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It is argued that a posteriori without conflicting with the dubbed no-hair theorem, a Kerr black hole acquires its own proper magnetosphere in the steady eigen-state. The angular frequency of field lines, |$ {\Omega_{\rm F}}$| , given as the eigenvalue in terms of the hole’s angular frequency, |$ {\Omega_{\rm H}}$| , couples with the frame-dragging angular frequency, |$ \omega$| , to create an inner general-relativistic domain of |$ {\Omega_{\rm H}} \gt \omega$|  |$ \gt {\Omega_{\rm F}}$| , in which the gradient of the electric potential is, when viewed by the fiducial observers (FIDOs), reversed in direction from that in the outer quasi-classical domain of |$ {\Omega_{\rm F}} \gt \omega$|  |$ \gt$| 0. The field lines are pinned down in the plasma source at the interface between the two domains (upper null surface S|$ _{\rm N}$| ), from which pair-particles well up, charge-separated into the ingoing and outgoing winds. The EMFs due to unipolar induction operate to drive the surface currents, following Ohm’s law, on the resistive membranes terminating the force-free domains (say, S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| ), to exert the surface torques onto the respective membranes, thereby extracting angular momentum from the hole and transferring to the domain of astrophysical loads.

1. Introduction

1.1. Pulsar Magnetospheres

It is generally accepted that at the final stage of their evolution massive stars undergo inexorable supernova explosion soon after the heavy iron cores gravitationally collapse at the center, with magnetic fluxes as well as angular momentum conserved. The strongly compressed core will be reborn as a rapidly rotating, strongly magnetized neutron star or a pulsar, and surviving magnetic fluxes will emanate from the star, to form a magnetosphere around it, rotating with the same angular frequency as that of the crust. It was at first believed that the neutron star possesses such a thin thermal atmosphere with a small pressure scale height that it is effectively surrounded by a vacuum. However, if it were in a vacuum, the rapid rotation of the enormously strong magnetic field would have produced a parallel component of electric field along the magnetic field, i.e., (⁠|$ \boldsymbol{E}\cdot\boldsymbol{B}$| ) |$ \propto$|  |$ E_{\parallel}$|  |$ \neq$| 0, so strong that electric charges of both signs would be pulled out of the surface (at least in principle), with subsequent pair cascade until the |$ E_{\parallel}$| component is completely screened, and hence vanishes in the steady state. The neutron star will congenitally possess a magnetosphere filled with dense plasma of electrodynamic origin (Goldreich & Julian 1969).

If the magneto-hydrodynamics (MHD) approximation with a perfect conductivity condition is applicable for the magnetospheric plasma in the steady state, the first degeneracy takes place, i.e., |$ E_{\parallel} =$| 0, so that no linear acceleration occurs in ideal MHD. However, one has instead the isorotation law of field lines pinned down at the stellar surface rotating around the fixed axis, denoted by |$ {\Omega_{\rm F}}$|  |$ =$|  |$ {\Omega_{\rm F}}$| (⁠|$ \Psi$| ) along each field line, where |$ \Psi$| denotes the field-stream function [see equations (3) and (24)]. Then, |$ {\Omega_{\rm F}}$| expresses not only the field-line angular frequency, leading to a magnetic slingshot effect (section 3), but also indicates the potential gradient with respect to |$ \Psi$| . It has, in fact, already been pointed out that a rotating, magnetized perfect conductor generates potential differences between the pole and the equator due to unipolar induction (Landau et al. 1984). Then, the poloidal component of the electric field associated with |$ {\Omega_{\rm F}}$| is thus given by
$${\boldsymbol{E}_{\rm p}}=- \frac{{\Omega_{\rm F}}}{2\pi c}\boldsymbol{\nabla}\Psi,$$
(1)
which indicates a potential difference, PD, between any two field lines, |$ \Psi_1$| and |$ \Psi_2$|⁠, given by
$${\rm\rm PD}=-\frac{1}{2\pi c}\int^{\Psi_2}_{\Psi_1} \Omega_{\rm F}(\Psi) d\Psi.$$
(2)

This PD will work as an electromotive force (EMF), if the electric current system is established from the unipolar inductor at the star surface to regions of astrophysical loads, and then will be the power source to carry the spin-down energy of the star, eventually turning to e.g., the kinetic energy of the pulsar wind in the inertial domain (Goldreich & Julian 1969, Okamoto & Sigalo 2006). Note that the PD is invariant along the field lines in the flat space [cf. equation (7) for the curved space].

A simple model to describe such a magnetosphere is the force-free model (e.g., Michel 1973, Okamoto 1974; see section 4). The spin-down energy of the star is conveyed as the Poynting-energy flux in the force-free domain to the “surface-at-infinity” terminating the force-free domain (say, S|$ _{{\rm ff}\infty}$| ), where astrophysical loads are supposed to be present; it is on this “force-free infinity surface,” S|$ _{{\rm ff}\infty}$| , that the EMF works, following Ohm’s law. One may thus make use of the concept of a DC circuit consisting of the EMF, given by equation (2), “current lines” (leads) and the resistive membrane terminating the force-free domain, S|$ _{{\rm ff}\infty}$| (see sections 4, 5, and 9).

1.2. Black-Hole Magnetospheres

After a more violent hypernova explosion, more massive stars will, on the other hand, continue to collapse forever, remaining the central core as a black hole, with its angular momentum conserved, and hence a born-again hole will rapidly be spinning (i.e., a Kerr black hole). One of crucial differences from a neutron star is that the growing horizon [with the membrane resistivity; see equation (9) later] will dissipate away pre-existent magnetic fluxes threading the core into the hole’s irreducible mass or entropy. Therefore, the hole magnetosphere must be constructed secondarily, through absorbing infalling matter containing plenty of magnetic fluxes from the surrounding debris, or disk, and then redistributing it over the “membrane” on the horizon (see e.g., Thorne et al. 1986). One may thus expect that a Kerr black hole will be surrounded by a rotating magnetosphere, extending from the horizon to infinity. Magnetic fluxes cannot, however, be pinned down on the horizon, because the horizon membrane is regarded as being resistive (Znajek 1977); also, there is no fixed material surface for magnetic fluxes to freeze in, and hence there is no unipolar inductor, nor dynamo process at work there.

Some important issues remaining to solve are (i) where these field lines that are supposed to compose the steady-state magnetosphere are pinned down at, and how |$ {\Omega_{\rm F}}$| can be determined in terms of the hole’s spin rate, |$ {\Omega_{\rm H}}$| ; (ii) how and where unipolar induction associated with the potential gradient, |$ {\Omega_{\rm F}}$| , is at work to drive the electric currents sustaining the magnetosphere; and (iii) where and how particles that are supposed to fill the magnetosphere be created and supplied into the force-free domains. It must furthermore be pointed out that the basic properties of force-freeness have not fully been elucidated, which would be helpful to solve these issues (see section 9). These issues are of course mutually closely related, and have been surviving to be clarified or detailed since the pioneering work by Blandford and Znajek (1977). It must be when these issues are resolved that one will succeed to accomplish a perfect marriage of electrodynamics with general relativity.

It has been argued within the framework of force-free degenerate electrodynamics (FFDE) (Okamoto 1992, 2006, 2009) that the |$ {\alpha}\omega$| mechanism combines with FFDE to divide a Kerr hole’s magnetosphere into the outer and inner domains bounded by the resistive membranes S|$ _{{\rm ff}\infty}$| and S|$ _{\rm ffH}$| , with the sources of electric currents being hidden under S|$ _{\rm N}$| between the two domains. Respective DC circuit-like current systems work wherein the Poynting flux flows in each force-free domain, and Joule-dissipates on S|$ _{{\rm ff}\infty}$| and S|$ _{\rm ffH}$| . This situation poses an eigenvalue problem for the two eigenfunctions of the current function |$ I$| ’s and field-line angular frequency, |$ {\Omega_{\rm F}}$| , with respect to the criticality-boundary conditions in the steady state.

It will be shown in this paper that by the |$ {\alpha}\omega$| mechanism, a rotating black hole will eventually acquire its own active magnetosphere, supported by internal electric currents in the steady axisymmetric state. The situation will be somewhat different from the view originally proposed (Blandford & Znajek 1977), that “when a rotating black hole is embedded in magnetic field lines supported by external electric currents in an equatorial disc, an electric potential difference will be induced.”

2. Magnetospheric Quantities in the Kerr Spacetime

The “3 |$ +$| 1” formalism of electrodynamics in general relativity is used throughout this paper (MacDonald & Thorne 1982, Thorne et al. 1986). Then, denoting the current function by |$ I$| (⁠|$ {\varpi}$| , |$ z$| ), as opposed to the stream function, |$ \Psi$| (⁠|$ {\varpi}$| , |$ z$| ), in the Boyer-Lindquist coordinates, the poloidal and toroidal components of the magnetic field, |$ \boldsymbol{B}$| , are given by
$${\boldsymbol{B}_{\rm p}}=- \frac{1}{2\pi{\varpi}} (\boldsymbol{t}\times\boldsymbol{\nabla}\Psi) , \quad B_{\rm t}=- \frac{2I}{\varpi \alpha c},$$
(3)
where |$ {\varpi}$| is the axial distance, |$ \boldsymbol{t}$| is the unit toroidal vector, and |$ {\alpha}$| is the lapse function, or red-shift factor. A line of |$ I=$| constant stands for the current line along which a poloidal electric current flows. In FFDE, next to the first one in ideal MHD with |$ E_{\parallel} =$| 0, the second degeneracy takes place, i.e., |$ I$|  |$ =$|  |$ I$| (⁠|$ \Psi$| ), and hence
$${\alpha}{\boldsymbol{j}_{\rm p}}=\frac{1}{2\pi \varpi} (\boldsymbol{t}\times\boldsymbol{\nabla} I) =-\frac{dI}{ d\Psi} {\boldsymbol{B}_{\rm p}}.$$
(4)

Therefore, there is no cross-field current appearing in the force-free domains, i.e., |$ j_{\perp} =$| 0, which means that no MHD acceleration takes place there.

The FFDE description of a pulsar or black-hole magnetosphere in the steady axisymmetric state would not be completed, unless two integral functions, |$ I$| and |$ {\Omega_{\rm F}}$| , are specified in some way or other. Even in the force-free magnetosphere the inertial effects of plasma particles must be taken into account somehow and somewhere to determine |$ B_{\rm t}$| or |$ I$| . Indeed, just as in MHD, |$ B_{\rm t}$|  |$ =$|  |$-$| (2|$ I/{\alpha} {\varpi} c$| ) is interpreted as indicating the swept-back component of |$ {\boldsymbol{B}_{\rm p}}$| due to inertial effects related to e.g., astrophysical loads, and |$ I$|  |$ =$|  |$-$| (⁠|$ {\alpha}{\varpi} cB_{\rm t}/$| 2) is here determined by the criticality condition, given later [see equations (28) and (29)].

