Energy self-extraction of a Kerr black hole with the force-free magnetosphere frame-dragged
Energy self-extraction of a Kerr black hole

Isao Okamoto1, Toshio Uchida2
Yoogeun Song3

Okamoto et al.

National Astronomical Observatory, 2-21-1 Osawa, Mitaka-shi, Tokyo 181-8588, Japan

School of Political Science and Economics, Meiji University, 1-9-1 Eifuku, Suginami-ku, Tokyo 168-8555

Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey RH5 6NT, United Kingdom

It is argued that the zero-angular-momentum-observers (ZAMOs) circulating with frame-dragging-angular-velocity $\om$ will see that the `null surface'  with $\omN=\OmF$ always exists in the force-free magnetosphere, when the condition $\OmF<\OmH$ is satisfied, where $\OmH$ and $\OmF$ are the horizon and field-line (FL) angular-velocities (AVs).
When $\OmFm\equiv \OmF-\om$ denotes the ZAMO-measured FLAV, this surface  where $\OmFm=0$ defines the gravito-magneto-centrifugal divider of the magnetosphere, with a kind of plasma-shed on it. The outer domain $\calDout$ outside  spins forward ($\OmFm>0$), whereas the inner domain $\calDin$ inside spins backward ($\OmFm<0$).

The force-free and freezing-in conditions break down on , thereby allowing the particle-current sources to be set up on . Because the electric field $\vcEp$ reverses direction there, the Poynting flux reverses direction as well from outward to inward, though the `positive' angular momentum always flows outwardly.

Electromagnetic self-extraction of energy will be possible only through the frame-dragged magnetosphere, with the inner domain $\calDin$ nested between the horizon and the surface , in order to comply with the 1st and 2nd laws of thermodynamics.

§ INTRODUCTION

More than four decades have passed since the pioneering paper by Blandford & Znajek, 1977 was published on the electromagnetic extraction of energy from Kerr black holes (BHs).  This controversial task, however, still remains a big challenge even in the latest new frontier in modern classical physics [Thorne & Blandford, 2017].  The purpose of this paper is to venture on challenging this formidable task.

Fundamental concepts and expressions, as well as the basic formulation of general relativity, thermodynamics and electrodynamics, necessary for elucidating extraction of energy from Kerr holes, have fortunately been given in the almost complete form already four decades ago  [Blandford & Znajek, 1977, Macdonald & Thorne, 1982, Znajek, 1977, Znajek, 1978, Phinney, 1983, Thorne et al., 1986]. Phinney (1983a,b) was the first who tried to develop a comprehensive model for `BH-driven hydromagnetic flows' or jets for active galactic nuclei (AGNs).
It seems to be regarded in these references that a `battery' exists on the event horizon, with magnetic field lines threading through the BH's horizon.
That is, a `magnetized' Kerr hole would possess not only a battery but also an internal resistance on the horizon. We refer to this model as the `single-pulsar model' with a `single electric circuit'. This is because the magnetosphere consists of double `wind structures' with a `negligible violation' of the force-free condition in-between for particle production and a single `series circuit', with a battery on the horizon and two resistances on the horizon and infinity surface.

Uchida (1997a,b) constructed a comprehensive theory of force-free electromagnetic fields and pointed out that the `breakdown of the force-free approximation' is not a `defect' proper to force-freeness. We show here that the `real' breakdown we meet with is not a defect but provides rather essential foundations necessary to rebuild an active `force-free' magnetosphere.

Based upon these pioneering works, we rebuild the pulsar force-free magnetosphere, with the dragging of inertial frames fully taken into account.
We introduce the central premise that the large-scale poloidal magnetic field $\vcBp$ will be trapped (i.e., frame-dragged) by the hole into circulation around the hole with the frame-dragging-angular velocity (FDAV) $\om$.  Magnetic FLs extend from near the horizon  to the infinity surface , with the field-line-angular-velocity (FLAV) $\OmF$ kept constant (Ferraro's law of iso-rotation). When the iso-rotation law holds for $\OmF$ in the absolute space, the zero-angular-momentum-observers (ZAMOs) circulating with the FDAV $\om$ will see that Ferraro's law does not hold for the ZAMO-FLAV $\OmFm=\OmFm-\om$ (see equation (<ref>)). The force-free magnetosphere consisting of charge-separated plasma is divided by the null surface  where $\OmFm=0$, i.e., $\om=\OmF\equiv\omN$, into the two domains, outer semi-classical (SC) and inner general-relativistic (GR), $\calDout$ and $\calDin$. The ZAMOs will soon realize that the breakdown of the force-free and freezing-in conditions is inevitable on the null surface  with $\OmFm\ggel 0$.

We think of three membranes in our `twin-pulsar model' (cf. Thorne et al., 1986). The first two `resistive' ones on the infinity and horizon surfaces, $\Sffinf$ and $\SffH$ terminate the two force-free domains $\calDout$ and $\calDin$ by particle acceleration on $\Sffinf$ and entropy production on $\SffH$, respectively. The third one is the `inductive' membrane $\SN$ on the null surface , which covers the particle-current sources.

Blandford & Globus, 2022 recently constructed the models for the ergo-magnetosphere, ejection disc, and magnetopause in M87. They argued that the force-free approximation is justifiable in the vicinity of the black hole.  There is actually good chemistry between thermodynamics for the Kerr hole with two hairs (e.g. entropy $S$ and angular momentum $J$) and electrodynamics for the pulsar magnetosphere with two conserved quantities along each FL ($\OmF$ and $I$).  This is thanks to the frame-dragging effect, which unifies force-free electrodynamics with thermodynamics. We argue that the force-free approximation is flexible and robust enough to accept the `breakdown' of the force-free condition (as well as the freezing-in one) on the `null surface' , which exists always when the hole loses energy [Blandford & Znajek, 1977].

A brief outline of each section is as follows.
Section <ref> points out that the Kerr holes are a rotating mass of `entropy matter', and hence without invoking the influx of negative angular momentum, it will be impossible to realize the outgoing flux of positive angular momentum, thereby extracting energy from the Kerr BH.
In section <ref>,
other than the `conserved' FLAV $\OmF$, we newly define the ZAMO-measured FLAV $\OmFm=\OmF-\om$, toward constructing gravito-thermo-electrdynamics. Then, from a thermodynamic point of view, we discuss the similarity and dissimilarity of the force-free magnetospheres between Kerr holes and pulsars in subsections <ref> and <ref>.

In section <ref>,
in order to elucidate the KBH's `complicated properties' of energy extraction, we use the wind-, flux- and circuit analyses expediently.
In particular, subsection <ref> argues that a pair of unipolar induction batteries with electromotive forces (EMFs), $\calEout$ and $\calEin$  will drive currents to flow through the circuits $\calCout$ and $\calCin$ in the outer SC domain $\calDout$ and the inner GR domain $\calDin$, respectively (see figure <ref>). There will be a huge voltage drop $\Dl V$ across  between the two EMFs for particle production.

Section <ref>
shows that the angular-momentum density of the electromagnetic field is positive ($\epsJ>0$) in $\calDout$ and negative ($\epsJ<0$) in $\calDin$. That is, the null surface  coincides with the zero-angular-momentum surface .   The energy density $\epsE$ changes positive to negative somewhere inward beyond the null surface .

In section  <ref>,
the ZAMOs will see that reversal of the electric field $\vcEp$ and the Poynting flux $\vcSEM$ on  leads to a breakdown of the force-free and freezing-in conditions. We show how critical the Constraint $\PN{\vcj}=I(\ell,\Psi)=0$ on  due to a  complete violation of the force-free condition is because this surface  will be widened to such a gap $\GN$ as filled with zero-angular-momentum-particles (ZAM-particles) pair-produced due to the voltage drop $\Dl V$ (see subsection <ref>).

In section <ref>,
the structure of a gap $\GN$ under the `inductive membrane' $\SN$ is discussed with respect to the pinning-down of threading FLs on the pair-created plasma
and parallel magnetization of it, as well as a magneto-centrifugal plasma-shed on the zero-angular-momentum surface , etc. (see also figure <ref> for a simple model for the function $I(\ell, \Psi)$).

In section <ref>,
we argue the `boundary condition' for determining the eigenvalue of $\OmF=\omN$ in the steady axisymmetric state. Some new pieces of evidence shown will be helpful in understanding an `enigmatic' flow of positive angular momentum from the horizon membrane $\SffH$, beyond the inductive membrane $\SN$ covering the Gap $\GN$, to the infinity membrane $\Sffinf$ (see Sec. <ref>). Thus, the ZAM-Gap $\SG$ will allow us to impose the conservation law of angular momentum as the `boundary condition' determining the eigenfunction $\OmF(\Psi)=\omN$; this means that the eigen-magnetosphere with $\omN=\OmF$ is `frame-dragged' by the hole's rotation (see Sec. <ref>).

Section <ref>
attempts to explain the null surface  in terms of a new kind of  (RTD) in the GR setting <cit.>. We conjecture that this RTD involving a voltage drop $\Dl V$ between the two EMFs will bring up a new mechanism of pair-particle creation at work on  toward widening to a GR gap $\GN$. As opposed to the `single-pulsar model' based on BH electrodynamics with a negligible violation of the force-free condition,
we propose the `twin-pulsar model' based on gravito-thermo-electrodynamics (GTED) because there will be “two pulsar-type magnetospheres" coexisting, outer prograde- and inner retrograde-rotating, respectively, with the RTD Gap in-between for the supply of electricity and particles.

Section <ref>
discusses the `energetics and structure' of the twin-pulsar model as opposed to that of the single-pulsar model (cf. e.g. IV D in Thorne et al., 1986).

The last section <ref> is devoted to discussions and conclusions with some remaining issues listed.

Appendix A discusses the position and shape of the null surface  in the force-free magnetosphere for the parameter $0\leq h=a/\rH\leq 1$.

§ THE KERR BLACK HOLE AS A THERMODYNAMIC OBJECT

The no-hair theorem says that Kerr holes possess only two hairs.  When one chooses the entropy $S$ and the angular momentum $J$ as two `extensive' variables, then all other thermodynamic quantities are expressed as functions of these two. For example, the BH's mass-energy $M$ is expressed in terms of $S$ and $J$, as follows;
M=√((ħcS/4πkG) + (πkcJ^2/ħGS)).

As one can, in principle, utilize a Kerr hole as a Carnot engine [Kaburaki & Okamoto, 1991], it may be regarded as a `thermodynamic object,' but not as an electrodynamic one because the Kerr hole by itself stores no extractable electromagnetic energy. Therefore, the Kerr hole may be regarded as a huge rotating mass of `entropy matter' (see, e.g., [Basu & Lynden-Bell, 1990]), fundamentally different from the magnetized rotating NS, which consists of `normal matter' with magnetic field lines emanating outside. Thus, its evolutionary behaviours, such as due to the extraction of angular momentum, are strictly governed by the four laws of thermodynamics (see, e.g., [Thorne et al., 1986] for a succinct summary).
The mass $M$ of the hole is divided into the `irreducible' and `rotational' masses, i.e.,

\begin{eqnarray}
M=M_{\rm irr}+M_{\rm rot},  \hspace{0.3cm}~~~~~ \label{massa} \\  % (2.2a)
M_{\rm irr}=\frac{M}{\sqrt{1+h^2}} =\sqrt{c^4 A_{\rm H}/16\pi G^2} =\sqrt{\hbar cS/4\pi kG}, \hspace{0.3cm} ~~~~~ \\ % (2.2b)
M_{\rm rot}= M[1- 1/\sqrt{1+h^2}], \hspace{0.3cm}  ~~~~~ 	% (2.2c)
\end{eqnarray}

where $A_{\rm H}$ is the horizon surface area, and $h$ is defined as the ratio of $a\equiv J/Mc$ to the horizon radius $\rH$, i.e.,
h=a/ =2πkJ/ħS=2GM/c^3.

The Kerr hole's thermo-rotational state is uniquely specified by its entropy $S$ and angular momentum $J$, or its mass-energy $M$ and the spin-parameter $h$.

The evolutional state of the BH losing energy is specified as the timeline of function $h(t)$ for the `outer-horizon in $0\leq h\leq 1$. We see $h=0$ for a Schwarzschild BH and $h=1$ for an extreme-Kerr hole [Okamoto & Kaburaki , 1990, Okamoto & Kaburaki , 1991, Okamoto, 1992].

The hole's irreducible mass $M_{\rm irr}$ and surface area $A_{\rm H}$ are functions of $S$ only, but the rotational mass $M_{\rm rot}$ may be a function of $S$ as well as $J$. Therefore, when the hole loses angular momentum by, e.g., an influx of negative angular momentum ($dJ<0$), the hole's total mass and rotational mass will decrease, i.e., $dM<0$ and $dM_{\rm rot}<0$, while $dM_{\rm irr}>0$ and $\Th dS>0$ always hold.

Different from a magnetized NS consisting of `normal matter', a Kerr hole with the mass function, $M=M(S,J)$ in equation (<ref>) will be the biggest rotating mass of `entropy matter'. Then, a naive question comes to mind: “how do magnetic field lines manage to thread and survive in `entropy matter' under the horizon?"   If a battery really existed in the horizon, this might indeed necessitate the threading of FLs into the matter under  such as e.g., the `imperfect conductor' [Blandford & Znajek, 1977, Znajek, 1978].

The zeroth law of thermodynamics indicates that two `intensive' variables, $\Th$ (the surface temperature) and $\OmH$, conjugate to $S$ and $J$, respectively, are constant on , e.g., $\om\to\OmH$ for $\al\to 0$.  In passing, the third law indicates that “by a finite number of operations, one cannot reduce the surface temperature to the absolute zero with $h=1$."  In turn, “the finite processes of mass accretion with angular momentum cannot accomplish the extreme Kerr state with $h=1$, $\Th=0$ and $\OmH=c^3/2GM$" [Okamoto & Kaburaki , 1991]. Incidentally, the `inner-horizon' thermodynamics can formally be constructed analogously to the `outer-horizon' thermodynamics [Okamoto & Kaburaki , 1993, Cvetič et al., 2018].

It is the first and second laws that govern the extraction process of energy, i.e.,

\begin{eqnarray}	  \hspace{2cm}
c^2 dM=\Th dS + \OmH dJ ,  \label{1st-l} \\  % (2.4a)
\Th dS \geq 0 ,  % (2.4b)
\end{eqnarray}

where $\Th$ and $\OmH$ are uniquely expressed in terms of $J$ and $S$ from equation (<ref>) or $M$ and $h$ [Okamoto & Kaburaki , 1990];

=c^2 (∂M/∂S)_J,

=c^2 (∂M/∂J)_S.

