PHYSICAL REVIEW D 99, 124013 (2019)
Blandford-Znajek process in vacuo and its holographic dual
Ted Jacobson* Center for Fundamental Physics, University of Maryland, College Park, Maryland 20742, USA † Maria J. Rodriguez Max Planck for Gravitational Physics, Potsdam 14476, Germany and Department of Physics, Utah State University, 4415 Old Main Hill Road, Utah 84322, USA
(Received 5 March 2019; published 11 June 2019)
Blandford and Znajek discovered a process by which a spinning black hole can transfer rotational energy to a plasma, offering a mechanism for energy and jet emissions from quasars. Here we describe a version of this mechanism that operates with only vacuum electromagnetic fields outside the black hole. The setting, which is not astrophysically realistic, involves either a cylindrical black hole or one that lives in 2 þ 1 spacetime dimensions, and the field is given in simple, closed form for a wide class of metrics. For asymptotically anti–de Sitter black holes in 2 þ 1 dimensions, the holographic dual of this mechanism is the transfer of angular momentum and energy, via a resistive coupling, from a rotating thermal state containing an electric field to an external charge density rotating more slowly than the thermal state. In particular, the entropy increase of the thermal state due to Joule heating matches the Bekenstein-Hawking entropy increase of the black hole.
DOI: 10.1103/PhysRevD.99.124013
I. INTRODUCTION Although the principles of the BZ process are fully Prodigious amounts of energy wind up in the rotation of understood (e.g., [3,4]), and it is well studied analytically, – spinning black holes. Although nothing can escape from semianalytically, and numerically (e.g., [1,5 9]), it remains inside a black hole, that energy can be extracted via a difficult to develop an intuitive picture of how the flow of dynamo effect, the “Blandford-Znajek (BZ) process” [1], electromagnetic energy and charge current is organized in when a black hole is threaded by a magnetic field and space and how and where the negative energy originates. The surrounded by plasma. The mechanism of the BZ effect is a difficulty is in part due to the three dimensionality, the number type of “Penrose process” [2]: near a spinning black hole is of different field quantities, and the complexity of the Kerr an “ergosphere” where it is impossible to remain stationary metric which describes the spacetime geometry of a spinning “ relative to a distant observer, because the local inertial black hole. In an effort to boil down the process to a toy ” 2 þ 1 frames are dragged around faster than the speed of light. model, we began looking for a version in a -dimen- A system in the ergosphere can carry negative values of the sional spacetime. We assumed that, as in the original BZ “ ” conserved global energy quantity, yet have positive ordi- solution, the plasma is force-free, i.e., that it has a (nonzero) nary energy relative to a local observer. (In this paper the charge 4-current density orthogonal to the field strength, but word “energy” will refer to the globally conserved quantity, discovered that there is no such energy-extracting stationary which agrees with “local energy” only far from the black axisymmetric solution (see Appendix A for a demonstration). hole, in its rest frame.) Total energy can therefore be Much to our surprise, however, we found that there is a purely globally conserved, while positive energy flows away and electromagnetic version of the BZ process, in which plasma negative energy flows into the black hole. plays no role. We begin here with cylindrical black hole (BH) exam- ples, since they are easier to visualize, and the analogy with the astrophysical case is closer. The 2 þ 1 version will then 3 þ 1 *[email protected] be obtained as a special case of the cylindrical one. † [email protected]; [email protected] Of particular interest is the spinning Banados-Teitelboim- Zanelli (BTZ) black hole background in 2 þ 1 dimensions Published by the American Physical Society under the terms of [10], since it is an exact solution of general relativity with a the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to negative cosmological constant, is locally maximally sym- the author(s) and the published article’s title, journal citation, metric, and is related by AdS/CFT duality [11] to a thermal and DOI. Funded by SCOAP3. state of a two dimensional conformal field theory. Thus one
2470-0010=2019=99(12)=124013(8) 124013-1 Published by the American Physical Society TED JACOBSON and MARIA J. RODRIGUEZ PHYS. REV. D 99, 124013 (2019) can also examine the conformal field theory (CFT) dual of a BZ process. A similar idea was previously pursued in Ref. [12], with force-free plasma in the Kerr-anti–de Sitter (AdS) background, but no energy extracting solution was found. It was suggested there that this might be due to the existence of a globally timelike Killing vector outside the event horizon, but this implies only that positive values of the corresponding conserved quantity cannot be extracted, and this need not be the relevant energy. Indeed, the spinning BTZ black hole also has a globally timelike Killing vector, yet we find that positive energy extraction is possible.