One needs to impose no net gain or loss of charges as one of global conditions in the steady state, which requires from equation (4) that
$$\begin{eqnarray} & \oint {\alpha}{\boldsymbol{j}_{\rm p}}\cdot d\!\boldsymbol{A}=- 2\pi \int_0^{\bar{\Psi}} \frac{dI}{d\Psi} d\Psi= 0, \nonumber \\ & {\rm\rm i.e.,} \quad I(0)=I(\bar{\Psi})=0, \end{eqnarray}$$
(5)
where |$ d\!\boldsymbol{A}$| is the surface element, and |$ \Psi =$| 0 and |$ = \bar{\Psi}$| are the first and last open field lines, assuming there is no line current there. This in turn requires the current-closure condition to hold globally, which needs a cross-field surface current on some terminating surfaces of the force-free domains, by breaking down the perfect conductivity condition [i.e., on S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| with the surface resistivity in (9); see section 5.
From the induction equation in MHD (see e.g., Landau et al. 1984), the perfect conductivity condition allows one to define the electric field related to |$ {\boldsymbol{B}_{\rm p}}$|⁠,
$${\boldsymbol{E}_{\rm p}}=- \frac{\Omega_{\rm F}-\omega}{2\pi{\alpha} c} \boldsymbol{\nabla} \Psi,$$
(6)
which is the general-relativistic version of the flat space expression in equation (1). Here, |$ \omega$| is the frame-dragging angular frequency. In this class of problem, the general-relativistic effects are condensed into two parameters, |$ {\alpha}$| and |$ \omega$| . Equation (6) for |$ {\boldsymbol{E}_{\rm p}}$| in curved space embodies how electrodynamics couples with general relativity or, more specifically, how dragging of the inertial frames affects the potential differences between the field lines. That is, just as equation (1) is so, equation (6) is a direct consequence from the fact that |$ {\Omega_{\rm F}}$| is the gradient of the electric potential with respect to the field-stream function, |$ \Psi$| , but modified by the |$ {\alpha}\omega$| effects. As usual, in ideal MHD, |$ ({\boldsymbol{E}_{\rm p}}\cdot{\boldsymbol{B}_{\rm p}}) =$| 0 from equations (3) and (6), which presumes that the parallel component of the electric field is fully screened, i.e., |$ E_{\parallel} =$| 0, by plenty of charged particles supplied in some way or another (see sections 8 and 9).
One can also formally define the potential difference, PD, between two field lines, |$ \Psi_1$| and |$ \Psi_2$| , on an arbitrary surface, |$ S$| , which each field line threads vertically, i.e., from equation (6),
$$\begin{equation} {\rm\rm PD}|_{S(\ell)}=-\frac{1}{2\pi c}\int^{\Psi_2}_{\Psi_1} [\Omega_{\rm F}(\Psi)-\omega(\ell;\Psi)] d\Psi, \end{equation}$$
(7)
where |$ \ell$| measures the distances outwardly along each field line. One may also regard |$ {\alpha}$| or |$ \omega$| as a continuous, monotonically increasing or decreasing function of |$ \ell$| for each field line. In particular, |$ \omega$|  |$ =$|  |$ \omega$| (⁠|$ \ell; \Psi)$| decreases from |$ {\Omega_{\rm H}}$| at |$ \ell$|  |$ =$|  |$ \ell_{\rm H}$| to 0 for |$ \ell \to \infty$| . For |$ 0\lt{\Omega_{\rm F}}\lt{\Omega_{\rm H}}$| , the FIDOs residing on the inertial frames always see that there is such a surface at |$ \ell$|  |$ =$|  |$ \ell_{\rm N}$| above the horizon (the upper null surface S|$ _{\rm N}$| ), yet fixed if |$ {\Omega_{\rm F}}$| is fixed in an eigen-state (see section 5) that satisfies |$ \omega$| (⁠|$ \ell;\Psi$| ) |$ =$|  |$ {\Omega_{\rm F}}$| (⁠|$ \Psi$| ), and hence |$ {\boldsymbol{E}_{\rm p}}$|  |$ =$| PD|$ \vert _{S(\ell_{\rm N})}$|  |$ =$| 0 in equations (6) and (7) (Okamoto 1992).

That is, the PD|$ \vert _{S(\ell)}$| depends on |$ \ell$| along field lines through |$ \omega = \omega$| (⁠|$ \ell$| ; |$ \Psi$| ), changing sign at S|$ _{\rm N}$| . This is crucially different from the pulsar case in equation (2). This potential difference does not drive any volume current across field lines on a non-resistive surface, |$ S$| (⁠|$ \ell$| ), in the force-free domains. It is on the resistive membranes only that the PD|$ \vert _S$| work as a kind of unipolar inductors to drive surface currents.

This S|$ _{\rm N}$| is nothing but the interface that electrodynamically divides the magnetosphere into the two domains; one is the outer quasi-classical (pulsar-type) domain with |$ \omega \lt {\Omega_{\rm F}}$| , and the other is the inner, general-relativistic domain with |$ \omega \gt {\Omega_{\rm F}}$| [the effective ergosphere as opposed to the ordinary ergosphere (Okamoto 1992)]. The key role of the frame dragging is thus to reverse, as viewed from FIDOs living in the inertial frame, the gradient of the electric potential, |$ {\Omega_{\rm F}}$| , to create an inner general-relativistic domain in an electrodynamic way so that the ingoing wind takes place in form of |$ \boldsymbol{v} = \boldsymbol{j}/\varrho_{\rm e}$|  |$ \lt$| 0 toward the horizon. The Poynting electromagnetic energy flux as well is reversed; that is, the Poynting flux, calculated from equations (3) and (6), i.e.,
$$\begin{equation} \boldsymbol{S}_{\rm EM}=\frac{\alpha c}{4\pi} (\boldsymbol{E}\times\boldsymbol{B})_{\rm p} =\frac{({\Omega_{\rm F}}-\omega)I(\Psi)}{2\pi{\alpha} c} {\boldsymbol{B}_{\rm p}} , \end{equation}$$
(8)
indeed changes sign at S|$ _{\rm N}$| ; |$ \boldsymbol{S}_{\rm EM}$|  |$ \gt$| 0 in the outer domain, where |$ \boldsymbol{v} = \boldsymbol{j}/\varrho_{\rm e} \gt$| 0, while |$ \boldsymbol{S}_{\rm EM}$|  |$ \lt$| 0 in the inner domain (see section 7). This means that there must be some current source as well as the particle source present hidden under S|$ _{\rm N}$| in FFDE.
Flow acceleration is in some sense completely prohibited in the force-free domains, because |$ E_{\parallel}$|  |$ =$|  |$ j_{\perp}$|  |$ =$| 0 in FFDE, and massless particles cannot be accelerated by any force. Any force-free magnetosphere is, however, not a closed physical system, unless it is terminated somewhere, but not at finite distances, but at near to the horizon and infinity surfaces, denoted rather vaguely by S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| , which play a role of substitutes of the inertial domains where force-freeness breaks down to allow a cross-field current to flow, and at which transfer of the Poynting to kinetic energy fluxes starts to take place (see section 9). The process of transfer on S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| is formally replaced with “Joule dissipation” of the surface currents, due to the surface resistivity (Znajek 1977),
$$\begin{equation} {\cal R}_{\rm ffH}={\cal R}_{{\rm ff}{\infty}}= \frac{4\pi}{c} = 377\ {\rm Ohm}. \end{equation}$$
(9)

This ensures that, provided that the EMF due to the PD in equation (7) is available, the surface currents flow on S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| , crossing the field lines threading there, to fulfill the current-closure condition, completing a DC circuit. Formally, Ohm’s law holds on S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| . On the other hand, the criticality condition at the fast magnetosonic surface leads to the determination of |$ B_{\rm t}$| or |$ I$| , corresponding to inertial effects on S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| [see equations (28), (29), (31), and (32) later].

For later convenience, one can introduce a dimensionless quantity useful in MHD as well as in FFDE from equations (3) and (6), i.e.,
$$\begin{equation} {\Lambda}(\ell,\Psi) = \frac{E_{\rm p}}{ |B_{\rm t}|}= \frac{E_{\rm p}}{(4\pi/c)(I/2\pi{\varpi})}=\frac{({\Omega_{\rm F}}-\omega) \Phi}{2I}, \end{equation}$$
(10)
where |$ \Phi\equiv B_{\rm p}{\varpi}^2$| . Roughly speaking, |$ {\Lambda}$| denotes a relation between the source and dissipation. In MHD, |$ I$|  |$ =$|  |$ I$| (⁠|$ \ell$| ,|$ \Psi$| ) and |$ \Phi$|  |$ =$|  |$ \Phi$| (⁠|$ \ell$| ,|$ \Psi$| ), and for |$ \ell \to \infty$| , e.g., |$ \omega \to$| 0, |$ I$|  |$ \to$| 0 and |$ \Phi \to$| 0, while |$ {\Lambda}\to \sqrt{1-(1/{\gamma_{\infty}}^2)} \approx$| 1 (see subsection 9.1). In FFDE, |$ I=I(\Psi)$| , and for |$ \ell \to \infty$| , |$ {\Lambda_{{\rm ff}\infty}} =$|  |$ +$| 1, while for |$ \ell\to \ell_{\rm H}$| , |$ {\Lambda_{{\rm ffH}}} =$|  |$-$| 1 [see equations (9) and (28)–(32)].

3. The Velocity of Plasma Particles in FFDE

Strong gravity by a black hole appears through the spacetime metric; mainly, |$ {\alpha}$| and |$ \omega$| in FFDE, because the particle mass is too small (by assumption) to feel gravitational and other forces. Thus, the particle velocity, |$ \boldsymbol{v}$| , is not determined dynamically, but must be done electrodynamically, i.e., from the current density divided by the charge density, |$ \boldsymbol{v} = \boldsymbol{j}/\varrho_{\rm e}$| [see equation (26) or (27) for |$ \varrho_{\rm e}$| ]. One restriction on |$ \boldsymbol{v}$| thus determined in FFDE is |$ \vert\boldsymbol{v}\vert \lt c$| , and the condition for inertial-free particles to restore feeling their inertia is |$ \vert\boldsymbol{v}\vert \to c$| or |$ {\gamma} \to \infty$| (the “plasma condition”), but this takes place not at finite distances in the force-free domains, but toward the membranes S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| (Okamoto & Sigalo 2006; Okamoto 2006).

Then, a physical role of massless particles in the force-free domain is just to carry electric charges, but the velocity given by |$ \boldsymbol{v} = \boldsymbol{j}/\varrho_{\rm e}$| is still regulated by indicating the presence of a strong magnetic field, to follow the following relations from the ideal MHD condition:
$$\begin{eqnarray} {\boldsymbol{v}_{\rm p}} & = {\kappa} {\boldsymbol{B}_{\rm p}}, \quad v_{\rm t}={\kappa}B_{\rm t}+ v_{\rm F},\end{eqnarray}$$
(11)
 
$$\begin{eqnarray} v_{\rm F}& \equiv & ({\Omega_{\rm F}}-\omega){\varpi}/{\alpha} ,\end{eqnarray}$$
(12)

(Mestel 1961), where |$ v_{\rm F}$| is the rotational velocity of each field line measured by the FIDOs in the inertial frames. Note that |$ v_{\rm F} = \pm c$| defines the outer and inner light surfaces, S|$ _{\rm oL}$| and S|$ _{\rm iL}$| , and |$ v_{\rm F} \to \pm\infty$| for |$ \ell \to \infty$| or |$ \ell_{\rm H}$| . It is then shown here that it is by terminating the force-free domain for massless particles to restore inertia (“plasma condition”) (see subsections 2.4 and 3.4 in Okamoto & Sigalo 2006; also subsection 9.1).

