Causal production of the electromagnetic energy flux and role of the negative energies in Blandford-Znajek process
aa r X i v : . [ a s t r o - ph . H E ] M a y Prog. Theor. Exp. Phys. , 00000 (28 pages)DOI: 10.1093 / ptep/0000000000 Causal production of the electromagneticenergy flux and role of the negative energies inBlandford-Zna jek process
Kenji Toma and Fumio Takahara Frontier Research Institute for Interdisciplinary Sciences, Tohoku University,Sendai 980-8578, Japan ∗ E-mail: [email protected] Astronomical Institute, Tohoku University, Sendai 980-8578, Japan Department of Earth and Space Science, Graduate School of Science, OsakaUniversity, Toyonaka 560-0043, Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Blandford-Znajek process, the steady electromagnetic energy extraction from a rotatingblack hole (BH), is widely believed to work for driving relativistic jets in active galacticnuclei, gamma-ray bursts and Galactic microquasars, although it is still under debatehow the Poynting flux is causally produced and how the rotational energy of the BH isreduced. We generically discuss the Kerr BH magnetosphere filled with a collisionlessplasma screening the electric field along the magnetic field, extending the arguments ofKomissarov and our previous paper, and propose a new picture for resolving the issues.For the magnetic field lines threading the equatorial plane in the ergosphere, we find thatthe inflow of particles with negative energy as measured in the coordinate basis is gener-ated near that plane as a feedback from the Poynting flux production, which appears tobe a similar process to the mechanical Penrose process. For the field lines threading theevent horizon, we first show that the concept of the steady inflow of negative electromag-netic energy is not physically essential, partly because the sign of the electromagneticenergy density depends on the coordinates. Then we build an analytical toy model ofa time-dependent process both in the Boyer-Lindquist and Kerr-Schild coordinate sys-tems in which the force-free plasma injected continuously is filling a vacuum, and suggestthat the structure of the steady outward Poynting flux is causally constructed by thedisplacement current and the cross-field current at the in-going boundary between theplasma and the vacuum. In the steady state, the Poynting flux is maintained withoutany electromagnetic source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index xxxx, xxx
1. Introduction
The driving mechanism of collimated outflows or jets with relativistic speeds which areobserved in active galactic nuclei (AGNs), gamma-ray bursts, and Galactic microquasarsis one of the major problems in astrophysics. A most widely discussed model is basedon Blandford-Znajek (BZ) process, the electromagnetic energy extraction from a rotatingblack hole (BH) along magnetic field lines threading it [1]. This process produces Poynting-dominated outflows, which may be collimated by the pressure of the surrounding mediumsuch as the accretion disk and the disk wind [e.g. 2, 3]. The particles in the outflow can be c (cid:13) The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited. radually accelerated depending on the geometrical structure [e.g. 4–6], which is consistentwith the recent observational implications from the radio jet in the giant elliptical galaxyM87 [7, 8] [see also 9].BZ process was proposed by a pioneering paper of [1], who found steady, axisymmetric,force-free solutions of Kerr BH magnetosphere in the slow rotation limit where the outwardangular momentum (AM) and Poynting fluxes are non-zero along the field lines threadingthe event horizon. This was followed by demonstrations of analytical and numerical magneto-hydrodynamic (MHD) solutions [e.g. 2, 3, 10–13] and other force-free solutions [e.g. 14–17].However, the physical mechanism how the fluxes are created in the electromagnetically-dominated plasma has not been clearly explained. In contrast, the origin of pulsar windsis identified definitely with the rotation of the matter-dominated central star. The rotationvelocity of the matters of the star V ϕ and the magnetic field threading the star B pro-vide the electromotive force V ϕ × B on charges, maintaining the electric field E and thepoloidal electric currents (with the toroidal magnetic field B ϕ ) which form the outwardPoynting flux E × B ϕ / π in the magnetosphere. As its feedback, the rotation of the stellarmatters slows down [18] [see also a review in 19, hereafter TT14]. BZ process, working in theelectromagnetically-dominated BH magnetosphere, does not include any matter-dominatedregion in which the poloidal magnetic field is anchored. One should also note that the toroidalmagnetic field cannot be produced in vacuum just by the rotation of the space-time [14, 20].Then how the electric field and currents forming the AM and Poynting fluxes are createdand how the BH rotational energy is reduced in BZ process are not simple problems andhave been still matters of debate. See recent discussions in [21] (hereafter K09) and [22].Among the numerous calculations, the force-free numerical simulations performed by [14](hereafter K04) are most insightful for investigating the essential points on the origin ofthe fluxes. TT14 extended the arguments in K04 and K09 and analytically showed thatfor open magnetic field lines threading the ergosphere in the steady, axisymmetric KerrBH magnetosphere, the situation of no electric potential difference with no poloidal electriccurrent (i.e. no outward AM or Poynting flux) cannot be maintained. The origin of theelectric potential differences is ascribed to the ergosphere. It was also shown that for theopen field lines threading the equatorial plane in the ergosphere , the poloidal currents aredriven by electric field (perpendicular to the magnetic field) stronger than the magnetic fieldin the ergosphere, where the force-free condition is violated (see also Section 3 below).In this paper, we mainly discuss the field lines threading the event horizon. Some theo-rists consider that the membrane paradigm [23] is useful for effectively understanding theproduction mechanism of the AM and Poynting fluxes for such field lines [e.g. 24, 25]. Thisinterprets the condition at the horizon as a boundary condition [1, 26] and the horizon asa rotating conductor which creates the potential differences and drives the electric currentsin an analogy with the unipolar induction for pulsar winds explained above. However, thehorizon does not actively affect its exterior, but just passively absorbs particles and waves[27]. The condition at the horizon should be interpreted as a regularity condition [10, K04].The mechanism of producing the steady AM and Poynting fluxes has to work outside thehorizon, making the physical quantities consistent with the regularity condition.For such a causal flux production, some other theorists proposed a scenario that certaintypes of negative energies (as measured in the coordinate basis, i.e. as measured at infinity) reated outside the horizon flows towards the horizon, resulting in the positive outwardenergy flux. This is an analogy with the mechanical Penrose process, in which the rotationalenergy of a BH is extracted by making it absorb negative-energy particles [28–30]. However,MHD simulations demonstrate that no regions of negative particle energy are seen in thesteady state [12], although a transient inflow of negative particle energy is possible as afeedback from generation of an outward MHD wave [11]. The role of negative electromagneticenergy density in the steady state has been discussed recently [K09; 31, 32], although theconcept of ‘advection of the steady field’ is ambiguous. Below we show that the sign ofthe electromagnetic energy density depends on the coordinates, and thus the negative fieldenergy is not physically essential (see Section 4.4 below).We argue that the causal production mechanism of the electromagnetic AM and Poyntingfluxes cannot be fully understood by investigating only the steady-state structure. We exam-ine a time-dependent process evolving towards the steady state with an analytical toy modelto clarify how the steady outward fluxes are created. In order to find the essential physics, ouranalysis is performed both in the Boyer-Lindquist (BL) and the Kerr-Schild (KS) coordinatesystems. Most of the previous analytical studies used the BL coordinates [e.g. 10, 17, 33–36][but see 37], most of the recent numerical simulations used the KS coordinates [e.g. K04;2, 3, 12, 13] [but see e.g. 16, 38], and both of them focused on the steady-state structure. Our new analytical studies of time-dependent process in the BL and KS coordinates will behighly helpful for understanding physics in BZ process.This paper is organized as follows. In Section 2, we explain our formulation of general rel-ativistic electrodynamics, set generic assumptions for Kerr BH magnetosphere, and reviewthe recent analytical understandings given by K04 and TT14. Section 3 concentrates on thefield lines threading the equatorial plane, for which we show the flux production mecha-nism and the role of the negative energy of particles. In Section 4, we explain differencesbetween the equatorial plane and the horizon, and then we focus on the field lines threadingthe horizon, discussing differences of the electromagnetic structures as seen in the BL andKS coordinates and the role of the negative electromagnetic energy density. In Section 5,we discuss the time-dependent process towards the steady state. Section 6 is devoted toconclusion.