When Kerr holes are regarded as a huge rotating mass of `entropy matter' confined by its self-gravity inside the event horizon, we conjecture that they will not allow the presence of magnetospheric or interstellar magnetic field lines threading or being anchored in the matter inside the horizon. Therefore, the Kerr hole itself will be unable to behave like a battery. It will nevertheless be argued here that Kerr holes can acquire and keep a force-free magnetosphere by making full use of frame-dragging.  It is indeed frame-dragging that acts as a bridge between (black hole) thermodynamics and (pulsar) electrodynamics across the event horizon.

§ THE FORCE-FREE APPROXIMATION AND THERMODYNAMICS

§.§ Kerr black hole force-free magnetospheres

The magnetosphere filled with perfectly conductive plasma is considered under the two basic presumptions of the force-free and freezing-in conditions around NSs and Kerr BHs in the stationary axisymmetric state with $\vcE_{\rm t} =0$. These two conditions are given by

\begin{eqnarray}
\vre\vcE+\vcj /c \times \vcB=0, \label{ff-fz}  \\  %  (3.1a)
\vcE+\vcv/c\times\vcB=0,  \label{fi-c}   % (3.1b)          %  \vcE\cdot\vcB=\vcj\cdot\vcE=\vcv\cdot\vcE =0. \label{deg} % % (IV.5c)
\end{eqnarray}

where $\vcB$ and $\vcE$ are the electromagnetic fields, $\vcj$ is the electric current and $\vcv$ is the velocity of `force-free', i.e., `massless' charged particles. All electromagnetic quantities are measured by the ZAMOs circulating with the FDAV $\om$. The magnetosphere is then characterized by the two quantities, $\OmF(\Psi)$ and $I(\Psi)$, which are conserved along each FLs,
where $\Psi$ is the stream function. Both of $\OmF$ and $\om$ are the AVs relative to absolute space [Macdonald & Thorne, 1982].

There are then the two pivotal FLAVs; one is naturally the conserved FLAV $\OmF$, and the other is the non-conserved, i.e., the ZAMO-measured gravito-FLAV $\OmFm$, given as follows;

\begin{eqnarray}
\OmFm=\OmF - \om,		\label{twoOmFa}  \\[1mm]  % (3.2a)
\OmF= \OmFm + \om.		\label{twoOmFb}  % (3.2b)
\end{eqnarray}

The former divides the force-free magnetosphere unequivocally into the two domains, i.e., the outer SC and inner GR domains, by the null surface , where $\OmFm\ggel 0$ for $\om\lleg\OmF$. This corresponds to the 2nd law of BH thermodynamics, through the Poynting flux $\vcSEM$ on the horizon  (see equation (<ref>) later). The latter will lead to the 1st law of BH thermodynamics on  where $\al\to 0$ and $\om\to \OmH$ (see equations (<ref>)). It seems that this is the first time to define the ZAMO-FLAV $\OmFm$ explicitly.

The poloidal and toroidal components of $\vcB$ and the electric field $\vcEp$ are defined by

\begin{eqnarray}  	%\hspace{1cm}
\vcBp=-(\uvt\times\vcnb\Psi/2\pi \vp), \ \ \vcBt=-(2I(\Psi)/\vp c\al)\uvt , \label{vcBp/Bt}	\\[1mm]  % (3.3a)
\vcEp=-(\OmFm/2\pi\al c)\vcnb\Psi , 	\label{ZAMO-Ep}  	 \\[1mm]  	% (3.3b)  \equiv  \vcEp^{\OmFm},
\vF=\OmFm\vp/ \al,	\label{ZAM0-vF}  % (3.3c)
\end{eqnarray}

where $\vcBt$ is regarded as the swept-back component of $\vcBp$ by `inertial loading' (i.e., particle acceleration and entropy production; see section  <ref>), and $\vF$ is the ZAMO-measured FLRV.  Because $\vF$ stands for the physical velocity of field lines relative to the ZAMOs, $\vcEp$ is entirely induced by the motion of the magnetic field lines, i.e., $\vcEp =-(\vF\uvt/c)\times \vcBp$. Equations (<ref>,b) correspond to Eqs. (5.2) and (5.3) for $v^{F}$ and $\vcE$ in Macdonald & Thorne, 1982, although the ZAMO-measured FLAV corresponding to $\OmFm$ was somehow undefined there.

The role of the FDAV $\om$ is thus to make the ZAMO-FLAV $\OmFm$ `violate' Ferraro's law of iso-rotation and change sign on , and hence $\vcEp$ in equation (<ref>) changes direction as well.  This leads to the elucidation of `how and where' the basic conditions given in equations (<ref>a,b) should be `broken down', thereby opening the route toward gravito-thermo-electrodynamics (GTED; see section <ref>).

The AV $\OmF$ possesses two-sidedness, i.e., the FLAV and the electric potential gradient, and $\OmF=-2\pi c(dA_0/d\Psi)=$constant along each FL. With respect to $I$, `current/angular-momentum duality' holds in the degenerate state.  From Eqs. (5.6a,b,c) in Macdonald & Thorne, 1982 for the current and charge densities, we have the poloidal and toroidal components of $\vcj$ and the charge density $\vre$

\begin{eqnarray}
\vcjp=-\frac{1}{\al}\dr{I}{\Psi}\vcBp,		\label{vcj-p}  \\[1mm]  % (3.4a)
\jt=\vre\vF + \frac{2 I}{\vp\al^2 c}\dr{I}{\Psi},   \label{vcj-t} \\[1mm]  % (3.4b)
\vre= -\frac{1}{8\pi^2 c} \nabla\cdot \left( \frac{\OmFm}{\al}\vcnb\Psi  \right). \label{bh/rho-e}   % (3.4c)
\end{eqnarray}

Because the hole's gravity produces a gravitational redshift of ZAMO clocks, their lapse of proper time $d\tau$ is related to the lapse of global time $dt$ by the lapse function $\al$, i.e., $d\tau/dt=\al$ [Macdonald & Thorne, 1982].

The angular momentum and energy fluxes are given in terms of two conserved quantities $\OmF(\Psi)$ and $I(\Psi)$;
=(Ψ),   =( I(Ψ)/2πc),

where the toroidal component of $\vcSE$ (and other fluxes) is and will be omitted throughout the paper.
These are apparently the same as the equations for the pulsar magnetosphere except for the redshift factor $\al$. The output power $\calPE$ and the loss rate of angular momentum $\calPJ$ observed by distant observers are given by

\begin{eqnarray}
\calPE= -c^2 \dr{M}{t} = \oint \al\vcSE\cdot d\vcA  = \frac{1}{c}\int^{\bar{\Psi}}_{\Psi_0}\OmF(\Psi) I(\Psi) d\Psi,  	\label{TotFluxE}  \\		% (3,6a)
\calPJ =-\dr{J}{t} = \oint \al\vcSJ \cdot d\vcA  = \frac{1}{c}\int^{\bar{\Psi}}_{\Psi_0} I(\Psi) d\Psi  %\hspace{0.8cm}
\label{TotFluxJ} 	% 	(3.6b)
\end{eqnarray}

(see Eqs. (3.89) and (3.90) in [Thorne et al., 1986]), where $\vcBp\cdot d\vcA=2\pi d\Psi$ and the integration is done over all open field lines in $\Psi_0\leq\Psi\leq \bar{\Psi}$.  We define the `overall' potential gradient, calculated from $\OmF(\Psi)$ weighted by $I(\Psi)$, i.e.,
= ∫^Ψ̅_Ψ_0 (Ψ) I(Ψ) dΨ/ ∫^Ψ̅_Ψ_0 I(Ψ) dΨ=/,

and then from equations (<ref>a,b) and (<ref>) we have

\begin{eqnarray}    \hspace{1cm}
c^2 dM=  \Th dS + \OmH dJ= \OmFb dJ,   \label{1st-(ii)} \\[1mm]   \hspace{1cm}   % (3.8a) <-----   (4.15a)
\Th dS= -(\OmH-\OmFb) dJ .  \label{2nd-(ii)}		 % % (3.8b)  <----  (4.15b)
\end{eqnarray}

Then, the 2nd law $\Th dS>0$ requires inequality $\OmH>\OmFb$ to be fulfilled always for the hole to lose energy through its force-free magnetosphere, i.e., $c^2 dM=\OmFb dJ<0$ (see equations (<ref>a,b)).

This situation indicates that the null surface  with $\omN=\OmF\approx \OmFb$ exists always, where $\OmFm=0$ (see equation (<ref>) later). Then, the outer SC domain $\calDout$ rotates forward ($\OmFm>0$), whereas the inner GR domain $\calDin$ rotates backward ($\OmFm<0$). Then, the influx of negative angular momentum into the horizon is equivalent to the positive angular momentum flux outwards to  together with positive energy with $c^2 dM=\OmFb dJ<0$. Thus the ZAMOs will see that each FL with the same $\OmF(\Psi)$ counter-rotates in between the two domains, $\calDout$ and $\calDin$, without losing a sense of congruity.

§.§ Pulsar force-free magnetospheres

It will be instructive to refer to the adiabatic-dragging of the pulsar force-free magnetosphere. The power and the rate of angular momentum loss in equations (<ref>a,b) are surely applicable to an NS with the surface AV $\OmNS$. The Poynting and angular-momentum fluxes are always outwards with no reversal of the direction of flows. The FLAV $\OmF$ is uniquely given by the `boundary condition' for FLs emanating from the surface  of a magnetized NS, i.e.,

and also the current function $I(\Psi)$ is specified by the `criticality condition' on the fast magneto-sonic surface S$_{\rm F}$ near infinity, i.e.,

(see equation (<ref>)), which is equivalent to Ohm's law for the surface current on the force-free infinity surface $\Sffinf$( $\lo$ ). Then, The total power is given by simply putting $\OmF=\OmNS$ and $I=I_{\rm NS}$ in equation (<ref>)

_E: NS =  1/c∫^Ψ̅_Ψ_0I_NS dΨ
=1/2c ∫_ S_ff∞  ^2(^2)_ff∞ dΨ.

The related EMF of the NS's battery is expressed in terms of potential gradient $\OmF$ by
which drives currents along the current-field-streamline $\Psit$ with $\vcjp>0$ and return currents along $\Psio$ with $\vcjp<0$, where $\Psiz<\Psio<\Psic<\Psit<\bar{\Psi}$ and $\vcjp\lleg 0$ for $\Psi\lleg\Psic$ (see equations (<ref>) and (<ref>)),  where $\bar{\Psi}$ is the last limiting field line satisfying $I(\Psiz)=I(\bar{\Psi})=0$ (see  figure 2 in Okamoto, 2006 for one example of $I(\Psi)$). The surface return currents flow from $I(\Psit)$ to $I(\Psio)$, crossing FLs between $\Psio$ and $\Psit$ on the resistive membrane $\Sffinf$, and the ohmic dissipation of surface current there formally represents particle acceleration taking place in the resistive membrane $ \Sffinf$ with resistivity $\calRinf$ (see equation (<ref>) later). That is, the conversion of field energy to kinetic energy takes place on $\Sffinf$ in the form of the MHD particle acceleration [Okamoto, 1974, Okamoto , 1978].

Now, we regard the toroidal field $\Bt=-(2I/\vp c)$ as the swept-back component of $\vcBp$ due to inertial loadings on the terminating surface $\Sffinf$ of the force-free domain. Thus, the behaviour of $I=I(\ell, \Psi)$ from the stellar surface to infinity will be described in the pulsar force-free magnetosphere as follows;

\begin{eqnarray}
I(\ell,\Psi) =  \left\{
\begin{array} {ll}
0 & ;\ell \lo \ell_{\rm NS},\ {\rm (no\ resistance)},  \\[1mm]
I_{\rm NS}(\Psi) &;\ \ell_{\rm NS} \lo \ell \lo \ell_{\rm F},  \ {\rm (FF\ region)}, \\[1mm]
\to 0 & ; \ell_{\rm F} \lo \ell \lo \ell_{\infty} \ {\rm (particle\ acceleration)}
\end{array}  \right.    \label{I/Pul-M}  \end{eqnarray}

(see equation (<ref>) for a Kerr hole's force-free magnetosphere).
We assume simply here that $I(\ell,\Psi)$ approaches zero for $\ell\to\infty$ or $\vp\to\infty$ along each FL. This means that all the Poynting energy is transferred eventually to the particle kinetic energy (see equation (<ref>) later).

NSs of `normal matter' are regarded as innately magnetized, and hence their magnetospheric  FLs will be anchored in the surface layer or crust.  When we think of the 1st law in `pulsar-thermo-electrodynamics' similar to the BH case, i.e.,
c^2 dM_NS=T_NSdS+_NSdJ,

we have $c^2 dM_{\rm NS}=\OmFb dJ$  by equation (<ref>),
for energy loss through the force-free magnetospheres. Then, we have for any entropy generation in the boundary layer

\[ T_{\rm NS}dS=-(\Om_{\rm NS}- \OmFb)dJ. \]

When $\OmF\approx \OmFb=\Om_{\rm NS}$, the `boundary condition' (<ref>) ensures $T_{\rm NS}dS=0$, that is, no entropy production near the boundary layer on . This is surely adiabatic extraction of energy from the NS through its force-free magnetosphere. There is naturally no necessity for the breakdown of the force-free condition anywhere (except but a negligible violation for a shortage of particles). In other words, the NS can `adiabatically' be dragging the force-free magnetosphere, through which the NS blows the magneto-centrifugal wind outward in the form of Poynting flux.

On the other hand, Kerr BHs consist of `entropy matter' confined within the event horizon, and there seem to be no theoretical nor observational pieces of evidence so far indicating that they are capable of being magnetized nor anchoring the magnetospheric FLs. The adiabatic extraction of energy $\OmF=\OmH$ will be excluded. Thus, for the force-free approximation to be applicable to the hole magnetosphere, it must and surely can break down, as argued so far in the null surface .

§ TOWARD BH GRAVITO-THERMO-ELECTRODYNAMICS

In order to clarify complicated properties of the force-free magnetosphere of Kerr holes, we use expediently the wind-, flux- and circuit-analyses, which must be complementary with each other. The key point is to keep the ZAMO's, i.e., the physical observer's viewpoint. The index toward GTEDn is, for example, given by the ZAMO-measured FLAV $\OmFm$ in equation (<ref>).