II. CYLINDRICAL BZ PROCESS In the BZ process [1] a stationary plasma, around a spinning black hole threaded by a magnetic field, radiates a Poynting flux of energy to infinity. To simplify the geometry, and to eliminate the need for charge outside FIG. 1. A line current driven opposite to an ambient electric the black hole, we enhance the axisymmetry to cylindrical field emits a Poynting flux of energy. This field configuration corresponds to (2). Its generalization to any stationary axisym- symmetry. Since the azimuthal circumference grows as the metric spacetime (1), allowing for a radial component of the cylindrical radius r, the radial component of the Poynting magnetic field, is given by (4). For spinning black cylinder ⃗ ⃗ vector E × B must then fall as 1=r to conserve energy. For spacetimes, the line current is behind the horizon, and the field is example, if there is a uniform electric field parallel to the nonsingular on the horizon provided the Znajek condition (9) cylindrical axis, then the magnetic field must have an holds. Energy is then extracted from the black cylinder if there is 0 Ω Ω azimuthal, circular component that falls as 1=r. Amp`ere’s magnetic flux through the horizon and < F < H (10). The law then implies that there must be an enclosed electric nonzero black hole spin is what makes this extraction possible. current parallel to the axis. In order for the Poynting flux to be radially outward, the flow of current must be opposite to A. Field configurations the external electric field. The energy required to drive this We adopt cylindrical spacetime coordinates ðt; r; ϕ;zÞ, current ends up in the outgoing Poynting flux. See Fig. 1 for an illustration of this scenario. and assume the spacetime line element takes the form Consider this model first in flat, empty spacetime. If 2 ¼ −α2 2 þ α−2 2 þ 2ð ϕ − Ω Þ2 þ 2 ð Þ current flows only along the axis, the Poynting flux ds dt dr r d dt dz ; 1 emerges from the axis. Of course a line source of α Ω Poynting flux does not much resemble the BZ process, where the functions and depend only on r, and we use units with c ¼ 1. The spacetime is then stationary and so let us introduce a cylindrical black hole and hide the line 2 source behind the event horizon. A solution essentially like cylindrically symmetric: ds is invariant under transla- the one just described can be placed on this black hole tion of t, ϕ, and z. We refer to the t coordinate as “time.” spacetime but, since nothing can escape from a black hole, This line element describes a wide class of spacetimes. αð Þ¼0 we expect that energy cannot be extracted from the source If there is a radius rH where rH , (1) describes behind the horizon. Indeed, if we try to construct such a a “black cylinder” or “black string,” i.e., a black hole solution, we find that the fields are singular on the horizon. with a cylindrical horizon at the outermost root rH.If Ω ≔ Ωð Þ ≠ 0 In effect, the horizon becomes a singular source of energy, H rH , the black cylinder is spinning with Ω which again does not resemble the BZ process. However, if angular velocity H and has an ergosphere where the time α2 2Ω2 the black hole is spinning with angular velocity ΩH, and if translation is spacelike, i.e., where 124013-2 BLANDFORD-ZNAJEK PROCESS IN VACUO AND … PHYS. REV. D 99, 124013 (2019) an index-free notation for the antisymmetric tensor Fab. dα ∧ dr ¼ 0 ¼ dΩ ∧ dr.) The field (4) remains degener- (For example, F ¼ Edx ∧ dt þ Bdx ∧ dy describes an ate, since it is the wedge product of two 1-forms. To recover electric field strength E in the x direction and magnetic (2) from (4) one should set α ¼ 1 and send ψ z to zero and field strength B in the z direction, in