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WDS'08 Proceedings of Contributed Papers, Part III, 198–203, 2008. ISBN 978-80-7378-067-8 © MATFYZPRESS Asymptotically Uniform Electromagnetic Test Fields Around a Drifting Kerr Black Hole O. Kopáček Astronomical Institute, Academy of Sciences, Prague, Czech Republic. Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. A test field solution describing the electromagnetic field around a Kerr black hole which is drifting in an arbitrary direction with respect to the asymptotically uniform magnetic field with the general orientation (with respect to the rotation axis) is constructed. Proper tetrad choice of a physically relevant observer is discussed. Force lines of electric and magnetic intensities as measured by the orbiting (freely falling, respectively) observer are explored and plotted for some special cases. Introduction Electromagnetic (EM) test field solutions of Maxwell equations in curved spacetime play an improtant role in astrophysics since we can usually suppose that astrophysically relevant EM fields are weak enough, so that their influence upon background geometry may be neglected. We are interested in the solutions describing an originally uniform magnetic field under the influence of the Kerr black hole. Since the Kerr metric is asymptotically flat, this EM field reduces to the original homogenous magnetic field in the asymptotic region. First, such a test field solution was given by Wald [1974] for the special case of perfect alignment of the asymptotically uniform magnetic field with the symmetry axis. More general solution for arbitrary orientation of the asymptotic field was given by Bičák et al. [1985]. We use this solution to construct the EM field around the Kerr black hole which is drifting through the asymptically uniform magnetic field. As we are motivated to investigate such a fields by the opportunity of making judgements about the motion of charged particles, we need to project acquired coordinate components onto a properly chosen tetrad. A realistic (though still simple enough) scenario of the motion of the test particles in the vicinity of the Kerr black hole is to bind it to the geodesic circular orbit in the equatorial plane above marginally stable orbit, rms , and let it freely fall below rms . This is roughly how we model the accretion process in the accretion discs. We are especially curious whether these fields would under certain circumstances allow the particle to escape from the equatorial plane and form an outflow of the matter from the accretion disc. Lines of force Kerr metric in Boyer-Lindquist coordinates t, r, θ, φ (Misner et al., 1973): ds2 = − sin2 θ Σ 2 ∆ [dt − a sin θ dφ]2 + [(r2 + a2 )dφ − a dt]2 + dr + Σdθ2 , Σ Σ ∆ (1) where ∆ ≡ r2 − 2M r + a2 , Σ ≡ r2 + a2 cos2 θ. (2) Natural definition of the lines of force (of magnetic and electric fields), as measured by given observer equipped with the orthonormal tetrad eµ(α) , is their identification with the lines along which magnetic/electric charge connected to this observer would start to move due to the presence of the EM field F µν . Coordinate components of the magnetic (electric) field intensities are determined by (coordinate components of) the Lorentz force felt by the unit magnetic (electric) charge are (Hanni et al., 1973): B µ =∗F µν uν , E µ = F µν uν , (3) where