§.§ The wind-flow analysis

§.§.§ The velocity $\vcv$ of `force-free' particles

Combining the two conditions (<ref>a,b), with use of equations (<ref>a,b,c) for $\vcj$ and $\vre$, we have $\vcv=\vcj/\vre$ for the velocity of the `force-free' particles
=/= -1/IΨ+,

which indicates that FLs, current lines and stream lines are thus parallel to each other, and hence they (FCSLs)  must be equipotentials in the force-free domains. And yet, the `force-free' plasma must be charge-separated, and the role of `massless' or `inertia-free' particles is just to carry charges, exerting no dynamical effect.

On the other hand, the axial symmetry $\vcE_{\rm t}=0$ imposes $\vcjp\times\vcBp=0$ and $\vcvp\times\vcBp=0$, respectively, in equations (<ref>a,b), and hence $\vcjp=\eta\vcBp$ and $\vcvp=\kappa\vcBp$, where $\eta$ and $\kappa$ are a scalar function of the position along each FL. The `force-free' plasma is yet charge-separated, i.e., $\vcjp=\vre\vcvp=\vre\kappa\vcBp\equiv \eta\vcBp$.

The ZAMO-measured particle-velocity $\vcv$ is summarized as follows;

\begin{eqnarray}
\hspace{3cm}		\vcv=\kappa\vcB+\vF \uvt, \label{vcv} \\  % (4.2a)
\hspace{3cm}		\vcvp=\kappa \Bp, \quad \vvt=\kappa\Bt +\vF, \label{vcvp/t} \\ % (4.2b)
%	\vF=\OmFm\vp /\al, \label{vp/vt} \\
\hspace{3cm}	\kp=-(1/\vre\al) (dI/d\Psi),  \label{kappa}  % (4.2c)
\end{eqnarray}

where $\vF$ is given by equation (<ref>).

§.§.§ The null surface  between two light surfaces

When the ZAMOs measures along each FL in case of $\OmH>\OmF>0$, $\OmFm$ increases from $-(\OmH-\OmF)$ on , beyond null on  where $\om=\omN$, to $\OmF$ on  on the same FL, because $\om$ decreases from $\OmH$ on  to zero on .  Thus, by putting $\vF=\mp c$ in equation (<ref>), we have
=+ c(/)_iL,    =- c (/)_oL,

that is, there are the two, inner and outer, light surfaces,  and , on both sides of  where $\vF=0$. For an arrangement of these characteristic surfaces, we see
>>=>> 0

from the horizon to infinity along the same FL,  where $\omiF$ and $\omoF$ are the values of $\om$ on the inner and outer fast-magnetosonic surfaces  and  (see section  <ref>).  We can thus `coordinatize' $\OmFm$ as well as $\om$ along each FL (see e.g. the horizontal axis of figure <ref>).

Each FL prograde-rotates in the outer domain $\calDout$ where $\omN \ggo \om\ggo 0$, i.e., $0 \lo \OmFm \lo \OmF$, and the same FL retrograde-rotates  in the inner domain $\calDin$ where $\OmH \ggo\om\ggo \omN $, i.e., $0\ggo \OmFm \ggo (\OmFm)_{\rm H}=-(\OmH-\OmF)$. This situation explains that in the former, the normal magnetocentrifugal wind of force-free particles flows passing outwardly through  toward  $\lo$ , while in the latter, the gravito-magnetocentrifugal wind of force-free particles flows passing inwardly through  towards  $\ggo$ .

It was already pointed out ([Blandford & Znajek, 1977]; the footnote at p.443) that “The outer light surface corresponds to the conventional pulsar light surface and physical particles must travel radially outwards beyond it. Within the inner light surface, whose existence can be attributed to the dragging of inertial frames and gravitational redshift, particles must travel radially inwards.”
It is because of the counter rotation of $\calDin$ by frame-dragging that the null surface  exists between the two light surfaces,  and  on the same FLs. Thus we see `$\om_{\rm oL} < \OmF <  \om_{\rm iL}$', which will correspond to an inequality `$\Omega_{\rm min}<\Omega< \Omega_{\rm max}$' given below equation (15) in Blandford & Globus, 2022. Also see equations (<ref>a,b) and (<ref>) for the behaviours of ,  and , and $\xiL\to 1$, $\xM\to 1.2599$ and $\xoL\to\infty$ for the slow-rotation limit of $h\to 0$.

We can find a related statement in Punsly & Coroniti, 1990 that “the magnetospheric plasma is produced in the spacial region between  and ; at the  (), the magnetic field rotates backwards (forwards) at the speed of light relative the plasma", and actually this is consistent with the existence between  and  of the null surface  where $\OmFm\ggel 0$.

This means that the breakdown of the force-free and freezing-in conditions must take place, i.e., $\vcv\ggel 0$ for $\OmF\ggel 0$ on  (see  section  <ref>).  We must also formulate the `boundary condition' to determine the final eigenvalue of $\OmF=\omN$ in some way or another (see section <ref>).

§.§.§ The eigen-functions $\Iout$ and $\Iin$ due to the criticality condition on the fast-magnetosonic surfaces

Let us determine the current/ angular momentum function $I(\Psi)$ for the two SC and GR domains. We `terminate' the force-free domains on the resistive membranes $\Sffinf$ and $\SffH$ near  and , respectively (exactly speaking, on the outer and inner fast-magnetosonic surfaces  and  (see, e.g., [Okamoto , 1978, Kennel, et al., 1983, Punsly & Coroniti, 1990]).

We have the behaviors of $\vcB$ and $\vcEp$ toward  and  by  equation (<ref>,b)
as follows;

\[ \vcB^2=\vcBp^2 +\vcBt^2=(2I/\vp\al c)^2+(\Bp\vp)^2/\vp^4     \]

\begin{eqnarray}
\simeq  \left\{
\begin{array} {ll}
(2\Iout /\vp c)^2 &; \Sffinf, \    \al \to1, \  \vp\to \infty, \\[1mm]
(2\Iin/\vp\al c)^2 &; \SffH, \  \al\to 0, \ \om\to \OmH,
\end{array}  \right.    \label{Iout/inA}  \end{eqnarray}

^2= (/c)^2 ^2
\begin{eqnarray}
\simeq  \left\{
\begin{array} {ll}
\left( \displaystyle{\frac{\OmF}{\vp c}}\right)^2 (\Bp\vp^2)^2 &; \Sffinf, \    \al\simeq 1, \  \vp\to \infty, \\[2mm]
\left( \displaystyle{\frac{\OmH-\OmF}{\al\vp c}}\right)^2 (\Bp\vp^2)^2 &; \SffH, \  \al\to 0, \ \om\to \OmH,
\end{array}  \right.    \label{Iout/inB}  \end{eqnarray}

where S$_{\rm oF}\lo\Sffinf$ and S$_{\rm iF}\ggo\SffH$.
Then $(\vcB^2-\vcE^2)\to 0$ reduces to the so-called `criticality condition'
for $\Iout$ and $\Iin$ (see Eqs. (6.6b) and (6.8b) in [Okamoto, 1992]; also [Znajek, 1977, Macdonald & Thorne, 1982]), as follows;

\begin{eqnarray}
\Iout =\displaystyle{ (1/2)\OmF(\Bp\vp^2)_{{\rm ff}\infty}}  &;\ \Sffinf, \label{Iout/in-a} \\[1mm]  %\hspace{1cm}
\Iin =	(1/2) (\OmH -\OmF)(\Bp\vp^2)_{{\rm ffH}}  &;\ \SffH.    \label{Iout/in-b}
\end{eqnarray}

The former $\Iout$ expresses the external resistance of particle acceleration on the resistive membrane $\Sffinf$ (see equation (<ref>) for $I_{\rm NS}(\Psi)$ in the pulsar case).  The latter $\Iin$ does another external resistance of entropy production on the resistive membrane $\SffH$ above the horizon, but this is not an internal resistance of a horizon battery (if any) (see section <ref> for the circuit-analysis).

By equations (<ref>a,b), the energy and angular momentum fluxes in equations (<ref>a,b) possess different forms in the outer and inner domains along each FL with FLAV $\OmF(\Psi)$;
= 1/c∫^Ψ̅_Ψ_0 dΨ,
= 1/c∫^Ψ̅_Ψ_0 dΨ
= 1/c∫^Ψ̅_Ψ_0 dΨ,   = 1/c∫^Ψ̅_Ψ_0 dΨ.

It must be on the null surface  that the influx of negative angular momentum in $\calDin$ (or equivalently the outward flux of positive one) must cancel out the outward flux of positive angular momentum in $\calDout$,
i.e., $\vcSJout= \vcSJin= -\vcSJinU$.  This condition $\Iout=\Iin$ turns out to yield the `boundary condition' to finally determine $\omN=\OmF$ for the whole magnetosphere frame-dragged into rotation with $\OmF=\omN$ (see section <ref>).

Next, the surface currents $\calI_{{\rm ff}\infty}$ and $\calI_{\rm ffH}$ flowing on the resistive membranes $\Sffinf$ and $\SffH$ with the surface resistivity $\RH=\calRinf=4\pi/c$ are defined as follows;

\begin{eqnarray}
\calI_{\rm ff{\infty}}=\left(\frac{\Iout}{2\pi \vp}\right)_{\rm ff{\infty}}
=\left(\frac{c}{4\pi} \frac{\OmF\vp}{c} \Bp\right)_{\rm ff{\infty}} =\left(\frac{E_{\rm p}}{\calRinf}\right)_{\rm ff{\infty}}, \nonumber \\[1mm]
\label{calIffinf}  \end{eqnarray}

\begin{eqnarray}
\calI_{\rm ffH}= \left( \frac{\Iin}{2\pi \vp} \right)_{\rm ffH} % \nonumber \hspace{3.5cm} \\[1mm]
=\left( \frac{c}{4\pi}  \frac{(\OmH-\OmF)\vp}{c} \Bp \right)_{\rm ffH}
=\left( \frac{E_{\rm p}}{\RH} \right)_{\rm ffH}.	 \nonumber \\[1mm]
\label{calIffH} 		% (4.10b)}
\end{eqnarray}

Ohmic dissipation of these two surface currents corresponds to particle acceleration and entropy production in each closed-circuit $\calCout$ and $\calCin$ (see equations (<ref>,b,c)).

The $\OmFm$-dependence of three energy fluxes $\vcSE$, $\vcSEM$ and $\vcSsd$ along each field-current-stream-lines (FCSLs) (modified from figure 3 in Okamoto, 2009).  The point is to `visualize' the effect of frame-dragging by `coordinatizing' $\OmFm$ as well as $\om$ (see  equation (<ref>)).
The conserved energy flux is shown by $\vcSE=\OmF\vcSJ$, whose two components are given by $\vcSEM=\OmFm\vcSJ$ for the Electro-Magnetic Poynting flux and $\vcSsd=\om\vcSJ$ for the frame-dragging Spin-Down energy flux, respectively, where $\vcSJ$ is the angular momentum flux (see equations (<ref>)$\sim$(<ref>)).  The ordinate $\vcEp=\OmFm=0$ expresses the null surface  ($\om=\OmF$) [Okamoto, 1992], which divides the force-free magnetosphere to the two, outer prograde-rotating and inner retrograde-rotating, domains, $\calDout$ and $\calDin$, with the respective light surfaces  and  (see  section  <ref> and equation (<ref>)).  The three energy fluxes are linked to the three terms of the first law of thermodynamics on the horizon under the resistive membrane $\SffH$.
It looks as if the hole were launching positive angular momentum $\vcSJ>0$ toward , when it is accepting an influx of negative angular momentum from the plasma-shed on  (see  section  <ref>).
Note that no reversal of the conserved flux $\vcSE=\OmF\vcSJ$ takes place, although $\vcSE=\vcSEM=\vcSsd =\vcSJ=0$ on , due to breakdown of the force-free condition (see section <ref> and  figure <ref>), while its two component fluxes $\vcSEM$ and $\vcSsd$ are not conserved. Indeed $\vcSEM$ reverses direction because the radiation condition for the Poynting flux is given by the sign of $\OmFm\ggel 0$. A pair of batteries as well as particle source must exist under the inductive membrane $\SN$ (see figures <ref> and <ref>).

§.§ The energy-flux analysis

§.§.§ Two `non-conserved' energy fluxes $\vcSEM$ and $\vcSsd$

As seen in equation (<ref>), the conserved FLAV $\OmF(\Psi)$ is resolved by frame-dragging to the ZAMO-FLAV $\OmFm$ and the FDAV $\om$ (both are a  `gravito-electric potential gradient', also). Substituting $\OmF$ in equation (<ref>) into equation (<ref>), we have

\begin{eqnarray}  \ \ \ 	%\hspace{1.5cm}
\vcSE=\OmF \vcSJ =(\OmF I/2\pi\al c) \vcBp, \label{Se}  \\[1mm]  		% (4.12a)
\vcSEM=\OmFm \vcSJ=(\OmFm I/2\pi\al c)\vcBp, 	 \label{Sem} 	\\[1mm]    % (4.12b)
\vcSsd =\om \vcSJ=(\om I/ 2\pi\al c) \vcBp    \label{Ssd}   		% (4.12c)
\end{eqnarray}

(see Eq. (4.13) in [Macdonald & Thorne, 1982][The two non-conserved energy fluxes $\vcSEM$ and $\vcSsd$ as well as the ZAMO-measured FLAV $\OmFm$ are somehow not defined explicitly. This unfortunate omission of these quantities may have blocked the path to GTED.]).

The Electromagnetic Poynting flux $\vcSEM$ can be simply derived from the vector product of $\vcBt$ and $\vcEp$ in equations (<ref>,b). The Spin-Down energy flux $\vcSsd$ is another non-conserved energy flux.  Just as the overall energy flux $\vcSE$ corresponds to $c^2 dM$ in equation (<ref>) for the 1st law, $\vcSEM$ and $\vcSsd$ correspond to $\Th dS$ and $\OmH dJ$ on the horizon.