The toroidal velocity becomes in terms of |$ v_{\rm p}/c$| and |$ {\Lambda}$|  
$$\begin{equation} v_{\rm t}=v_{\rm F}\left(1-\frac{1}{\Lambda}\frac{v_{\rm p}}{c}\right), \quad {\rm or} \quad \Lambda=\frac{v_{\rm F}}{v_{\rm F}-v_{\rm t}}\frac{v_{\rm p}}{c} \end{equation}$$
(13)
with |$ {\Lambda}$| given in equation (10). Inserting this into an equality,
$$\begin{equation} \frac{v_{\rm p}^2}{c^2}+\frac{v_{\rm t}^2}{c^2} =\frac{v^2}{c^2} =1-\frac{1}{{\gamma}^2}, \end{equation}$$
(14)
one has
$$\begin{equation} \left(1+\Lambda^2 \frac{c^2}{v_{\rm F}^2} \right) \frac{v_{\rm p}^2} {c^2}-2{\Lambda} \frac{v_{\rm p}}{c} +\Lambda^2 \left(1-\frac{v^2}{v_{\rm F}^2} \right)=0, \end{equation}$$
(15)
which shows that there are the two solutions for a given |$ v$| or |$ {\gamma}$| . Then, solving for |$ v_{\rm p}/c$| in the ranges of |$ v_{\rm F}^2 \gt c^2$|⁠,
$$\frac{v_{\rm p}}{c}={\Lambda} \displaystyle {\frac{1 \mp (c/v_{\rm F}) \sqrt{ D} }{1+\Lambda^2 (c^2/v_{\rm F}^2) }},$$
(16)
 
$$\frac{v_{\rm t}}{c}= \displaystyle {\frac{ \Lambda^2(c/v_{\rm F}) \pm \sqrt{D}}{1+\Lambda^2 (c^2/v_{\rm F}^2) }},$$
(17)
 
$$D \equiv \frac{v^2}{c^2}- \Lambda^2 \left(1-\frac{v^2}{v_{\rm F}^2} \right),$$
(18)
where |$ \mp$| signs express the so-called physical and unphysical solutions (see e.g., Okamoto 1974, 1978). Obviously, |$ D$| must be positive, and also |$ v_{\rm t} \gt$| 0 in equation (13), so that one has some important constraints for |$ {\Lambda}$|⁠,
$$\begin{equation} \frac{v_{\rm p}}{c} \lesssim {\Lambda} \lesssim \displaystyle {\frac{\displaystyle {\frac{v}{c}}}{\sqrt{1-\displaystyle {\frac{v^2}{v_{\rm F}^2}}}}} , \end{equation}$$
(19)
including the signs of |$ v_{\rm p}$| , |$ {\Lambda}$| , |$ v$| and |$ v_{\rm F}$| . In the far regions of |$ v_{\rm F}^2 \gg c^2$| from S|$ _{\rm N}$| where |$ v_{\rm F} =$| 0 in equation (12), one has
$$\begin{eqnarray} & \frac{v_{\rm p}}{c}\approx {\Lambda} \left[1 \mp (c/v_{\rm F}) \sqrt{ D}\right], \quad \frac{v_{\rm t}}{c}\approx \Lambda^2(c/v_{\rm F}) \pm \sqrt{D} , \nonumber \\ & D \approx \frac{v^2}{c^2}- \Lambda^2 =1-\frac{1}{{\gamma}^2}-\Lambda^2. \end{eqnarray}$$
(20)
It is when |$ {\gamma} \to \infty$| , or |$ v\to c$| toward S|$ _{{\rm ff}\infty}$| or S|$ _{\rm ffH}$| that particles recover their inertia lost in the force-free domain. One can also apply criticality theory for wind theory to the present case, that is, requiring the two solutions in (16) and (17) to cross each other like an X, and hence demanding the square-root terms to vanish, i.e., |$ D \to$| 0 in equation (20), one has for |$ \ell \to \infty$| and |$ \ell \to \ell_{\rm H}$| (i.e., |$ v_{\rm F} \to \pm\infty$| )
$$\begin{eqnarray} & |v_{\rm p}| \approx |v| \approx c, \quad v_{\rm t}\approx 0, \quad {\gamma} \to \infty, \quad \nonumber \\ & {\Lambda_{{\rm ff}\infty}}=1, \quad {\Lambda_{{\rm ffH}}}=-1, \end{eqnarray}$$
(21)
which leads to the determination of the first eigenvalues of |$ I$| ’s in terms of |$ {\Omega_{\rm F}}$| ; we then equate the two |$ I$| ’s at S|$ _{\rm N}$| , to the second one for |$ {\Omega_{\rm F}}$| (see section 5). The eigenvalues |$ I$| ’s in the eigenstate mean fixing the swept-back component, |$ B_{\rm t}$| , by inertial (i.e., astrophysical) loads upon the poloidal component, |$ {\boldsymbol{B}_{\rm p}}$| ; thereby, |$ v_{\rm t}$|  |$ \to$| 0 and |$ v_{\rm p}$|  |$ \to$|  |$ c$| in equation (11) (see subsection 9.1).
Related to the “plasma condition” mentioned above, it is in passing worthwhile to consider the “radiative boundary condition” (see e.g., Thorne et al. 1986). Using equations (3), (6), and (10), one has
$$\begin{eqnarray} 0 \leq B^2- E^2=B_{\rm t}^2\left[\! 1-\left(\!1-\frac{c^2}{v_{\rm F}^2}\!\right) \Lambda^2 \right] \approx B_{\rm t}^2(1-\Lambda^2 ) \end{eqnarray}$$
(22)
in the region of |$ v_{\rm F}^2 \gg c^2$| , and hence it is confirmed by the |$ B^2$|  |$-$|  |$ E^2$|  |$ \to$| 0 that |$ {\Lambda_{{\rm ff}\infty}} =$| 1 for |$ \ell \to \infty$| or |$ {\Lambda_{{\rm ffH}}} =$|  |$-$| 1 for |$ \ell \to \ell_{\rm H}$|⁠.

4. The Force-Free Field Structure

It will be helpful to briefly look back on how the force-free magnetospheric theory has been constructed for such compact stars as neutron stars and black holes, with similarity and dissimilarity between them taken into account (Okamoto 1992, 2006, 2009). The dissimilarity naturally comes from general-relativistic effects due to the |$ {\alpha}\omega$| mechanism.

The FFDE condition indicates that the electromagnetic energy be so large that the particle energy including the rest-mass energy is negligible, or equivalently the Lorenz force dominates the inertial forces in the momentum equation, i.e.,
$$\begin{equation} \rho_{\rm e}\boldsymbol{E} +(\boldsymbol{j}/c)\times \boldsymbol{B}= {\rm Inertial forces} \approx 0, \end{equation}$$
(23)
on the presumption of the existence of charged particles (though massless) plentiful enough to carry electric currents, and thereby to sustain the magnetosphere.
The dubbed stream equation for the force-free pulsar magnetosphere is given by the transfield component of the special-relativistic version of equation (23), i.e.,
$$\begin{eqnarray} & \boldsymbol{\nabla}\cdot\left[\frac{1}{{\varpi}^2}\boldsymbol{\nabla}\Psi\right] \nonumber \\ & = \boldsymbol{\nabla}\cdot\left[\frac{{\Omega_{\rm F}}^2}{c^2} \boldsymbol{\nabla}\Psi\right]- \frac{{\Omega_{\rm F}}}{c^2}\frac{{\Omega_{\rm F}}}{\Psi} |\boldsymbol{\nabla}\Psi|^2-\frac{8\pi^2}{{\varpi}^2 c^2}\frac{I^2}{\Psi}. \end{eqnarray}$$
(24)

It can easily be seen that a split-monopolar field, |$ \Psi =$|  |$ \Psi_0$| (1 |$-$| cos |$ {\theta}$| ) with |$ \Phi =$| (⁠|$ \Psi_0/2\pi$| )sin|$ ^2{\theta}$| , satisfies equation (24) for |$ I = \frac12 {\Omega_{\rm F}}\Phi_{{\rm ff}\infty}$|  |$ =$| (⁠|$ {\Omega_{\rm F}}\Psi_0/$| 4|$ \pi$| )sin|$ ^2{\theta}$| as the criticality condition at S|$ _{{\rm ff}\infty}$| and |$ {\Omega_{\rm F}} =$| constant as the boundary condition (Michel 1973, Okamoto 1974, 2006). Then, |$ \Lambda =$|  |$ v_r/c$|  |$ =$| 1 everywhere according to equation (10). The poloidal electric current driven by the unipolar inductor expressed in equation (2) flows along the current line given by |$ I$| (⁠|$ \Psi_1$| ) |$ =$|  |$ I$| (⁠|$ \Psi_2$| ) |$ =$| constant (say |$ = I_{\overline{12}}$| ) in the force-free domain, and then crosses the field lines between |$ \Psi_1\lt\Psi\lt\Psi_2$| in the form of a surface current |$ {\cal I}_{{\rm ff}\infty}$|  |$ =$|  |$ I_{\overline{12}}/$| 2|$ \pi{\varpi}\vert _{{\rm ff}\infty}$| on the resistive membrane, S|$ _{{\rm ff}\infty}$| , Joule-dissipating there in accordance with Ohm’s law [see equations (29) and (32)]. Thus, the force-free magnetosphere may be thought of as the piling-up of an infinite number of DC circuits.