2. Formulation and Assumptions decomposition of space-time
The space-time metric can be generally written as ds = g µν dx µ dx ν = − α dt + γ ij ( β i dt + dx i )( β j dt + dx j ) , (1)where α is called the lapse function, β i the shift vector and γ ij the three-dimensional metrictensor of the space-like hypersurfaces. Those hypersurfaces are regarded as the absolutespace at different instants of time t [cf. 23]. We focus on Kerr space-time with fixed BHmass M and angular momentum J . (The electromagnetic field with outward fluxes whichwe consider below is a test field for Kerr space-time.) We adopt the units of c = 1 and The force-free electrodynamics without decomposition of tensors into spatial and temporalcomponents has also been developed [39–42]. 3/28 M = 1, for which the gravitational radius r g = GM/c = 1. We use the dimensionless spinparameter a ≡ J/ ( M r g c ).Kerr space-time has two symmetries, i.e. ∂ t g µν = ∂ ϕ g µν = 0. These correspond to theexistence of the Killing vector fields ξ µ and χ µ . In the coordinates ( t, ϕ, r, θ ), ξ µ = (1 , , , , χ µ = (0 , , , . (2)The event horizon, where g rr = 0, is located at r H = 1 + √ − a . The ergosphere is theregion r < r es = 1 + √ − a cos θ , where the Killing vector ξ µ is space-like, ξ = g tt = − α + β >
0. At infinity, this space-time asymptotes to the flat one.The local fiducial observer [FIDOs; 23, 29], whose world line is perpendicular to theabsolute space, is described by the coordinate four-velocity n µ = (cid:18) α , − β i α (cid:19) , n µ = g µν n ν = ( − α, , , . (3)The AM of this observer is n ϕ = 0, and thus FIDO is also a zero AM observer [ZAMO; 23].Note that the FIDO frame is not inertial, but it can be used as a convenient orthonormalbasis to investigate the local physics [23, 43, 44]. It should also be confirmed that the FIDOsare time-like, physical observers (i.e. n µ n µ = − α = r ̺ ∆Σ , β ϕ = − ar Σ ,γ ϕϕ = Σ ̺ sin θ, γ rr = ̺ ∆ , γ θθ = ̺ , (4)where ̺ = r + a cos θ, ∆ = r + a − r, Σ = ( r + a ) − a ∆ sin θ. (5)BL FIDOs rotate in the same direction as the BH with the coordinate angular velocityΩ ≡ dϕdt = − β ϕ = 2 ar Σ . (6)The BL coordinates have the well-known coordinate singularity ( α = 0 and γ rr = ∞ , where∆ = 0) at the horizon. The BL FIDOs are physical observers only outside the horizon.The KS coordinates have no coordinate singularity at the event horizon. The coordinates t and ϕ are different from those in the BL coordinates. The non-zero metric components inthis coordinate system are: α = 1 √ z , β r = z z , γ rϕ = − a (1 + z ) sin θ,γ ϕϕ = Σ ̺ sin θ, γ rr = 1 + z, γ θθ = ̺ , (7)where z = 2 r/̺ [K04; 37]. The KS spatial coordinates are no longer orthogonal ( γ rϕ = 0).From the space-time symmetries, g µν ξ µ ξ ν = g tt = − α + β ,g µν ξ µ χ ν = g tϕ = γ ϕj β j = β ϕ ,g µν χ µ χ ν = g ϕϕ = γ ϕϕ (8)are the same in the BL and KS coordinates. We should note that the KS FIDOs are differentfrom the BL FIDOs (K04). .2. The electrodynamics We follow the definitions and formulations of K04 for electrodynamics in Kerr space-time(except for keeping 4 π in Maxwell equations), in a similar way to TT14 [see also K09,references therein, and 45]. The covariant Maxwell equations ∇ ν ∗ F µν = 0 and ∇ ν F µν = 4 πI µ are reduced to ∇ · B = 0 , ∂ t B + ∇ × E = 0 , (9) ∇ · D = 4 πρ, − ∂ t D + ∇ × H = 4 π J , (10)where ∇ · C and ∇ × C denote (1 / √ γ ) ∂ i ( √ γC i ) and e ijk ∂ j C k , respectively, and e ijk =(1 / √ γ ) ǫ ijk is the Levi-Civita pseudo-tensor of the absolute space. The condition of zeroelectric and magnetic susceptibilities for general fully-ionized plasmas leads to the followingconstitutive equations, E = α D + β × B , (11) H = α B − β × D , (12)where C × F denotes e ijk C j F k . At infinity, α = 1 and β = 0, so that E = D and H = B .Here D µ = F µν n ν and B µ = − ∗ F µν n ν are the electric and magnetic fields as measured bythe FIDOs, while E µ = γ µν F να ξ α and H µ = − γ µν ∗ F να ξ α are the electric and magnetic fieldsin the coordinate basis, where γ µν = g µν + n µ n ν . The current J is related to the current asmeasured by the FIDOs, j , as J = α j − ρ β . (13)See Appendix A on the relation between convective current and particle velocity.The covariant energy-momentum equation of the electromagnetic field ∇ ν T νµ = − F µν I ν gives us the AM equation as ∂ t l + ∇ · L = − ( ρ E + J × B ) · m , (14)and the energy equation as ∂ t e + ∇ · S = − E · J , (15)where C · F denotes C i F i , m = ∂ ϕ , l = αT tϕ = 14 π ( D × B ) · m (16)is the AM density, L = − ( E · m ) D − ( H · m ) B + 12 ( E · D + B · H ) m (17)is the AM flux ( L i = αT iϕ ), e = − αT tt = 18 π ( E · D + B · H ) (18)is the energy density, and S = 14 π E × H (19)is the Poynting flux ( S i = − αT it ). .3. Kerr BH magnetosphere We study the axisymmetric electromagnetic field in Kerrspace-time which is filled with a plasma. (The steadiness of the field is partly discardedin Section 5.) We put the additional assumptions similarly to TT14: (1) The poloidal B field produced by the external currents (whose distribution is symmetric with respect to theequatorial plane) is threading the ergosphere. We call the field lines threading the ergosphere‘ergospheric field lines’. (2) The plasma in the BH magnetosphere is dilute and collisionless,but its number density is high enough to screen the electric field along the B field lines, i.e. D · B = 0 . (20)The energy density of the particles is much smaller than that of the electromagnetic fields.(3) The gravitational force is negligible compared with the Lorentz force. (The gravitationalforce overwhelms the Lorentz force in a region very close to the event horizon [44], but thephysical condition in that region hardly affects its exterior.)The condition D · B = 0 and equation (11) lead to E · B = 0. In the steady state, we have ∇ × E = 0, which means that E is a potential field, and the axisymmetry leads to E ϕ = 0.Then one can write E = − ω × B , ω = Ω F m . (21)Substituting this equation into ∇ × E = 0, one obtains B i ∂ i Ω F = 0 . (22)That is, Ω F is constant along each B field line. The E field is also described by E i = − Ω F ∂ i Ψin terms of the magnetic flux function Ψ, so that each B field line is equipotential and Ω F corresponds to the potential difference between the field lines.In the steady, axisymmetric state, the equations (14) and (15) are reduced to ∇ · (cid:18) − H ϕ π B p (cid:19) = B i ∂ i (cid:18) − H ϕ π (cid:19) = − ( J p × B p ) · m , (23) ∇ · (cid:18) Ω F − H ϕ π B p (cid:19) = B i ∂ i (cid:18) Ω F − H ϕ π (cid:19) = − E · J p , (24)where the subscript p denotes the poloidal component. Here one sees that the poloidal AMand Poynting fluxes are described by L p = − H ϕ π B p , S p = Ω F − H ϕ π B p , (25)respectively. It should be noted that H ϕ = ∗ F µν ξ µ χ ν and Ω F = − F tθ /F ϕθ are the same inthe BL and KS coordinates (K04).TT14 shows that the condition Ω F > F > , H ϕ = 0 (26)has to be maintained, i.e. the poloidal AM and Poynting fluxes are steadily non-zero (TT14).The following discussion in this paper focuses on how their values are causally determinedand the role of the negative energies as measured in the coordinate basis. .3.2. Particle motions and light surfaces. Under the assumption (2) for the magne-tospheric plasma stated in Section 2.3.1, the force-free condition ρ E + J × B = 0 (or ρ D + j × B = 0) is satisfied when D < B (e.g. K04; TT14; see also Appendix A). Thenequation (23) indicates B i ∂ i H ϕ = 0 (for D < B ) . (27)Equations (23) and (24) mean that no AM or energy is exchanged between the particles andthe electromagnetic fields.In this case, in the BL coordinates, the particles drift in the azimuthal direction withangular velocity Ω F when B ϕ = 0 (TT14). The light surfaces are thus defined as wherethe four-velocity of a particle with the coordinate angular velocity Ω F becomes null, i.e. f (Ω F , r, θ ) = 0, where f (Ω F , r, θ ) ≡ ( ξ + Ω F χ ) = − ̺ ∆Σ + γ ϕϕ (Ω F − Ω) . (28)There can be two light surfaces; the outer light surface (outside which f >
0) and the innerlight surface (inside which f > < Ω F < Ω H , Ω F = Ω + p ̺ ∆ / Σ γ ϕϕ > Ωat the outer light surface, and Ω F = Ω − p ̺ ∆ / Σ γ ϕϕ < Ω at the inner light surface, whichis located in the ergosphere [33, K04; TT14]. The condition 0 < Ω F < Ω H is satisfied whenthe outward Poynting flux is non-zero either for the field lines threading the horizon [1] orthe ergospheric field lines threading the equatorial plane (TT14). Note that f (Ω F , r, θ ) is ascalar, so that the location of each light surface is the same in the BL and KS coordinates.If D > B is realized, the cross-field current flows, i.e. J p × B p = 0 (TT14). The force-freecondition is violated, and H ϕ varies along a field line.