flat spacetime with ΩF to infinity, with E ≔ ψ zΩF held fixed. The constant Cartesian coordinates.) Maxwell’s equations are dF ¼ 0 parameters (and their notation) in (4) are directly analogous (absence of magnetic monopoles and Faraday’s law) and to functions that appear in standard treatments of stationary d F ¼ J (Gauss and Amp`ere-Maxwell laws), where d is axisymmetric plasma magnetospheres (e.g., [14,15] and the (metric independent) exterior derivative operator, J is references therein). the current 3-form, and * is the Hodge dual operation that The singularity on the axis carries a line current I in the sends a p form ω to the orthogonal 4 − p form ω with the −z direction, sourcing the azimuthal magnetic field, and a same magnitude. We adopt the orientation of dt ∧ dr ∧ magnetic monopole line charge density 2πψz, sourcing the 2 2 dϕ ∧ dz for defining the dual (ω ∧ ω ¼ ω ϵ, where ω is radial magnetic field. (2πψz is the z derivative of the the squared norm of ω and ϵ is the unit 4-form with the magnetic flux through a cylinder ending at coordinate z.) adopted orientation). Note that the metric (1) is a sum of The quantity ΩF is the “angular velocity of the magnetic squares of 1-forms, hence those 1-forms are orthonormal, field lines,” in the sense that the electric field vanishes in the ð∂ þ Ω ∂ Þ ¼ 0 which facilitates the computation of duals and norms. local frame that rotates that way: t F ϕ · F . Let us begin with the simple example in flat spacetime, This is a standard concept in ideal plasma physics, where the plasma determines the preferred frame. It is meaningful I F ¼ dr − Edt ∧ dz; ð2Þ here without the plasma only because the field is 2πr degenerate. We are interested in the solution (4) on a rotating black where I and E are constants. This describes a uniform hole background, but it is worth noting that it could also be electric field in the zˆ direction, together with a magnetic terminated on a cylindrical conductor rotating with angular field circulating around the origin in planes of constant velocity Ω . t and z, sourced by a line current at r ¼ 0 flowing in the −zˆ F direction. The dual of the field (2) is B. Energy extraction R I 1 ¼ ϕ ∧ − ϕ ∧ ð Þ The Maxwell action is given by − F ∧ F, from F 2π d dt Erd dr: 3 2 which it follows that the conserved Noether current It is clear by inspection that Maxwell’s equations dF ¼ associated with the t-translation symmetry of the spacetime 0 ¼ d F are satisfied (since d2 ¼ 0 and dr ∧ dr ¼ 0), so is [14] this is indeed a vacuum solution, except at the axis where 1 dϕ is singular. The field (2) is the wedge product of two J E ¼ −ð∂t · FÞ ∧ F þ ∂t · ðF ∧ FÞ: ð6Þ 1-forms, which implies that F ∧ F ¼ 0. Equivalently, the 2 ⃗ ⃗ Lorentz-invariant scalar E · B vanishes, as it does for ideal As is usual, we call this conserved quantity simply energy. ⃗ plasmas since E vanishes in the rest frame of a perfect The outgoing energy flux is given by the integral of the conductor. A field with this algebraic property is called 3-form J E over a surface of constant r, which by virtue of “degenerate.” the conservation law is the same as the flux across any other The field (2) can be easily generalized to the case of any surface of constant r. Replacing ∂t → −∂ϕ in (6), one background spacetime with a metric of the cylindrical form obtains the angular momentum current J L associated with (1). Allowing also for a radial component of the magnetic the ϕ-translation symmetry. field, the result is given by The energy current outward through a surface of constant r arises from the part of J E that contains no dr factor [so I the second term on the right-hand side of (6) does not F ¼ dr þ ψ zðdϕ − ΩFdtÞ ∧ dz; ð4Þ 2πrα2 