The general-relativistic version of equation (23) in the transfield direction yields
$$\begin{eqnarray} \boldsymbol{\nabla}\cdot\left[\frac{{\alpha}}{{\varpi}^2}\boldsymbol{\nabla}\Psi\right] &=& \boldsymbol{\nabla}\cdot\left[\frac{({\Omega_{\rm F}}-\omega)^2}{{\alpha} c^2}\boldsymbol{\nabla}\Psi\right] \quad\quad \nonumber \\\quad\quad\quad\quad &- \frac{({\Omega_{\rm F}}-\omega)}{{\alpha} c^2}\frac{{\Omega_{\rm F}}}{\Psi} |\boldsymbol{\nabla}\Psi|^2-\frac{8\pi^2}{{\alpha}{\varpi}^2 c^2}\frac{I^2}{\Psi} \end{eqnarray}$$
(25)
(MacDonald & Thorne 1982, Thorne et al. 1986), which of course reduces for |$ {\alpha}\to 1$| and |$ \omega$|  |$ \to$| 0 to equation (24). Conversely, incorporating the |$ {\alpha}\omega$| mechanism into the flat-spacetime equation (24) leads to the equation describing the outer quasi-classical domain and the inner general-relativistic domain for a force-free magnetosphere of a rotating black hole (Okamoto 1992, 2006, 2009).
The charge distribution is given in the same way as the stream equation (25), as follows:
$$\begin{eqnarray} 8\pi^2 c\varrho_{\rm e}&=& \displaystyle {\frac{ \displaystyle {\frac{{\varpi}({\Omega_{\rm F}}-\omega)}{{\alpha} c}}}{1-\left[ \displaystyle {\frac{{\varpi}({\Omega_{\rm F}}-\omega)}{{\alpha} c}} \right]^2 }} \cdot \left[\frac{8\pi^2}{{\alpha}^2 {\varpi} c}\frac{I^2}{\Psi} \right. \quad \nonumber \\ & + \displaystyle {\frac{{\varpi}({\Omega_{\rm F}}-\omega)}{{\alpha}^2 c}} \boldsymbol{\nabla}\Psi\cdot\boldsymbol{\nabla}\omega \nonumber\\ & \left.-\frac{c}{{\varpi}}\boldsymbol{\nabla}\Psi\cdot\boldsymbol{\nabla} {\rm ln} \displaystyle {\frac{{\varpi}^2 ({\Omega_{\rm F}}-\omega)}{{\alpha}^2 c}} \right] \end{eqnarray}$$
(26)
(see Okamoto 2009 for derivation). On the null surface, S|$ _{\rm N}$| , where |$ \omega$|  |$ =$|  |$ {\Omega_{\rm F}}$| , and hence |$ {\boldsymbol{E}_{\rm p}}$|  |$ =$|  |$ \boldsymbol{S}_{\rm EM}$|  |$ =$| 0, |$ \varrho_{\rm e}$| may not vanish. It is now not clear whether or not possible disagreement of the surface of |$ \varrho_{\rm e} =$| 0 with S|$ _{\rm N}$| in FFDE poses any significant constraint on processes of pair-creation, and subsequent charge-separation within the gap hidden under S|$ _{\rm N}$| (see section 9). One can obtain the charge density for a force-free pulsar magnetosphere,
$$ \begin{eqnarray} && 8\pi^2 c\varrho_{\rm e}= \displaystyle {\frac{ {{\varpi}{\Omega_{\rm F}}/c}}{1 \!-\! ({\varpi}{\Omega_{\rm F}}/ c)^2 }} \!\left(\frac{8\pi^2}{{\varpi} c}\frac{I^2}{\Psi} - \frac{c}{{\varpi}}\boldsymbol{\nabla}\Psi\cdot\boldsymbol{\nabla} {\rm ln} \displaystyle {\frac{{\varpi}^2 {\Omega_{\rm F}}}{ c}} \!\right) , \end{eqnarray} $$
(27)
by putting |$ {\alpha}\to 1$| and |$ \omega \to$| 0 (Okamoto 1974).

Just as equation (24) describes the steady-state structure of a force-free pulsar magnetosphere, equation (25) describes the steady-state force-free magnetosphere of a black hole, yet double-structured, with the interface, S|$ _{\rm N}$| , at |$ \omega={\Omega_{\rm F}}$| , and the two, inner and outer, light surfaces S|$ _{\rm iL}$| and S|$ _{\rm oL}$| . The primary energy source feeding the pulsar magnetosphere is the spin-down energy of a neutron star, provided in the form of the Poynting energy, through the unipolar inductor expressed in equation (2). Quite likewise, one must find any EMF(’s) of driving the electric current maintaining the steady state of a black-hole magnetosphere. Note that the magnetic field lines in the steady state will not be supported by external electric currents (if any) in an equatorial disc, but the internal electric currents due to unipolar induction.

It will then be reasonable to presume that the physical roles of |$ {\Omega_{\rm F}}$| and |$ I$| appearing in equation (25) must be the same as in equation (24). In particular, |$ {\Omega_{\rm F}}$| denotes not only the field-line angular frequency of each field line, but also the potential gradient with respect to |$ \Psi$| . Then, if |$ {\Omega_{\rm F}}$| is specifiable in terms of |$ {\Omega_{\rm H}}$| somewhere, this must indicate the location of field lines pinning down as well as the existence of unipolar induction at work there, as can be seen in the pulsar case. Also, |$ I$| ’s in the inner and outer domains define the current lines along which the poloidal electric currents flow in the force-free domains at finite distances, and the surface currents on the resistive membranes, S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| , in the DC circuits. Thus, equation (25) describes such a steady-state field structure that can be maintained in the balance of the source coming from |$ {\Omega_{\rm F}}$| and the dissipation expressed by |$ I$| ’s. Therefore, the magnetosphere must not be external, but internal to a rotating black hole in the sense that the rotating black hole born probably immersed initially in the external magnetic fields has now acquired its own internal magnetosphere as an inseparable partner. The magnetic field lines pass smoothly throughout the magnetosphere, rotating with |$ v_{\rm F}$|  |$ =$| [(⁠|$ {\Omega_{\rm F}} - \omega$| )|$ {\varpi}]/{\alpha}$| ; |$ v_{\rm F} = \mp c$| at S|$ _{\rm iL}$| or S|$ _{\rm oL}$| , and |$ v_{\rm F}\to \mp\infty$| toward the inner or outer magnetosonic surface S|$ _{\rm iF}$| or S|$ _{\rm oF}$| [see equations (28) and (29)].

It may thus be interpreted that the superluminal rotation of field lines beyond S|$ _{\rm iL}$| and S|$ _{\rm oL}$| helps to maintain causal contact between S|$ _{\rm iF}$|  |$ \approx$| S|$ _{\rm ffH}$| and S|$ _{\rm oF}$|  |$ \approx$| S|$ _{{\rm ff}\infty}$| , in the steady eigen-magnetosphere, in which one will be allowed to impose the boundary condition at S|$ _{\rm N}$| , to connect the two |$ I$| ’s obtained from the criticality condition on S|$ _{\rm iF}$|  |$ \approx$| S|$ _{\rm ffH}$| and S|$ _{\rm oF}$|  |$ \approx$| S|$ _{{\rm ff}\infty}$| and to finally fix |$ {\Omega_{\rm F}}$| (see section 5).

5. The Eigen-Functions

The force-free domains of which the field structure for a black hole is described by equation (25) are just the domains through which the angular momentum and energy fluxes pass smoothly, but with no source nor dissipation within. Thus, the domains by themselves alone cannot be such a physically closed system as a DC circuit. This is because equation (25) contains such unknown integral functions of |$ \Psi$| , |$ {\Omega_{\rm F}}$| and |$ I$| ’s, that cannot be specified within the force-free domains. This means that the force-free domains for a black hole must be terminated by some surfaces on which |$ {\Omega_{\rm F}}$| and |$ I$| ’s are specifiable as the eigenvalues due to the criticality-boundary conditions.

For a pulsar, the “boundary condition” yields |$ {\Omega_{\rm F}} = \Omega_0$| at the stellar surface, S|$ _0$| , where field lines are regarded as anchored at, as already stated in subsection 1.1. Then, the “criticality condition” at S|$ _{{\rm ff}\infty}$|  |$ \approx$| S|$ _{\rm F}$| determines |$ I = \frac12 {\Omega_{\rm F}}$| (⁠|$ B_{\rm p}{\varpi}^2$| )|$ _{{\rm ff}\infty}$| , which may be regarded as being almost the same as that for the outgoing wind from a black hole [see equation (29)]. A force-free pulsar magnetosphere through which the massless (by definition) wind flows outwardly is enclosed by the two bounding surfaces, S|$ _0$| and S|$ _{{\rm ff}\infty}$| . Thus, the single-eigenvalue problem is the case for a force-free pulsar magnetosphere.

On the other hand, for a force-free black-hole magnetosphere, the double-eigenvalue problem due to the criticality-boundary condition is the case. That is, similarly and also dissimilarly to the pulsar case, a force-free black hole magnetosphere must be not only terminated by the two membrane surfaces, S|$ _{\rm ffH}$| at |$ \omega\approx{\Omega_{\rm H}}$| in the inner side and S|$ _{{\rm ff}\infty}$| at |$ {\omega}$|  |$ \approx$| 0 in the outer side, but also divided by S|$ _{\rm N}$| at |$ \omega={\Omega_{\rm F}}$| into the two domains. The massless (by definition) ingoing wind flows from S|$ _{\rm N}$| toward S|$ _{\rm ffH}$| in the inner domain, while the massless outgoing wind flows from S|$ _{\rm N}$| outward to S|$ _{{\rm ff}\infty}$| in the outer domain. Then, at first applying the criticality condition for massless ingoing and outgoing winds at the inner and outer magnetosonic surfaces S|$ _{\rm iF}$| and S|$ _{\rm oF}$| , which are almost the same as S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| in FFDE, one has
$$I_{\rm in} = (1/2) ({\Omega_{\rm H}}-{\Omega_{\rm F}}) (B_{\rm p}{\varpi}^2)_{\rm ffH} \ {\rm ;\quad S_{\rm IF}},$$
(28)
 
$$I_{\rm out}= (1/2) {\Omega_{\rm F}} (B_{\rm p}{\varpi}^2)_{\rm ff{\infty}},$$
(29)
as the first eigen-functions (Okamoto 1992, 2006, 2009). These current functions must satisfy condition (5). |$ I_{\rm out}$| is effectively the same as that for a force-free pulsar wind, and in reality gives a current function corresponding to the EMF given in equation (2) [see equation (79)].
The surface currents on S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| , defined by |$ {\alpha}{\cal I}$|  |$ =$|  |$ I/$| 2|$ \pi{\varpi}$| (⁠|$ I = I_{\rm in}$| , |$ I_{\rm out}$| ) crossing field lines in the range of |$ \Psi_1\lt\Psi\lt\Psi_2$| , is driven by the electric field on S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| from equation (6):
$$ \begin{equation} {\alpha} {\boldsymbol{E}_{\rm p}}= \left\{ \begin{array} {ll} + [({\Omega_{\rm H}} -{\Omega_{\rm F}}) /2\pi c] \boldsymbol{\nabla}\Psi & ; \ {\rm\rm S$_{\rm ffH}$}, \\ -({\Omega_{\rm F}} /2\pi c) \boldsymbol{\nabla}\Psi & ; \ {\rm\rm S$_{{\rm ff}\infty}$}, % {\rm ;\ \Sffinf} . \end{array} \right. \end{equation} $$
(30)
where |$ {\Omega_{\rm F}}$| will be determined as the eigenvalue self-consistently by the “boundary condition” later [see equation (33) and (34)]. It turns out from equations (28)–(30) that Ohm’s law holds on the resistive membranes, S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| , i.e.,
$$\begin{eqnarray}{\cal I}_{\rm ffH}= \frac{c}{4\pi } \left[ \frac{ ({\Omega_{\rm H}}-{\Omega_{\rm F}... ...arpi}}{c} B_{\rm p}\right]_{\rm ffH}=\frac{E_{\rm ffH}} { {\cal R}_{\rm ffH}} ,\end{eqnarray}$$
(31)
 
$$\begin{eqnarray} {\cal I}_{{\rm ff} \infty} & = & \frac{c}{4\pi} \left(\frac{{\Omega_{\rm F}}{\varpi}}{c}B_{\rm p} \right)_{{\rm ff} \infty} =\frac{E_{{\rm ff}\infty}} { {\cal R}_{{\rm ff}{\infty}}} ,\end{eqnarray}$$
(32)
and |$ {\Lambda_{{\rm ff}\infty}} =$| 1, |$ {\Lambda_{{\rm ffH}}}$|  |$ =$|  |$-$| 1 from equations (21). It is these surface currents satisfying the global current-closure condition that play a crucial role in extracting angular momentum from the hole to the inner force-free domain, and transferring it from the outer force-free domain to the inertial domain of astrophysical loads beyond, by the surface torques on S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| , respectively (see section 7).
As the “boundary condition” at the interface |$ \ell = \ell_{\rm N}$| between the two domains, one may demand continuity of the electric current as well as that of the energy and angular-momentum fluxes, i.e., |$ I_{\rm in} = I_{\rm out}$| , which yields for the second eigenfunction, |$ {\Omega_{\rm F}}$| , and the final eigenfunction, |$ I$|⁠, from equations (28) and (29)
$${\Omega_{\rm F}}(\Psi)=\frac{ {\Omega_{\rm H}}}{1+\zeta}, \quad ... ...ac{{\Omega_{\rm H}}}{2}\frac{ \zeta}{1+\zeta}(B_{\rm p}{\varpi}^2)_{{\rm ffH}},$$
(33)
 
$$\begin{eqnarray}\zeta(\Psi) \equiv \frac{ (B_{\rm p}{\varpi}^2)_{{\rm ff}{\infty}}}{(B_{\rm p}{\varpi}^2)_{\rm ffH} }=\frac{{\Omega_{\rm H}}}{{\Omega_{\rm F}}} - 1.\end{eqnarray}$$
(34)