uν represents a 4-velocity of the charge. Tetrad components of the vector field determining desired lines of force are given as the spatial part of the projection onto eµ(α) : ∗ µ ν (i) µ ν B (i) = B(i) = e(i) = E(i) = e(i) µ F νu , E µF νu , 198 (4) KOPÁČEK: ELECTROMAGNETIC FIELDS AROUND A DRIFTING KERR BLACK HOLE (α) where e µ are 1-forms dual to the tetrad vectors eµ(α) . Lowering/rising spatial tetrad indices doesn’t matter since the tetrad is supposed to be orthonormal – g(µ)(ν) = η(µ)(ν) . Electromagnetic field We start with Fµν describing the test field with asymptotic form of a general (ie. not necessarily parallel) uniform magnetic field (Bičák et al., 1985). Due to the axial symetry of Kerr space-time only two components of asymptotic field were considered in that paper without any loss of generality (asymptotic components B0 (parallel) and B1 (equatorial) to be specific). We rewrite components of EM tensor (eq. (A3) of Bičák et al., 1985) denoting Bx ≡ B1 , Bz ≡ B0 and splitting the result into two parts according to the asymptotic component. We obtain the asymptotically perpendicular part of the field: Bx Ftr =Bx aM rΣ−2 ∆−1 sin θ cos θ[(r3 − 2M r2 + ra2 (1 + sin2 θ) + 2M a2 cos2 θ) cos ψ − a(r2 − 4M r + a2 (1 + sin2 θ)) sin ψ], Bx Ftθ =Bx aM Σ−2 (r2 cos 2θ + a2 cos2 θ)(a sin ψ − r cos ψ), Bx Ftφ =Bx aM Σ−1 sin θ cos θ(a cos ψ + r sin ψ), Bx Frθ = − Bx (a cos ψ + r sin ψ) − Bx a∆−1 (M r − a2 sin2 θ) cos ψ − a(r sin2 θ + M cos2 θ) sin ψ , Bx Frφ = − Bx sin θ cos θ (r − M a2 ∆−1 ) cos ψ − a(1 + rM ∆−1 ) sin ψ Bx Fθφ (5) Bx + a sin2 θFtr , 2 2 =Bx (r sin θ + M r cos 2θ) cos ψ − a(r sin2 θ + M cos2 θ) sin ψ + (r2 + a2 )Bx M Σ−2 (r2 cos 2θ + a2 cos2 θ)(a sin ψ − r cos ψ) and the part which approaches uniform field aligned along the axis: Bz Ftr =Bz aM Σ−2 (r2 − a2 cos2 θ)(1 + cos2 θ), Bz Ftθ =2Bz aM rΣ−2 sin θ cos θ(r2 − a2 ), Bz Frφ Bz Fθφ (6) =Bz r sin2 θ + Bz a2 sin2 θM Σ−2 (r2 − a2 cos2 θ)(1 + cos2 θ), =Bz ∆ sin θ cos θ + 2(r4 − a4 )Bz M rΣ−2 sin θ cos θ, where we use the azimuthal coordinate ψ of Kerr ingoing coordinates, which is related to Boyer–Lindquist coordinates as follows: r − r+ a ln , (7) ψ =φ+ r+ − r− r − r− √ with r± ≡ M ± M 2 − a2 denoting the outer and the inner horizon. We notice that limr→∞ ψ = φ. As we shall introduce a drift of the black hole in the general direction we lose axial symmetry B and need to consider all spatial components of the asymptotic magnetic field. We obtain Fµνy (which B Bx may only appear due to nonzero drift) by rotating Fµν along the z-axis by angle π2 - i.e. Fµνy = Bx Fµν φ → φ − π2 , Bx → By which only causes sin ψ → − cos ψ and cos ψ → sin ψ. Since the drift shall induce uniform electric field in the asymptotic region we need to have appropriate E B Fµνx,y,z handy. We get them easily by performing dual transformation of Fµνx,y,z . Dual transformation is carried out as follows: 1 ∗ Fαβ = F µν εµναβ , (8) 2 where εµναβ is the Levi-Civita tensor whose components are given as (Misner et al., 1973): p √ (9) εµναβ = −det||gσω ||[µναβ] ≡ −g[µναβ], with [µναβ] denoting completely antisymmetric symbol. Determinant of the Kerr metric is g = 2 gtt grr gθθ gφφ − gφt grr gθθ = − sin2 θ Σ2 . Performing the dual transformation we immediately obtain EM tensors with desired asymptotics of uniform electric field: Ex,y,z Bx,y,z Fµν =∗ Fµν (Bx,y,z → −Ex,y,z ). (10) 199 