contribute], which for the field (4) is given by I ψ Ω 1 where , z, and F are constants. The dual of (4) is J j ¼ Ω ψ ∧ ϕ ∧ ð Þ E r 2π F zIdt d dz: 7 I F ¼ dϕ ∧ dt 2π The outward flux of this current over a constant r cylinder Δ Δ Ω ψ Δ Δ 1 rðΩ − Ω Þ of length z and time interval t is F zI t z, so the þ ψ dt þ F ðdϕ − ΩdtÞ ∧ dr: ð5Þ z r α2 energy and angular momentum fluxes per unit proper length per unit t-coordinate time at any radius are given by As with (2), Maxwell’s equations are satisfied by inspec- E ¼ Ω ψ L ¼ ψ ð Þ tion. (The functions α and Ω depend only on r, so that r F zI; r zI: 8 124013-3 TED JACOBSON and MARIA J. RODRIGUEZ PHYS. REV. D 99, 124013 (2019) The outgoing energy flux is positive if I has the same sign In the remainder of this article, we shall focus on this as ΩFψ z, which means that the current is opposite to the example. Maintaining the convention that the vector electric field. This corresponds to the situation described in potential is an inverse length, in three spacetime dimen- the Introduction to this section, where the line current sions the Maxwell action written at the beginning of driven against the electric field produces an outgoing Sec. II B should be multiplied by a constant with dimen- Poynting flux. sions of length 1=g2. We adopt units here with g ¼ 1. The line element (1) indicates that αdt and dr=α are unit Dualizing interchanges electric and magnetic quantities, 1-forms, so that dt and dr=α2 are both singular at α ¼ 0 in a so we change our notation for the parameters accordingly: Ω black cylinder spacetime. Therefore, the I and F terms in Q ≔−2πψz is the electric charge of the solution, and the field (4) are both singular at the horizon. However, the Φ_ ≔−I is the t derivative of the magnetic flux through a forms dt and dr become proportional on the horizon, and loop encircling the origin (or the magnetic monopole their divergences can cancel. To analyze this divergence current). The constant ΩF is now the “angular velocity balancing act, one can either transform to a new coordinate of the electric field lines,” that is, the angular velocity of the system that is regular on the horizon, or construct scalars by frame in which the magnetic field (which in 2 þ 1 contracting F with a basis of finite vectors. One finds (see dimensions is a spatial pseudoscalar) vanishes (presuming Appendix B) that a necessary condition for finiteness at the the vector ∂t þ ΩF∂ϕ is timelike, which is here the case for future horizon is large enough r). The three-dimensional field and its dual are given by1 I ¼ 2πrHψ zðΩH − ΩFÞ: ð9Þ Φ_ Q α2 F3 ¼ − dϕ ∧ dt − dt ∧ dr This condition is also sufficient, provided has non- 2π 2πr vanishing first derivative at r (i.e., provided the black hole H QrðΩ − Ω Þ is not extremal). The relation (9) is known as the “Znajek − F ð ϕ − Ω Þ ∧ ð Þ 2πα2 d dt dr; 11 condition” [16]. If the regularity condition (9) holds, then the outgoing Φ_ Q 3F3 ¼ dr þ ðdϕ − Ω dtÞ: ð12Þ energy flux (8) becomes 2πα2r 2π F E ¼ 2πðψ Þ2r Ω ðΩ − Ω Þ: ð10Þ The Znajek horizon regularity condition (9) in the present r z H F H F _ notation is Φ ¼ QrþðΩH − ΩFÞ. The dual of the regular If the black cylinder is not rotating (ΩH ¼ 0), then the field is thus outward energy flux is always negative. This means that, as ðΩ − Ω Þ ¼ Q rþ H F þ ϕ − Ω ð Þ expected, while energy can flow inward, it cannot be 3F3 2 dr d Fdt : 13 extracted. However, if the black cylinder is rotating, and 2π α r if 0 < Ω < Ω , then positive energy can be radiated F H This solution can be placed in particular on the rotating outward provided there is a nonzero magnetic flux ψ z BTZ black hole background [10], which is a three-dimen- through the horizon. The energy