#  The essence of the Blandford–Znajek process

Shunichiro Kinoshita<EMAIL_ADDRESS>Department of Physics, Chuo
University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan  Takahisa Igata
<EMAIL_ADDRESS>Department of Physics, Rikkyo University, Toshima, Tokyo
175-8501, Japan

###### Abstract

From a spacetime perspective, the dynamics of magnetic field lines of force-
free electromagnetic fields can be rewritten into a quite similar form for the
dynamics of strings, i.e., dynamics of “field sheets”. Using this formalism,
we explicitly show that the field sheets of stationary and axisymmetric force-
free electromagnetic fields have identical intrinsic properties to the world
sheets of rigidly rotating Nambu–Goto strings. Thus, we conclude that the
Blandford–Znajek process is kinematically identical to an energy-extraction
mechanism by the Nambu–Goto string with an effective magnetic tension.

††preprint: RUP-17-21

##  I Introduction and summary

The rotational energy of rotating black holes is a promising energy source for
the formation of relativistic jets, which are ubiquitous in astrophysics. The
Blandford–Znajek process Blandford:1977ds , which is an energy-extraction
mechanism by force-free electromagnetic fields, can efficiently achieve
powerful energy fluxes and thus has been widely believed to be a viable
mechanism to extract the rotational energy of a black hole. There have been a
number of analytical and numerical investigations from various aspects during
the four decades since this process was proposed. A detailed analysis of the
extraction mechanism was in terms of the membrane paradigm MacDonald:1982zz .
In recent years interpretations and explanations of this mechanism have been
discussed (see, e.g., Refs. Komissarov:2008yh ; Koide:2014xpa ; Toma:2016jmz
). Moreover, various numerical simulations have been developed and
demonstrated McKinney:2004ka ; McKinney:2012vh ; Penna:2013rga . In this paper
we will analytically reveal the essence of the energy-extraction mechanism in
the Blandford–Znajek process from an alternative perspective.

Recently, it has been elucidated that rigidly rotating Nambu–Goto strings
Frolov:1996xw  twining around a rotating black hole can highly efficiently
extract the rotational energy from the black hole Kinoshita:2016lqd . The
order of the possible energy flux, namely the energy-extraction rate, is given
by the so-called Dyson luminosity111For historical details, see Ref.
Barrow:2017atq . $\sim c^{5}/G_{\mathrm{N}}\simeq 10^{59}\mathrm{erg/s}$
Dyson:1963 , consisting of only fundamental constants: the speed of light $c$
and Newton’s constant $G_{\mathrm{N}}$, and a dimensionless string tension
$G_{\mathrm{N}}\mu/c^{2}$ as a coefficient. Such energy flux and angular-
momentum flux are locally determined by the locus where the string intersects
the light surface associated with its angular velocity, at which the velocity
of the corotating frame coincides with the speed of light. Moreover, a
necessary condition for the energy extraction to occur is that the light
surface enters into the ergoregion and the angular velocity of the string is
less than that of the black hole.

In this mechanism, if we replace the tension of the Nambu–Goto string with a
typical magnetic tension of electromagnetic fields surrounding a rotating
black hole $\sim B^{2}r_{\mathrm{h}}^{2}$, we can reproduce the energy-
extraction rate of the Blandford–Znajek process while assuming conventional
values of the magnetic field and the black hole mass. This fact suggests that
both the energy-extraction mechanisms are closely related and the magnetic
field lines with magnetic tension play an essential role in the
Blandford–Znajek process. The purpose of this paper is to exhibit that such
observations are exactly true in a quantitative and theoretical sense as well
as a qualitative and intuitive sense. We show that the energy-extraction
mechanism by stationary, axisymmetric force-free electromagnetic fields is
essentially identical to that by rigidly rotating Nambu–Goto strings. It
follows that the Blandford–Znajek process, i.e., the energy-extraction
mechanism from rotating black holes via force-free electromagnetic fields, is
the Penrose process for magnetic field lines with angular-momentum and energy
transport mediated by their magnetic tension.