KOPÁČEK: ELECTROMAGNETIC FIELDS AROUND A DRIFTING KERR BLACK HOLE In the explicit form we get: E B B Ftr x,y,z = sin θ Σ Fθt x,y,z (Bx,y,z → −Ex,y,z )g φt + Fθφx,y,z (Bx,y,z → −Ex,y,z )g φφ g θθ , E B B Ftθ x,y,z = sin θ Σ Ftr x,y,z (Bx,y,z → −Ex,y,z )g φt + Fφrx,y,z (Bx,y,z → −Ex,y,z )g φφ g rr , E B Ftφx,y,z = sin θ Σ Frθx,y,z (Bx,y,z → −Ex,y,z )g rr g θθ , (11) Frθx,y,z = sin θ Σ Fφtx,y,z (Bx,y,z → −Ex,y,z ) (g φt )2 − g φφ g , E B B Frφx,y,z = sin θ Σ Fθφx,y,z (Bx,y,z → −Ex,y,z )g φt + Fθt x,y,z (Bx,y,z → −Ex,y,z )g tt g θθ , E B B Fθφx,y,z = sin θ Σ Fφrx,y,z (Bx,y,z → −Ex,y,z )g φt + Ftr x,y,z (Bx,y,z → −Ex,y,z )g tt g rr . E B tt Now we are fully equipped to construct any asymptotically uniform test field on the Kerr background just by linear superposing of the above EM tensors. As we are concerned in constructing Fµν which describes the test field around the black hole drifting through asymptotically uniform magnetic field in general direction, we shall employ Lorentz transformation to find the correct asymptotic components of such a field. Once obtained we just use them to replace the original “nondrifting” quantities Ex , Ey , Ez , Bx , By and Bz . Matrix of general Lorentz transformation is (Jackson, 1998): γ ′ −γvx ||Λνµ || = −γvy −γvz 1 γvy −γvx (γ−1)v 2 1 + v2 x −γvz (γ−1)vx vy v2 (γ−1)vy2 1 + v2 (γ−1)vy vz v2 (γ−1)vx vy v2 (γ−1)vx vz v2 (γ−1)vx vz v2 (γ−1)vy vz , v2 (γ−1)v 2 1 + v2 z (12) 1 where v = (vx2 + vy2 + vz2 ) 2 and γ = (1 − v 2 )− 2 . Bx Bz Our original field, Fµν = Fµν + Fµν , has a simple asymptotic form in Minkowskian coordinates, 0 0 asymptotic ||Fµν || = 0 0 0 0 −Bz 0 0 0 . Bx 0 0 Bz 0 −Bx (13) ′ To transform the covariant tensor Fµν , the inverse Lorentz transformation Λµν ′ = (Λνµ )−1 would be used. But we realize that the Boyer–Lindquist coordinate system which we use to perform all the calculations (and also to express the EM tensor of the final field) is centered around black hole and the rest frame of the black hole is thus our “laboratory” reference frame. As we consider a drift of the black hole against the field eq. (13), we need to perform inverse Lorentz transformation. Quantities Bx and Bz appearing therein would be primed in standard notation. For inverse transformation of covariant tensors we use ′ original Λνµ . Thus for “drifting” Fµν we have (denoting Fµ′ ν ′ that of eq. (13)): asymptotic Fµν = Fµasymptotic Λµµ Λνν , ′ν′ ′ ′ (14) which may be written in the matrix formalism as follows: asymptotic || = ||Λµµ ||t ||Fµasymptotic || ||Λνν || = ||Λµµ || ||Fµasymptotic || ||Λνν ||, ||Fµν ′ν′ ′ν′ ′ ′ ′ ′ (15) and results in: 0 vy γBz −v γB 0 y z asymptotic ||Fµν || = γ(vx Bz − vz Bx ) −γBz + vz M vy γBx −vy M 2 −γ(vx Bz − vz Bx ) −vy γBx γBz − vz M vy M , 0 γBx − vx M −γBx + vx M 0 (16) γ (vz Bz + vx Bx ). with M ≡ γ+1 Final step of the derivation is thus substitution of “nondrifting” quantities Ex,y,z and Bx,y,z in the E B tensors Fµνx,y,z and Fµνx,y,z by Lorentz transformed values from eq. (16) and superposing the components to acquire general EM tensor describing the field around the Kerr black hole drifting through 200 KOPÁČEK: ELECTROMAGNETIC FIELDS AROUND A DRIFTING KERR BLACK HOLE asymptotically uniform magnetic field of general orientation: Ex Ey Fµν =Fµν (Ex → −vy γBz ) + Fµν (Ey → γ(vx Bz − vz Bx ))+ Ez Bx + Fµν (Ez → vy γBx ) + Fµν (Bx → γBx − vx M )+ (17) By Bz + Fµν (By → −vy M ) + Fµν (Bz → γBz − vz M ). Choice of the tetrad As we are primarily interested in astrophysically relevant situations we choose the orthonormal tetrad carried by the observer on the circular Keplerian orbit around the black hole. Such an orbit is specified by the values of constants of motion – by specific angular momentum L̃ ≡ uφ and specific energy Ẽ ≡ −ut which are expressed as follows (Bardeen et al., 1972): √ √ √ ± M (r2 + a2 ∓ 2a M r) r2 − 2M r ± a M r p q , L̃(r) = , (18) Ẽ(r) = √ √ r r2 − 3M r ± 2a M r r(r2 − 3M r ± 2a M r) where the upper signs are valid for the prograde (direct) orbits and the lower ones for the retrograde (counter–revolving) orbits. Such a tetrad is physical only above marginally stable orbit rms which represents a radial boundary for the stationary geodesic motion in the equatorial plane: p (19) rms = M 3 + Z2 ∓ (3 − Z1 )(3 + Z1 + 2Z2 ) , where Z1 ≡ 1 + 1 − a2 M2 1/3 h 1+ a 1/3 M + 1− a 1/3 M i and Z2 ≡ orthonormal tetrad of the orbiting frame Yokosawa et al. [2005]: eµ(t) = u = γ̃ µ A ∆Σ 1/2 q 3a2 M2 + Z12 . Above r = rms we use [1, 0, 0, ΩKep], 1/2 ∆ [0, 1, 0, 0], = Σ 1 eµ(θ) = √ [0, 0, 1, 0], Σ " √ 1/2 1/2 # A Σ A µ √ + v(φ) Ω e(φ) = γ̃ v(φ) , 0, 0, , ∆Σ ∆Σ sin θ A eµ(r) (20) where we define A ≡ (r2 + a2 )2 − a2 ∆ sin2 θ to express the coordinate angular velocity of LNRF (i.e. angular velocity of the frame dragging) Ω = 2a A M r. Angular velocity of a circular orbit is ΩKep = ±1 where the upper signs are for prograde orbits and the lower ones for the retrograde orbits. −1/2 3/2 M r ±a A sin θ (Ω v(φ) = Σ∆ − Ω) stands for the linear velocity of the orbiting tetrad as measured by ZAMO Kep 1/2 2 −1/2 observer in LNRF. Finally γ̃ = (1 − v(φ) ) is the relevant Lorentz factor of this motion. For r < rms there are no more circular orbits. Thus we suppose that the orbiting observer who reaches this limit performs a free fall to the black hole keeping the values of the constants of motion corresponding to the marginally stable orbit at rms . He is falling with Ẽms ≡ Ẽ(rms ) and L̃ms ≡ L̃(rms ) given by eqs. (18) – (19). Having fixed ut (r < rms ) = −Ẽms , uφ (r < rms ) = L̃ms and uθ = 0 we get radial component ur easily from the normalisation condition uµ uµ = −1. Contravariant components of the 4-velocity are then: 1 [(r[r2 + a2 ] + 2M a2 )Ẽms − 2M aL̃ms ], r∆ q 1 2 − 4M aẼ L̃ 2 ur = − 3/2 [r(r2 + a2 ) + 2M a2 ]Ẽms ms ms − (r − 2M )L̃ms − r∆, r uθ = 0, 1 [2M 2 aẼms + M (r − 2M )L̃ms ]. uφ = r∆ ut = 201 (21) KOPÁČEK: ELECTROMAGNETIC FIELDS AROUND A DRIFTING KERR BLACK HOLE Spatial 1-forms of the tetrad of this falling observer may be expressed as follows Dovčiak [2004]: r (ur [ut , ur , 0, uφ ] + [0, 1, 0, 0]), e(r) µ = p ∆(1 + ur ur ) eµ(θ) = [0, 0, r, 0], r ∆ (φ) [−uφ , 0, 0, ut]. eµ = 1 + ur ur (22) Structure of the electromagnetic field We shall make just a brief qualitative overview of the possible structures of the lines of force of magnetic and electric intensities felt by the above specified observer which may be either co-rotating or counter-rotating with the background geometry. In all figures we show a portion of the equatorial plane. The circle signifies the upper event horizon of the black hole. Original asymptotically uniform magnetic field is restricted to be perpendicular to the symmetry axis (i.e. it lies in the equatorial plane). Without any loss of generality we