following integral on thetwo regions V +3 and V − :0 = Z V +3 ∪ V − √ g √ g (cid:8) Z ∂ µ ( √ g e − U g µν ∂ ν V ) −− V ∂ µ ( √ g e − U g µν ∂ ν Z ) (cid:9) dx dx dx == Z ∂V +3 ∪ ∂V − dS e − U (cid:18) Z dVdn − V dZdn (cid:19) (10)Assuming that V may be discontinuous on S and calculating the integral atinfinity with the information given by the asymptotic expansions, the followingexpression is obtained:0 = − π d + Z S dS e − U (cid:20) V dZdn − Z dVdn (cid:21) (11)This equation yields the expression for the dipole density on S , as it hap-pened in the non-relativistic case, in terms of the discontinuities of V : σ = 14 π e − U (cid:20) V dZdn − Z dVdn (cid:21) (12)This formula is again similar to the classical one (3), except for a metricfactor. As it happened then, there is a contribution of a sheet of dipoles andalso of the distribution of monopoles. This last term was not considered in [3].
As an example of how this formalism works, the magnetic source for Bonnor’smagnetic dipole [8] will be calculated. This solution is the Bonnor transform ofthe Kerr metric [9]: ds = − (1 − mrr − a cos θ ) dt ++(1 − mrr − a cos θ ) − (cid:8) ( r − a − mr ) sin θdφ +( r − a cos θ − mr ) [( r − m ) − ( a + m ) cos θ ] ( dθ + dr r − mr − a ) (cid:27) (13)4 = 2 am cos θr − a cos θ (14)which describes the field around a magnetic dipole of mass equal to 2 m andmagnetic moment 2 am .As it was done in [3] the radial coordinate will be taken to be nonnegative.Events on the surface r = 0 with collatitude θ will be identified with thosewith π − θ and therefore the range of θ will be restricted to [0 , π/
2) to avoiddouble-counting them. This means that the magnetic potential is discontinuouson r = 0: [ V ] = − ma cos θ (15)Since the solution satisfies the required asymptotic behaviours, only a Z function satisfying (9) is needed for constructing the magnetic source: Z = ( r − m ) cos θ − a m cos θr − a cos θ (16)Applying (8) and (12) to this solution, the expressions for the magneticdensity are obtained: σ = mπ a | a cos θ − m sin θ | / cos θ (17)Obviously there is no monopole density. Integrating σ on the surface r = 0,the correct result for the magnetic dipole moment is obtained: d = Z S σ dS = Z π Z π/ dθ ( m a sin θ ) = 2 m a (18)This is the same result that was obtained in [3] since there is no contributionfrom monopole sheets, as was to be expected. It has been shown in this lecture a new method for constructing electromagneticsources for static axially simmetric spacetimes. With this new approach theinclusion of asymptotically monopolar electric fields has been achieved. Thiswas not possible in a previous formalism [3] because vector potentials were usedand therefore Dirac string singularities would be present if monopoles wereincluded. Also the influence of the charge distribution on the dipole density hasbeen considered. A further generalization to stationary nonstatic axisymmetricspacetimes [10] is in preparation. 5 he present work has been supported in part by DGICYT Project PB92-0183;L.F.J. is supported by a FPI Predoctoral Scholarship from Ministerio de Ed-ucaci´on y Ciencia (Spain). The author wishes to thank F. J. Chinea, L. M.Gonz´alez-Romero and J. A. Ruiz Mart´ın for valuable discussions.
References [1]
El Escorial Summer School on Gravitation and General Relativity 1992:Rotating Objects and Other Topics (eds.: F. J. Chinea and L. M. Gonz´alez-Romero), Springer-Verlag, Berlin-New York (1993)[2] W. Israel,
Phys. Rev. D2 , 641 (1970)[3] L. Fern´andez-Jambrina and F.J. Chinea, Class. Quantum Grav. , 1489(1994) [arXiv: gr-qc/0403118][4] L. Fern´andez-Jambrina Ph.D. thesis , Universidad Complutense de Madrid(1994) ISBN: 84-699-0403-4[5] O. D. Kellogg:
Foundations of Potential Theory.
Dover, New York, 1954[6] L. Fern´andez-Jambrina, F. J. Chinea,
Phys. Rev. Lett. , 2521 (1993)[arXiv: gr-qc/0403102][7] L. Fern´andez-Jambrina, Class. Quant. Grav. , 1483 (1994) [arXiv: gr-qc/0403113][8] W. B. Bonnor, Z. Phys. , 444 (1966)[9] W. B. Bonnor,
Z. Phys.161