For this purpose, we will first reformulate the dynamics of force-free
electromagnetic fields in accordance with a “field-sheet” formalism
Carter:1979 ; Uchida:1997a ; Uchida:1997b ; Gralla:2014yja . In this
formalism, magnetic field lines are fundamental objects to describe dynamics
rather than electric and magnetic fields. A time evolution of a magnetic field
line can be regarded as a 2D extended object in a spacetime. We will call such
objects “field sheets”, a name that was adopted in Ref. Gralla:2014yja , with
a similar connotation to the term “world sheet” of strings in a spacetime. In
the case of magnetically dominated force-free electromagnetic fields
($F_{\mu\nu}F^{\mu\nu}>0$) in particular, the field sheets become 2D timelike
surfaces characterized by the electromagnetic field strength $F_{\mu\nu}$. We
can recast the equations of motion for force-free electromagnetic fields,
equivalent to the Maxwell equations with the force-free condition, in the
equations of motion in terms of the field sheets. Thus, it turns out that the
dynamics of field sheets for force-free electromagnetic fields is similar to
the dynamics of a world sheet for a string, and they both belong to the same
category of dynamics of a 2D surface in a spacetime (a similar approach to
magnetohydrodynamics and its phenomenological applications were discussed in,
e.g., Refs. Christensson:1999tp ; Semenov:2000vb ; Semenov:2004ib ). This
provides an insight into a correspondence between extraction mechanisms by
force-free electromagnetic fields and rigidly rotating strings.

Needless to say, electric and magnetic fields are not components of vector
fields, but some of the components of a tensor field, i.e., the
electromagnetic field strength $F_{\mu\nu}$. Therefore, depending on
coordinate systems or reference frames, their physical interpretations can
change as well as their values. One may, for instance, move to a frame at
which only magnetic fields can be observed even if there exists a net energy
flow. Thus, the Poynting flux is only an interpretation of the energy flow in
another frame. Furthermore, in a strong gravitational field, namely, in a
curved spacetime, the spacetime metric affects conventions or interpretations
of the electric and magnetic field without physical significance. This means
that explanations based on electric and magnetic fields will vary depending on
the choice of frames. This seems to interfere with the concise understanding
of the mechanism. What we should emphasize is that the field sheet is a
geometrical object irrelevant to any coordinate system as well as a physically
intelligible object, thought of as the time evolution of a magnetic field
line. It is expected that we can grasp the essence without suffering from
coordinate systems.

With the above perspective in mind, in Sect. III, we focus on stationary and
axisymmetric force-free electromagnetic fields in a stationary and
axisymmetric spacetime to examine the energy-extraction process from a
rotating black hole. Since the basic properties of such electromagnetic fields
have been widely examined and are well known in the literature (see, e.g.,
Ref. Frolov:1998wf  and references therein), we translate those into
expressions based on the field sheet. We demonstrate that the field sheet of
the electromagnetic field and the world sheet of the Nambu–Goto string have
identical intrinsic properties such as their induced geometries. In
particular, the specific angular-momentum and energy fluxes per unit tension,
flowing on each field sheet, are determined by local configurations of the
magnetic field lines in the same manner as the string configurations. Most
importantly, on the light surface these specific quantities depend only on its
locus without global configurations determined by the equations of motion and
must satisfy there identical relations regardless of whether magnetic field
lines or strings; if the angular velocity of a magnetic field line is less
than that of a black hole, the angular momentum can be extracted, and in
addition if the locus where the magnetic field line crosses the light surface
enters into the ergoregion, the energy can be extracted. (The importance of
the light surface and other characteristic surfaces such as the Alfvén surface
has been discussed in the literature Takahashi:1990bv ; Contopoulos:2012py ;
Thoelecke:2017jrz .) However, because the dynamics of magnetic field lines and
strings are quite similar but different, both configurations cannot be
identical in general. This fact indicates that global configurations of the
magnetic fields have little significance for this energy-extraction mechanism
and a local, kinematical process in the ergoregion should govern the energy-
extraction mechanism. Hence, we can conclude that the essence of both the
energy-extraction mechanism by the force-free electromagnetic fields and the
rigidly rotating strings is identical.

In contrast with the tension of Nambu–Goto strings, the magnetic tension can
vary and should be determined by solving the equations of motion. This means
that global configurations of the magnetic fields can affect the value of the
magnetic tension. However, a role of the magnetic tension proportionally
provides efficiency of extraction rate. Roughly speaking, the larger the
magnetic tension is, the more efficient the extraction rate becomes. In this
process, the magnetic tension transports the angular-momentum and energy
fluxes on the magnetic lines.

The fact that the essence of the extraction mechanism is local kinematics in
the ergoregion irrelevant to global configurations offers some instructive
insights to clarify the whole picture of the Blandford–Znajek process. It is
not so significant whether magnetic field lines can penetrate the event
horizon or they reach the outer-light cylinder. These cannot be necessary
conditions for the Blandford–Znajek process to work. Moreover, the event
horizon rather than the ergoregion does not play an essential role in the
energy extraction (this issue was addressed in, e.g., Refs. Komissarov:2002dj
; Ruiz:2012te  on the basis of numerical simulations).