flux formula (10) is sional solution of Einstein’s equations with a negative precisely analogous to that for a force-free plasma on a −2 cosmological constant, Λ3 ¼ −l . The corresponding line Kerr black hole spacetime [1,14,15]. element (in Boyer-Lindquist-like coordinates) is (1) with- The vacuum Maxwell equations are invariant under 2 electric-magnetic duality, F → F, as is the energy momen- out dz , and with tum tensor and Noether current (6), so the dual (5) of the 2 2 2 2 ðr − rþÞðr − r−Þ r−rþ field (4) provides another solution that extracts energy from α2 ¼ ; Ω ¼ ; ð14Þ r2l2 r2l the black cylinder. The dual solution has a magnetic field in the z direction and azimuthal and radial electric field where r− and rþ are the inner and outer horizon radii, components. The radial electric field is sourced by an respectively, with 0 124013-4 BLANDFORD-ZNAJEK PROCESS IN VACUO AND … PHYS. REV. D 99, 124013 (2019) Φ_ ¼ Q ð∂ þ Ω ∂ Þ ¼ ϕ ð Þ j 2πl t F ϕ ;a2π td : 17 One dynamically consistent boundary condition fixes j [25], which fixes the black hole charge and the asymptotic magnetic field. The transfer of energy between the CFT and the (unspecified) degrees of freedom carrying the current then entails Ohmic dissipation. The conductivity σ of the CFT is the ratio of the current to the electric field, figured in the rest frame of the thermal state. That is, σ ¼ðs · jÞ=ðu · s · fÞ, where f ¼ da ¼ ðΦ_ =2πÞdt ∧ dϕ is the boundary electromagnetic field strength, u ¼ γð∂t þ ΩH∂ϕÞ is the velocity 2-vector of this −1 rotating frame, and s ¼ γðlΩH∂t þ l ∂ϕÞ is the orthogo- nal, spatial unit 2-vector (with respect to the boundary 2 ¼ − 2 þ l2 ϕ2 γ ¼ð1 − l2Ω2 Þ−1=2 metric, ds dt d ), with H . After using the Znajek horizon regularity condition, the FIG. 2. Electric field lines for the field (13), with rþ ¼ 0.3, dependence on the electromagnetic field parameters drops r− ¼ 0.2, and ΩF ¼ ΩH=2 ¼ 1=3, in units with l ¼ 1.As explained in Appendix C, the field lines are intersections of out, and we find the electric field sheets with a surface of constant ingoing Kerr ϕ¯ γl 1 ℏ coordinate v, plotted with r and treated as polar coordinates on σ ¼ ¼ ¼ ; ð18Þ the Euclidean plane. The field lines for r 124013-5 TED JACOBSON and MARIA J. RODRIGUEZ PHYS. REV. D 99, 124013 (2019) exponentially approaches the condition ΩH ¼ ΩF, for freedom and the external source that couples to them which the fluxes vanish. [19,21,25]. ¼ 4ℏ The black hole entropy SBH A= G evolves as ACKNOWLEDGMENTS M_ − Ω J_ ðΩ − Ω ÞJ_ Φ_ QðΩ − Ω Þ _ ¼ H ¼ F H ¼ H F ð Þ We thank S. Hartnoll, G. Horowitz, N. Iqbal, K. Jensen, SBH 2π ; 19 TH TH TH D. Marolf, A. Speranza, and S. Theisen for helpful discussions and correspondence. This research was initi- where the overdot signifies a derivative with respect to the ¼ ℏκ 2π ated at the Peyresq Physics 21 meeting and partially Killing time t, and TH = is the Hawking temper- supported there by OLAM. T. J. was also supported in ature. [The first equality invokes the first law of black hole part by NSF Grants No. PHY-1407744 and No. PHY- mechanics, while the second and third make use of (8) 1708139 at the University of Maryland, and by Perimeter with the notational change indicated in the paragraph Institute for Theoretical Physics. Research at Perimeter containing (11).] Institute is supported by the Government of Canada The rate of entropy generation per unit spacetime through Industry Canada and by the Province of Ontario volume, due to Ohmic dissipation in the holographic dual through the Ministry of Research and Innovation. The work field theory process, is given by the rate of energy of M. J. R. was supported by the Max Planck Gesellschaft dissipation divided by