One has thus solved the double eigenvalue problem due to the “criticality-boundary conditions,” to determine the “current eigen-function” |$ I=I_{\rm in}=I_{\rm out}$| for each domain, and the “field line angular frequency eigen-function” |$ {\Omega_{\rm F}}$| for each magnetic flux. This is accomplished by relating the two criticality conditions on the two membranes S|$ _{\rm ffH}$|  |$ \approx$| S|$ _{\rm iF}$| and S|$ _{{\rm ff}\infty}$|  |$ \approx$| S|$ _{\rm oF}$| connected by magnetic field lines, without violating the principle of causality

6. The Slow-Rotation Limit

As shown above, the force-free approximation allows one to formally decouple the criticality-boundary value problem for |$ I$| and |$ {\Omega_{\rm F}}$| from solving a notoriously nonlinear second-order elliptic-hyperbolic partial differential equation (25) containing |$ I$| (⁠|$ \Psi$| ) and |$ \Psi(\Psi)$| , for which one must use the eigenfunction given in equations (33) and (34). Only one known analytic solution of equation (25) was obtained by using the perturbation method of the slow-rotation limit for a split-monopolar field (Blandford & Znajek 1977, Okamoto 2009), as reproduced succinctly in the following.

By the dubbed no-hair theorem, a Kerr black hole has only two hairs, e.g., the total mass energy, |$ Mc^2$| , and the angular momentum, |$ J$| , or equivalently |$ M$| and the spin parameter, |$ h$| , where |$ h \equiv a/r_{\rm H}$| (⁠|$ r_{\rm H}$| is the horizon radius, and |$ a = J/Mc$| ). All spin-thermodynamic variables are then expressible in terms of |$ M$| and |$ h$| (Okamoto 1992, 2006). Thus, slow-rotation limit is expressed by |$ h^2\ll 1$| , and hence one may use
$$\Psi=\Psi_0 [1-{\rm cos} {\theta} + h^2 f(r) {\rm sin}^2{\theta} {\rm cos} {\theta} ]$$
(35)
for a perturbed split-monopolar structure, and then the stream equation (25) to solve for |$ f$| (⁠|$x$| ) with |$ x=r/r_{\rm H}$| reducing to
$$x(x-1) f”+f'- 6f=-\frac{2}{x}\left(1+\frac{1}{x}\right).$$
(36)
Solving equation (36) for |$ f$| (⁠|$ x$| ) along with the double eigenvalue problem, one has
$$\begin{eqnarray} & f(x) = 8x^3 \left\{\left(1-\frac{3}{4x}\right)\left[I_{\rm A}(x)-{\rm ln}\left(1-\frac{1}{x}\right) {\rm ln} x \right. \right. \quad\quad\quad \nonumber \\ & \phantom{f(x)=}\left. \left.-\frac{1}{x}\left(\! 1+ \frac{1}{4x} +\frac{1}{9x^2} \!\right)\!\right]-\frac{{\rm ln} x}{x} \!\left(\! 1-\frac{1}{4x}-\frac{1}{24x^2} \!\right) \!\right\},\end{eqnarray}$$
(37)
 
$$I_{\rm A}(x) = \sum_{n=1}^{\infty}\frac{1}{n^2 x^n} =-\int^1_0\frac{1}{t}{\rm ln}\left(1-\frac{t}{x}\right) dt.$$
(38)
It does not seem that the physical properties of this mathematical solution (37) have been clarified fully so far. The solution is of course non-singular, and remains finite for both |$ x \to$| 1 and |$ x \to \infty$| , and indeed yields
$$f(1)\equiv f_{\rm H}=0.5676, \quad f(\infty)= 0$$
(39)
(see figure 1, and also Okamoto 2009 for details). Then, a field line emanating from the horizon at |$ x =$| 1 with colatitude |$ \theta_{\rm H}$| has at |$ x$| above the horizon
$$ \begin{eqnarray} {\theta} &=& \theta_{\rm H}+h^2 [f_{\rm H}- f(x)] {\rm sin}\ \theta_{\rm H}{\rm cos}\ \theta_{\rm H}, \end{eqnarray}$$
(40)
 
$$\begin{eqnarray}\Phi & = & B_{\rm p}{\varpi}^2=\frac{\Psi_0{\rm sin}^2\theta_{\rm H}}{2\pi} \left\{ 1+h^2f_{\rm H}(2-3 {\rm sin}^2\theta_{\rm H}) \right. \nonumber \\ && \left. + h^2 \left[ f_{\rm H}-f(x) + \frac{1}{2x^2}\left(1+\frac{1}{x}\right)\right] {\rm sin}^2\theta_{\rm H}\right\}.\end{eqnarray}$$
(41)
[See equations (102) and (103) in Okamoto 2009 for |$ \Phi$| at |$ x =$| 1 and |$ \infty$| ]. The frame-dragging angular frequency, |$ \omega$| , along a perturbed field line with |$ {\theta} = \theta_{\rm H}$| at |$ x =$| 1 varies as follows:
$$\begin{equation} \omega = \frac{{\Omega_{\rm H}}}{x^3} \left\{1+h^2\left(1-\frac{1}{x}\right) \left[2\left(1+\frac{1}{x}\right)+\frac{{\rm sin}^2\theta_{\rm H}}{x^2}\right]\right\} . \end{equation}$$
(42)
Then, the eigen-functions become
$$I = \frac{I_0{\rm sin}^2\theta_{\rm H}}{ 2} \!\left\{\!1+\frac{h^2}{2}[4f_{\rm H}-(5f_{\rm H}-1){\rm sin}^2\theta_{\rm H}] \!\right\},$$
(43)
 
$${\Omega_{\rm F}} = \frac{{\Omega_{\rm H}}}{2} \left[1+\frac{h^2}{2} (1-f_{\rm H}){\rm sin}^2\theta_{\rm H}\right] ,$$
(44)
 
$$\zeta(\theta_{\rm H}) = 1-h^2 (1- f_{\rm H}){\rm sin}^2\theta_{\rm H},$$
(45)
 
$$\displaystyle I_0 \equiv \frac{\Psi_0 {\Omega_{\rm H}}}{4\pi}=\frac{c^3}{4GM} \frac{\Psi_0}{2\pi} h.$$
(46)
Note that |$ {\Omega_{\rm F}}\geq({\Omega_{\rm H}}/2)$| and |$ \zeta$| (⁠|$ \theta_{\rm H}$| ) |$ \leq$| 1. From equations (10), (41)–(44), and (46), one has
$$\Lambda(x,\theta_{\rm H}; h^2) = \Lambda_0(x) + h^2 F(x,\theta_{\rm H}),$$
(47)
where
$$\Lambda_0(x) \equiv \left(1-\frac{2}{x^3}\right),$$
(48)
 
$$F(x,\theta_{\rm H}) \equiv G_1(x)+ G_2(x) {\rm sin}^2\theta_{\rm H},$$
(49)
 
$$G_1(x) = -\frac{4}{x^3} \left(1-\frac{1}{x^2}\right),$$
(50)
 
$$\begin{equation}G_2(x)= \frac{1-f_{\rm H}}{x^3}-\frac{2}{x^5} \left(1-\frac{1}{x}\right) \\ + \left(1-\frac{2}{x^3}\right)\left[ \frac{1}{2x^2} \left(1+\frac{1}{x}\right)-f(x) \right].\end{equation}$$
(51)
Note that |$ F$| (⁠|$ x$| ,0) |$ = G_1$| (⁠|$ x$| ), and |$ F$| (⁠|$ x$| ,|$ \pi/$| 2) |$ = G_1$| (⁠|$ x$| ) |$ + G_2$| (⁠|$ x$| ). For the location of S|$ _{\rm N}$| , where |$ \omega = {\Omega_{\rm F}}$| and |$ \Lambda$| (⁠|$ x_{\rm N}$| , |$ \theta_{\rm H}$| , |$ h^2$| ) |$ =$| 0, one has in order of |$ h^2$|  
$$\begin{eqnarray} \frac{x_{\rm N}}{2^{1/3}} & = 1-\frac{h^2}{3} F(2^{1/3},\theta_{\rm H}) \quad\quad\quad \nonumber \\ & = 1+\frac{h^2}{3} \left[ \left(1- \frac{1}{2^{2/3}} \right) (1+{\rm cos}^2\theta_{\rm H}) +\frac{f_{\rm H}}{2} {\rm sin}^2 \theta_{\rm H}\right].\end{eqnarray}$$
(52)
It can be seen that |$ F(1,\theta_{\rm H})=F(\infty,\theta_{\rm H}) =$| 0, and hence
$$\begin{equation} \left\{ \begin{array} {ll} \Lambda(1,\theta_{\rm H}; h^2)= \Lambda_0(1) =-1, \\[4mm] \Lambda(x_{\rm N},\theta_{\rm H}; h^2)= 0, \\[4mm] \Lambda(\infty,\theta_{\rm H}; h^2)=\Lambda_0(\infty) =1 \end{array} \right. \end{equation}$$
(53)
(see figure 1). For a pulsar case with a split-monopolar field, one has |$ \Phi$|  |$ =$| (⁠|$ \Psi_0/$| 2|$ \pi$| )sin|$ ^2{\theta}$| and |$ I$|  |$ =$| (⁠|$ {\Omega_{\rm F}}/$| 2)|$ \Phi$| , and hence |$ \Lambda =$| 1 everywhere. This big difference is mainly due to the |$ {\alpha}\omega$| efffect.
In the limit of |$ h^2 \to$| 0, the hole still retains some traces of the null surface, S|$ _{\rm N}$| , dividing the magnetosphere at |$ x_{\rm N}=2^{1/3}$| into the two domains (Okamoto 2009)
$$\begin{equation} \zeta=1, \quad {\Omega_{\rm F}}={\Omega_{\rm H}}/2, \quad \Lambda(x,\theta_{\rm H}; 0) =\Lambda_0(x)\end{equation}$$
(54)
[see also equation (73)].
Behaviors of $ \Lambda_0$ ($ x$ ) and $ F$ ($ x$ ,$ \theta_{\rm H}$ ) for $ \theta_{\rm H} =$ 0 and $ \pi/2$ [see equations (47)-(51)]. Note that $ \Lambda$ ($ x$ ,$ \theta_{\rm H}; h^2$ ) is obtained by adding $ F$ ($ x,\theta_{\rm H}$ ) multiplied by factor $ h^2$ to $ \Lambda_0$ ($ x$ ), and $ x_{\rm N}$ is given by adding $-h^2$ (2$ ^{1/3}$ )$ F$ (2$ ^{1/3}$ ,$ \theta_{\rm H}$ ) to 2$ ^{1/3}$ , as seen in equation (52). $ f_{\rm H} = f$ (1) $ =$ 0.5676. (Color online)
Fig. 1.