Generally speaking, the most important issue for extracting the rotational
energy from black holes is angular-momentum transport to gain the energy in
the ergoregion. The outward energy flux is just a by-product of the angular-
momentum transport. In the Einstein gravity the spacetime metric can couple to
energy–momentum tensors for any matters or fields. If one wishes to extract
the energy and angular momentum from the (black hole) spacetime, any other
method does not exist except for the method via the energy–momentum tensor
(for gravitational waves, the energy–momentum pseudotensor). To elucidate an
extraction mechanism of the rotational energy, we should examine what contents
of the energy–momentum tensor mainly contribute to the angular-momentum
transport (the extraction mechanism for a general energy–momentum tensor is
discussed in Ref. Lasota:2013kia ). Such angular-momentum and energy transfer,
i.e., local conservation of the energy–momentum tensor, should be governed by
local physics causally connected. Thus, the extraction process can be roughly
dissected into three parts: generating an energy by angular-momentum transfer
in the ergoregion, transporting the gained energy to a region far away from
the black hole, and disposing of garbage that has lost its angular momentum
and energy to the black hole. The essence of the energy extraction mentioned
previously is nothing but this generating process. On the other hand, global
configurations of the magnetic fields at the event horizon and at a far region
are related to the disposing process and the transporting one, respectively.

After all, “boundary conditions” at the event horizon such as whether the
magnetic field lines can penetrate the horizon cannot be a necessary condition
for the energy extraction. In fact, even if the magnetic field lines failed to
penetrate the horizon and could never get drawn into the black hole, the
energy extraction by the Blandford–Znajek process can succeed. One may say
that the energy is not extracted from any black hole in that case. However,
the fact remains that the gravitational energy measured in an asymptotic
region is extracted from the total system including the spacetime. Moreover,
even though the magnetic field lines cannot reach the outer-light cylinder, we
can say that the Blandford–Znajek process is at work if the magnetic field
lines extend to a region sufficiently far away from the black hole and the
gained energy in the ergoregion can be transferred there.

In this paper, we show that the essential mechanism of the Blandford–Znajek
process is determined by local physics in the neighborhood of the ergoregion.
Of course, global configurations of the magnetic field and the electric
current are important to make the Blandford–Znajek process efficiently and
successfully sustaining. However, this is just a stage rather than a principal
role. We stress that, in general, how to extract rotational energy from the
spacetime and how to arrange appropriate configurations of the magnetic field
or electric current for extracting the energy are different questions.
Furthermore, we should separately consider the kinematical properties without
globally solving the equations of motion and dynamical properties.

The rest of the paper is organized as follows. We first review the field-sheet
formalism for force-free electromagnetic fields. Then, we explicitly show a
correspondence between the energy-extraction mechanisms by stationary,
axisymmetric force-free electromagnetic fields and rigidly rotating strings.
In Appendix A we briefly summarize some results for the rigidly rotating
strings shown in Ref. Kinoshita:2016lqd .

##  II Field-sheet formalism for force-free electromagnetic fields

In this section, in order to elucidate the similarity between Nambu–Goto
strings and magnetic field lines of force-free fields, we will rewrite the
equations of motion for force-free electromagnetic fields according to a
“field-sheet” formalism Carter:1979 ; Uchida:1997a ; Uchida:1997b ;
Gralla:2014yja . Unless otherwise specified, Newton’s constant
$G_{\mathrm{N}}$ and the speed of light $c$ are set to unity hereafter.

Let $F_{\mu\nu}$ be the electromagnetic field strength and let $j_{\mu}$ be
the current density four-vector of electric charge. In terms of $F_{\mu\nu}$,
the Maxwell equations are given by

$\displaystyle\nabla_{\alpha}F^{\mu\alpha}=4\pi
j^{\mu},\quad\nabla_{[\>\\!\mu}F_{\nu\lambda\>\\!]}=0.$ (1)

In general, when an electromagnetic field interacts with charged matter such
as plasma, the energy–momentum tensor for the electromagnetic field,

$\displaystyle T_{\mu\nu}\equiv
F_{\mu\alpha}F_{\nu}{}^{\alpha}-\frac{1}{4}\>\\!F^{\alpha\beta}F_{\alpha\beta}\>\\!g_{\mu\nu},$
(2)

satisfies

$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu}=-4\pi F_{\nu\alpha}\>\\!j^{\alpha},$
(3)

where the right-hand side of the above equation means the Lorentz force. In
the situation where the force density four-vector
$F_{\nu\alpha}\>\\!j^{\alpha}$ can be neglected, i.e.,

$\displaystyle F_{\mu\alpha}j^{\alpha}=0,$ (4)

which is known as the force-free condition, the energy–momentum tensor of the
electromagnetic field is individually conserved
$\nabla_{\mu}T^{\mu}{}_{\nu}=0$, so that neither angular momentum nor energy
is exchanged between the electromagnetic field and the other matter. The
Maxwell equations (1) together with the force-free condition (4) yield the
equations

$\displaystyle
F_{\mu\alpha}\nabla_{\beta}F^{\alpha\beta}=0,\quad\nabla_{[\>\\!\mu}F_{\nu\lambda\>\\!]}=0.$
(5)

The dynamics described by these equations is force-free electrodynamics (FFE).