Note that $\vcSE$ is directed always outward, that is, “the direction of energy flow cannot reverse on any given field line unless the force-free condition breaks down" [Blandford & Znajek, 1977], but the ZAMOs or the physical observers are aware that $\vcSE$ consists of $\vcSEM=\OmFm\vcSJ$ and $\vcSsd=\om\vcSJ$. In particular, the Poynting flux $\vcSEM$ reverses direction on the null surface , where $\OmFm\ggel 0$ (see figure <ref>), and hence the force-free condition must break down there. Indeed, it will be argued that breakdown takes place on  (see sections <ref>), and a relevant pair of `surface' unipolar induction batteries on both sides of  will be set up there, back-to-back, each other and yet oppositely directed. In addition,  a particle source related can be excavated under the null surface  between the two light surfaces $\omoL$ and $\omiL$ (see subsections <ref> and <ref>).

By equations (<ref>) and (<ref>) for $\vcEp$ and $\vcSEM$, we can confirm that “Physical observers will see the electric field reverse direction on the surface $\omN=\OmF$. Inside this surface, they see a Poynting flux of energy going toward the hole. (..., when the hole is losing energy electromagnetically, this surface always exists.)” (see the caption of Figure 2, [Blandford & Znajek, 1977]). Also, outside this surface, they will see another Poynting flux going toward infinity (see figure <ref>). They will then understand that “a sufficiently strong flux of angular momentum leaving the hole" will be equivalent to “a sufficiently strong influx of negative angular momentum leaving this surface  inwardly".

§.§.§ Entropy production

On entropy production on the resistive membrane $\SffH$, utilizing $\Iin$ in equation (<ref>) and $\vcSEM$ in equation (<ref>), we have

\begin{eqnarray}
\TTH \frac{dS}{dt} = - \oint_{{\rm S}_{\rm ffH}} \al\vcSEM\cdot d\vcA  	 % \\[1mm]  % (4.13a)
= \frac{1}{c}\int^{\bar{\Psi}}_{\Psi_0}(\OmH-\OmF) I(\Psi) d\Psi  	  \\[1mm]  % (4.13a)
=-(\OmH-\OmFb) \dr{J}{t}>0, \label{ThdS/dt}    % (4.13b)
\end{eqnarray}

where $\vcBp\cdot d\vcA=2\pi d\Psi$. This is equivalent to the ohmic dissipation of the surface current $\calI_{\rm ffH}$ on the resistive membrane with the resistivity of $\RH=4\pi/c=377$ohm, i.e.,
dS/dt = ∫_S_ffH _ffH^2 dA
=1/2 c ∫_Ψ_0^ (-)^2(^2)_ffH dΨ

(see Eq. (3.99) in [Thorne et al., 1986]).

Thus, when the hole loses angular momentum by $dJ$, the hole loses energy $\OmFb dJ$, which is carried by the conserved flux $\vcSE\approx \vcSEM$ beyond   toward the resistive membrane $\Sffinf$, to convert to the kinetic energy of particles. The amount of energy paid back to the hole as its cost of extraction is given by $\Th dS=-(\OmH-\OmFb)dJ$ (see equation (<ref>a,b)).

§.§.§ The efficiency of energy extraction

The `overall' efficiency $\epsGTEb$ is defined by the ratio of `actual energy extracted to maximum extractable energy' when unit angular momentum is removed (cf. [Blandford & Znajek, 1977]), i.e. from equations (<ref>), (<ref>a,b) and (<ref>)
=(dM/dJ)/ (∂M/∂J)_S=/ =/.

The constraint of the energy extraction and its efficiency become by equation (<ref>)

\begin{eqnarray}    \hspace{1cm}
\OmH>\OmF\approx \OmFb > 0 ,  \label{1st-(iii)} \\[1mm]   \hspace{1cm}   % (4.17a)
\epsGTE=\frac{\OmF}{\OmH} \approx \epsGTEb=\frac{\calPE}{\OmH\calPJ}=\frac{\OmFb}{\OmH} < 1.\label{2nd-(iii)} % (4.17b)
\end{eqnarray}

These inequalities ensure that “when the hole is losing energy electromagnetically, the null surface  on $\om=\OmF$ always exists" (see Eqs. (4.6) and (4.7) in [Blandford & Znajek, 1977]).

A schematic picture illustrating a pair of circuits $\calCout$ and $\calCin$ closed in the force-free domains $\calDout$ and $\calDin$.
These domains are separated by the non-force-free Gap $\SG$, where the breakdown of the force-free condition severs current- and stream-lines, i.e., $\PG{\vcj}=\PG{\vcv}=0$ (see equation (<ref>a,b)).
There will be dual unipolar inductors with EMFs, $\calEout$ and $\calEin$, at work related to the magnetic spin-axes oppositely directed.
The AVs of the magnetic axes are $\OmF$ and $-(\OmH-\OmF)$, respectively, and the former is parallel to the hole's spin axis, and the latter is anti-parallel. The difference of the two is $\OmH$ (see  equation (<ref>);
figures <ref> and <ref>).
Note that $\vcvp=\vcjp/\vre>0$ in the prograde-rotating $\calDout$ and $\vcvp <0$ in the retrograde-rotating $\calDin$, because the magnetic-sling shot works outwardly in the former ($\OmFm>0$) and inwardly in the latter ($\OmFm<0$).  Along the FCSL $\Psio$ where $\vcjp<0$, electrons flow out in $\calDout$, and positrons flow in $\calDin$, each charge-separated when these particles flow out from the plasma-shed to both directions (see section <ref>)).  The opposite is true along the FCSL $\Psit$ where $\vcjp>0$; positrons flow outward in $\calDout$ and electrons flow inward in $\calDin$. It is a huge voltage drop of $\Dl V\propto \OmH$ (see equation (<ref>)) that will lead to viable particle production of ample plasma particles towards the development of a dense Gap with the half-width $\Dlom$ (see figure <ref>). The particles with $\PG{\vcv}=0$, circulating around the hole's axis with $\omN$, are the ZAM-particles, dense enough to pin down magnetic field lines, to fix $\OmF=\omN$ and to make the Gap magnetized, thereby enabling the dual batteries to drive currents in each circuit (see figure <ref>). It is conjectured in the twin-pulsar model (see section <ref>) that the outer half of the Gap in $0\lo\OmFm\lo\Dlom$ plays a role of a `normal' magnetized NS spinning with $\OmF$, while the inner half in $0\ggo\OmFm\ggo -\Dlom$ behaves like an `abnormal' magnetized NS counter-spinning with $-(\OmH-\OmF)$. Relaxation of the RTD (see section <ref>) in-between will lead to widening of the null surface  to the Gap $\SG$ with width $\simeq$ 2$\Dlom$.

§.§ The circuit analysis

§.§.§ The current-closure condition

We impose no net gain nor loss of charges over any closed surface in the force-free domains in the steady axisymmetric state. For the closed surface from the first open FL $\Psi=\Psiz$ to the last open FL $\Psi=\bar{\Psi}$ in the poloidal plane, we have by equation (<ref>)
∮·d∝[I()-I()]=0, i.e.,

when there is no line current at $\Psi=\Psiz$, nor $=\Psib$. This requires that function $I(\Psi)$ has at least one extremum at $\Psi=\Psic$ where $(dI/d\Psi)_{\rm c}=0$ (see Fig. 2 in [Okamoto, 2006] for one example of $I(\Psi)$), and hence
== -1/IΨ{

<0;   <Ψ< ,

=0;    Ψ=,

>0;    <Ψ<,

where $\Psiz<\Psio<\Psic<\Psit<\Psib$  (see figure <ref> and section <ref> for the two circuits $\calCout$ and $\calCin$ in the outer and inner domains).

§.§.§ A pair of batteries for the dual circuits

The breakdown of the two conditions (<ref>a,b) in-between the two domains will provide an arena of setting up a pair of batteries and the voltage drop between their EMFs for particle production [Okamoto, 2015]. The covering surface is referred to as the inductive membrane $\SN$, as opposed to the two resistive membranes $ \Sffinf$ and $\SffH$ in both ends of the force-free domains (see figures <ref> and <ref>).

Let us then pick up such field-current-stream-lines (FCSLs) $\Psio$ and $\Psit$ for the circuits $\calCout$ and $\calCin$ as the two roots of an algebraic equation  $I(\Psi)=\Iot$, i.e., $I(\Psi_1)=I(\Psi_2)\equiv \Iot$ in the range of $0<\Psio<\Psic<\Psit<\bar{\Psi}$ (see  equation (<ref>)), where $(dI/d\Psi)_{\rm c}=0$ and $\vcjp\lleg 0$ for $\Psi\lleg\Psic$ (see  figure <ref>).

The Faraday path integrals of $\vcEp$ in equation (<ref>)

along two circuits, $\calCout$ and $\calCin$, yield

\begin{eqnarray}
\calEout= \oint_{{\cal C}_{\rm (out)}} \al\vcEp\cdot d\vcell
=-\frac{1}{2\pi c}\int_{\Psi_1}^{\Psi_2} \OmF(\Psi)d \Psi, \hspace{0.5cm} \label{EMF-out} \hspace{0.3cm}  \\ % (4.19a)
\calEin=\oint_{{\cal C}_{\rm (in)}} \al\vcEp\cdot d\vcell
=+\frac{1}{2\pi c}\int_{\Psi_1}^{\Psi_2} (\OmH-\OmF)d \Psi .\hspace{0.5cm}  \label{EMF-in} \hspace{0.33cm} % (4.19b)
\end{eqnarray}

There is no contribution to EMFs from integrating along $\Psio$ and $\Psit$ and on the null surface  because of $\vcEp\cdot d\vcell=\PN{\vcEp}=0$.
These batteries on both outer and inner surfaces of  with no internal resistance provide electricity to the outer resistances on the resistive membranes $\Sffinf$ and $\SffH$.
That is, the EMFs for the two DC circuits $\calCout$ and $\calCin$ drive the volume current $\vcj$ (see  equation (<ref>)) in the force-free domain $\calDout$ and $\calDin$ and the surface membrane currents $\calI_{\rm ff{\infty}}$ and $\calI_{\rm ffH}$ on $\Sffinf$ and $\SffH$ (see equations (<ref>a,b)).  No volume current is allowed to cross the null surface  (or the Gap $\GN$) by the breakdown of the force-free condition from one circuit to another, i.e., $\PN{\vcj}=0$ (see equation (<ref>)).

§.§.§ Two outer and inner resistances

The outer resistive membrane $\Sffinf$ may also be interpreted as possessing the same surface resistivity $\calRinf=4\pi/c=377$ Ohm as on the inner membrane $\SffH$ above , and Ohm's law holds on $\Sffinf$, i.e., $\calRinf \calI_{\rm ff{\infty}}=(E_{\rm p})_{\rm ff{\infty}}$. Thus similarly to equation (<ref>) and by equation (<ref>b)
we have

\begin{eqnarray}
\int_{ {\cal S}_{{\rm ff}\infty } } \calRinf \calI_{\rm ff{\infty}} ^2 dA
=  \int_{ {\cal S}_{{\rm ff}\infty } }  \vccalI_{\rm ff{\infty}}\cdot\vcEp  dA
=  \int_{ {\cal S}_{{\rm ff}\infty } } \vcSEM\cdot d\vcA  \nonumber  \\[1mm]
=\frac{1}{2c} \int_{ {\cal S}_{{\rm ff}\infty } } \OmF^2(\Bp\vp^2)_{{\rm ff}\infty} d\Psi =\calPEout ,   \hspace{1cm}
\label{Sffinf-M}  \end{eqnarray}

where $\vcSEM=\vcSE$ and $\vcSsd=0$ on $\Sffinf$ for $\om\to 0$. On the other hand, from equation (<ref>), one has on the horizon resistive membrane $\SffH$

\begin{eqnarray}
\int_{{\cal S}_{\rm ffH}} \RH \calI_{\rm ffH}^2 dA= \TTH \frac{dS}{dt}  =\OmH\calPJin -\calPEin.
\label{ThdS/dt-b}  % (4.20b))
\end{eqnarray}

The above two expressions sum up to

\begin{eqnarray}
\int_{{\cal S}_{\rm ffH}} \RH \calI_{\rm ffH}^2 dA + \int_{ {\cal S}_{{\rm ff}\infty } } \RH \calI_{\rm ff{\infty}} ^2 dA = \OmH\calPJin
\end{eqnarray}

because $\calPEout=\calPEin=- c^2(dM/dt)$ and $\calPJin=\calPJout= -(dJ/dt)$ hold across the Gap $\GN$ with $\PG{I}=0$ by the `boundary conditions' (see equations (<ref>)), and hence we have $c^2 dM=\Th dS+ \OmH dJ$. It turns out thus that the first law of thermodynamics participates directly in ohmic dissipation of the surface currents for entropy production and particle acceleration on the two resistive membranes $\SffH$ and $\Sffinf$ (see equation (<ref>)).

The two EMFs in equations (<ref>a,b) for circuits $\calCout$ and $\calCin$ are also responsible for launching the Poynting energy fluxes in both outward and inward directions, i.e., $\vcSEM\ggel 0$ for $\OmFm\ggel 0$.  Equation (<ref>) for $\calEout$ coincides `almost exactly' with  equation (<ref>) for a pulsar magnetosphere, because of $\om\ll \omN=\OmF$ and hence $\vcSEMout\approx\vcSE$ in the outer SC domain $\calDout$.

This ohmic dissipation implies particle acceleration. The rate per unit $\tau$ time at which electromagnetic fields transfer redshifted energy to particles is by Eq. (4.14) in Macdonald & Thorne, 1982;

\begin{eqnarray}
-\frac{1}{\al} \vcnb\cdot\al\vcSE=\al\vcj\cdot\vcE+(\om/c)(\vcj\times\vcB)\cdot \vcm   \hspace{0.5cm}   \nonumber \\
=\frac{\OmF\vp}{c}\jvl\Bp  %  \nonumber \\
=- \frac{\OmF\vp}{c}\frac{\Bp}{2\pi\vp\al}\LPPlDr{\Iout}{\ell}>0.
\label{DivvcSE} \end{eqnarray}

This shows that when the current function $I(\ell,\Psi)$ is continuously decreasing with $\ell$ in the resistive membrane $\Sffinf$ from near  to , the MHD acceleration will take place (see figure <ref>), but the force-free magnetosphere regards the `force-free' domain with $\jvl=0$ formally as extending to the force-free infinity surface $\Sffinf$ where $|\jvl|\gg |\jpl|$. By doing so, the circuit $\calCout$ closes so as to fulfil the current-closure condition in the steady state.

§.§.§ Particle production due to voltage drop

The difference between the two EMFs in equations (<ref>a,b) across   is

where $\Dl\Psi=\Psio-\Psit$, and the difference in a quantity $X$ across  is denoted with
X=(X)_N(out)  - (X)_N(in).