3. Field lines threading the equatorial plane
The mechanism of Ω F and H ϕ being determined along the ergospheric field lines threadingthe equatorial plane has been already clarified (K04; TT14). Generally in the BL coordinates,one has from equations (11), (12), and (21)( B − D ) α = − B f (Ω F , r, θ ) + 1 α (Ω F − Ω) H ϕ . (29)The key point is that H ϕ = 0 on the equatorial plane because of the symmetry. Therefore, theregion of f (Ω F , r, θ ) > D > B around the equatorial plane, driving the poloidal currentto flow across the field lines. (Note that B − D = F µν F µν / D > B also in the KS coordinates.) This leads to H ϕ = 0 outside the region where D >B , which we call ‘current crossing region’. The value of Ω F will be regulated so that thecurrent crossing region is finite (see Figure 4 of TT14), and thus it is expected to dependon the microphysics in the ergosphere. The values of Ω F and H ϕ will be determined by theconditions around the equatorial plane and at infinity. Finite particle mass may cause some inertial drift currents to flow across the field lines, whichtransfer the AM and energy between the particles and the electromagnetic fields [46]. We assumethat this effect is negligible for the flux production.7/28 ! B ! r ! ! θ ! ! φ ! " ! ! ! Fig. 1
Motion of the positively (negatively) charged particle near the equatorial plane inthe BL coordinates. This schematic picture is applicable both in the BL coordinate basisand in the BL FIDO orthonormal basis.In the current crossing region, D is in the opposite direction of E , i.e. D · E <
0, as seen inthe BL coordinates (see Figure 3 of TT14). This leads to ( J p × B p ) · m < E · J p < D · E < F > H ϕ < H ϕ > r = r es , has Ω F = H ϕ = 0. This means thatthe current flows inward along the ergospheric field lines and outward along the last ergo-spheric field line (see Figure 3 below). Correspondingly, the current crossing region extendsover r H < r < r es . Such a poloidal current structure prevents the BH from charging up. Equations (23) and (24) imply that the particles in the current crossing region lose theirAMs and energies by the feedback, +( J p × B p ) · m and + E · J p , from the production of theelectromagnetic AM and Poynting fluxes. We find that this feedback can make the particleshave negative energy as measured in the coordinate basis.The production of the particle negative energy can be explained by showing the particlemotions in the local orthonormal basis carried with the BL FIDOs, in which the equation ofa particle motion with four-velocity u , three-velocity v , charge q , and mass m is written as d ˆ u i d ˆ t = qm ( ˆ D i + ǫ ijk ˆ v j ˆ B k ) , (30)where ˆ C i denotes the vector component in respect of the FIDO’s orthonormal basis [23,TT14]. In this basis one can investigate local, instantaneous particle motions under theLorentz force as special relativistic dynamics. (The FIDO frame is not inertial and a particlefeels the gravitational force, although we neglect it compared with the Lorentz force, basedon the assumption (3) set in Section 2.3.1.) The AM and energy per mass of a particle asmeasured in the coordinate basis are l p = u µ χ µ = γ ϕϕ ( v ϕ − Ω) u t = √ γ ϕϕ ˆ v ϕ ˆ u t , (31) e p = − u µ ξ µ = [ α + γ ϕϕ Ω( v ϕ − Ω)] u t = ( α + √ γ ϕϕ Ωˆ v ϕ )ˆ u t , (32) − . . . ˆ v r , ˆ v ϕ ˆ t/ τ gy D/B = 1 . D/B = 1 . D/B = 1 . v r ˆ v ϕ Fig. 2
Velocity components ˆ v r (upper three lines with positive values) and ˆ v ϕ (lowerthree lines with negative values) of the posively charged particle in the fixed BL FIDO’sorthonormal basis as functions of time normalized by gyration time scale τ gy = m/q | ˆ B | . Thesolid, dashed, and dot-dashed lines are calculation results for | ˆ D | / | ˆ B | = 1 . , . , and 1 . v r = ˆ v ϕ = 0.where we have used ˆ v ϕ = ( √ γ ϕϕ /α )( v ϕ − Ω) [cf. 44].Near the equatorial plane, the ˆ B field is approximately perpendicular to that plane, because B ϕ = H ϕ /α = 0 at that plane, and then the ˆ D field is radial in that plane (see Figure 1). Themotion of a test particle can be easily solved in such fields [45]. For the case of D ≥ B whichwe focus on, the positively (negatively) charged particles are accelerated in the directionsof ˆ D ( − ˆ D ) and ˆ D × ˆ B . In Figure 2, we show the calculation results for | ˆ D | / | ˆ B | = 1 . , . , and 1 .
3, where we fix the basis and assume that the electromagnetic fields are uniform.For D = B (i.e. | ˆ D | / | ˆ B | = 1 .