An important property for the force-free electromagnetic fields satisfies

$F_{\alpha\beta}{}^{*}\\!F^{\alpha\beta}=0,$ (6)

where ${}^{*}\\!F_{\mu\nu}$ is the dual of $F_{\mu\nu}$, defined by
${}^{*}\\!F_{\mu\nu}\equiv F^{\alpha\beta}\epsilon_{\alpha\beta\mu\nu}/2$.
These fields are called degenerate. Note that the force-free condition for
nonzero $j^{\mu}$ implies that $F_{\mu\nu}$ is degenerate,222 In four
dimensions, the relation
$F_{[\alpha\beta}F_{\mu\nu]}=-\epsilon_{\alpha\beta\mu\nu}F_{\lambda\rho}{}^{*}\\!F^{\lambda\rho}/12$
is satisfied. While $F_{\mu\nu}j^{\mu}=0$, we have
$F_{\lambda\rho}{}^{*}\\!F^{\lambda\rho}=0$ if $j^{\mu}$ is nonzero.  but
degeneracy does not always lead to force-freeness. To clarify the physical
meanings, let $t^{\mu}$ be a four-velocity of a timelike observer. Then, the
electric and magnetic fields measured by this observer are given by
$\mathsf{E}^{\mu}=F^{\mu\nu}t_{\nu}$ and
$\mathsf{B}^{\mu}=-{}^{*}\\!F^{\mu\nu}t_{\nu}$, respectively. The degenerate
condition physically means that $\mathsf{E}^{\alpha}\mathsf{B}_{\alpha}=0$,
where this relation holds even for an arbitrary observer $t^{\mu}$ because
$F_{\alpha\beta}{}^{*}\\!F^{\alpha\beta}$ is scalar. Now, the fact that
$F_{\mu\nu}$ is closed together with the degeneracy implies that
${}^{*}\\!F_{\mu\nu}$ is tangent to a 2D submanifold, $\mathscr{S}$.
Furthermore, we assume that $F_{\mu\nu}$ is magnetically dominated, i.e.,

$\displaystyle
F_{\alpha\beta}F^{\alpha\beta}=2\>\\!(\mathsf{B}^{\alpha}\mathsf{B}_{\alpha}-\mathsf{E}^{\alpha}\mathsf{E}_{\alpha})>0.$
(7)

Naively, this condition implies that the magnetic field should be stronger
than the electric field. Because this condition is also described by a scalar,
there is a notion independent of observers. The degeneracy and magnetically
dominated condition for $F_{\mu\nu}$ guarantee the existence of a pure
magnetic frame in which $\mathsf{E}^{\mu}=0$. Therefore,
$F_{\alpha\beta}F^{\alpha\beta}>0$ states that the electromagnetic field
$F_{\mu\nu}$ is purely magnetic with its magnitude defined by

$\displaystyle B\equiv\sqrt{\frac{F_{\mu\nu}F^{\mu\nu}}{2}}.$ (8)

Namely, we can take ${}^{*}\\!F_{\mu\nu}$ in the form

${}^{*}\\!F_{\mu\nu}=B\sigma_{\mu\nu},$ (9)

where $\sigma_{\mu\nu}$ denotes the 2D volume element on $\mathscr{S}$. It
turns out that the magnetically dominated condition for $F_{\mu\nu}$ is
equivalent to $\sigma_{\mu\nu}\sigma^{\mu\nu}=-2$, and then $\mathscr{S}$
becomes timelike. We call such $\mathscr{S}$ a field sheet. The scalar
function $B(>0)$ irrelevant to coordinate systems means the proper magnitude
of the magnetic field, which can be observed at the rest frame of the magnetic
field lines. Moreover, the magnetic tension and pressure are given by this
quantity, so that they are also proper quantities independent of coordinate
systems.