the temperature in the corotating through the Gravitation and Black Hole Theory frame of the thermal state. The energy dissipation rate is the Independent Research Group and by NSF Grant inner product of the electric field with the charge current in No. PHY-1707571 at Utah State University. that frame, hence the total rate of entropy generation is ð Þ2πl APPENDIX A: STATIONARY AXISYMMETRIC _ ¼ u · j · F ð Þ SFT : 20 FORCE-FREE SOLUTIONS IN 2 + 1 DIMENSIONS γTH In three spacetime dimensions, the force-free condition This agrees with (19) when the previously given expres- F ja ¼ 0 is equivalent to the condition sions for u, j, and F are inserted. (This is special case of a ab very general relation between boundary and black hole d F ¼ ζF; ðA1Þ entropy generation that has been established in [29].) It is amusing to note that this dual description is a perfect for some function ζ.IfF is stationary and axisymmetric electrical analogy for the mechanical analogy originally then F has the form F ¼ αðrÞdt þ βðrÞdϕ þ γðrÞdr,so 0 0 described by Blandford and Znajek [1]. That analogy d F ¼ α dr ∧ dt þ β dr ∧ dϕ. The force-free condition involved a thermally conducting disk spinning with angular (A1) thus implies that either the charge current vanishes, or F ∝ α0dr ∧ dt þ β0dr ∧ dϕ, in which case F has no dt ∧ velocity ΩH, coupled by friction to a concentric, thermally insulating ring spinning with a smaller angular velocity. dϕ term (i.e., no azimuthal electric field). But this implies that the energy current 1 IV. DISCUSSION J ¼ −ð∂ Þ ∧ þ ∂ ð ∧ ÞðÞ E t · F F 2 t · F F A2 We have thus far treated the electromagnetic field as a test field on the background of a spinning BTZ black hole has no dt ∧ dϕ term, so there is no radial energy background and inferred adiabatic evolution from the flux. (Equivalently, the Poynting vector has no radial conservation laws. To more fully probe the duality component.) one should solve the coupled Einstein-Maxwell equations. A similar study was carried out in [22] for the case of APPENDIX B: DERIVATION OF THE electromagnetic energy flowing into a nonrotating black ZNAJEK CONDITION hole, allowing the nonlinear conductivity of the dual field theory to be found. In our case, a first step would be to use To characterize the regularity condition for the field strength for the background the charged, spinning black hole solution [17,30], which also has a magnetic field with I Ω ¼ Ω F ¼ dr þ ψ ðdϕ − Ω dtÞ ∧ dz ðB1Þ F H in our notation. This is the solution to which the 2πrα2 z F black hole would relax, and the azimuthal electric field alone could be treated as a test field. The next step would be on the horizon, one can either transform to a coordinate to attempt to find an Einstein-Maxwell solution including system that is regular on the horizon or characterize F by the azimuthal electric field, which could describe nonlinear the scalars obtained when contracting it with a basis of effects in the conductivity. Another avenue to investigate regular vectors. Here we use the second method. For three would be alternate boundary conditions at infinity, which of the four basis vectors we can use the Killing vectors, ∂t, can modify the nature of the dual field theory degrees of ∂ϕ, and ∂z, which are all tangent to the horizon. (Note that, 124013-6 BLANDFORD-ZNAJEK PROCESS IN VACUO AND … PHYS. REV. D 99, 124013 (2019) ϕ 1 Ω although the coordinates t and are singular on the horizon ¼ þ ϕ¯ ¼ ϕ þ ð Þ dv dt 2 dr; d d 2 dr: C1 of a spinning black hole, the vectors ∂t and ∂ϕ are the time α α translation and rotation Killing vectors and, regardless of what coordinate system is used, these vectors are regular on In terms of these regular differentials, the