0) in particular, the particles are strongly accelerated in thedirection of ˆ D × ˆ B , and then one obtainsˆ v ϕ ≈ − ℓ gy = m/q | ˆ B | (not normalized by the gravitational radius).As a consequence, from equations (31) and (32), one has l p ≈ −√ γ ϕϕ ˆ u t < , (34) e p ≈ ( α − p β )ˆ u t < , (35)in the ergosphere, where α < β = γ ϕϕ Ω . For D > B , ˆ v ϕ does not approach −
1, so that e p > α = β . However, α → r → r H implies that e p < l p and e p arescalars, and thus l p < e p < ℓ gy is expected to be ∼
10 orders of magnitude smaller than
GM/c [cf. 47], so that the distance which a particle travels until it achieves the asymptotic azimuthalvelocity is tiny compared to the size of the ergosphere. This justifies our calculations of theparticle motion in the fixed orthonormal basis with the uniform electromagnetic fields, andthe asymptotic velocities can be interpreted as the local velocities of the test particles. ince the current crossing region is bounded at r < r es , the positively charged particlesdo not cross the last ergospheric field line and will gyrate around this field line. When theyemerge out of the ergosphere, they contribute to the current flowing outward along the lastergospheric field line (see Figure 3). The particles outside the ergosphere generally havepositive energies. We argue that BZ process for the ergospheric field lines threading the equatorial planeis similar to the mechanical Penrose process, in which the rotational energy of a BH isextracted as mechanical energy by making the BH absorb negative-energy particles [28,29]. For simplicity, let us consider the positively and negatively charged particles in thegeometrically thin current crossing region as a one-fluid. The energy equation for this fluidin the steady state is written as ∂ r √ γ ( − αT r p ,t ) = E · J p < , (36)where T ν p ,µ is the energy-momentum tensor of the fluid. The boundary condition at r = r es is T r p ,t = 0. Therefore, the solutions of equation (36) should be F r ≡ − αT r p ,t > − T r p ,t = − ρ m U t U r >
0, where ρ m and U µ are the comovingmass density and the four-velocity of the fluid, respectively. Since all the particles may havenegative energy, it is reasonable to estimate − U t < , U r < . (37)This means that the current crossing region generates the inflow of the negative-energy fluidand the outward Poynting flux, which appears to be a similar process to the mechanicalPenrose process.As a result, the BH loses its rotational energy by the poloidal particle energy flux F p = − αρ m U t U p . We summarize our argument in Figure 3 (see Section 4 for the field linesthreading the horizon).However, it is too simple to treat the charged particles in the current crossing region asa one-fluid, since the average velocities of the positively and negatively charged particlesshould be different. Furthermore, Figure 2 is just the result of the test particle calculations.More detailed studies of the plasma particle motions are required to confirm whether thecondition of equation (37) is realistic in the current crossing region. MHD numerical simulations treat the energy of particles (while force-free simulations not), sothat they can observe the negative particle energy in principle. However, the MHD simulationresults of the dilute Kerr BH magnetosphere with cylindrical magnetic field at the far zonein [12] do not show any negative particle energy in the steady state. This is just due tothe disappearance of the ergospheric field lines threading the equatorial plane, although thereason of this disappearance has not been identified. Such behavior is also seen in the threedimensional MHD simulations including the dense accretion flow [3, 13][but see 48]. p ! J p ! J p ! J p ! S p ! !" ! !),(-.’ ! F p ! J p ! Fig. 3
Schematic picture of the poloidal currents J p ( open arrows ), the poloidal Poyntingflux S p ( filled arrows ), and the poloidal particle energy flux F p near the equatorial plane (i.e.the inflow of the particle negative energies; striped arrow ) in the steady state in the northernhemisphere in the KS coordinates. The BH loses its rotational energy directly by S p alongthe field lines threading the horizon (see Sections 4 and 5) and by F p near the equatorialplane which is associated with S p along the field lines threading the equatorial plane in theergosphere (see Section 3).
4. Field lines threading the event horizon
In contrast to the equatorial plane where H ϕ = 0 from the symmetry, one generally has H ϕ = 0 at the horizon. Thus the above argument on the field lines threading the equatorialplane is not applicable for the field lines threading the horizon. At the horizon, the regularitycondition should be satisfied [23, 26]:ˆ B ϕ = − ˆ D θ (in BL coordinates) . (38)For the BZ split-monopole solution [1] as an example, in which ˆ B r = 0, ˆ B θ ≈
0, and ˆ D r ≈ a ), so that one has B − D > . (39)Therefore the force-free condition is satisfied at the horizon.We confirm this fact more generally in the KS coordinates. From the calculation shown inAppendix B, we obtain( B − D ) α = − B θ B θ f (Ω F , r, θ ) + ( B ϕ B ϕ + B r B r ) ̺ ∆Σ+4 r sin θ (Ω F − Ω) B r B ϕ + 4 r Σ " − (cid:18) Ω F Ω (cid:19) (cid:18) − ̺ Σ (cid:19) ( B r ) , (40)where Σ > ̺ is generally satisfied (see equation B2). For the BZ split-monopole solutionas an example, in which B r > B ϕ < !" Ω ! $%&’(")*+,-*’ ! D ! D ! B p ! J p ! E × H φ ! $%&’(")*+,-*’ ! D ! D ! B p ! E × H φ ! J p ! ./"0**+1,’2(&3 ! ! | ˆ B r | ≫ | ˆ B ϕ | ∼ | ˆ D θ | | ˆ B r | ≪ | ˆ B ϕ | ∼ | ˆ D θ | Fig. 4
Electromagnetic field structures of the BZ split-monopole solution as measured inthe BL ( left ) and KS ( right ) coordinates.below), B θ ≈
0, and Ω F ≈ Ω H /
2, one has B − D > > Ω F . We seethat B − D > B θ is weak, B r B ϕ <
0, and Ω > Ω F . (Notethat for the field lines threading the equatorial plane, B θ is the dominant field componentnear that plane, where B − D < Here we focus on the electromagnetic structure of the BZ split-monopole solution, and showthat some properties are measured differently in the BL and KS coordinates. This analysis isuseful for finding the essential physics in BZ process for the field lines threading the horizon,which should be independent of the adopted coordinate systems.The split-monopole field is given by B r = const . × sin θ √ γ , B θ ≈ , (41)which satisfies ∇ · B = 0. (Note that sin θ/ √ γ → /r for r → ∞ .) In the BL coordinates,one has B ϕ = 1 α H ϕ , (42) D θ = − α (Ω F − Ω) B r √ γ. (43)As is well known, D θ changes its sign at the point where Ω = Ω F (see Figure 4, left ). Wecan see that B ϕ and D θ diverge as r → r H , while B r √ γ is finite, and one has | ˆ B r | ≪ | ˆ B ϕ | ∼ | ˆ D θ | (44)near the horizon. n the other hand, in the KS coordinates, one has B ϕ = αH ϕ − B r sin θ (2 r Ω F − a )∆ sin θ , (45) D θ = − α (Ω F B r − β r B ϕ ) √ γ. (46)Equation (45) is derived by rewriting H ϕ in equation (12) with equation (11) and (B3)(K04). The regularity condition at the horizon (∆ = 0) for the steady flow to pass with nodiverging physical quantities is given by αH ϕ = B r sin θ (2 r Ω F − a ) , (47)which is equivalent to equation (38). We calculate B ϕ and D θ from r = r H towards infinityfor small values of a and Ω F = Ω H /
2, and find that B ϕ <