We describe the dynamics of field sheets as string world sheets. In terms of
${}^{*}\\!F_{\mu\nu}$, the equations of FFE are rewritten as

$\displaystyle{}^{*}\\!F^{\alpha\beta}\>\\!\nabla_{[\>\\!\mu}{}^{*}\\!F_{\alpha\beta\>\\!]}=0,\quad\nabla_{\alpha}{}^{*}\\!F^{\alpha\mu}=0.$
(10)

Substituting Eq. (9) into the above, we obtain the equations of FFE in terms
of $B$ and $\sigma_{\mu\nu}$ given by

$\displaystyle\sigma^{\alpha\beta}\nabla_{\alpha}\sigma_{\beta\mu}=N_{\mu}{}^{\alpha}\>\\!\nabla_{\alpha}\ln
B,$ (11) $\displaystyle\nabla_{\alpha}(B\>\\!\sigma^{\alpha\mu})=0,$ (12)

where $N_{\mu}{}^{\nu}$ is defined by

$\displaystyle N_{\mu}{}^{\nu}\equiv g_{\mu}{}^{\nu}-h_{\mu}{}^{\nu},$ (13)
$\displaystyle\
h_{\mu}{}^{\nu}\equiv\sigma_{\mu\alpha}\>\\!\sigma^{\alpha\nu}.$ (14)

Note that $h_{\mu\nu}$ is the induced metric on the field sheet and
$N_{\mu}{}^{\nu}$ is the projection tensor onto directions normal to the field
sheet. The former equation (11) describes dynamics of field sheets acted on by
a force associated with $B$, i.e., magnetic pressure; the latter equation (12)
does conservation of the magnetic flux. This expression tells us that the
dynamics of field sheets in FFE is similar to the dynamics of world sheets for
Nambu–Goto strings. In fact, it is known that the equations of motion for a
Nambu–Goto string can be written as

$\sigma^{\alpha\beta}\nabla_{\alpha}\sigma_{\beta\mu}=0,$ (15)

where $\sigma_{\mu\nu}$ denotes the volume element of the world sheet in this
case Schild:1976vq . Comparing the above equation with Eqs. (11) and (12), we
can easily notice that the only difference between the dynamics of field
sheets and world sheets is the addition of the extra scalar quantity $B$,
which describes the magnetic tension and pressure. (In the flat spacetime such
a correspondence was discussed in Refs. Kastrup:1979fz ; Rinke:1979mw ;
Nambu:1981gt .)

From a geometrical point of view, their meanings are so clear. In general, the
extrinsic curvature of a submanifold is defined by

$K^{\lambda}{}_{\mu\nu}\equiv-
h_{\mu}{}^{\alpha}h_{\nu}{}^{\beta}\nabla_{\beta}h_{\alpha}{}^{\lambda},$ (16)

where $h_{\mu\nu}$ is the induced metric on the submanifold. If the
submanifold is 2D, we have

$K^{\lambda}{}_{\mu\nu}h^{\mu\nu}=-\sigma^{\alpha\beta}\nabla_{\alpha}\sigma_{\beta}{}^{\lambda}.$
(17)

Thus, the dynamics of field sheets and world sheets belong to the same class
of dynamics of a 2D surface (see Ref. Carter:2000wv  and references therein).
Because every degenerate, closed two-form defines a foliation of
spacetime,333See Appendix A in Ref. Compere:2016xwa . the field sheets
represented by $F_{\mu\nu}$ can define a 2D timelike foliation with a
coordinate transformation $x^{\mu}=X^{\mu}(\tau,\sigma,\alpha,\beta)$, where
$(\tau,\sigma)$ denote local coordinates on field sheets $\mathscr{S}$ defined
by $\alpha=\mathrm{const.}$ and $\beta=\mathrm{const}$. Note that, once
$\alpha$ and $\beta$ are fixed, $X^{\mu}$ give embedding functions of
$\mathscr{S}$. Let $\partial_{a}$ be coordinate derivatives with respect to 2D
local coordinates on the field sheet. The Latin indices denote intrinsic
components on the field sheet. The volume element $\sigma_{\mu\nu}$ is related
to $X^{\mu}$ as

$\displaystyle\sigma^{\mu\nu}=\sigma^{ab}h_{a}{}^{\mu}h_{b}{}^{\nu},$ (18)

where $h_{a}{}^{\mu}\equiv\partial_{a}X^{\mu}$, and $\sigma_{ab}$ is the
intrinsic volume element on $\mathscr{S}$. Moreover, the induced metric can be
intrinsically written as

$h_{ab}=g_{\mu\nu}(X)h_{a}{}^{\mu}h_{b}{}^{\nu}=h_{\mu\nu}h_{a}{}^{\mu}h_{b}{}^{\nu}.$
(19)