metric takes the the horizon.) The contractions of the three Killing vectors form with F are finite as long as the constants ψ and Ω are z F ds2 ¼ −α2dv2 þ 2dvdr þ r2ðdϕ¯ − ΩdvÞ2; ðC2Þ finite. For the fourth regular basis vector we use the ¼ 0 4-velocity k of null (k · k ), zero angular momentum which is regular except at r ¼ 0. This is called the ingoing ∂ ¼ 0 ∂ ¼ 0 (k · ϕ ), and zero longitudinal momentum (k · z ) Kerr coordinate system. geodesics, with the affine parameter fixed by the choice When expressed using these regular coordinates, the dual ∂ ¼ −1 k · t . These conditions (which are conserved along field (12) takes the form the geodesic) determine k uniquely up to the choice of lð þ lΩ Þ whether the geodesic is ingoing or outgoing. The ingoing ¼ Q r− Fr þ ϕ¯ − Ω ð Þ 3F3 2 2 dr d Fdv : C3 one is 2π ðr þ rþÞðr − r−Þ −2 k ¼ α ð∂t þ Ω∂ϕÞ − ∂r: ðB2Þ (We write the dual 1-form since it is simpler than the field strength 2-form F3.) Since there is no globally defined, The additional nonzero scalar that can be formed using k is natural time coordinate relative to which one might define “ ” ψ ðΩ − Ω Þ − I=2πr the electric field vector, we capture the shape of the field ∂ ¼ z F ð Þ k · z · F α2 : B3 (C3) as follows. The kernel of 3F3 at each point is two- dimensional and is timelike outside the inner horizon since A necessary condition for finiteness at the future horizon is F3 is electric dominated there (15). Since there is no charge therefore current, 3F3 is closed (d 3 F3 ¼ 0), so this distribution of ¼ 2π ψ ðΩ − Ω Þ ð Þ planes is integrable, i.e., they are tangent to two-dimen- I rH z H F : B4 sional submanifolds, which can be called the “electric field ðα2Þ sheets” (see [14] for a discussion of the magnetic case in This is also sufficient, provided ;r is nonvanishing at r , as is the case as long as the black hole is not extremal. four spacetime dimensions). The intersection of an electric H field sheet with a spacelike surface is an electric field line APPENDIX C: INGOING KERR COORDINATES on that surface. However, since we do not have preferred FOR THE BTZ BLACK HOLE spacelike surfaces in the BTZ spacetime, we employ the intersection with a constant v null surface to generate a The spinning BTZ black hole metric is given in Boyer- picture of the electric field lines. They are tangent to a 2 Lindquist coordinates by (1) less the dz term and with α vector field X satisfying X · 3F3 ¼ 0 ¼ X · dv. One more and Ω given by (14). The coordinates t and ϕ are singular at condition is needed to specify the magnitude of X, although the horizon in this spacetime. In their place, we can this magnitude will not affect the field lines. If we choose ¯ introduce new coordinates v ¼ t þ fðrÞ and ϕ ¼ for this condition X · dr ¼ 1, then ϕ þ gðrÞ, which will be regular if the functions fðrÞ and ð Þ ϕ¯ lð þ lΩ Þ g r are defined so that v and are constant on the ¼ ∂ − r− Fr ∂ ð Þ X r 2 2 ϕ¯ : C4 infalling, zero angular momentum null geodesics, whose ðr þ rþÞðr − r−Þ affine 4-velocity k is given by (B2). The requirements k · dv ¼ 0 and k · dϕ ¼ 0 then impose f0 ¼ α−2 − 1 and In Fig. 2 we plot the streamlines of this vector field, treating g0 ¼ α−2, respectively, hence we have r and ϕ¯ as polar coordinates on the Euclidean plane. [1] R. D. Blandford and R. L. Znajek, Electromagnetic extrac- [3] I. Okamoto, Electromagnetic extraction of energy from tion of energy from Kerr black holes, Mon. Not. R. Astron. Kerr black holes, Publ. Astron. Soc. Jpn. 58, 1047 Soc. 179, 433 (1977). (2006). [2] R. Penrose, Gravitational collapse: The role of general [4] J.-P. Lasota, E. Gourgoulhon, M. A. Abramowicz, A. relativity, Riv. 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