Behaviors of |$ \Lambda_0$| (⁠|$ x$| ) and |$ F$| (⁠|$ x$| ,|$ \theta_{\rm H}$| ) for |$ \theta_{\rm H} =$| 0 and |$ \pi/2$| [see equations (47)-(51)]. Note that |$ \Lambda$| (⁠|$ x$| ,|$ \theta_{\rm H}; h^2$| ) is obtained by adding |$ F$| (⁠|$ x,\theta_{\rm H}$| ) multiplied by factor |$ h^2$| to |$ \Lambda_0$| (⁠|$ x$| ), and |$ x_{\rm N}$| is given by adding |$-h^2$| (2|$ ^{1/3}$| )|$ F$| (2|$ ^{1/3}$| ,|$ \theta_{\rm H}$| ) to 2|$ ^{1/3}$| , as seen in equation (52). |$ f_{\rm H} = f$| (1) |$ =$| 0.5676. (Color online)

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The poloidal current and the charge density become from equations (4) and (26)
$$\begin{eqnarray} {\boldsymbol{v}_{\rm p}} & = & \frac{2\pi r_{\rm H}^2 c{\boldsymbol{B}_{\rm p}}}{\Psi_0} \displaystyle {\frac{1}{1-(2/x^3)}}.\end{eqnarray}$$
(55)
 
$$\varrho_{\rm e}= - \frac{I_0}{2\pi r_{\rm H}^2 c}\frac{1-(2/x^3)}{\sqrt{1-(1/x)}} {\rm cos} \theta_{\rm H},$$
(56)
 
$${\boldsymbol{v}_{\rm p}} = \frac{2\pi r_{\rm H}^2 c{\boldsymbol{B}_{\rm p}}}{\Psi_0} \displaystyle {\frac{1}{1-(2/x^3)}}.$$
(57)

Note that equation (55) for |$ {\boldsymbol{j}_{\rm p}}$| contains terms up to an order of |$ h^3$| , but equations (56) and (57) for |$ \varrho_{\rm e}$| and |$ {\boldsymbol{v}_{\rm p}}$| are up to an order of |$ h$| , since the third-order terms need much more careful calculations.

As can be seen in equation (55), |$ {\boldsymbol{j}_{\rm p}}$| is continuous across S|$ _{\rm N}$| . However, for the upper hemisphere with |$ {\boldsymbol{j}_{\rm p}} \lt$| 0, |$ \varrho_{\rm e} \lt$| 0, |$ {\boldsymbol{v}_{\rm p}} \gt$| 0 and |$ \Lambda \gt$| 0 in the outer domain, |$ \varrho_{\rm e} \gt$| 0, |$ {\boldsymbol{v}_{\rm p}} \lt$| 0 and |$ \Lambda \lt$| 0 in the inner domain, and vice versa in the lower hemisphere. In the force-free domains, the velocity of charged particles is given by |$ {\boldsymbol{v}_{\rm p}} = {\boldsymbol{j}_{\rm p}}/\varrho_{\rm e}$| , and hence the interface between the two, inner and outer, domains, S|$ _{\rm N}$| , yet infinitely thin is not a kind of stagnation surface, but indeed |$ \varrho_{\rm e} \to \mp$| 0 and hence |$ {\boldsymbol{v}_{\rm p}}\to \pm \infty$| toward S|$ _{\rm N}$| , which indicates that the source must be existent hidden under S|$ _{\rm N}$| for a pair-plasma of electrons and positrons welling up, with electrons outgoing in the outer domain and positrons ingoing in the inner domain for 0 |$ \lt$|  |$ \theta_{\rm H}$|  |$ \lt$|  |$ \pi/$| 2 (see section 8, subsection 9.2). The divergence of |$ {\boldsymbol{v}_{\rm p}}$|  |$ \to$|  |$ \pm\infty$| does not imply a breakdown of the relation |$ {\boldsymbol{v}_{\rm p}} = {\boldsymbol{j}_{\rm p}}/\varrho_{\rm e}$| , but that of force-freeness, requiring the inclusion of inertial effects, i.e., the existence of the particle source under S|$ _{\rm N}$|⁠.

7. Black-Hole Thermodynamics

To understand the extraction process correctly, one needs some fundamental knowledge about black-hole thermodynamics. It is the first and second laws of black-hole thermodynamics that govern the process of extracting its rotational energy as the output power from a Kerr hole, and hence its evolution. The power is defined by the minus of the time variation of the total mass-energy, for which the first law, |$ c^2 dM=TdS+{\Omega_{\rm H}} dJ$| , yields
$${\cal P}_E=c^2 \left\vert\frac{dM}{dt}\right\vert={\Omega_{\rm H}} \left\vert\frac{dJ}{dt}\right\vert-T \left\vert\frac{dS}{dt}\right\vert,$$
(58)
where |$ T$| is the Hawking temperature and |$ S$| is the Bekenstein–Hawking entropy. Note that both |$ T$| and |$ S$| are derived by a quantum-mechanical consideration of the vacuum state due to the strongest gravity (see e.g., Thorne et al. 1986).
Because the hole cannot swallow a raw Poynting flux as it is, it must be cooked through Joule heating of the surface current on S|$ _{\rm ffH}$| into a digestible form of irreducible mass or entropy, i.e., from equations (9) and (28)
$$\begin{eqnarray} T\frac{dS}{dt} & = \oint {\cal R}_{\rm ffH}({\alpha} {\cal I}_{\rm ffH})^2 dA \quad\quad\quad\quad \nonumber \\\quad\quad\quad\quad & = \frac{1}{2\pi c} \oint[{\Omega_{\rm H}}- {\Omega_{\rm F}}(\Psi)] I (\Psi) d\Psi, \end{eqnarray}$$
(59)
which is naturally equal to the total ingoing Poynting flux flowing onto S|$ _{\rm ffH}$|⁠, i.e., from equation (8)
$$\begin{eqnarray} &\left. \oint \alpha \boldsymbol{S}_{\rm EM} \cdot d\boldsymbol{A} \right|_{\rm ffH}=- \frac{1}{2\pi c}\oint [{\Omega_{\rm H}}- {\Omega_{\rm F}}(\Psi)] I(\Psi) d\Psi . \end{eqnarray}$$
(60)

For the second law, |$ dS\geq 0$| , to hold, the condition |$ {\Omega_{\rm H}} \geq {\Omega_{\rm F}}$| must be satisfied in equation (59) (Blandford & Znajek 1977).

At the cost of Joule heating, the surface current exerts surface torque against S|$ _{\rm ffH}$| , thereby extracting angular momentum of the hole,
$$\begin{eqnarray} &\frac{dJ}{dt}=-\oint \left( \frac{\alpha \boldsymbol{\cal{I}}_{\rm ffH}}{c} \times {\boldsymbol{B}_{\rm p}}\right) \cdot {\varpi} \boldsymbol{t} dA =- \frac{1}{2\pi c} \oint I (\Psi) d\Psi, \end{eqnarray}$$
(61)
by which one may define the outward angular momentum flux, |$ \boldsymbol{S}_J$| , in such a way that
$$\oint {\alpha}\boldsymbol{S}_J \cdot d\boldsymbol{A} =-\frac{dJ}... ...\quad \boldsymbol{S}_J =\frac{I(\Psi)}{2\pi{\alpha} c}{\boldsymbol{B}_{\rm p}}.$$
(62)
Associated with |$ \boldsymbol{S}_J$| , there is the spin-down energy flux of a purely general-relativistic origin (MacDonald & Thorne 1982, Thorne et al. 1986)
$$\boldsymbol{S}_{\rm SD} = \omega \boldsymbol{S}_J =[\omega(\ell, \Psi) I(\Psi)/2\pi{\alpha} c] {\boldsymbol{B}_{\rm p}} ,$$
(63)
and the surface integral of |$ \boldsymbol{S}_{\rm SD}$| over S|$ _{\rm ffH}$| yields |$-{\Omega_{\rm H}}(dJ/dt)$| , i.e.,
$$ \begin{equation}\left. \oint {\alpha} \boldsymbol{S}_{\rm SD} \cdot d\boldsymbol{A} \right|_{\rm ffH} =\frac{{\Omega_{\rm H}}}{2\pi c}\oint I(\Psi)d\Psi=-{\Omega_{\rm H}}\frac{dJ}{dt} .\end{equation}$$
(64)
Note that from equation (8), |$ \boldsymbol{S}_{\rm EM}$|  |$ =$| (⁠|$ {\Omega_{\rm F}} - \omega$| )|$ \boldsymbol{S}_J$| . Then, the total energy flux consists of two terms, i.e.,
$$\begin{eqnarray}&&\boldsymbol{S}_E = \boldsymbol{S}_{\rm SD} +\boldsymbol{S}_{\rm EM} ={\Omega_{\rm F}}\boldsymbol{S}_J=[ {\Omega_{\rm F}}(\Psi) I(\Psi)/2\pi{\alpha} c] {\boldsymbol{B}_{\rm p}}, \end{eqnarray}$$
(65)
and hence the |$ \boldsymbol{S}_{\rm EM}$| , |$ \boldsymbol{S}_{\rm SD}$| and |$ \boldsymbol{S}_E$| in terms of |$ \boldsymbol{S}_J$| are, respectively, proportional to (⁠|$ {\Omega_{\rm F}} - \omega$| ), |$ \omega$| , and |$ {\Omega_{\rm F}}$| (see figure 3 in Okamoto 2009). The Poynting flux reaching S|$ _{{\rm ff}\infty}$| is equal to the total output power there, because of |$ \boldsymbol{S}_{\rm SD} \to$| 0, and also to the Joule dissipation rate of the surface current on the resistive membrane S|$ _{{\rm ff}\infty}$| , i.e.,
$$\begin{equation} \left. {\cal P}_E = \oint {\alpha}\boldsymbol{S}_E \cdot d\boldsymbol{A} \right|_{{\rm ff}\infty} =\frac{1}{2\pi c} \oint {\Omega_{\rm F}}(\Psi) I (\Psi) d\Psi . \end{equation}$$
(66)

The spin-down energy (the source) is thus shared between the power (gain) and dissipation (loss), by the ratio of |$ {\Omega_{\rm F}}$| : (⁠|$ {\Omega_{\rm H}} - {\Omega_{\rm F}}$| ) |$ \propto$| 1|$ :\zeta$| per unit flux tube.