The difference of the ZAMO-FLAV $\OmFm$ between  and  becomes from equation (<ref>)

\begin{eqnarray}
(\OmFm)_{\infty}-(\OmFm)_{\rm H}  \nonumber \hspace{4cm} \\[1mm]   \hspace{0.5cm}
=\OmF- [-(\OmH-\OmF)]=\OmF + (\OmH-\OmF)= \OmH.
\label{DOmFm}	\end{eqnarray}

The `spark' models so far used for pair-production discharges in the literature are based mainly on an extension from a `negligible violation' of the force-free condition [Blandford & Znajek, 1977, Macdonald & Thorne, 1982, Phinney, 1983, Beskin et al., 1992, Hirotani & Okamoto, 1998, Song et al., 2017, Hirotani et al., 2018, Sitarz et al., 2024]; also see Ruffini, et al., 2010, Chen & Fiuza, 2023 for general review.
It is argued here that the `complete violation' of the force-free condition due to frame-dragging on the null surface  leads to a unique gap model for the particle-current sources. It was emphasized already in  Okamoto, 2015 that “the present gap model with a pair of batteries and a strong voltage drop is fundamentally different from any existing models based on pulsar outer-gap models."  The significant differences from the previous particle production mechanism come mainly from the existence of the counter-rotating inner domain $\calDin$ due to frame-dragging, with $\epsJ\leq 0$ inside  (see section <ref>).

The null surface  dividing the force-free magnetosphere into the two, GR and SC, domains seems to be genetically endowed with the discontinuity $\Dl(\OmFm)_{\rm N}\approx (\OmFm)_{\infty}-(\OmFm)_{\rm H}=\OmH$ and $\DN{\calE}=-\Dl V$, to widen the surface  to a gap $\SG$ under $\SN$, thereby constructing a magnetized `' and `charge-neutral' Gap between the two force-free domains with $\epsJ\ggel 0$ (see  section   <ref>).
Thus, the voltage drop $\Dl V$ in equation (<ref>) reveals that the null surface  will be a kind of rotational-tangential-discontinuity (RTD) due to the two, inner and outer, magnetic rotators, namely between the two, counter-rotating each other, outer and inner domains $\calDout$ and $\calDin$,  although $\OmFm$ and $\vcEp$ seem to change sign smoothly through zero (see  section   <ref>; cf. <cit.>).
It turns out thus that the maximum available voltage drop $\Dl V=(\OmH/2 \pi c)\Dl\Psi$ will be utilizable in-between the two circuits $\calC_{\rm out}$ and $\calC_{\rm in}$ for particle production (see section <ref>).

§ THE ENERGY AND ANGULAR-MOMENTUM DENSITIES OF THE ELECTROMAGNETIC FIELDS

For the densities of the field energy and angular momentum, substituting $\vcB$ and $\vcEp$ from Eqs. (<ref>,b) into equations (2.30a) and (2.31a) in Macdonald & Thorne, 1982, we have

\begin{eqnarray}
\epsE= \frac{\al\Bp^2}{8\pi} \left[1+\frac{\Bt^2}{\Bp^2} +\frac{\vp^2}{\al^2 c^2}(\OmF^2 -\om^2 )\right] ,  \label{epsEba} \\ % (5.1a)
\epsJ=\frac{\vp\vF}{c}\Bp^2 = \frac{\OmFm (\vp\Bp)^2}{\al c}  \label{epsJbb}  % (5.1b)
\end{eqnarray}

(also see Eq. (2.17a) in Okamoto, 1992 and Eq. (55) in Komissarov, 2009), where $\epsE$ and $\epsJ$ are regarded as an `explicit' function of $\om$ along each field FL labeled with $\Psi$. The angular momentum density $\epsJ$ changes sign on the null surface , and we refer to the zero-angular-momentum-density surface as , which accords with .

Near the null surface  where $\om\approx\OmF$ and $\vp\Bt\propto I\approx 0$ (see  section  <ref>), we see
≈(^2/8π)_N>0,   ≈0,

The field energy will be strong enough to magnetize the plasma pair-produced with the voltage drop $\DN{\calE}=-\Dl V$ between the two EMFs in  equation (<ref>). Conversely, the plasma density will be large enough to keep the field $\vcBp$ frozen-in to ensure the magnetosphere frame-dragged by $\OmF=\omN$.

It is the frame-dragging term $\om^2$ in equation (<ref>) that builds a negative-energy region with $\epsE<0$ in the inner domain $\calDin$.
At the inner light surface , where $\vF=-c$ and $\omiL$ is given by equation (<ref>). When $(\Bp\vp^2)_{\rm ffH}/ (\Bp\vp^2)_{\rm iL}\approx 1$ and $\OmF\approx 0.5\OmH$, we have analytically from equation (<ref>)

≈-( ^2/8π/c )_iL   [2- (/c )_iL ],

and then we see $(\epsE)_{\rm iL}<0$, if $(\OmF\vp/c\al)_{\rm iL}<2$, i.e., $\omN<\omiL/2$.

Also we see from (<ref>) that there will be such a surface $\SepsE$ that divides the inner domain $\calDin$ farther into the two regions by $\epsE(\om,\Psi)\ggel 0$ for $\om \lleg \omepsE$ between  and $\SffH\approx$, where $\omepsE$ is a solution of equation for $\epsE=0$ in  equation (<ref>), i.e.,
^2 = ^2 +^2 c^2/^2 (1+^2/^2 ),

where $\al/\vp$ and $\Bt/\Bp$ are thought of as functions of $\om$ and $\Psi$.
This obviously indicates $\OmH> \omepsE>\omN=\OmF$ (see equation (<ref>) and $\al\to 0$ for $\om\to\OmH$). Therefore, it is the frame-dragging that produces not only the inner domain $\calDin$ of $\OmFm \leq 0$ with but also a region of the negative-energy density of $\epsE\leq 0$ in $\OmH\geq\om\geq \omepsE$. We guess $\omepsE\approx \omiL$.

Near the the resistive horizon membrane $\SffH$ where $(\Bp\vp^2)_{\rm ffH}/(\Bp\vp^2)$ $\approx 1$ and hence $\Bt^2/\Bp^2\approx
(2\Iin/\vp c\al\Bp)^2\approx ((\OmH-\OmF)\vp/(\al c))^2$ by equation (<ref>), we have
$$ \epsE \approx -\left(\frac{\Bp^2}{4\pi \al} \frac{\OmF\OmH\vp^2}{c^2}\right)_{\rm ffH} \left[\left(1-\frac{\OmF}{\OmH}\right) -\left( \frac{c^2\al^2}{2\OmF\OmH\vp^2}\right)_{\rm ffH} \right] $$

≈- (1-/)(^2/4π ^2/c^2 )_ffH <0

for $\al\to 0$ toward the resistive horizon membrane $\SffH$.

For the density of angular momentum near the horizon, we have
=-(-)  ()^2c_ffH<0.

It will be evident in equation (<ref>) that $\OmH> \omepsE >\omN=\OmF$, which does not lead to any more robust condition upon $\OmF$ than those from the second law and the radiation condition toward the horizon.

The above result suggests that the `negative-energy region' will surely extend from near  to  in the inner domain $\calDin$. What is essential for extraction of energy is the `inflow of negative angular momentum' from  (equivalent to the outflow of positive angular momentum from ) because this is followed by the `inflow of negative energy' (equivalent to the outflow of positive energy). Therefore, the existence itself of the `negative energy region' near the horizon necessarily may not be an exact or direct indicator of energy extraction from a Kerr hole. Important is the evidence that the force-free magnetosphere is divided into the two, prograde-rotating SC and retrograde-rotating GR, domains by the null surface , which coincides with the `Zero-Angular-Momentum-Density surface'  where $\epsJ=\vF=\OmFm=0$ and $\epsE>0$.

§ BREAKDOWN OF THE FORCE-FREE AND FREEZING-IN CONDITIONS

The force-free condition must be broken somewhere in the active force-free magnetosphere of a Kerr BH [Uchida, 1997].
The ZAMOs see that the electric field $\vcEp$ and hence the Poynting flux $\vcSEM=\OmFm\vcSJ$ reverse direction on every FLs threading the null surface , because of the counter-rotation of the inner GR domain ($\OmFm<0$) against the outer SC domain ($\OmFm>0$). Also, they will understand why a pair of unipolar-induction batteries $\calEout$ and $\calEin$ must be set up on the outer and inner sides of the null surface , oppositely directed (see  figure <ref>).
These evidences obviously require the particle source as well as the current source under the null surface  with $\omN=\OmF$ between the outer and inner light surfaces $\omoL$ and $\omiL$. That is, both of the force-free and freezing-in conditions (<ref>a,b) must `break down' on this surface  with $\vcEp=0$.[We can confirm the pieces of evidence showing presence of vector $\vec{E}$'s reversal of direction, in Figure 2 and its caption in Blandford & Znajek, 1977,

and in figure 38 in Thorne et al., 1986, as well as in each ordinate of figures <ref>, <ref>, <ref>, <ref> at the null surface  where $\vcEp=0$ in this paper. ]

Then, when FLs thread this surface , i.e., $\PN{\vcBp}\neq 0$ and $\PN{\OmF}\neq 0$, the following quantities must necessarily vanish on ;

\begin{eqnarray}
\PN{\vcEp} =\PN{\vcSEM}=\PN{\vre}=\PN{\vF}=\PN{\epsJ} =0,  \hspace{0.3cm} \ \ \ \ \ \ \label{SNa} \\  % (6.1a)
\PN{\vcj}=\PN{I}  =\PN{\Bt} =\PN{\vcSJ} =\PN{\vcSsd}=\PN{\vcSE}=0, \hspace{0.3cm} \ \ \ \ \ \ \label{SNb} \\  % (6.1b)
\PN{\calPE}=\PN{\calPJ}=0,  \hspace{1cm}  	\label{SNd} \\  	% (6.1c)
\PN{\vcv}=\PN{\vcj/\vre}=0,  \hspace{1cm}  	\label{SNc}   	% (6.1d)
\end{eqnarray}

where equations (<ref>a,b), (<ref>a,b,c), (<ref>), (<ref>,b,c) and (<ref>) are used, We denote the value of any function $X(\OmFm,\Psi)$ on  with $\OmFm=0$;

The first group in equation (<ref>) contains the quantities that reverse direction or change sign across . The reversal of these quantities comes directly from the change of $\OmFm$ in sign
($\OmFm\ggel 0$). The second ones in equation (<ref>) contain quantities that vanish but do not reverse direction nor change sign at first sight (see, e.g., figure <ref> for $I$). The third ones vanish because these contain $\PN{I}=0$ in the integrand (see equations (<ref>)).
Finally, the fourth one in equation (<ref>), i.e., $\PN{\vcv}=\PN{\vcj/\vre}=\PN{dI/d\Psi)/\vre}=0$ stem from the breakdown of freezing-in-ness and, similarly to those in the first group, reverse and changes sign due to the existence of $\vre$.

The above Constraints unequivocally require us to rebuild the whole force-free magnetosphere from a `single-pulsar model' to a `twin-pulsar model', as follows:

The most important of the above Constraints will be that $\PN{\vcv}=\PN{\vcj}=0$ requires current- and stream-lines to be severed on the null surface . That is, the current-wind system is separated into the two, outer SC and inner GR, domains by the null surface , i.e., the magneto-centrifugal divider with $\epsJ=\vF=\OmFm=0$.  It is to accommodate the particle-current sources where the force-free and freezing-in conditions `must' break down completely, but not negligibly [Blandford & Znajek, 1977, Macdonald & Thorne, 1982].

Constraint $\PN{\vcj }=0$ plays the role of a perfect `circuit breaker' as a safety device to block `acausal' currents across  from a horizon battery (if any) to external resistances such as particle acceleration. This instead indicates the necessity of a pair of `surface batteries' back-to-back at both sides of , yet oppositely directed, with the particle source in-between, in which the voltage drop between the two EMFs will produce pair-particles (see  section <ref>, figure <ref>).

Constraint $\PN{\vcv}=0$ means that the particles pair-created under  with $\epsJ=0$ are `' particles circulating with $\omN=\OmF$. That is, Constraints $\PN{I}=\PN{\vcSJ}=\PN{\vcSE}=0$ mean that no angular momentum nor energy is transported across , even when the FLs are continuous. It is helpful to remind that the toroidal field $\Bt$ is a swept-back component of the poloidal component $\vcBp$ due to inertial loadings in the `resistive membranes' $\Sffinf$ and $\SffH$ (see  figure <ref>).
$\PN{I}=0$ means that there must be a jump of $I(\Psi)$ from $\Iin$ to $\Iout$, just like in the NS surface (see equations (<ref>) and (<ref>)). Thus, we may impose the `boundary condition' $\DG{I}=0$ (see equation (<ref>)).

Although the current $\vcj$ does not reverse direction, the velocity $\vcv$ does, i.e., $\vcv\ggel 0$ and $\epsJ\ggel 0$ in  equation (<ref>) for $\OmFm\ggel 0$ (see section <ref>, figure <ref>).
Thus, the surface  will behave like a watershed in a mountain pass (i.e. `plasma-shed') between outflows and inflows of `force-free' and `charge-separated' particles pair-created by the voltage drop (see equations (<ref>a,b) and (<ref>)), and yet both flows are due to the magneto-centrifugal forces at work toward the opposite directions, inward and outward by $\OmF\ggel 0$, respectively.
As the outer pulsar-type magneto-centrifugal wind flows through  in $\calDout$ with $\vF> 0$, the inner anti-pulsar-type wind will pass through  in $\calDin$ with $\vF< 0$ (see  figure <ref>;  section  <ref>).

Since $\vre\vcEp$ vanishes but does not change sign on , this reacts back on the force-free condition in  equation (<ref>), producing just $\PN{\vcj }=0$, whereas the change in direction of $\vcEp\propto\OmFm$ across  is taken over the particle velocity $\vcv$ as it is, i.e., $\PN{\vcv}\ggel 0$ for $\OmFm\ggel 0$ in the freezing-in condition (<ref>).
This is because axial symmetry $\vcE_{\rm t}=0$ will lead to $\vcvp=\kappa \vcBp$ and  $\kappa=-(1/\vre \al)(dI/d\Psi)=0$ on  (see equation (<ref>)),
and hence the ZAMOs will see that the particle velocity $\vcv$ behaves like $\OmFm\ggel 0$ across , contrary to $\vcjp$. Note that when $\vcEp\propto \OmFm$ across , this nature must straightly be succeeded to the particle velocity, i.e., $\vcv\propto \OmFm$ as well.