align it with the horizontal axis of presented figures (only nonzero asymptotical component is thus Bx ). Since the impact of the rotation of the black hole upon the structure of the field lines is most apparent for extreme Kerr black hole (a = M ) we confine ourselves to this case from now on. Marginally stable orbit rms thus coincides with the horizon at r = M for prograde orbits while it lies at rms = 9M for retrograde orbits. First we present the field lines around a nondrifting black hole and then the drift is also introduced rsin φ M 3 3 2 2 1 1 rsin φ M 0 0 −1 −1 −2 −2 −3 −3 −2 −1 0 rcos φ M 1 2 −3 −3 3 −2 −1 0 rcos φ M 1 2 3 Figure 1. Magnetic field as felt by co-rotating (on the left) and counter-rotating (on the right) observers around a nondrifting extreme Kerr black hole. rsin φ M 3 3 2 2 1 1 rsin φ M 0 0 −1 −1 −2 −2 −3 −3 −2 −1 0 rcos φ M 1 2 −3 −3 3 −2 −1 0 rcos φ M 1 2 3 Figure 2. Projection of the electric field as felt by co-rotating (on the left) and counter-rotating (on the right) around a nondrifting extreme Kerr black hole. 202 KOPÁČEK: ELECTROMAGNETIC FIELDS AROUND A DRIFTING KERR BLACK HOLE 1.75 1 1.7 1.65 0.95 1.6 0.9 rsin φ 1.55 M rsin φ M 0.85 1.5 1.45 0.8 1.4 0.75 1.35 −1.8 −1.75 −1.7 −1.65 −1.6 −1.55 −1.5 −1.45 rcos φ M 0.65 0.7 0.75 0.8 rcos φ M 0.85 0.9 Figure 3. Projection of the electric field as felt by counter-rotating observer around a nondrifting extreme Kerr black hole. In the left panel we zoomed the area around a null point of the electric field, the right panel shows the structure of the electric field near the horizon. rsin φ M 3 1.7 2 1.6 1 1.5 0 rsin φ M 1.4 −1 1.3 1.2 −2 −3 −3 1.1 −2 −1 0 rcos φ M 1 2 3 −1.5 −1.4 −1.3 −1.2 rcos φ M −1.1 −1 −0.9 Figure 4. Magnetic field as felt by counter-rotating observer around a drifting (vx = 0.5 c and vy = −0.7 c) Kerr black hole. Null point of the magnetic field is noticed and zoomed. Conclusion In this paper we presented initial steps of our investigation of the asymptotically uniform electromagnetic test fields around a drifting Kerr black hole. Components of the electromagnetic tensor Fµν describing such a field were given explicitly (in a symbolic way regarding its length) in the terms of the former nondrifting solution. Structure of the resulting fields has been briefly overviewed in the special case of the original uniform magnetic field (in which the Kerr black hole is then immersed) being perpendicular to the symmetry axis. Acknowledgments. The author thanks Dr. Vladimı́r Karas for his kind guidance and helpful advices. References Bardeen J. M., Press W. H., Teukolsky S. A.: Rotating Black Holes: Locally Nonrotating Frames, Energy Extraction, and Scalar Synchrotron Radiation, ApJ, 178, 347-370, 1972 Bičák J. and Janiš V.: Magnetic fluxes across black holes, MNRAS, 212, 899-915, 1985 Dovčiak M.: Radiation of accretion discs in strong gravity, PhD thesis, 2004 Hanni, R. S., Ruffini, R.: Lines of force of a point charge near a Schwarzschild black hole, PhysRevD 8, 3259-3265, 1973 Jackson J. D.: Classical electrodynamics - 3rd ed, John Wiley & Sons, INC., 1998 Misner C. W., Thorne K. S. and Wheeler J.A.: Gravitation, Freeman, San Francisco, 1973 Wald, R. M.: Black hole in a uniform magnetic field, PhysRevD, 6, 1680, 1974 Yokosawa M., Inui T.: Magnetorotational Instability around a Rotating Black Hole, ApJ, 631, 1051-1061, 2005 203