For a split-monopolar field structure, one has for the losses of the total angular momentum and energy:
$${\cal P}_J= -\frac{dJ}{dt}={\cal S}_J \left[1+\frac{2h^2}{5}(1-f_{\rm H})\right],$$
(67)
 
$${\cal P}_E = - c^2 \frac{dM}{dt}= {\cal S}_E \left[1+\frac{4h^2}{5}(1-f_{\rm H}) \right],$$
(68)
 
$${\cal S}_J = \frac{2\Psi_0 I_o}{3c}, \quad {\cal S}_E \equiv ({\Omega_{\rm H}}/2){\cal S}_J,$$
(69)
where |$ I_0$| is given in equation (46), and then |$ {\cal S}_J =$| O(⁠|$ h$| ) and |$ {\cal S}_E = {\rm O}$| (⁠|$ h^2$| ). The total values of the electromagnetic and spin-down energy fluxes passing through any surafce are
$$\begin{eqnarray} {\cal P}_{\rm EM} & = {\cal S}_E \left[\left(1-\frac{2}{x^3}\right)+\frac{4h^2}{5}(1-f_{\rm H}) \left(1-\frac{1}{x^3}\right) \right. \nonumber \\ & \left.-\frac{4h^2}{x^3} \left(1-\frac{1}{x^3}\right) \left(1+\frac{1}{x} +\frac{2}{5x^3}\right) \right],\end{eqnarray}$$
(70)
 
$$\begin{eqnarray} {\cal P}_{\rm SD} & = \frac{2{\cal S}_E}{x^3}\left[1+\frac{2h^2}{5}(1-f_{\rm H}) \right. \nonumber \\ & & \left. +2h^2\left(1-\frac{1}{x}\right) \left(1+\frac{1}{x} +\frac{2}{5x^3}\right) \right] , \quad\quad\quad \end{eqnarray}$$
(71)
which naturally satisfy |$ {\cal P}_E = {\cal P}_{\rm EM} + {\cal P}_{\rm SD}$| . The “average” location of S|$ _{\rm N}$| , |$ \overline{x_{\rm N}}$| , is defined by |$ {\cal P}_{\rm EM} =$| 0 as follows:
$$\frac{\overline{x_{\rm N}}}{2^{1/3}}=1+\frac{2h^2}{5}\left(1-\frac{1}{2^{2/3}}+\frac{f_{\rm H}}{3}\right)$$
(72)
(cf. equation (52) for |$ x_{\rm N}$| ). Then, up to the order of |$ h^2$| , |$ {\cal P}_{\rm EM} \gtreqqless$| 0 for |$ x \gtreqqless \overline{x_{\rm N}}$| (see figure 3 in Okamoto 2009).
Even in the limit of |$ h^2 \to$| 0, in addition to that of S|$ _{\rm N}$| at |$ x =$| 2|$ ^{1/3}$| , some vestige of the Kerr state remains, related to the results in equations (54); that is,
$$\frac{{\cal P}_E}{ {\cal S}_E}=1, \quad \frac{{\cal P}_{\rm EM}}{... ..._E}=\Lambda_0(x), \quad \frac{{\cal P}_{\rm SD}}{ {\cal S}_E}=1- \Lambda_0(x).$$
(73)

8. Electromotive Force

The Poynting flux, |$ \boldsymbol{S}_{\rm EM}$| , in equation (8), together with the electric field, |$ {\boldsymbol{E}_{\rm p}}$| , in equation (6) and the potential difference, PD, in equation (7) change their direction at S|$ _{\rm N}$| . This indicates that hidden under the interface, S|$ _{\rm N}$| , between the domains of inflow (⁠|$ {\boldsymbol{v}_{\rm p}} = {\boldsymbol{j}_{\rm p}}/\varrho_{\rm e} \lt$| 0) and outflow (⁠|$ {\boldsymbol{v}_{\rm p}} = {\boldsymbol{j}_{\rm p}}/\varrho_{\rm e} \gt$| 0) [see e.g., equation (57) for a split-monopolar field] must be situated both sources of electromagnetic energy as well as of charged particles carrying electric currents.

Let us then cut open along S|$ _{\rm N}$| , to look for any sign of pair-creation and charge-separation under it within the framework of force-freeness. Expanding |$ \omega$| (⁠|$ \ell$| ) like |$ \omega$| (⁠|$ \ell$| ) |$ \approx \omega$| (⁠|$ \ell_{\rm N}$| )|$ \pm\Delta\omega$| , at |$ \ell = \ell_{\rm N}\mp \Delta \ell \equiv \overline{\ell}_{\rm in}$| or |$ \overline{\ell}_{\rm out}$| , where |$ \Delta\ell $| is the half-width of the gap and |$ \Delta\omega = \vert$| (⁠|$ \partial\omega/\partial\ell$| )|$ \vert _{\rm N} \Delta\ell $| , and substituting into |$ {\boldsymbol{E}_{\rm p}}$| in equation (6) and the PD in equation (7), one has
$${\alpha}{\boldsymbol{E}_{\rm p}}\approx \pm [\Delta\omega/2\pi c]\boldsymbol{\nabla} \Psi,$$
(74)
 
$${\rm PD}_{\rm in} {\rm or} {\rm PD}_{\rm out} = \pm \displaystyle {\frac{1}{2\pi c}}\int^{\Psi_2}_{\Psi_1} \Delta\omega d\Psi,$$
(75)
retaining first-order terms only, where |$ \Psi_1$| and |$ \Psi_2$| are chosen in such a way that |$ I$| (⁠|$ \Psi_1$| ) |$ =$|  |$ I$| (⁠|$ \Psi_2$| )(⁠|$ \equiv I_{\overline{12}}$| ). The condition for expansion in equation (75) to be valid will be
$$\begin{equation} \Delta\ell \ll 2 \left| \frac{\partial\omega}{\partial\ell}\right|_{\rm N}\Big/ \left|\frac{\partial^2 \omega}{\partial \ell^2}\right|_{\rm N}. \end{equation}$$
(76)

It must be remarked again that the major premise in this FFDE model is that field lines are smooth and continuous across the gap, accompanied by no discontinuity of |$ {\Omega_{\rm F}}$| (see subsection 9.3).

Then, the gap with the PD’s existent at the two surfaces will give rise to such a parallel component of the electric field across 2|$ \Delta\ell $| at |$ \Psi$|  |$ =$|  |$ \Psi_1$| and |$ \Psi_2$| as
$$\begin{eqnarray} (E_{\parallel})_{\rm N} & \approx & \mp \frac{|{\rm PD}_{\rm in}| + |{\rm PD}_{\rm out}|}{2\Delta\ell } \nonumber \\ & \approx & \mp \frac{1}{2\pi c} \int^{\Psi_2}_{\Psi_1} \left| \frac{\partial\ {\rm ln} \omega}{\partial\ell}\right|_{\rm N} {\Omega_{\rm F}} \ d\Psi . \end{eqnarray}$$
(77)

The gap width will be determined by the condition that a necessary and sufficient number of pair-particles must be created and provided into the inner and outer force-free domains to support the electric currents, and thereby the field structure described by equation (25). An important factor is that (⁠|$ E_{\parallel}$| )|$ _{\rm N}$| is not transient, nor sporadic, but stationary and stable for promoting pair-creation (cf. Beskin et al. 1992, Hirotani & Okamoto 1998). Note that even if the gap is closed in the limit of force-freeness, i.e., 2|$ \Delta\ell \to$| 0, (⁠|$ E_{\parallel}$| )|$ _{\rm N}$| will remain as a vestige of the gap.1

The EMF’s at work on the two resistive membranes S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| from equations (7) or (30), (33) and (34) are given by
$${\rm EMF}_{{\rm S}_{\rm ffH}} = + \frac{1}{2\pi c}\int^{\Psi_2}_{\Psi_1} ({\Omega_{\rm H}}-{\Omega_{\rm F}}) d\Psi,$$
(78)
 
$${\rm EMF}_{ {\rm S}_{{\rm ff} \infty}} \approx -\frac{1}{2\pi c}\int^{\Psi_2}_{\Psi_1}{\Omega_{\rm F}} d\Psi,$$
(79)
which drive the surface currents, to exert surface torques on S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| (see sections 5 and 7). Also, the presence of the two EMF’s, oppositely directed, on S|$ _{{\rm ff}\infty}$| and S|$ _{\rm ffH}$| at both ends of the volume current lines will give rise to voltage differences, VD, between S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| along the current lines |$ \Psi_1$| and |$ \Psi_2$|⁠,
$${\rm VD}\approx \vert {\rm EMF}_{{\rm S}_{\rm ffH}}\vert+ \vert {... ...\rm ff} \infty\vert \approx \frac{{\Omega_{\rm H}} (\Psi_2-\Psi_1)}{2\pi c},$$
(80)
which will be helpful to drive charged particles inputted from the source under S|$ _{\rm N}$| , to circulate smoothly along the current lines |$ I$| (⁠|$ \Psi_1$| ) |$ =$|  |$ I$| (⁠|$ \Psi_2$| ) in the force-free domains, connected by the surface currents, |$ {\cal I}_{{\rm ff}\infty}$| and |$ {\cal I}_{{\rm ffH}}$| , on S|$ _{{\rm ff}\infty}$| and S|$ _{\rm ffH}$| in a closed DC circuit in the steady state.

It was presumed in previous studies (Okamoto 2006, 2009) that the two unipolar inductors (double EMF’s) existed back-to-back, oppositely directed, with S|$ _{\rm N}$| between. However, it will now have to be correctly interpreted that the PD associated with |$ {\Omega_{\rm F}}$| works to produce the two EMF’s oppositely directed on the membrane surfaces S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$|⁠.

9. Discussion

It goes without saying that FFDE is the inertia-free limit of MHD, or conversely MHD is recovered by supplementing FFDE with inertial effects, corresponding to astrophysical loads, such as particle acceleration. The “physics” of FFDE is elucidated in connection with unipolar induction, and resistive membranes in the general-relativistic setting.