0, that D θ does not change itssign (see Figure 4, right ), and that | ˆ B r | ≫ | ˆ B ϕ | ∼ | ˆ D θ | (48)near the horizon. That is, the D field in the KS coordinates is not only so weak that itcannot drive the cross-field current but also it does not change its direction, i.e. D · E > D field with D · E < F and D = 0 appears special, and it wasconsidered as a key in some previous analytical discussions [e.g. 35, K09]. However, theelectromagnetic quantities are clearly continuous or seamless in the KS coordinates, as shownin Figure 4.Below we generically consider the cases in which the force-free condition is satisfied alongthe field lines threading the horizon (see Section 4.1). In those cases, an essential pointis that the outward AM and Poynting fluxes, L p = − H ϕ B p / π and S p = − Ω F H ϕ B p / π ,are seamless along each field line from the event horizon to infinity in the steady state(from equation 27), with no transfer of AM and energy from the particles. This situation isdescribed in Figure 3. Now we discuss how L p and S p are created along the field lines threading the horizon. Bland-ford & Znajek [1] show that Ω F and H ϕ in the steady state are determined mathematically from equations (21) and (27) with the conditions at the horizon and at infinity (see alsoK04). This mathematics and the seamless property shown above may lead to an incorrectconsideration that the fluxes are created at the horizon. The conditions at the horizon andat infinity are not boundary conditions but regularity conditions [10, K04], as stated inSection 1. The place where the fluxes are created must not be the horizon, but outside thehorizon.We note that the non-zero outward AM and Poynting fluxes at the horizon in the KScoordinates do not violate causality, because the steady fluxes carry no information. It shouldbe also noted that the steady Poynting flux S p = − Ω F H ϕ B p / π is not a product of a certainenergy density and its advection speed like steady particle energy flux F p = ( − αρ m U t ) U p (see Section 4.4 for a related discussion). The Poynting flux is just a result of the currentsflowing in the plasma with the potential differences. onsequently, the issue on the field lines threading the horizon is well defined as “How isthe steady current structure causally built?” We consider that this issue may not be resolvedby investigating only the steady-state structure. The phenomena at the horizon should be aresult from those having occurred outside the horizon in the prior times t . In Section 5, weaddress this issue by discussing a time-dependent state evolving towards the steady state. Lasota et al. [31] and Koide & Baba [32] argue that the outward Poynting flux is mediated by‘inflow of the negative electromagnetic energy’ (see also K09). Although this interpretationanalogous to the mechanical Penrose process looks attractive for causal production of thePoynting flux, it is difficult to consider the flow of the steady field (rather than waves).Furthermore, we find that the sign of the electromagnetic energy density depends on thecoordinates.In the BL coordinates, the electromagnetic AM and energy densities can be written downby (K09) l = 14 πα γ ϕϕ (Ω F − Ω)( B θ B θ + B r B r ) , (49) e = 18 πα h α B + γ ϕϕ (Ω − Ω )( B θ B θ + B r B r ) i . (50)Thus l and e is negative (and diverges) near the horizon when Ω F < Ω H . This condition issatisfied in the BZ split-monopole solution.On the other hand, in the KS coordinates, the calculations shown in Appendix B lead to4 παl = Σ sin θ̺ (Ω F − Ω) B θ B θ − r sin θB r B ϕ + Ω F ( ̺ + 2 r ) sin θ ( B r ) (51)8 παe = (cid:20) Σ sin θ̺ (Ω F + Ω)(Ω F − Ω) + ̺ ∆Σ (cid:21) B θ B θ + ∆ sin θ ( B ϕ ) − a sin θB r B ϕ + (cid:2) ( ̺ + 2 r ) sin θ (cid:3) ( B r ) . (52)In the BZ split-monopole solution as an example, in which B θ ≈ B ϕ < l > , e > . (53)This condition is generally valid when B θ is weak and B r B ϕ < l = αT tϕ = − T µν n µ χ ν , e = − αT tt = T µν n µ ξ ν (54)depend on the coordinates, while T tϕ = T µν ξ µ χ ν and T tt = T µν ξ µ ξ ν are scalars. The conceptof the negative electromagnetic energy density depends on the coordinates, and thus it isnot physically essential. Lasota et al. [31] argue that the electromagnetic energy density calculated in the KS coordinatesis negative near the horizon, but they define the electromagnetic energy density as T µν l µ ξ ν where l µ = αn µ and n µ is the four-velocity of the BL FIDO.14/28 r ! !" ! &’( ! V<0 ! J r ! S r ! Fig. 5
Schematic picture of a time-dependent process evolving towards the steady state.The plasma particles keep injected between the inner and outer light surfaces, and thevacuum is being filled with those plasma. This picture focuses on the inflow. The innerboundary of the force-free region propagates towards the event horizon, producing the steadypoloidal current structure and the outward AM and Poynting fluxes.
5. Process towards the Steady State
As stated in Section 4.3, we address the issue how the steady poloidal current structure isbuilt causally, by discussing a time-dependent state evolving towards the steady state.In the steady state, the plasma has the inner and outer light surfaces (see Section 2.3.2).The particles flow in across the inner light surface and flow out across the outer light surface.Therefore, new particles have to keep injected between the two light surfaces, as discussed inmany literatures [e.g. 2, 10, 33, 49]. In this paper we have assumed that the plasma particleskeep injected from outside the magnetosphere through electron-positron pair creation bycollisions of two photons [2, 50, 51] and/or diffusion of high-energy hadrons [47] , and thatthose particles maintain D · B = 0 and carry the currents.Now let us first consider a vacuum in Kerr space-time, and then begin the continuous injec-tion of force-free plasma particles between the two light surfaces as a gedankenexperiment.The inflow (outflow) will fill the vacuum near the horizon (at infinity). Simultaneously wewill see a process building the poloidal current structure. Hereafter we will call the (inflow+ outflow) region filled with the force-free plasma ‘force-free region’. Figure 5 is a schematicpicture of this process focusing on the inflow.We show the space-time diagrams of the inner and outer boundaries of the force-free regionin the BL and KS coordinates in Figure 6, in which the radial light signals are representedby the small arrows. The outflow continues to propagate into the vacuum, i.e. the radius ofthe outer boundary r → ∞ for t → ∞ . In the BL coordinates, the inflow also continues to In the geometrically thick accretion disk the particles can be non-thermally accelerated anddiffused out of the disk. The amount of those high-energy hadrons does not appear to be sufficientfor the total mass loading of AGN jets which provides the observationally inferred Lorentz factorΓ ∼ − D · B = 0 [52, 53].15/28 i ! r i ! r ! r ! r H ! r H ! ! ! t ! t ! !" ! ./ ! t H ! Fig. 6
Space-time diagram of the inner and outer boundaries of the force-free region inthe BL and KS coordinates. In each diagram the left and right long arrows correspond tothe motions of the inner and outer boundaries, respectively, while the small arrows to thepropagation of light.propagate towards the horizon, r → r H for t → ∞ . In the KS coordinates, the inflow can passthe horizon in a finite time of t = t H . In both of the coordinates, when the inner boundaryapproaches the horizon, the outward signal from it becomes slower and slower and it canhardly affect the force-free region. This will lead to the steady state. Although such a time-dependent state should be analyzed numerically, we use a toy modelto qualitatively illustrate the process of building the poloidal current structure. This modelassumes that (1) B p is fixed to be split-monopole ∂ r ( √ γB r ) = 0 , B θ = 0 (55)in the whole region, and that (2) the Kerr BH magnetosphere is separated into the force-free region and the vacuum by geometrically thin boundaries moving radially. For furthersimplicity, (3) we assume that the force-free region and the vacuum have their steady-statestructures, but the values of the physical quantities, particularly Ω F and H ϕ , keep updatedas determined by the varying conditions of the inner and outer boundaries.Some of these assumptions would be violated in realistic experiments. Nevertheless weconsider that our toy model is useful to suggest the key points for resolving the issue on thecausality in the coordinate basis (Section 5.1.4), which also allows us to understand how thesteady state is maintained (Section 5.3). The electromagnetic quantities in the force-freeregion are given as follows. The condition D · B = 0 and ∇ × E = 0 lead to E ff ϕ = E ff r = 0 , E ff θ = −√ γ Ω F B r , (56) In some MHD simulations, a static plasma (not a vacuum) is initially given and then a centralstar starts rotating [54] or a BH starts rotating [55]. They show that a switching-on wave propagatesoutward and that the outflow region settles down to the steady state after it passes the outer fastmagnetosonic point [22]. 16/28 here ∂ r Ω F = 0 . (57)Hereafter we will put the subscript and superscript ‘ff’ on the quantities in the force-freeregion. Equations (11) and (12) give us D ff ϕ = D ff r = 0 , D ff θ = √ γα (Ω − Ω F ) B r , (58) H ff ϕ = αB ff ϕ , H ff r = αB r − √ γ Ω D θ ff , H ff θ = 0 . (59)Equation ∇ × H = 4 π J and the force-free condition lead to ∂ r H ff ϕ = − π √ γJ θ ff = 0 , (60) ∂ θ H ff ϕ = 4 π √ γJ r ff , (61)These two equations imply that ∂ r ( √ γJ r ff ) = 0. We focus on the northern hemisphere, where J r ff < H ff ϕ <