There will be no `single' circuit allowed, with such a current as crossing  due to a `single' battery at any plausible position [Blandford, 1979, Macdonald & Thorne, 1982, Thorne et al., 1986]. Each electric circuit must close within its respective force-free domain ($\calDout$ or $\calDin$), with each EMF ($\calEout$ or $\calEin$) in equation (<ref>), and with each eigenvalue $I(\Psi)$ ($\Iout$ or $\Iin$) in equation (<ref>).

Two vectorial quantities $\vcjp$ and $\vcSJ$ are closely related to each other through the current/angular-momentum function $I(\Psi)$, i.e., two-sidedness in the force-free domains. It seems that both do not reverse direction, despite that $\PN{\vcjp}=\PN{\vcSJ}=0$. This is because an outflow of negative charges means the ingoing current, and an inflow of negative angular momentum means an outflow of positive one (see figures <ref> and  <ref>). Despite that the null surface  exists always, and yet Constraints $\PN{\vcSJ}=\PN{\vcSE}=0$ hold, the overall energy flow $\vcSE=\OmF\vcSJ$ seems to flow outwards all the way along each open FL, apparently as if crossing   where the force-free condition `breaks down' on   i.e., $\PN{I}=0$, and indeed the Poynting flux $\vcSEM$ reverses direction. These are quite natural tactics stemming from the supreme order of keeping the pair-producing Gap in zero-angular-momentum and the charge-neutral state.

A plausible behavior of the angular-momentum-flux/current function, $I(\OmFm,\Psi)$ (see  equation (<ref>)). The abscissa is the `coordinatized' $\OmFm$ along a field line $\Psi=$constant, and the ordinate on the null surface  divides the magnetosphere into the two domains of pro- and retro-grade rotation by $\OmFm\ggel 0$.  Breakdown of the force-free and freezing-in conditions on the null surface  leads to Constraints $\PG{\vcv}=\PG{\vcj}=0$ in equations (<ref>a,b), which mean that current- and stream-lines no longer thread $\SG$ (see  figures  <ref>, <ref>, <ref>).
The voltage drop $\Dl V$ between the two EMFs in equation (<ref>) will induce steady particle production, thereby developing a Gap with $\PG{I}=0$ in a finite zone $|\OmFm|\lo \Dlom$ between the two force-free domains with $I=\Iout(\Psi)$ and $=\Iin(\Psi)$, respectively (see equations (<ref>a,b)).
When the rate of positive angular momentum conveyed outwardly in $\calDout$is equal to that of negative angular momentum conveyed inwardly in $\calDin$, the `ZAM-state' of the Gap $\PG{\epsJ}=0$ is maintained, i.e., $\DG{I}=\Iout+\IinU=0$. The `boundary condition' $\DG{I}=0$ in  equation (<ref>) yields the eigenfunction $\OmF(\Psi)=\omN$ in  equation (<ref>).
Just as the watershed at a mountain range produces two down-streams to both sides by the gravitational force, the plasma-shed amid the Gap at $\OmFm\approx 0$ will divide `pair-produced and then charge-separated' particles into outflows and inflows $\vcv\ggel 0$ by the magneto-centrifugal force due to $\OmFm\ggel 0$.
There may be a kind of boundary layers in the vicinity $\OmFm\simeq\pm\Dlom$ where non-force-free ZAM-particles pair-created in $\vre\simeq 0$ are changing to force-free charge-separated plasma with $\vre\approx\mp e n^{\pm}$, and $I$ increases (or decreases) to $\Iout$ (or $\IinU$) rather steeply. The non-force-free, matter-dominated Gap, filled with ZAM-particles, will ensure the pinning-down of poloidal field lines $\vcBp$ with $\OmF=\omN$, and the pinning-down conversely ensure magnetization of ZAM-particles with $\PG{I}=0$ in the Gap within $|\OmFm|\lo \Dlom$ (see section <ref>). Thus the whole force-free magnetosphere will be kept in the frame-dragged state by the hole's rotation.

§ THE ZERO-ANGULAR-MOMENTUM AND CHARGE-NEUTRAL GAP $\GN$

The ZAM-gap $\GN$ under the inductive membrane $\SN$ will play a hub of the Kerr hole's magneto-spherical activities.

§.§ A plausible gap structure with $I=\OmFm=0$

We presume that the RTD with the voltage drop $\DN{\calE}=-\Dl V$ in the force-free limit will be relaxed as a result of pair-particle production to a ZAM-Gap $\GN$ with a half-width $\Dlom$. For the `widened' null Gap $\SG$ in $|\OmFm|\lo\Dlom$, we replace Constraints in equation (<ref>) on  as follows;

\begin{eqnarray}
\PG{\OmFm}= \PG{\vcEp}=\PG{\vre} = \PG{\vcj}=\PG{\vcv}= \PG{I} \hspace{0.6cm}   \label{SGa} \\ % = \PG{\vp\Bt}
=\PG{\epsJ}=\PG{\vcSJ}=\PG{\vcSEM}=\PG{\vcSsd}=\PG{\vcSE} \hspace{0.6cm}   \label{SGb} \\
=  \PG{\calPE}= \PG{\calPJ} =0.	 \hspace{0.6cm}   \label{SGc}
\end{eqnarray}

As before, $\DG{X}$ denotes the difference of $X(\OmFm, \Psi)$ across the Gap width $2\Dlom$ (cf. $\DN{X}$ in equation (<ref>));
X =X(,Ψ)-X(-,Ψ) ≡(X)_G(out)  - (X)_G(in) .

We think of such a simple form of $I=I(\OmFm,\Psi)$ along a typical open field line in $\Psiz\leq\Psi\leq\Psib$ as follows;

\begin{eqnarray}
I(\OmFm) =  \left\{
\begin{array} {ll}
\to 0 & ;\ \Sffinf\ ( \OmFm\to\OmF), \\ [1mm]
\Iout & ;\ \calDout\ ( \Dlom \lo \OmFm \lo \OmF),  \\[1mm]
0 & ;\ \SG\ ( |\OmFm|\lo \Dlom) , \\[1mm]
\Iin &;\ \calDin\ (-\Dlom \ggo \OmFm\ggo-(\OmH-\OmF)),  \\[1mm]
\to 0 & ;\ \SffH\  (\OmFm\to -(\OmH-\OmF))
\end{array}  \right.    \label{OL-I}  \end{eqnarray}

(see  figure <ref>), where $\Iout$ and $\Iin$ are given by equations (<ref>a,b).
The behaviour of $I(\OmFm,\Psi)$ in the outer domain $\calDout$ will be similar to that of the force-free pulsar magnetosphere (see equation (<ref>)).
When the Gap $\GN$ of the particle source will be situated well inside the light surfaces, we have by equation (<ref>)
|| ≪c(/)_oL ≈c(/)_iL.

It is not clear now how helpful or rather indispensable is the above condition in constructing a reasonable gap model, but we presume that the particle production will eventually take place intensely by the voltage drop across the Gap $\GN$, $\Dl V=-(\OmH/2\pi c)\Dl\Psi$, almost independently of the presence of the two light surfaces in wind theory.

The Gap $\GN$ under the inductive membrane $\SN$ must be in the ZAM-state $\PG{\epsJ}=0$, so that the particles and the field carry no angular momentum nor energy across the Gap, i.e., $\PG{\vcSJ}=\PG{\vcSE}=0$ in equation (<ref>), despite that the Gap is threaded by poloidal field lines, i.e., $\PG{\vcBp}\neq 0$.
Also, the Gap must be magnetized (i.e., $\PN{\epsE}\neq 0$; see equation (<ref>)), in almost the same sense as a magnetized NS because the poloidal magnetic field lines (with no toroidal component) threading the Gap will naturally be pinned down in the ZAM-particles pair-created and circulating round the hole with $\omN=\OmF$ (see section <ref>).  Therefore, the magnetized ZAM-particles will ensure $\OmF=\omN$ (see  section <ref>). The non-force-free magnetized ZAM-Gap $\SG$ where $\PG{\vcj}=\PG{\vcv}=0$ but $\PG{\vcBp}\neq 0$ will therefore be formed in the steady-state, with its `surfaces'  and , respectively, at $\OmFm\approx \pm\Dlom$ (see  figure <ref>), where $\Dlom\approx|\PN{\partial\om/\partial\ell}| \Dlell$ stands for the Gap half-width (see equation (<ref>)), and for $\Dlom\to 0$, $\SG\to$ (see figure 4 in Okamoto, 2015 for the interplay of microphysics with macrophysics in the magnetized, matter-dominated Gap). Thus we conjecture that the voltage drop, $\Dl V=-\DN{\calE}$ on  in equation (<ref>), will produce pair-particles copious enough, and the plasma pressure in the steady-state will expand  to $\GN$ with a half-width $\Dlom$ in $|\OmFm|\lo\Dlom$.

§.§ Pinning-down of threading field lines on ZAM-particles and magnetization of the matter-dominated Gap

When we regard the Gap surfaces  and  as being equipped with EMFs $\calEout$ and $\calEin$, respectively, these EMFs will not only drive currents in the respective circuits $\calCout$ and $\calCin$ but also produce a strong voltage drop $\Dl V=-\DG{\calE}$ across the Gap, which will create copious ZAM-particles necessary to pin threading FLs down on. The ZAM-Gap filled with ZAM-particles will then circulate the hole with $\omN=\OmF$, and the poloidal field lines threading the Gap $\SG$ will surely be pinned down on ZAM-particles with $\OmF=\omN$. Thus, the ZAM-Gap will be in the perfectly magnetized state, with no electric current and no angular momentum flux allowed to cross, i.e., $\PG{\Bt}=\PG{I}=\PG{\vcj}=0$. The physical state of the ZAM-Gap $\GN$ will be analogous to that of the NS inside, ensuring the boundary condition $\OmF=\OmNS$ for the FLs emanating from the NS surface.

The two circuits $\calCout$ and $\calCin$ are connected to each other by the FLs $\Psio$ and $\Psit$, and hence the iso-rotation law $\OmF(\Psi)=$constant is still obeyed along each FL threading the particle production Gap $\SG$, but current- and stream-lines are severed by the breakdown of the force-free and freezing-in conditions in the Gap. The ZAMOs will see that FLs circulate forward in the outer domain $\calDout$ but backwards in the inner domain $\calDin$.  The `boundary condition' of no jump of the angular momentum transport rate across the Gap, i.e., $\DG{I}=\PG{I}=0$, will be maintained to determine the eigenfunction $\OmF$ (see  section   <ref>).

§.§ Magneto-centrifugal plasma-shed on the ZAM-surface

The outer force-free domain $\calDout$ behaves like a pulsar force-free magnetosphere rotating forward, whereas the inner domain $\calDin$ does like an anti-pulsar-type magnetosphere rotating backwards. Then, plasma particles pair-created by the voltage drop $\Dl V=- \DG{\calE}$ circulate at $\omN=\OmF$ but may not behave as `force-free' particles with negligible inertia within the Gap. These ZAM-particles with $\PG{\vcv}=0$ will soon become charge-separated like inside in `two' batteries, to flow from the Gap out to the force-free domains as `electric charges' along current-lines as well as `wind particles' along stream-lines (see figures  <ref> and <ref>).

The null surface $=$ midst the ZAM-Gap $\GN$ redefines quite a new general-relativistic type of divider due to magneto-centrifugal force modified by frame-dragging, for particles pair-created in the ZAM-Gap, outward and inward ($\vcv\ggel 0$).  That is, `this surface'  will play the role of a magneto-centrifugal plasma-shed, akin to a gravitational water-shed of a mountain ridge for heavy rainfalls on the Earth.
This will be quite a natural way to launch `magneto-centrifugal' winds from the ZAM-Gap for both directions toward infinity ($\vcv>0$) and the horizon ($\vcv<0$) (the wind-flow analysis; section <ref>), similarly to the Poynting flux ($\vcSEMout>0$) for particle acceleration on the resistive membrane $\Sffinf$ and the one ($\vcSEMin<0$) for entropy production on another membrane $\SffH$ (the energy-flux analysis; section <ref>).

§.§  Pair-creation, charge-separation and pair-annihilation

One of the important properties of the `force-free' plasma will be `charge-separateness', which we will be able to utilize in another way of discharging a pair of batteries on this surface  in the two resistive membranes $\Sffinf$ and $\SffH$.

Let us consider that strong enough pair-creation due to the voltage drop between the two EMFs will be at work in producing copious pair-particles (i.e., $\gamma+\gamma\to e^{-}+e^{+}$).  These mixed charges of $e^{\pm}$ in the Gap $\GN$ will then be charge-separated in the presence of the battery EMF $\calEout$ in the outer circuit $\calCout$, as the $e^{-}$-stream flowing from under  outward into the `FCS-line $\Psio$', and similarly as the $e^{+}$-stream flowing from under  outward into the `FCS-line' $\Psit$ (see the arrows of $\vcjp$ in figure <ref>). This is because the wind blows always outward in the outer domain ($\OmFm>0$ and $\vcv>0$). Also, by the effect of the battery $\calEin$, $e^{+}$- and $e^{-}$-streams from  will flow inward into the same FCSLs, $\Psio$ and $\Psit$, respectively, in the inner circuit $\calCin$ ($\OmFm<0$ and $\vcv<0$).

These two $e^{\pm}$-streams in the two circuits $\calCout$ and $\calCin$ will collide with each other in the restive membranes, $\Sffinf$ and $\SffH$,  respectively, and then pair-annihilation will take place, thereby liberating energy of the same order of magnitude as consumed when pair-particles are created in the Gap (i.e., $e^{-}+e^{+}\to \gamma+\gamma$).  This will be comparable to the amount of energy due to Ohmic dissipation of the surface current $\calIffinf$ and $\calIffH$ in order of magnitude (see equations (<ref>,b,c)). Also, the sum of the energy liberated in $\Sffinf$ and $\SffH$ will be similar to the amount due to the sum of the Poynting flux $\vcSEM$, inward and outward (see equation (<ref>) later).

§ THE EIGEN-MAGNETOSPHERE

For a viable force-free magnetosphere, we refer to the condition by which to finally determine the eigenfunction $\OmF(\Psi)=\omN$ as the `boundary condition,' distinguishing from the `criticality condition' for another eigenfunction $I(\Psi)$ in equations (<ref>a,b).