9.1. Is the Membrane Paradigm “Artificial”?

The Alfvénic surface, S|$ _{\rm A}$| , at the axial distance, |$ {\varpi_{\rm A}}$| , revives as one of the important critical surfaces in a pulsar MHD wind, and then the condition |$ {\varpi_{\rm A}}^2/{\varpi}^2\ll 1$| discriminates between the asymptotic and sub-asymptotic domains of the wind zone, with the dividing surface being vaguely denoted by S|$ _{\rm a}$| . In the former domain, coordinate variables |$ {\varpi}$| and |$ z$| do not appear explicitly in expressions for flow-field variables, with the magnetization parameter, |$ {\sigma} \propto \Phi$| (⁠|$ \ell$| ,|$ \Psi$| ), where |$ {\sigma} = {\Omega_{\rm F}}^2\Phi/$| 4|$ \pi\eta c^3$| and |$ \eta$| (⁠|$ \Psi$| ) |$ =$|  |$ \rho\kappa$| (see e.g., Okamoto & Sigalo 2006), instead playing the role of an independent variable; it can be shown that there appears no fast magnetosonic surface, S|$ _{\rm F}$| , in that domain. This implies that S|$ _{\rm F}$| must exist between the two domains; that is, S|$ _{\rm F}$|  |$ \approx$| S|$ _{\rm a}$| . The flow properties can be completely determined by the criticality conditions at S|$ _{\rm A}$| and S|$ _{\rm F}$| . Also, the flow upstream from S|$ _{\rm F}$| is causally disconnected from that downstream, and MHD acceleration is relatively ineffective in the sub-asymptotic domain of |$ S$|  |$ \lesssim $| S|$ _{\rm F}$| , where |$ j_{\perp} \approx$| 0 and |$ {\gamma} \lesssim {\gamma_{\rm F}}$|  |$ \ll {\gamma_{\rm F}}^3$|  |$ = {\gamma_{\infty}}$| , whereas it is in the asymptotic domain (⁠|$ S$|  |$ \gt$| S|$ _{\rm F}$|  |$ \approx$| S|$ _{\rm a}$| ) that the main MHD acceleration takes place, where |$ j_{\perp} \gt$| 0 and |$ {\gamma_{\rm F}}$|  |$ \lesssim $|  |$ {\gamma}$|  |$ \lesssim $|  |$ {\gamma_{\infty}}$|  |$ =$|  |$ {\gamma_{\rm F}}^3$| , and |$ I$| (⁠|$ {\sigma}$| ) |$ \to$| 0 for |$ {\sigma} \propto \Phi$|  |$ \to$| 0 (Okamoto 2002, 2003, 2006). It can then be shown that in the asymptotic domain of a MHD outgoing wind,
$$\Lambda\approx \sqrt{1-\frac{1}{{\gamma}^2}}=\frac{v}{c}$$
(81)
(I. Okamoto in preparation), as guessed from equation (13) for |$ \vert v_{\rm F}\vert \gg \vert v_{\rm t}\vert$| . It is conversely quite a small deviation of |$ {\Lambda}$| from unity that yields the extremum attainable value of |$ {\gamma}$| by MHD acceleration at infinity, i.e.,
$${\gamma_{\infty}}= \mu_{\varepsilon}= {\gamma_{\rm F}}^3 \approx \frac{1} {\sqrt{2(1-\Lambda_\infty ) }},$$
(82)
where |$ \mu_{\varepsilon}$| (⁠|$ \gg$| 1) is one of the Bernoulli integrals (see Okamoto & Sigalo 2006).

Just as MHD is so, FFDE as its inertia-free limit is also capable of describing the magnetosphere-wind systems self-consistently within its framework. The “sub-asymptotic domain” with |$ j_{\perp} \approx$| 0 for a MHD outflow reduces to the “force-free domain” with |$ j_{\perp} =$| 0 in FFDE, and the “asymptotic domain” with cross-field volume currents does to the “resistive membrane,” S|$ _{{\rm ff}\infty}$| , with surface currents. No current line can cross field lines within the “force-free domain”, and it is meaningless to discuss flow acceleration there, as anticipated from the fact that little acceleration takes place in the sub-asymptotic domain in MHD. The criticality condition at S|$ _{\rm F}$| survives in the inertial-free limit (Okamoto 1978), to yield the current function, |$ I$| , as the eigenvalue given in (29), which takes care of astrophysical loads in FFDE.

Then, the “asymptotic domain” of the main MHD acceleration where |$ j_{\perp} \gt$| 0, |$ \partial I$| (⁠|$ \Psi$| ,|$ \ell$| )|$ /\partial \ell \lt$| 0, |$ I$|  |$ \to$| 0 for |$ \ell \to \infty$| , and hence |$ \partial {\gamma}/\partial \ell \gt$| 0, with an increase of |$ {\gamma}$| from |$ {\gamma}$|  |$ =$|  |$ {\gamma_{\rm F}}$| at S|$ _{\rm F}$| to |$ {\gamma} = {\gamma_{\rm F}}^3$|  |$ \gg {\gamma_{\rm F}}$| at S|$ _{\infty}$| may seem in FFDE to be as if compressed and projected onto the membrane S|$ _{{\rm ff}\infty}$| with the surface currents flowing, |$ {\cal I}_{{\rm ff}\infty}$|  |$ \gt$| 0 on there. This will look for the inhabitants in the force-free domain, like a mirage of a real ship by chance at sea over the horizon on Earth, but this is never a hallucinatory image like a hot-road mirage.

Mentioning in FFDE terminology, the force-free domains must be terminated by taking into account inertial effects at the membrane surfaces, S|$ _{\rm ffH}$| and S|$ _{{\rm ff}\infty}$| , so that the force-free versions of the current functions |$ I$| (⁠|$ \Psi$| )’s can be determined as the eigenvalues by the criticality conditions (S|$ _{{\rm ff}\infty}$|  |$ \approx$| S|$ _{\rm oF}$| and S|$ _{\rm ffH}$|  |$ \approx$| S|$ _{\rm iF}$| ), to allow the surface currents to flow, and to ensure the current-closure condition to hold. The transition from a FFDE massless outgoing wind to a MHD one will be made by returning the resistive membrane, S|$ _{{\rm ff}\infty}$| , to the original size of the asymptotic domain from S|$ _{\rm F}$| to a real S|$ _{\infty}$| , by including particle mass. If the Membrane Paradigm2 were “artificial” (e.g., Komissarov 2009), FFDE itself would be “more artificial” in that the particle inertia is artificially regarded as being negligible everywhere (except S|$ _{\rm ffH}$| or S|$ _{{\rm ff}\infty}$| ). Some kind of “virtual” surfaces, S|$ _{{\rm ff}\infty}$| and S|$ _{\rm ffH}$| , are indispensable as one of the basic properties of FFDE, to thereby render that physics works soundly in FFDE as well. Also, observers resident in the inner force-free domain would see the surface cross-field current on the resistive membrane, S|$ _{\rm ffH}$| , which indicates Joule dissipation of the ingoing Poynting flux, leading to an increase of the hole’s entropy as the cost of exerting the surface torque to extract the hole’s angular momentum and rotational energy from the hole (see sections 7 and 8). If one is content with this self-consistent, macroscopic description of the membrane S|$ _{\rm ffH}$| , one may not necessarily get deeply involved in quantum processes under the carpet [see also item (4) in Concluding Remarks, in Okamoto 2009].

9.2. Particle Supply from Under SN

As for particle supply from under S|$ _{\rm N}$| , it may be helpful to remember a similar phenomena taking place on Earth. It is found that magmas are welling out from e.g., the Central Ocean Ridges in the Mid Atlantic Ocean. According to widely accepted theory of plate tectonics, the driving forces for rising magmas, ocean-floor spreading and resulting continental drifts ultimately originate from mantle convection under the crust. The present situation may seem to be quite different from up-welling magmas, but some analogy may be helpful in this simple model based on FFDE: Pair particles of electron-positron will be thought of as welling out from beneath the surface, S|$ _{\rm N}$| , with pair particles charge-separated toward the, respective, inner and outer, directions. Just as the primary force of pulling out charged particles from a neutron star crust is the parallel component of the electric field, |$ E_{\parallel} \propto$| (⁠|$ \boldsymbol{E}\cdot\boldsymbol{B}$| ), which if unshielded in a vacuum would be most effective in the particle supply of electrons and positive ions from the crust, so (⁠|$ E_{\parallel}$| )|$ _{\rm N}$| across S|$ _{\rm N}$| in equation (77) would pull out pair particles of electrons and positrons from the gap under S|$ _{\rm N}$| , to charge-separate and to input them into the ingoing and outgoing winds. Although |$ E_{\parallel}\propto$| (⁠|$ \boldsymbol{E}\cdot\boldsymbol{B}$| ) will be shielded to vanish for a pulsar magnetosphere in the steady state, (⁠|$ E_{\parallel}$| )|$ _{\rm N}$| at S|$ _{\rm N}$| for a Kerr hole magnetosphere will not be shielded completely, even in the steady state, being at work without a break to supply charged particles into the force-free domains.

It may be said that unipolar induction will be the ultimate mechanism for driving electric currents, and thereby sustaining magnetospheres around rotating, magnetized compact stars as a body.

9.3. Peeking in on the Gap inside through SN

Under the important premise that |$ {\Omega_{\rm F}}$| as well as the stream function |$ \Psi$| , itself, be continuous across S|$ _{\rm N}$| (see also footnote 1), that is, making use of expression (6) as it is, (⁠|$ E_{\parallel}$| )|$ _{\rm N}$| across S|$ _{\rm N}$| in equation (77) is derived rather phenomenologically within the framework of FFDE, at first making the gap open by |$ 2\Delta \ell$| forcibly, and then closing it in the force-free limit. This will be expected to promote pair-creation due to e.g., cascades by discharge with the number of particles being numerous enough to pin down the magnetic fluxes so as to make them rotate with |$ {\Omega_{\rm F}} = \omega$| (⁠|$ \ell_{\rm N}$| ) |$ \approx {\Omega_{\rm H}}/$| 2, and to produce simultaneous charge-separation of any charged particles created.

Important to note here is the interplay of microphysics and macrophysics. Unipolar induction due to the potential gradient generated by magnetic field lines rotating with |$ {\Omega_{\rm F}}$| must be supported by the pinning-down of field lines onto charged particles created by quantum processes; the parallel electric field at S|$ _{\rm N}$| due to unipolar induction has conversely to promote microphysical processes of particle creation to induce macrophysical processes involving the pinning-down of field lines. This interplay will take place in an environment where force-freeness as well frozen-inness (the perfect conductivity condition) are broken down, and it is not obvious how to define a macroscopic concept of magnetic field lines as well as the angular frequency of them |$ {\Omega_{\rm F}}$| inside the gap. One may calculate them by taking some ensemble average of micro-processes.

In any cases, this simple model for the force-free magnetosphere of a rotating black hole will allow us to sweep the details of the gap physics under the carpet, S|$ _{\rm N}$| , and to make exploring the gap inside under S|$ _{\rm N}$| in earnest be out of the scope in this paper.

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1

Under the present major premiss that magnetic-field lines thread the null surface, S|$ _{\rm N}$| , between the force-free domains continuously, with constant |$ {\Omega_{\rm F}}$| (⁠|$ \Psi$| ) for each field line, it is not possible to interpret the presence of (⁠|$ E_{\parallel}$| )|$ _{\rm N}$| across S|$ _{\rm N}$| as due to any difference of the field line angular frequency |$ {\Omega_{\rm F}}$| across the gap, because |$ {\Omega_{\rm F}}$| (⁠|$ \Psi$| ) leads to the corotational electric field as seen in equations (1) and (6), but to no parallel component, |$ E_{\parallel}$| , and so one cannot connect (⁠|$ E_{\parallel})_{\rm N}$| directly to any gap of |$ {\Omega_{\rm F}}$|

2

The membrane paradigm, originally formulated by Thorne et al. (1986), endowed the membrane at the horizon with a function of battery, along with thermodynamic properties, etc., consequently inviting critiques of causality violation (Punsly 1991). This is however not simply wrong, because one may define the electromotive force driving the membrane current on S|$ _{\rm ffH}$| originated from unipolar induction [see equation (78)].

© 2012 Astronomical Society of Japan
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