0. The current flowing outward J r ff >
0, which prevents the BH fromcharging up, is assumed to be concentrated on the equatorial plane. The poloidal AM andPoynting fluxes are L r ff = − H ff ϕ π B r , S r ff = Ω F − H ff ϕ π B r , (62)which satisfy ∂ r ( √ γL r ff ) = 0 and ∂ r ( √ γS r ff ) = 0.In the vacuum region, one has ρ = J = 0. Equations ∇ × E = 0 and ∇ × H = 0 lead to E vac ϕ = 0 , H vac ϕ = B vac ϕ = 0 , (63)which indicates L r vac = S r vac = 0 . (64)Hereafter we will put the subscript and superscript ‘vac’ on the quantities in the vacuumregion. Let us focus on the inner boundary ofthe force-free (inflow) region, and derive the conditions on the boundary, i.e. the junctionconditions between the force-free and vacuum regions. The similar analysis can be done forthe outer boundary. For equation − ∂ t D r + 1 √ γ ∂ θ H ϕ = 4 πJ r , (65)we substitute D r = D r vac H ( − R ) , (66) H ϕ = H ff ϕ H ( R ) , (67) J r = J r ff H ( R ) + η r δ ( R ) , (68)where H ( R ) and δ ( R ) are the Heaviside step function and the Dirac delta function,respectively, and R = r − r i − Z t V dt, (69)where r i and V are the initial radius and the velocity of the boundary. The location of theboundary is represented by R = 0. We have introduced η r in equation (68), i.e. possible ontribution to J r from moving surface charges at the boundary. The assumption (3) statedin the first part of this section implies that the timescale for the quantities in the force-freeand vacuum regions becoming adjusted for steady-state structure is much smaller than thetimescale of the boundary propagation. We focus on the latter timescale, considering thatonly R = R ( t ) depends on t in equation (65). Then we have − D r vac V δ ( R ) + 1 √ γ ( ∂ θ H ff ϕ ) H ( R ) = 4 πJ r ff H ( R ) + 4 πη r δ ( R ) . (70)Taking account of equation (61), we obtain η r = − D r vac π (cid:12)(cid:12)(cid:12)(cid:12) R =0 V, (71)which implies that the surface charge density on the boundary σ = − D r vac | R =0 / π . This canbe confirmed by integrating ∇ · D = 4 πρ over the infinitesimally thin (in the r direction)region enclosing the small area on the boundary and taking account of D r ff = 0.For equation − ∂ t D θ − √ γ ∂ r H ϕ = 4 πJ θ , (72)we substitute D θ = D θ vac H ( − R ) + D θ ff H ( R ) , (73) J θ = η θ δ ( R ) , (74)and equation (67). We have introduced η θ , possible contribution to J θ from the surfacecurrent flowing on the boundary. Then we have − D θ vac V δ ( R ) + D θ ff V δ ( R ) − √ γ H ff ϕ δ ( R ) = 4 πη θ δ ( R ) , (75)which leads to V = 1 √ γ H ff ϕ + 4 π √ γη θ D θ ff − D θ vac (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R =0 . (76)The last one of Maxwell equations nontrivial for the present problem is ∂ t B ϕ + 1 √ γ ( ∂ r E θ − ∂ θ E r ) = 0 , (77)for which we substitute B ϕ = B ϕ ff H ( R ) , (78) E θ = E vac θ H ( − R ) + E ff θ H ( R ) , (79) E r = E vac r H ( − R ) . (80)Then we have − B ϕ ff V δ ( R ) + 1 √ γ h − E vac θ δ ( R ) + E ff θ δ ( R ) − ( ∂ θ E vac r ) H ( − R ) i = 0 . (81)Integrating equation (81) over − ǫ < R < ǫ and take a limit of ǫ →
0, the last term vanishes,and we obtain V = 1 √ γ E ff θ − E vac θ B ϕ ff (cid:12)(cid:12)(cid:12)(cid:12) R =0 , = α √ γ D ff θ − D vac θ B ϕ ff (cid:12)(cid:12)(cid:12)(cid:12) R =0 , (82) here we have used equation (11) for the last equality. Eliminating D θ ff − D θ vac from equations(76) and (82) leads to V = ± α √ γ rr s π √ γη θ H ff ϕ . (83)Here we take the minus sign, since we have assumed that the inner boundary keeps mov-ing inward. In Section 5.1.3, we will confirm that this assumption is consistent with theelectromagnetic structure which we found.Let us consider the case of η θ = 0. Then we have V = − α √ γ rr , (84)and H ff ϕ = − α r γ ϕϕ γ θθ ( D ff θ − D vac θ ) (cid:12)(cid:12)(cid:12)(cid:12) R =0 = − r γ ϕϕ γ θθ [(Ω − Ω F ) √ γB r − αD vac θ ] (cid:12)(cid:12)(cid:12)(cid:12) R =0 . (85)Substituting dr = V dt for equation (1), we find ds = γ ϕϕ ( dϕ − Ω dt ) + γ θθ dθ ≥ , (86)which has to be ds = 0. This means that the four-velocity of the boundary is null. In reality,however, the particles at the boundary cannot propagate with this speed, and thus one canconclude η θ > , (87)i.e., the cross-field current must flow on the boundary. Note that equation (85) with αD vac θ → In our toy model of the time-dependent state, we have nottaken into account equations of the particle motions, using the force-free approximation forthe force-free region, but we have assumed that the inner boundary keeps moving inward,i.e.
V <
0. Here we examine the direction of the Lorentz force exerted on the particles at theboundary, and confirm that it is consistent with the assumption of
V <
0. It is reasonablethat the force-free approximation is not applicable for the boundary between the force-freeand vacuum regions, and indeed we have seen that the cross-field current flows there, η θ > n ff of the force-free region is high enough to screen theelectric field along the B field lines, i.e. D r ff = 0. We may even assume that n ff ≫ ρ ff /e ,where ρ ff is the charge density of the force-free region, and then the distribution of n ff isnot directly related to that of ρ ff . On the other hand, n approaches zero at the boundarytowards the vacuum region, where n ≫ ρ/e is not valid, and non-zero surface charge density σ just implies non-zero surface mass density σ m . Thus we can write the equation of theparticle motions in the r direction as ∇ ν [ σ m U r U ν δ ( R ) + ρ mff U r ff U ν ff H ( R )] = F r ν I ν and thecontinuity equation as ∇ ν [ σ m U ν δ ( R ) + ρ mff U ν ff H ( R )] = 0, where ρ mff is the mass density ofthe force-free region. We combine these two equations, use U ν ff ∂ ν H ( R ) = U t ff ( V r ff − V ) δ ( R ), nd integrate the equation over R (i.e., keep the components including δ ( R ) as done inSection 5.1.2) to have σ m U ν ∇ ν U r + ρ mff U t ff ( V r ff − V )( U r ff − U r ) = σD r | R =0 + γ θθ α √ γ η θ B ϕ | R =0 . (88)At the boundary D r = 0 and n ∼ ρ/e , and then the Lorentz force will be much strongerthan the gravitational and inertial forces. We neglect the latter forces as in Section 3.2, sothat the first term in the left-hand side of equation (88) can be rewritten as σ m U ν ∂ ν U r . Inequation (88), D r | R =0 should have a value between D r ff = 0 and D r vac = − πσ , and B ϕ | R =0 between H ff ϕ /α < H vac ϕ /α = 0. We also found η θ >
0. These mean that the right-handside of equation (88) is negative, i.e., the Lorentz force exerted on the boundary is in thedirection of − r .The second term in the left-hand side of equation (88) represents momentum change ofthe boundary layer due to its mass exhange with the force-free region, and this term is zerowhen V = V r ff . In the other case, we have V > V r ff since the boundary and the force-freeregion do not separate. In our toy model, the particles are continuously injected betweenthe two light surfaces, and the particles flow outward across the outer light surface andflow inward across the inner light surface. For the inflowing force-free region, the continuityequation ∇ ν ( ρ mff U ν ff ) = 0 and its assumed steady axisymmetric structure mean U r ff <
0. If U r > U r ff , the acceleration σ m U ν ∂ ν U r is negative, and we continue to have U r < V <
0. The acceleration could be positive when U r < U r ff <
0, but this case means
V <
V >
0. Therefore, the inner boundary keeps
V <
V > η r → − η r and η θ → − η θ , and thus one obtains the conditions D r vac | R =0 = 4 πσ and η θ <
0. These indicate F r ν I ν >
0. Equation (88) is valid with change V r ff − V → V − V r ff .The same argument as for the inner boundary leads to the conclusion V >
Since
V < B ϕ ff = B ff ϕ /γ ϕϕ = H ff ϕ /αγ ϕϕ <