§.§ The `boundary condition' for the eigenfunction $\OmF$

One of the vital roles of the ZAM-Gap is to anchor the poloidal field $\vcBp$ onto the ZAM-particles pair-created in there and to accomplish magnetization of the ZAM-Gap, thereby ensuring $\OmF=\omN$ for threading field lines. Accordingly, the ZAM-state of the Gap must always be maintained in the magnetosphere frame-dragged by the hole into circulation with $\omN=\OmF$. Nevertheless, the actual position of the Gap and the specific value $\OmF=\omN$ per se remain still undetermined. That is, we must formulate the `boundary condition' with Constraints (<ref>) or (<ref>) and  with a postulate (<ref>) appropriately taken into account, that is, $\PG{I}=\DG{I}=0$ at the place of the ZAM-Gap ;

\begin{eqnarray}
\DG{I}=\Iout(\Psi) -\Iin(\Psi) \quad \quad  \label{DN/SNa} \\   % (8.1a)
=\Iout(\Psi) +\IinU(\Psi)=0  \label{DN/SNb}.       % (8.1b)
\end{eqnarray}

This ensures the continuity of overall energy flux as well as angular momentum flow across the Gap, thereby keeping the charge-neutral ZAM-state

along each FL threading the Gap (see figure <ref>; equations (<ref>a,b)).

Condition (<ref>) shows that the outward transport rate of positive angular momentum leaving  into the SC domain $\calDout$ must be equal to that entering  from the GR domain $\calDin$. Note that the energy-angular momentum flow does not take place actually inside the ZAM-Gap with $\PG{I}=0$. Condition (<ref>) implies equivalently that the outward rate of positive angular momentum leaving  is offset by the inward rate of negative angular momentum leaving  toward the hole.

Now, by equations (<ref>), we have

which apparently shows that the overall energy and angular momentum fluxes flow outward continuously across the Gap $\GN$, regardless of $\PG{\vcSE}=\PG{\vcSJ}=0$. Likewise, the `boundary condition' (<ref>a,b) ensures no discontinuity of the power $\calPE$ and the loss rate of angular momentum $\calPJ$ across the ZAM-Gap, i.e., as shown by equations (<ref>a,b) and (<ref>) and (<ref>);

\begin{eqnarray}
\DG{\calPE}=\calPEout- \calPEin  =\calPEout + \calPEinU =0,   \label{DGcalE}   \\
%  \label{DGcalE/a}  \\  	% (8.3a)		=\calPEout + \calPEinU =0,   \label{DGcalE/b}		% (8.3b)
% \end{eqnarray} \label{DGcalEa,b} %   \end{subequations}	% (8.3a,b)    \begin{subequations}  \begin{eqnarray}
\DG{\calPJ}=\calPJout -\calPJin  =\calPJout + \calPJinU =0.        \label{DN/calPJ} % \\ %   % (8.4b)
\end{eqnarray}

Analysis of wave propagation of linear perturbations both outward and inward from the null surface  may be of interest with regard to the causality question of the boundary condition (<ref>a,b) on the null surface  in the `force-free' magnetosphere [Uchida, 1997, Uchida, 1997, Punsly, 2008].

§.§ The final eigenfunctions $I(\Psi)$ and $\OmF(\Psi)$ in the force-free magnetosphere

From equations (<ref>a,b) and (<ref>a,b), we have

\begin{eqnarray}
\OmF(\Psi)=\omN= \frac{\OmH}{1+\zeta}, 	\label{EigenOmFI}      \\  % (8.5a) 		\hspace{1cm}
I=\Iout=\Iin=- \IinU=\frac{\OmH}{2(1+\zeta)} (\Bp\vp^2)_{\rm ffH},  	 \label{EigenOmF}  \\  % (8.5b)  \hspace{1cm}
\zeta(\Psi) \equiv (\Bp\vp^2)_{{\rm ff}\infty}/(\Bp\vp^2)_{\rm ffH}    \label{Zeta}    % (8.5c)     \hspace{1cm}
\end{eqnarray}

[Okamoto, 2015]. In this eigenstate, the null surface $=$ will be the magneto-centrifugal plasma-shed, from which the angular momentum and the Poynting fluxes, positive and negative, flow out both ways toward $\Sffinf$ and $\SffH$,  (ii) their related AVs are given by $(\OmFm)_{\rm out}=\OmF$ in the outer domain $\calDout$, and by $(\OmFm)_{\rm in}=-(\OmH-\OmF)$ in the inner domain $\calDout$, and (iii) the difference $\OmH$ of the two AVs corresponds to the voltage drop between a pair of batteries $\Dl V$ in equation (<ref>), and this drop will lead to sustainable particle production.

Constraints $\PG{\vcj}=\PG{\Bt}=\PG{I}=\PG{\vcv}=0$ in equation (<ref>) imply that no transport of angular momentum and energy is possible within the ZAM-Gap, i.e., $\PG{\vcSJ}=\PG{\vcSE}=0$. These indicate a disconnection of current- and stream-lines between the two force-free domains, and hence the necessity of the current-particle sources and related EMFs in the Gap.  It will be ensured in equation (<ref>) that the copious charged ZAM-particles pair-produced in $|\OmFm|\lo\Dlom$ serve to connect and equate both $\Iout$ and $\IinU$ across the Gap $\SG$, despite $\PG{\vcv}=\PG{\vcj}=0$. Also, the overall flow of energy-angular momentum is continuous across the ZAM-Gap as seen in equations (<ref>), regardless of $\PG{\calPE}=\PG{\calPJ}=0$
from equations (<ref>a,b)
as far as the boundary condition $\DG{I}=0$ in equation (<ref>) is satisfied.

The eigen-efficiency of extraction is given from equation (<ref>) by

(see section <ref>).
When the plausible field configuration allows us to put $\zeta\approx 1$ and hence $\epsGTE\approx 0.5$, we have from equations (<ref>),

\begin{eqnarray}
\OmF\approx \OmFb \approx \frac12\OmH,  \label{matching} \\  % (8.7a)
\hspace{1.5cm}	  c^2 |dM| \approx  \Th dS \approx \frac12 \OmH |dJ|  \label{therfirst-eig} % (8.7b)
\end{eqnarray}

by equations (<ref>a,b).

A pair of batteries's EMFs become for $\zeta \approx 1$ by equations (<ref>)

\begin{eqnarray}
\calEout  =-\frac{\OmH}{2\pi c} \int_{\Psi_1}^{\Psi_2}   \frac{1}{1+\zeta} d\Psi
\approx - \frac{\OmH \Dl\Psi}{4\pi c} 	\approx -\frac{\Dl V}{2},   \hspace{0.7cm}
\label{eigEMF-out} 	 \\ % (8 8a)
\calEin	%=\oint_{{\cal C}_{\rm in}} \al\vcEp\cdot d\vcell
=\frac{\OmH}{2\pi c}\int_{\Psi_1}^{\Psi_2}  \frac{\zeta}{1+\zeta} d\Psi
\approx   \frac{\OmH \Dl\Psi}{4\pi c}	 \approx  \frac{\Dl V}{2}.   \hspace{0.5cm}
\label{eigEMF-in} \hspace{0.33cm} % (8.8b)
\end{eqnarray}

Thus, the impedance matching between particle acceleration and entropy-production yields good agreement with the eigen-magnetosphere for $\zeta\approx 1$.

A twin-pulsar model with three (two resistive and one inductive) membranes with $\OLOmFm$ in equation (<ref>).  The two magnetospheres are anti-symmetric to each other with respect to the ZAM-surface $=$ ; the outer domain $\calDout$ behaves like a `normal pulsar-type' magnetosphere rotating with the FLAV $(\OLOmFm)_{\rm (out)}=\OmF$, whereas the inner domain $\calDin$ does like an `anti-pulsar-type' magnetosphere counter-rotating with the FLAV $(\OLOmFm)^{\rm (in)}= -(\OmH-\OmF)$. The inductive membrane $\SN$ covers the magnetized ZAM-Gap in $|\OmFm| \lo\Dlom$ at $\PG{\OLOmFm}=0$.
The ingoing flux of negative energy in $\calDin$ from $\GN$ is equivalent to the outgoing flux of positive energy from the hole, just as the ingoing flux of negative angular momentum is so to the outgoing flux of positive angular momentum.  A steady pair-production mechanism due to the voltage drop $\Dl V$ will be at work to supply ZAM-particles dense enough to anchor threading field lines, thereby ensuring  $\OmF=\omN$. The two batteries supply electricity to `external resistances', such as Joule heating leading to `particle acceleration' and `entropy production', respectively, in the resistive membranes $\Sffinf$ and $\SffH$. The plasma-shed on the ZAM-Gap will easily divide wind particles charge-separated into both directions ($\vcv\ggel 0$) by the magnetic sling-shot ($\OmFm\ggel 0$) (see  figure <ref>).
The ZAMOs
will see that the ZAM-Gap with $\PG{\epsJ}=0$ will be frame-dragged into rotation with the AVs $\omN=\OmF$, together with the two GR and SC domains (though spinning oppositely each other).

§ A TWIN-PULSAR MODEL WITH ROTATIONAL-TANGENTIAL DISCONTINUITY

Path integrals of $\vcEp$ in equation (<ref>a,b) along the two closed circuits $\calCout$ and $\calCin$ reveal a sharp potential drop $\Dl V$ between the EMFs for the two circuits in the SC and GR domains, thereby producing pair-particles of charges $e^{\pm}$.
This discontinuity RTD comes from the differential rotation of the outer prograde-rotating domain with $\OmF$ and the inner retrograde-rotating one with $-(\OmH-\OmF)$ and will differ distinctly from any ordinary tangential and/or rotational discontinuities in classical magnetohydrodynamics  (see e.g. <cit.>).

We attempt to briefly analyze a fundamental feature of this RTD on this surface  in the force-free limit.  Important is the evidence that the above results, including the voltage drop $\Dl V$ across  comes from the `continuous' function of the FDAV $\om$, and yet all these results seem also to be obtainable by assuming a `discontinuous' step function $\OLom$ for the FDAV $\om$;
= {

0     ;  (>0),

≡    ;        (=0),

;  (<0 ),
which means that $\om\approx 0$ in the SC domain and $\approx \OmH$ in the GR domain, but $\om=\omN=\OmF$ on the null surface , i.e., the ZAM-dividing surface .
Likewise, $\OmFm$, $\vF$ and $\epsJ$ are also replaced by the following step-functions ($\OLOmFm\equiv \OmF-\OLom$);
= {

≡()_(out)     ; ⇑  (>0),

0 ≡()_(N)      ;      (=0),

-(- ) ≡()^(in)     ; ⇓  (<0),
$\OLvF=\OLOmFm\vp/\al$ and $\OLvarepsJ$ $=\OLvF(\vp\Bp^2 /c)$, where ${\bf \Uparrow}$ and ${\bf \Downarrow}$ show that the $\calDout$ prograde-rotates with $\OmF$, while the $\calDin$ retrograde-rotates with $-(\OmH-\OmF)$, respectively (see the two arrows in figures <ref>, <ref>, and <ref>).
The differences of $\OLOmFm$ and the EMFs across  become

\begin{eqnarray}
\DN{\OLOmFm}=(\OLOmFm)_{\rm (out)}- (\OLOmFm)^{\rm (in)}=\OmH,  \label{DN-OmFm} \\ % (9.3a)
\DN{\calE}=- \frac{ \DN{\OLOmFm}}{2\pi c}\Dl \Psi=- \Dl V     \label{Dl-V2} % (9.3b)
\end{eqnarray}

(see equation (<ref>)).

The related electric field $\OLEp$ and its discontinuity at  become from equation (<ref>)

\begin{eqnarray}
\OLEp=-  \frac{\OLOmFm}{2\pi\al c}\vcnb\Psi, \hspace{1.0cm}     \label{OLEp} \\  % (9.4a)
\hspace{0.5cm}	\DN{\OLEp}=-  \frac{\DN{\OLOmFm}}{2\pi c} \LPfrac{\vcnb\Psi}{\al}_{\rm N}
= -  \frac{\OmH}{2\pi c} \LPfrac{\vcnb\Psi}{\al}_{\rm N}.  \hspace{1.0cm}	\label{DNOLEp}  % (9.4b)
\end{eqnarray}

The expression for $\OLEp$ naturally reproduces the same results for $\calEout$ and $\calEin$ in Faraday path integrals of $\vcEp$ along the circuits $\calCout$ and $\calCin$, respectively, as given in expressions (<ref>a,b). Also, the discontinuity of the EMF is given already in equation (<ref>).

The energy fluxes $\vcSEM$ and $\vcSsd$ are also replaced with the step-functions $\OLvcSEM$ and $\OLvcSsd$, respectively, i.e.,
=,   =.

There is naturally no discontinuity in the `overall' energy and angular momentum fluxes across  with $\DN{\vcBp}=0$, i.e., similarly to equation (<ref>)

We compute likewise the differences of the Poynting flux $\OLvcSEM$ and the spin-down flux $\OLvcSsd$ across
from equations (<ref>) and (<ref>);

\begin{eqnarray}
\DN{\OLvcSEM}=  \OLvcSEMout - \OLvcSEMinU,  \label{energyN}  \\ % (9.7a)
\DN{\OLvcSsd}  = - \OLvcSsdin,     \label{energyG}  \ \  % (9.7b)
\end{eqnarray}

where $\OLvcSsdout=0$ and $\OLvcSEMout= \vcS_{\rm E,(out)}$, because $\om$ is regarded as negligible in the outer SC domain. Then, by equation (<ref>) we have $-\DN{\OLvcSsd}= \DN{\OLvcSEM}$, i.e.,

+= =,

and hence, this reduces to equation (<ref>), because $\OLvcSEMinU =-(\OmH-\OmF)\vcSJin <0$ and $\OmF \vcSJin=\OmF \vcSJout$.