By using these intrinsic quantities, we can rewrite Eq. (11) in terms of
$X^{\mu}$ as

$D^{2}X^{\mu}+\Gamma^{\mu}{}_{\alpha\beta}D_{a}X^{\alpha}D^{a}X^{\beta}=N^{\mu\alpha}\partial_{\alpha}\ln
B,$ (20)

where $D_{a}$ denotes the covariant derivative with respect to $h_{ab}$ and
$\Gamma^{\mu}{}_{\alpha\beta}$ is the Christoffel symbol associated with
$g_{\mu\nu}$. Note that, intriguingly, this equation is derived from the
following action: $S[X^{\mu}]=-\int d^{2}\sigma B(X)\sqrt{-h}$, where $h$
denotes the determinant of the induced metric $h_{ab}$.

It is instructive to compare a perfect fluid in terms of conservation of the
energy–momentum tensor. As is well known, the energy–momentum tensor of a
perfect fluid is given by

$T^{\mu\nu}=\rho u^{\mu}u^{\nu}+p(g^{\mu\nu}+u^{\mu}u^{\nu}),$ (21)

where $u^{\mu}$ is a four-velocity of the fluid, and $\rho$ and $p$ are the
energy density and pressure in its rest frame, respectively. Decomposing the
conservation law $\nabla_{\mu}T^{\mu\nu}=0$ into tangential components and
normal ones with respect to $u^{\mu}$ yields

$\displaystyle(\rho+p)u^{\mu}\nabla_{\mu}u^{\nu}$
$\displaystyle=-(g^{\mu\nu}+u^{\mu}u^{\nu})\nabla_{\mu}p,$ (22) $\displaystyle
u^{\mu}\nabla_{\mu}\rho+(\rho+p)\nabla_{\mu}u^{\mu}$ $\displaystyle=0.$

If the fluid is pressureless (namely, a dust fluid), the four-velocity of the
fluid obeys geodesic equations. In a parallel manner, we consider an
energy–momentum tensor given by

$\displaystyle T^{\mu\nu}=$ $\displaystyle-\mu
h^{\mu\nu}+\tilde{p}(g^{\mu\nu}-h^{\mu\nu})$ (23) $\displaystyle=$
$\displaystyle(\mu+\tilde{p})\sigma^{\mu\alpha}\sigma^{\nu}{}_{\alpha}+\tilde{p}g^{\mu\nu},$

where $\mu$ is a tension equal to its energy density and $\tilde{p}$ is a
normal pressure. Note that Nambu–Goto strings have a constant $\mu$ and
$\tilde{p}=0$, and magnetically dominated electromagnetic fields have
$\mu=\tilde{p}=B^{2}/2$. Decomposing the energy–momentum conservation into
components tangential and normal to the field sheet yields

$\displaystyle(\mu+\tilde{p})\sigma^{\alpha\beta}\nabla_{\alpha}\sigma_{\beta\mu}$
$\displaystyle=(g_{\mu}{}^{\alpha}-h_{\mu}{}^{\alpha})\nabla_{\alpha}\tilde{p},$
(24)
$\displaystyle\sigma^{\alpha\mu}\nabla_{\alpha}\mu+(\mu+\tilde{p})h^{\mu}{}_{\nu}\nabla_{\alpha}\sigma^{\alpha\nu}$
$\displaystyle=0.$

Thus, it turns out that the world line and its volume element $u_{\mu}$ for a
fluid element of perfect fluids correspond to the world sheet (field sheet)
and its volume element $\sigma_{\mu\nu}$ for a Nambu–Goto string or a magnetic
field line. Moreover, in each pressureless case the world lines of dusts
become geodesics and the world sheets of Nambu–Goto strings are extremal
surfaces.

If the system admits some symmetries and such a symmetry is characterized by a
Killing vector field $\xi^{\mu}$, the conservation law of the energy–momentum
tensor in terms of $\xi^{\mu}$ yields

$\displaystyle 0$ $\displaystyle=\nabla_{\mu}(T^{\mu\nu}\xi_{\nu})$ (25)
$\displaystyle=-\nabla_{\mu}[(\mu+\tilde{p})h^{\mu\nu}\xi_{\nu}]+\xi^{\mu}\nabla_{\mu}\tilde{p}$
$\displaystyle=-\nabla_{\mu}[(\mu+\tilde{p})h^{\mu\nu}\xi_{\nu}],$

where we have used $\xi^{\mu}\nabla_{\mu}\tilde{p}=0$ because of the symmetry.
This conservation law is equivalent to that of a string with an effective
tension $\mu+\tilde{p}$. Thus, even though the pressure does not vanish,
$\tilde{p}\neq 0$, kinematic properties such as the conservation law can be
identical thanks to the symmetry.