0, equation (82) means D ff θ | R =0 > D vac θ | R =0 . (89)The electromagnetic AM density is given as l = − D θ B r √ γ/ π . Then equation (89) indicates l ff | R =0 < l vac | R =0 . (90)That is, the inner boundary of the force-free region converts the vacuum with larger AMdensity into the force-free plasma with smaller AM density. Now equation (14) can be writtenas B r ∂ r (cid:18) − H ϕ π (cid:19) = − ∂ t l + √ γJ θ B r . (91)Substituting l = l vac H ( − R ) + l ff H ( R ) (92)and equations (67) and (74) for equation (91), we obtain L r ff = h V ( l ff − l vac ) + √ γη θ B r i R =0 . (93) aking account of equations (87) and (90), we find that the electromagnetic AM flux inthe force-free region is produced by the conversion of the electromagnetic AM density fromthe vacuum to the force-free plasma through the boundary and the torque of the cross-fieldcurrent at the boundary.Equation (93) can also be derived from equation (76). These equations also mean that H ff ϕ is produced by the displacement current √ γV ( D θ ff − D θ vac ) and the cross-field current − π √ γη θ . None of these two contributions appears in the steady state (see Sections 4.1 and4.2).Equation (15) can be reduced to B r ∂ r (cid:18) Ω F − H ϕ π (cid:19) = − ∂ t e − E r J r − E θ J θ . (94)Substituting e = e vac H ( − R ) + e ff H ( R ) (95)and equations (67), (68), and (74) for equation (94), we obtain S r ff = h V ( e ff − e vac ) − E r η r − E θ η θ i R =0 . (96)By using the expressions e ff = 18 π ( E ff θ D θ ff + B ϕ ff H ff ϕ + B r H r ) , (97) e vac = 18 π ( E vac r D r vac + E vac θ D θ vac + B r H r ) , (98)we find that E r | R =0 = E ff r + E vac r (cid:12)(cid:12)(cid:12)(cid:12) R =0 , E θ | R =0 = E ff θ + E vac θ (cid:12)(cid:12)(cid:12)(cid:12) R =0 (99)satisfy equation (96). In equation (96), the term − E r η r | R =0 = αD r vac D vac r V / π <
0. One cansee that the Poynting flux in the force-free region is produced by the electromagnetic energyconversion V ( e ff − e vac ) | R =0 and the work of the cross-field current − E θ η θ | R =0 . We can obtain the same conclusions as above in the KS coordinates, where the calculationsare complicated compared to those in the BL coordinates due to γ rϕ = 0. Equations havingdifferent shapes from those in the BL coordinates are D ff θ = 1 α ( −√ γ Ω F B r + √ γβ r B ϕ ) , (100) H ff ϕ = αB ff ϕ − √ γβ r D θ ff , H ff r = αB r , (101)for the force-free region, and H vac ϕ = αB vac ϕ − √ γβ r D θ vac = 0 , (102)for the vacuum region. From equations (65), (72), and (77), we obtain η r = − D r vac π (cid:12)(cid:12)(cid:12)(cid:12) R =0 V, (103) = 1 √ γ H ff ϕ + 4 π √ γη θ D θ ff − D θ vac (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R =0 (104)= " √ γ α ( B ff ϕ − B vac ϕ ) + 4 π √ γη θ D θ ff − D θ vac − β r R =0 , (105) V = 1 √ γ E ff θ − E vac θ B ϕ ff − B ϕ vac (cid:12)(cid:12)(cid:12)(cid:12) R =0 (106)= (cid:18) α √ γ D ff θ − D vac θ B ϕ ff − B ϕ vac − β r (cid:19) R =0 , (107)where we have used equations (101) and (102) to derive equation (105). Eliminating D θ ff − D θ vac from equations (105) and (107) leads to V = ± α √ γ r γ ϕϕ γ θθ s π √ γη θ αγ ϕϕ ( B ϕ ff − B ϕ vac ) − β r . (108)The sign of the first term in the right-hand side is not determined in this analysis. However,we have assumed V <
0, and will confirm that the assumption
V < η θ = 0, we have V = ± α √ γ r γ ϕϕ γ θθ − β r , (109)and ( B ϕ ff − B ϕ vac ) R =0 = ± s γ θθ γ ϕϕ ( D ff θ − D vac θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R =0 . (110)Substituting dr = V dt for equation (1), we find ds = √ γ ϕϕ dϕ ± αγ rϕ p γγ θθ dt ! + γ θθ dθ ≥ , (111)which has to be ds = 0, indicating that the four-velocity of the boundary is null. Becausethis velocity cannot be realized, we can conclude η θ = 0.The force-free region always satisfies equation (45) and has no diverging quantities, andthus the regularity condition at the horizon (equation 47) automatically becomes satisfiedafter the boundary crosses the horizon.We have shown that η θ > √ γJ θ is the same in the BL andKS coordinates, we have η θ > η θ / ( B ϕ ff − B ϕ vac ) | R =0 < ds <
0. Then we have ( B ϕ ff − B ϕ vac ) | R =0 < H ff ϕ is produced by the displacement current and the cross-fieldcurrent at the boundary. If the former is dominant, D θ ff | R =0 > D θ vac | R =0 is realized so that H ff ϕ <
0. This means l ff | R =0 < l vac | R =0 . The production of the electromagnetic AM flux can lso be understood by the same equation as equation (93) in the BL coordinates. In thiscase, equation (107) with ( B ϕ ff − B ϕ vac ) | R =0 < V + β r <
0, which means that theminus sign should be taken in equation (108).The production of the Poynting flux can also be understood by equation (96) similarly tothe case in the BL coordinates. Here the electromagnetic energy densities are e ff = 18 π ( E ff θ D θ ff + B ϕ ff H ff ϕ + B r H ff r ) , (113) e vac = 18 π ( E vac r D r vac + E vac θ D θ vac + B r H vac r ) , (114)where we should note that H ff r = αB ff r is different from H vac r = αB vac r , since B r = γ rr B r + γ rϕ B ϕ . We confirmed that equation (99) satisfies equation (96) also in the KS coordinates.We confirm that the assumption V < σ m U ν ∇ ν U r + ρ mff U t ff ( V r ff − V )( U r ff − U r ) = (cid:18) ∆ ̺ − β r Vα (cid:19) γ rr σD r | R =0 + γ θθ α √ γ η θ H ϕ | R =0 . (115)We should have σD r | R =0 < , H ϕ | R =0 < η θ >
0. Thus the Lorentz force is in the − r direction while V <
0, and willoverwhelm the gravitational and inertial forces. The inner boundary layer starts with
V <
V > V ∼ , U r ∼
0. That is, theinner boundary layer keeps
V < As shown above, H ff ϕ and Ω F , or the electromagnetic AM and Poynting fluxes, are createdat the inner boundary which propagates towards the horizon. This is a causal mechanism ofthe flux production as measured in the coordinate basis.After the inner boundary becomes very close to the horizon in the BL coordinates or itcrosses the horizon in the KS coordinates, it does not affect the exterior, and H ff ϕ and Ω F are fixed to be consistent with the regularity condition at the horizon (and at infinity).This implies that no source of H ff ϕ and Ω F or the AM and Poynting fluxes is required inthe steady state. The steady poloidal currents are just flowing along the field lines withoutcrossing them, and no force is required to drive the currents in the steady state, partlybecause the force-free plasma is assumed to have no resistivity. This situation is essentiallydifferent from that in a steady pulsar wind, in which the electromotive force V ϕ × B drivesthe cross-field current in the rotating star, and the fluxes definitely have the electromagneticsources, i.e. ∇ · L p = − ( J p × B p ) · m and ∇ · S p = − E · J p (see Section 1 and TT14).As a result, we see that the BH loses its rotational energy directly by S p along the fieldlines threading the horizon, as described in Figure 3.The plasma may have a finite resistivity in more realistic BH magnetosphere or in thenumerical simulations. In this case a certain force is required to maintain the steady-statecurrents. K04 and K09 suggest that a weak D field component parallel to the B field line isinduced and drives the currents in the steady state without violating the regularity conditionsignificantly. he inner boundary approaching the horizon as seen in the BL coordinates looks similar tothe stretched horizon in the membrane paradigm at first sight. However, they are essentiallydifferent. In the membrane paradigm, H ff ϕ is produced by the fictitious cross-field currentflowing on the stretched horizon with Joule dissipation. On the other hand, we have shownthat H ff ϕ is produced not only by the cross-field current but also by the displacement current.We should note that mechanism of driving the cross-field current on the inner boundarymight be different from that near the equatorial plane which is discussed in Section 3. Inthe latter case, D > B can be realized due to the property of the ergosphere, which drivesthe cross-field current. On the other hand, the mechanism of driving the cross-field currentbetween the force-free and vacuum regions (and its relation to the property of the ergosphere)may not be understood in our toy model, where D · B = 0 could be violated there. Morestudies on the plasma physics as measured by the FIDOs would be required.