In the pulsar force-free magnetosphere, the conserved energy flux $\vcSE=\OmF\vcSJ$ alone flows outward from  to . In contrast, in the hole's force-free magnetosphere, the conserved energy flux $\vcSE$ is forcibly split into the two non-conserved fluxes $\vcSEM$ and $\vcSsd$ in $\calDin$, to comply with the 1st and 2nd laws of thermodynamics, but these energy fluxes flow along the same equipotential FCSLs, so that the Kerr hole will be unable to discriminate between the sum of $\vcSEM= - (\om-\OmF) \vcSJ$ and $\vcSsd=\om\vcSJ$ and that of $\OLvcSEM=-(\OmH-\OmF)\vcSJ$ and $\OLvcSsd=\OmH\vcSJ$ in the inner GR domain, while $\OLvcSEM=\vcSE$ and $\OLvcSsd= 0$ hold in the outer SC domain (see equations (<ref>), (<ref>) and (<ref>)). Therefore, the basic properties of the energy fluxes in the curved space with $\om$ and $\OmFm(\om,\Psi)$ will be fully reproduced in the pseudo-flat space with $\OLom$ and $\OLOmFm(\OLom,\Psi)$ [Okamoto, 2015].

A kind of inevitable relaxation of RTD will occur due to the pair-particle production by the voltage drop $\Dl V$, thereby leading to widening from the ZAM-surface to a ZAM-Gap $\SG$ with a finite thickness.
This Gap may be regarded as effectively consisting of two halves of `fictitious' magnetized NSs consisting of 'shell-like structures', e.g., the outer one forward-rotating with $(\OmFm)_{\infty}=\OmF\approx (\OmH/2)$ and the inner one backward-rotating with $(\OmFm)_{\rm H}= -(\OmH-\OmF)\approx -(\OmH/2)$, and yet the two structures are reversely packed together, and threaded by the poloidal field $\PG{\vcBp} \neq 0$ with no toroidal component and pinned down in the ZAM-particles pair-produced in the Gap, to ensure $\OmF=\om_{\rm G}$.

The above conjecture is depicted schematically in  figure <ref>; the pulsar-type wind is slung outward from the outer magnetized `fictitious' shell-like structure spinning with $(\OLOmFm)_{\rm (out)} =\OmF$ through the outer domain $\calDout$ with the outer , in which the related Poynting flux $\OLvcSEMout$ is equal to $\OmF \vcSJout>0$, with negligible frame-dragging spin-down energy flux followed. In contrast, the anti-pulsar-type wind is reversely slung inward from the inner magnetized `fictitious' shell-like structure counter-spinning with $(\OLOmFm)^{\rm (in)} =-(\OmH-\OmF)$ through the inner domain $\calDin$ with inner , in which the positive-valued Poynting flux is directed inward, i.e., $\overline{\vcSEMinU} =(\OmH-\OmF) \vcSJinU<0$,
while the negative-valued frame-dragging spin-down flux is directed inward as well, and hence $\OLvcSsdin = -\OmH\vcSJinU>0$, which may be understood as an equivalent to an in-flow of the negative energy, i.e. $\OLvcSsdinU =\OmH \vcSJinU=- \OLvcSsdin<0$, related to the in-flow of negative angular momentum. So, the `overall' energy flux becomes
= +=-= >0,

which will be equal to $\vcSEout=\OmF\vcSJout$ across the Gap (see equations (<ref>) and (<ref>); section <ref>).

§ ENERGETICS AND STRUCTURE IN THE TWIN-PULSAR MODEL

§.§ Energetics of the hole's self-extraction of energy

A variant of the first law, $\OmH |dJ|=\Th dS+c^2 |dM|$, seems to indicate that the energy extracted through the spin-down energy flux will be shared at the inductive membrane $\SN$ between the two Poynting fluxes toward the two resistive membranes $\SffH$ and $\Sffinf$.  By integrating equation (<ref>) over all the open field lines from $\Psi_0$ to $\Psib$, we have
∫_Ψ_0^ ·d= -  ∫_Ψ_0^
α·d+ ∫_Ψ_0^

which explains that the power $\OmH\calPJ$ self-extracted from the horizon is allocated between entropy production in $\SffH$ and particle acceleration in $\Sffinf$ (cf. [Macdonald & Thorne, 1982], $ 7.3; [Thorne et al., 1986], Ch. IV D).
The two terms of the right-hand side of equation (<ref>) actually become, by equations (<ref>a,b), (<ref>) and (<ref>), respectively;

\begin{eqnarray}
\TTH \frac{dS}{dt}
=\frac{1}{2 c}   \int_{\Psi_0}^{\Psib} 	  % \int_{{\cal S}_{\rm ffH}}
(\OmH-\OmF)^2(\Bp\vp^2)_{\rm ffH} d\Psi,	 \hspace{0.5cm}	\label{H/resistanc}   \ %(10.2a)
\end{eqnarray}

\begin{eqnarray}
-c^2 \dr{M}{t} = \calPEout
= \frac{1}{2c}   \int_{\Psi_0}^{\Psib}    % \int_{ {\cal S}_{{\rm ff}\infty }}
\OmF^2 (\Bp\vp^2)_{{\rm ff} \infty} d\Psi \hspace{1.3cm}  \ \  \label{L/res/a} \\  				% (10.2b)
= \frac{1}{2c}  \int_{\Psi_0}^{\Psib} 	%	\int_{\Psi_1}^{\Psi_2}    % \int_{ {\cal S}_{{\rm ffH}} }
\OmF(\OmH-\OmF) (\Bp\vp^2)_{{\rm ffH}} d\Psi \ \hspace{0.9cm}   \label{L/res/b} 			% (10.2c)
\end{eqnarray}

(see equation (<ref>)), where $c^2 dM=\OmFb dJ= -\calPEin dt= -\calPEout dt$.
The left hand-side of equation (<ref>) reduces by equations (<ref>) and (<ref>)  to

\begin{eqnarray}
- \OmH\dr{J}{t}  =\OmH\calPJout= \frac{\OmH}{2c}  \int_{\Psi_0}^{\Psib}
\OmF (\Bp\vp^2)_{{\rm ff} \infty} d\Psi 	 \label{J/res/a} \\  % (10.3a)
=\OmH \calPJin = \frac{\OmH}{2c} \int_{\Psi_0}^{\Psib} 	%\int_{ {\cal S}_{{\rm ffH}}}
(\OmH-\OmF) (\Bp\vp^2)_{{\rm ffH}} d\Psi, \ \ 	 \label{J/res/b} 	 % (10.3b)
\end{eqnarray}

(see equation (27) in [Blandford & Globus, 2022]). Summing up equations (<ref>) and (<ref>) or (<ref>) with use of the `boundary condition' $\Iout=\Iin$ yields $-\OmH (dJ/dt)=\OmH\calPJ$. Also, equations (<ref>) and (<ref>) show $\DG{\calPE}=\DG{\calPJ}=0$, despite of the RTD of the EMFs $\DG{\calE}=-\Dl V$ existent  in the Gap $\GN$ (see equation (<ref>)).

The point is that the ZAM-particles created inside the Gap $\GN$ are spinning with $\omN=\OmF$ dragged by the hole's rotation, literally with no angular momentum. This means that the particles will effortlessly flow out of the Gap, flung outwards or inwards from the surfaces  or  on the `plasma-shed', with positive or negative angular momentum by the respective magneto-centrifugal forces, thus keeping the ZAM-state of the Gap.
This corresponds to the situation where the outgoing Poynting flux $\vcSEM> 0$ is related to the outer EMF $\calEout$, whereas the ingoing Poynting flux $\vcSEM<0$ is related to the inner EMF $\calEin$.  Then, the distant observers may think as if the spin-down energy `self-extracted' through the resistive horizon membrane $\SffH$ were shared between the out- and in-going Poynting fluxes reaching the two resistive membranes $\Sffinf$ and $\SffH$, respectively, to dissipate in particle acceleration and entropy generation as seen in  equation (<ref>).
Kerr holes can use the role of an acceptor of negative angular momentum and an emitter of positive angular momentum properly and simultaneously.

§.§ The stream equation for the twin-pulsar model

We derive two expressions for $\jt$ ; firstly, from equations (<ref>,c)
= -/8π^2 c·(/Ψ) +1/^2 cI^2Ψ ,

and secondly, from Eq. (10.4) in [Macdonald & Thorne, 1982]
= - c/8π^2 [·(/^2Ψ)+/c^2(Ψ·)].

Equating the above two expressions for $\jt$ leads to the `stream equation' in the force-free limit for the `field-current-stream-line' structure in terms of the FLAV $\OmF(\Psi)$ (also ZAMO-FLAV $\OmFm$) and the current/angular-momentum function $I(\Psi)$ as follows;

\begin{eqnarray}
\nabla\cdot\left\{ \frac{\al}{\vp^2} \left[1-\frac{\OmFm^2\vp^2}{\al^2 c^2} \right] \nabla\Psi \right\} \hspace{1cm}	\nonumber \\
\hspace{0.5cm}	+ \frac{\OmFm}{\al c^2}\dr{\OmF}{\Psi}(\nabla\Psi)^2  +\frac{16\pi^2}{\al\vp^2 c^2} I\dr{I}{\Psi}=0
\label{stream/MT} \end{eqnarray}

(see equation (6.4) in [Macdonald & Thorne, 1982]). This contains not only the two light surfaces  and , but also the null surface  in-between, by $\vF=\pm c$ and $=0$.  The breakdown of the force-free and freezing-in conditions appears in a complicated form of severance of both current- and stream-lines, $\vcj=\vcv=0$, and emergence of a gap $\GN$, $I=\OmFm=0$, on the null surface  (sections <ref> and <ref>; figure <ref>).
All these complications allow the particle-current sources to be inserted into the (pulsar) force-free magnetosphere by particle production due to the voltage drop between a pair of batteries under the null surface .

When the null surface  develops into a gap $\GN$ with any finite width $|\OmFm|\lo\Dl\om$ where $I(\ell,\Psi)=0$, then the stream equation (<ref>) will relevantly be modified. We conjecture now that the poloidal component $\vcBp$ with no toroidal one, $\Bt=I(\ell, \Psi)=0$, will be robust enough to thread the particle-production Gap $\GN$ due to the voltage drop $\Dl V$, with the ZAM-state of the Gap maintained to keep circulation with $\OmF=\omN$ around the hole. Probably, this situation will be compatible with the solution of the stream equation  $\nabla\cdot ((\al/\vp^2) \nabla\Psi)=0$ for the particle production Gap within $|\OmFm|\lo \Dl \om$ where $\OmFm=I=0$.

§ DISCUSSION AND CONCLUSIONS

§.§ Astrophysical roles of frame-dragging in energy extraction

The frame-dragging effect plays an indispensable role in reforming the pulsar force-free magnetosphere into the specifications suitable for that of a Kerr hole, in particular, so as to be adaptive to the 1st and 2nd laws of thermodynamics and to include the current-particle sources. The observance of the two laws demands a breakdown of the force-free condition.  But the issue, “where does the breakdown take place in the force-free magnetosphere?”, seems to have remained almost untouched for more than four decades since Blandford & Znajek, 1977, probably because the astrophysical roles of frame-dragging remain so far ill-understood (see, e.g., [Uchida, 1997, Blandford, 2002]).  This may explain why the topic of extracting energy from Kerr holes has continued to be a big challenge so far [Thorne & Blandford, 2017].

The ZAMOs circulating with $\om$ around the hole will be certain that the Kerr hole can self-extract energy if and only if frame-dragging is correctly taken into account. The coupling of FDAV $\om$ with FLAV $\OmF$ begins with the ZAMO-FLAV, $\OmFm=\OmF-\om$. Then, they will see for $0<\OmF<\OmH$ that the coupling necessarily leads to nesting the inner domain $\calDin$ counter-rotating ($\OmFm<0$) inside the outer domain $\calDout$. The magnetic sling-shot effect works inwardly through  towards the hole in the former domain, oppositely to in the latter domain through , and hence the ZAMOs will see “a sufficiently strong flux of negative angular momentum leaving the null surface ", and this does not contradict with “a Poynting flux going towards the hole" [Blandford & Znajek, 1977]. They will understand that the overall energy flux $\vcSE=\OmF\vcSJin$ always flows outward. It is on the null surface $=$ that the spin-axis of the hole's force-free magnetosphere changes from positive in $\calDout$ to negative in $\calDin$, and the Poynting flux emitted changes direction from outward to inward, and hence the complete violation must take place of the freezing-in-ness and force-freeness.

The force-free magnetosphere is `magneto-centrifugally' divided by this surface $=$ into the SC and GR domains, spinning reversely each other.  The inner domain $\calDin$ with negative angular momentum density $\epsJ<0$ may be referred to as the `effective ergosphere' [Okamoto, 1992]
because FCSLs there are negative-angular-momentum orbits and hence correspond to negative-energy orbits in the ergosphere in the Penrose process.
The counter-rotating inner domain thus nested inside  will be designed so that the electrodynamical process of self-extraction of energy can surely obey the thermodynamic laws. The null surface  is the key surface where the eigen-FLAV $\OmF$ and the eigen-FDAV $\omN$ can simultaneously be determined uniquely, thereby dragging the force-free magnetosphere into circulation round the hole with the FDAV $\omN=\OmF$, and hence enabling the Kerr hole to self-extract its rotational energy, though `non-adiabatically'. It is recently pointed out that “what is dragged by the Kerr hole are the ZAMOs and the compass of inertia" [Costa & Natário, 2021].  The force-free magnetospheres circulating with $\omN=\OmF$ around the Kerr hole may be included among them as well.

Thanks to frame-dragging, “a physical observer will see not only a Poynting flux of energy from  entering the hole" [Blandford & Znajek, 1977], but also he will see another Poynting flux from  outward, to transform into kinetic energy through the particle acceleration zone in the outer resistive membrane $\Sffinf$, and then to evolve to a high-energy gamma-ray jet beyond [Okamoto , 2015]. The force-free theory, inclusive of `complete violation' of its force-freeness, will yield a self-consistent, unique theory for extracting energy from Kerr holes.  These ingenious actions of frame-dragging may seem to be due to a wire puller manoeuvring behind the scene, and we may so far never have seen them as their real astrophysical effects.  The ZAMOs will not think of these rather modest actions of frame-dragging as `spooky' because they are the `physical observers'.

§.§ Concluding remarks

If observed large-scale high-energy $\gamma$-ray jets from AGNs really originate from quite near the event horizon of the central super-massive BHs, it appears to be plausible that these jets are a magnificent manifestation of the trinity of general relativity, thermodynamics, and electrodynamics (GTED);  precisely speaking,  frame-dragging, the first and second laws, and unipolar induction. The heart of the black hole's central engine may lie in the Gap $\GN$  between the two, outer and inner, light surfaces just above the horizon, and the embryo of a jet will be born in the Gap $\GN$ under the inductive membrane $\SN$.
The confirmation of this postulate awaits further illumination of the BH Gap physics.