##  III Energy extraction by force-free electromagnetic fields

###  III.1 Stationary and axisymmetric force-free electromagnetic fields

In this section, we consider stationary and axisymmetric force-free
electromagnetic fields in a rotating black hole, and reinterpret energy-
extraction processes via the force-free electromagnetic fields from a
perspective of the field-sheet formalism. Such stationary and axisymmetric
force-free electromagnetic fields in a stationary and axisymmetric spacetime
have been comprehensively investigated in the literature, and their basic
properties are well known Frolov:1998wf . Note that the above assumptions are
conventional ones when the Blandford–Znajek process has been discussed. In the
field-sheet approach, we will follow Ref. Gralla:2014yja .

In order that we will later focus on the Kerr spacetime as an explicit
example, we suppose that the spacetime is stationary and axisymmetric and
admits two Killing vector fields $(\partial_{t})^{\mu}$ and
$(\partial_{\phi})^{\mu}$, which respectively represent time translation
symmetry and axisymmetry of the spacetime as
$\mathcal{L}_{\partial_{t}}g_{\mu\nu}=\mathcal{L}_{\partial_{\phi}}g_{\mu\nu}=0$.
Here, $\mathcal{L}$ denotes the Lie derivative. Its metric can be written as

$g_{\mu\nu}dx^{\mu}dx^{\nu}=g_{tt}dt^{2}+2g_{t\phi}dtd\phi+g_{\phi\phi}d\phi^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2},$
(26)

where we assume that $g_{tt}g_{\phi\phi}-g_{t\phi}^{2}=0$ characterizes the
event horizon and $g_{tt}g_{\phi\phi}-g_{t\phi}^{2}<0$ implies the outside of
the black hole. Here, we have taken a coordinate system in which the Killing
vector fields $(\partial_{t})^{\mu}$ and $(\partial_{\phi})^{\mu}$ are
manifestly orthogonal to 2D surfaces spanned by $(r,\theta)$ such as the
Boyer–Lindquist coordinates for the Kerr spacetime. Moreover, we suppose that
the electromagnetic fields share the same Killing symmetries such that
$\mathcal{L}_{\partial_{t}}F_{\mu\nu}=\mathcal{L}_{\partial_{\phi}}F_{\mu\nu}=0$.

It is known that stationary and axisymmetric force-free electromagnetic fields
are given by the following field strength:

$F=\frac{1}{2}F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}=d\psi\wedge(\eta-d\varphi),$
(27)

where $\eta\equiv d\phi-\omega(\psi)dt$, and both $\psi$ and $\varphi$ are
scalar functions independent of $t$ and $\phi$. This yields a common component
representation of the field strength:

$F_{\mu
t}=F_{\mu\nu}(\partial_{t})^{\nu}=-\omega(\psi)\partial_{\mu}\psi,\quad
F_{\mu\phi}=F_{\mu\nu}(\partial_{\phi})^{\nu}=\partial_{\mu}\psi,\quad
F_{r\theta}=\frac{\partial\varphi}{\partial
r}\frac{\partial\psi}{\partial\theta}-\frac{\partial\varphi}{\partial\theta}\frac{\partial\psi}{\partial
r},$ (28)

which are related to an electric field, and poloidal and toroidal components
of the magnetic field, respectively. Note that we can easily confirm that
$dF=0$ is satisfied. The form of this field strength (27) can be expressed as
$F=d\psi\wedge d\hat{\phi}$ in terms of the so-called Euler potentials given
by the following scalar functions: $\psi$ and
$\hat{\phi}\equiv\phi-\omega(\psi)t-\varphi$ (see, e.g., Ref. Gralla:2014yja
). This means that the field sheets are represented by the intersections of
the hypersurfaces of constant $\psi$ and $\hat{\phi}$. Each magnetic field
line given by the field sheets is rotating with angular velocity
$\omega(\psi)$. Obviously, the field sheets described by the same $\psi$ have
the same angular velocity because of the axisymmetry. Such a $\psi$-constant
surface is a so-called magnetic surface, and $2\pi\psi=\int F$ gives the
magnetic flux across the region enclosed by the magnetic surface.

