Magnetic Moment Density from Lack of Smoothness of the Ernst Potential
aa r X i v : . [ g r- q c ] J un Magnetic Moment Density from Lack of Smoothness of the ErnstPotential
L. Fern´andez-Jambrina
Departamento de Geometr´ıa y Topolog´ıa,Facultad de Ciencias Matem´aticas,Universidad Complutense de MadridE-28040-Madrid, Spain
In this talk it is shown a way for constructing magnetic surface sources for station-ary axisymmetric electrovac spacetimes possessing a non-smooth electromagneticErnst potential. The magnetic moment density is related to this lack of smooth-ness and its calculation involves solving a linear elliptic differential equation. Asan application the results are used for constructing a magnetic source for the Kerr-Newman field.
The Darmois’ junction conditions provide a way for constructing surfacesources for non-smooth spacetimes. Whenever the extrinsic curvature of a hy-persurface contained in the spacetime is discontinuous, a distributional stresstensor can be assigned to it . Furthermore, if the electromagnetic curvatureis discontinuous across a hypersurface, a distributional electromagnetic sourcecan be located on it . The existence of a timelike (axial) Killing field allows thecalculation of the mass (angular momentum) surface density in terms of thediscontinuity of the extrinsic curvature of the hypersurface . An expressionfor the electric charge density is also obtained for stationary fields , but sofar, no expression has been provided for the magnetic moment density. In thenext section we shall follow a different approach. Instead of considering thediscontinuities of the electromagnetic curvature we shall study the discontinu-ities of the electromagnetic Ernst potential , restricting ourselves to stationaryaxisymmetric electrovac spacetimes. This approach follows closely the classicaltheory of potential, where a discontinuous (non-smooth) scalar potential givesrise to a dipole (monopole) source for the field that derives from the potential.As an example, we shall apply the formalism to the Kerr-Newman spacetime. As it has already been stated, we shall just study stationary axisymmetricelectrovac spacetimes. Let us consider the metric, ds = − e U ( dt − A dφ ) + e − U { e k ( dρ + dz ) + ρ dφ } , (1)1ritten in Weyl coordinates. U , A , k are functions of ρ and z .The electric, E , and magnetic field, B , as viewed from an orthonormalcoframe whose timelike one-form is given by θ = e U ( dt − Adφ ), can bewritten, according to Maxwell equations, in terms of a complex function, Φ, f = E + i B = − e − U d Φ , (2)the Ernst electromagnetic potential . This scalar potential Φ satisfies one ofthe Ernst equations , which in this notation can be written as, L Φ ≡ √ g ∂ µ (cid:26) N √ g (cid:18) e − U g µν − iρ A ǫ µν (cid:19) ∂ ν Φ (cid:27) = 0 , (3)where g is the metric induced by (1) on the hypersurfaces t = const. , N =( − g tt ) − is the lapse function and ǫ is the Levi-Civit`a tensor on the surfacesof constant time, t , and azimuthal angle, φ . For simplicity the whole equationhas been written as the action of a differential operator, L , on the potential.This is a consequence only of the Maxwell equations in the curved space-time whose metric is given by Eq. 1, regardless of whether the electromagneticfield is the source of the gravitational field.Since we will be interested in compact sources, we shall only considermetrics which are asymptotically flat in some coordinates ( t, r, θ, φ ), ds = − (cid:18) − mr (cid:19) (cid:18) dt + 2 J sin θr dφ (cid:19) ++ (cid:18) mr (cid:19) (cid:8) dr + (cid:0) r + c r (cid:1) (cid:0) dθ + sin θdφ (cid:1)(cid:9) + O (cid:18) r (cid:19) , (4)where m is the total mass of the source and J is the total angular momen-tum and c is just a constant (its value is − m for Kerr-Newman metrics).The electromagnetic potential of the compact source will have the followingasymptotic expansion,Φ = er + M cos θr + c r + O ( r − ) , (5)where e is the total charge, M is a complex constant whose real and imaginaryparts are, respectively, the electric and magnetic dipole moment and c is aconstant that may arise in some choices of coordinates.As it was done in , , , , we shall introduce a function Z that satisfies thefollowing elliptic differential equation out of the surface source and behaves atinfinity like the cartesian coordinate z ,2 + Z = 0 Z = ( r + c ) cos θ + O ( r − ) , (6)where + denotes the complex conjugate and c is a constant.The region where the electromagnetic surface source is located will betaken as the surface, S , where the electromagnetic potential (or its deriva-tive), Φ, is discontinuous, just as it is done in the classical theory of potential.Without loss of generality we shall assume that this surface is closed. Thethree-space, Ω, defined by any of the hypersurfaces t = const. , excluding S ,will be divided into two regions, Ω + and Ω − , respectively the outer and innerparts of Ω with respect to S .Taking into account Eq. 3 and Eq. 6 the following Green identity can bewritten, 0 = Z Ω √ g ( Z L Φ − Φ L + Z ) dx dx dx == Z ∂ Ω dS N (cid:26) e − U (cid:18) Z d Φ dn − Φ dZdn (cid:19) + i Aρ ( Z ∗ d Φ( n ) + Φ ∗ dZ ( n )) (cid:27) , (7)using the Stokes theorem. The two-dimensional Hodge dual on the surfaces ofconstant time and azimuthal angle is denoted by ∗ and n is the unitary outernormal to S .The boundary ∂ Ω + is formed by S and the sphere at infinity whereas ∂ Ω − is the surface S . The integral at infinity can be performed with the informationwe get from the asymptotic behaviours. The discontinuity of the integrand on S , σ M = 14 π (cid:20) N (cid:26) e − U (cid:18) Φ dZdn − Z d Φ dn (cid:19) − i Aρ ( Z ∗ d Φ( n ) + Φ ∗ dZ ( n )) (cid:27)(cid:21) , (8)denoted by squared brackets, can be interpreted as the electromagnetic dipolemoment surface density of the source for this field. Its real part is the electricdipole density and its imaginary part is the magnetic moment density. As an example for this formalism, the magnetic dipole density for a surface inthe Kerr-Newman spacetime will be calculated. We shall follow and restrictthe range of the Boyer-Lindquist coordinate r to positive values. Points on thehypersurface r = 0 with coordinates ( t, φ, , θ ) and ( t, φ, , π − θ ) are identified.Hence the Ernst potential, 3 = er − i a cos θ , (9)will be discontinuous on r = 0.The Kerr-Newman metric in Boyer-Lindquist coordinates induces the lineelement, ds = a cos θ dθ + sin θ ( a − e tan θ ) dφ , (10)on the surface r = 0. We just need a solution of Eq. 6, Z = ( r − m ) cos θ + e cos θ + i a m cos θr + i a cos θ , (11)in order to calculate the magnetic moment surface density, σ M = (cid:0) e cos θ + e + a cos θ (cid:1) i e π a cos θ √ a − e tan θ , (12) Acknowledgments
The present work has been supported by DGICYT Project PB92-0183. Theauthor wishes to thank F. J. Chinea and L. M. Gonz´alez-Romero for valuablediscussions.
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