6. Conclusion
We have generically discussed the axisymmetric Kerr BH magnetosphere in which a colli-sionless plasma satisfies D · B = 0 (i.e. there is no gap in the plasma region), and clarifiedthe causal production mechanism of the electromagnetic AM and Poynting fluxes (i.e. H ϕ and Ω F ) along the ergospheric field lines crossing the outer light surfaces and the role of thenegative energies as measured in the coordinate basis. Our conclusion is the following.For the field lines threading the equatorial plane, as shown in K04 and TT14, H ϕ isproduced by the cross-field current flowing in the region where D > B near the equatorialplane, and Ω F will be regulated so that the current crossing region is finite. In this paper, wehave shown that the particles in that region can have negative AM and negative energy asmeasured in the coordinate basis by a feedback from the flux production, and shown by usingthe one-fluid approximation that those particles flow towards the horizon (see Figure 3). ThusBZ process for these field lines appears to be a similar process to the mechanical Penroseprocess. We have also compared our arguments to the recent MHD numerical simulationresults briefly in Section 3.4.For the field lines threading the horizon, the structure of the outward electromagnetic AMand Poynting fluxes (or the poloidal currents and the electric potential differences) must notbe created by the horizon, but must be a result from phenomena having occurred outside thehorizon in the prior coordinate times. To illustrate this concept, we have built a toy modelof a time-dependent state in which the force-free plasma injected continuously between thetwo light surfaces is filling a vacuum (see Figure 5). As a result, we have seen that thefluxes are produced by the contributions from the displacement current and the cross-fieldcurrent at the in-going boundary (see equations 93 and 96). In the steady state, the in-going boundary does not affect the force-free region, and the fluxes are maintained withoutany electromagnetic source (if the resistivity is negligible). H ϕ and Ω F are maintained tobe consistent with the regularity condition at the horizon and at infinity. The force-freecondition is satisfied along the field lines threading the horizon in the steady state, andthen conversion of the AM and energy from the particles is negligible. Thus we supportthe mathematical treatments of [1] and [17] for determining H ϕ and Ω F of the steady-stateforce-free plasma.We have shown that the concept of the inflow of negative electromagnetic energy alongthe field lines threading the horizon is not physically essential. The steady outward Poynting ux should be interpreted just as a result of the currents flowing in the plasma with theelectric potential differences. The outward Poynting flux at the horizon in the KS coordinatesdoes not violate causality, because the steady fluxes carry no information. The BH loses itsrotational energy directly by this outward Poynting flux without being mediated by anyinfalling negative-energy objects, as described in Figure 3.Finally, we should emphasize that our analysis is based on several assumptions (see Sec-tions 2.3, 3.3, and 5). Our arguments for the particle motions near the equatorial plane andin the toy model are required to be justified by numerical simulations. As for the electro-magnetic field, the principal assumption is D · B = 0 in the steady state. It is still debatedwhether this condition is satisfied in the steady state at the boundary between the inflowand the outflow (i.e. at r = r i in our toy model) [10, 35, 47, 49–51] and even in the whole BH(or pulsar) magnetosphere [56–58]. This issue is closely related to radiation physics, whichshould be resolved for validating theories on BZ process by observations [7, 8, 49]. Acknowledgment
We thank the referee for his/her useful comments. A part of this paper has already beenpresented by K.T. in the workshop “Relativistic Jets: Creation, Dynamics, and InternalPhysics” held at Krakow, 20-24 April 2015. K.T. thanks its organizers for the wonderfulhospitality and its participants for stimulating him to improve the discussion in this paper.K.T. also thanks T. Harada, S. Koide, Y. Kojima, K. Nakao, and Y. Sekiguchi for usefuldiscussions. This work is partly supported by JSPS Grants-in-Aid for Scientific Research15H05437 and also by JST grant “Building of Consortia for the Development of HumanResources in Science and Technology.”
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A. Convective currents and the force-free condition
The relation between convective current and velocity of particles is summarized as follows.Let us consider the case in which the positively and negatively charged particles have thesame velocity v as measured in the coordinate basis. Generalization to other cases is easy.The local physics as measured by the FIDOs indicatesˆ j = ρ ˆ v , (A1)where we note that ρ = − I µ n µ is a scalar. Each spatial component in respect of the BLFIDO’s orthonormal basis can be rewritten as (TT14) √ γ ϕϕ j ϕ = ρ √ γ ϕϕ α ( v ϕ + β ϕ ) , √ γ rr j r = ρ √ γ rr α v r . (A2)(The form of the θ component is the same as the r component.) Note that j µ = γ µν I ν is afour-vector, while the particle velocity is v i = u i /u t in terms of the four-velocity u µ . Thenwe can write j = 1 α ρ ( v + β ) . (A3)This relation is valid also in the KS coordinates. Equation (13) leads to J = ρ v . (A4) s an example, the velocity of the D × B drift is (TT14)ˆ v d = ˆ D × ˆ B ˆ B , v d = α D × B B − β . (A5)Then we have the drift current as j d = ρ D × B B . (A6)Under the assumptions in Section 2.3.1, the current is generally j = j d + C B for D < B ,where C is a scalar factor. This corresponds to the force-free condition, ρ D + j × B = 0 , (A7)which is equivalent to ρ E + J × B = 0. B. Calculations in the KS coordinates
Here we explain how equations (40), (51), and (52) are derived, and examine the sign of D · E in the KS coordinates. The following identities are useful for such calculations:Σ − r = ∆( ̺ + 2 r ) , (B1)Σ − ( ̺ + 2 r ) a sin θ = ̺ , (B2)Σ( ̺ − r ) + 4 r a sin θ = ̺ ∆ . (B3)From equations (11) and (21), generally one has D = 1 α (cid:2) ( ω + β + 2 ω i β i ) B − ( ω i B i + β i B i ) (cid:3) . (B4)In the KS coordinates, this equation is reduced to( B − D ) α = − B f (Ω F , r, θ ) + (Ω F B ϕ + β r B r ) , (B5)where Ω F B ϕ + β r B r = ( γ rϕ Ω F + β r ) B r + γ ϕϕ (Ω F − Ω) B ϕ . (B6)By using equation (28) and the above identities, we derive equation (40).The electromagnetic AM density is written by using equations (11) and (21) as l = 14 π e ϕjk D j B k = − πα (cid:2) ( ω i + β i ) B i B ϕ − ( ω ϕ + β ϕ ) B i B i ) (cid:3) . (B7)In the KS coordinates, one has4 παl = − ( ω r + β r ) B r B ϕ + ( ω ϕ + β ϕ )( B r B r + B θ B θ ) , (B8)which can be straightforwardly rewritten as equation (51).The electromagnetic energy density is written by using equations (11) and (12) as e = 18 π ( E i D i + B i H i ) = α π ( D + B ) + 14 π D i e ijk β j B k . (B9)Then by using equation (B4), one has8 παe = ( α − β + ω ) B − ( ω i B i ) + ( β i B i ) , (B10)which can be straightforwardly rewritten as equations (52). or the field lines threading the equatorial plane, B − D < B r = 0, H ϕ = 0 [which lead to B ϕ = 0 by equation (45)], and Ω F < Ω (TT14).Let us confirm D · E < D · E = E i D i = − e ijk ω j B k D i = ω ϕ e ϕjk D j B k = Ω F α h γ ϕϕ (Ω F − Ω)( B θ B θ + B r B r ) − ( ω r + β r ) B r B ϕ i . (B11)The conditions B r = 0 and Ω F < Ω lead to D · E <
0. Such a D field drives the poloidalcurrent to flow in the direction of − E ..