The “proper” magnetic field $B$ is given by

$B^{2}\equiv\frac{1}{2}F_{\mu\nu}F^{\mu\nu}=\eta^{2}|\nabla\psi|^{2}+|\nabla\psi|^{2}|\nabla\varphi|^{2}-(\nabla\psi\cdot\nabla\varphi)^{2}.$
(29)

Here, we have written the contractions for vector fields $u$ and $v$ as
$u\cdot v=u_{\mu}v^{\mu}$ and $u^{2}=u_{\mu}u^{\mu}$ in the abbreviated
notation. In addition, we have

$F_{r\theta}F^{r\theta}=g^{rr}g^{\theta\theta}F_{r\theta}^{2}=|\nabla\psi|^{2}|\nabla\varphi|^{2}-(\nabla\psi\cdot\nabla\varphi)^{2}.$
(30)

As we mentioned in the previous section, $B>0$ is a scalar function irrelevant
to coordinate systems, which gives the magnetic tension and pressure.

Now, $\chi$ is a corotating vector defined by

$\chi^{\mu}=\left(\frac{\partial}{\partial
t}\right)^{\mu}+\omega(\psi)\left(\frac{\partial}{\partial\phi}\right)^{\mu},$
(31)

which satisfies $\chi^{\mu}\eta_{\mu}=0$ and
$\chi^{2}=g_{tt}+2g_{t\phi}\omega+g_{\phi\phi}\omega^{2}=-(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})\eta^{2}$.
Since the corotating vector $\chi$ satisfies $F_{\mu\nu}\chi^{\nu}=0$, $\chi$
is tangential to the field sheet. It turns out that $\chi$ represents a
corotating frame with the angular velocity $\omega(\psi)$ for each field sheet
labeled by constant $\psi$. If $\chi$ (also, $\eta$) becomes null, the
velocity of such a corotating frame has reached the speed of light. The locus
characterized by $\chi^{2}=0$ is called the light surface.444 In general, two
light surfaces can exist in a black hole spacetime: one is the inner-light
surface near the black hole, and the other is the outer-light surface (light
cylinder) in the far region. Since the inner-light surface is important for
extracting the rotational energy from the black hole Kinoshita:2016lqd , we
will focus only on the inner-light surface hereafter. Meanwhile, because a
field sheet is a 2D surface, there is another tangential vector linearly
independent of $\chi$. We will introduce the other tangential vector as

$\displaystyle\lambda^{\mu}$
$\displaystyle\equiv{}^{*}\\!F^{\mu\nu}\nabla_{\nu}t$ (32)
$\displaystyle=\frac{1}{\sqrt{-g}}\left[F_{\theta\phi}\left(\frac{\partial}{\partial
r}\right)^{\mu}+F_{\phi
r}\left(\frac{\partial}{\partial\theta}\right)^{\mu}+F_{r\theta}\left(\frac{\partial}{\partial\phi}\right)^{\mu}\right].$

By construction, this vector is tangential to the field sheet and normal to
$\nabla_{\mu}t$, so that $\lambda$ becomes a generator of the intersection of
the field sheet and a constant-$t$ surface. This means that the integral
curves of $\lambda$ are magnetic field lines at a time $t$. The direction of
$\lambda$ is radially outward for $F_{\theta\phi}=\partial_{\theta}\psi>0$,
while its direction is radially inward for $\partial_{\theta}\psi<0$. It turns
out that the direction of $\lambda$ corresponds to that of the magnetic field.
Note that, since $[\lambda,\chi]=\mathcal{L}_{\lambda}\chi=0$ are satisfied,
$\chi$ and $\lambda$ can be coordinate bases on the field sheet as
$\chi=\partial/\partial\tau$ and $\lambda=\partial/\partial\sigma$, where
$\tau$ and $\sigma$ denote local coordinates on the field sheet. The dual of
$F_{\mu\nu}$ is expressed as

${}^{*}\\!F^{\mu\nu}=\lambda^{\mu}\chi^{\nu}-\chi^{\mu}\lambda^{\nu}.$ (33)

This expression manifestly shows that it is a two-form tangential to the field
sheet, i.e., proportional to the volume form of the field sheet as
${}^{*}\\!F_{\mu\nu}=B\sigma_{\mu\nu}$.

###  III.2 Induced geometry on the field sheet

As we mentioned, $\chi$ and $\lambda$ can constitute the tangential coordinate
bases on the field sheet, so that components of the induced metric on the
field sheet in terms of the coordinate system $(\tau,\sigma)$ are given by

$\displaystyle h_{\tau\tau}$
$\displaystyle\equiv\chi^{\mu}\chi^{\nu}g_{\mu\nu}=g_{tt}+2g_{t\phi}\omega+g_{\phi\phi}\omega^{2},$
(34) $\displaystyle h_{\tau\sigma}$
$\displaystyle\equiv\chi^{\mu}\lambda^{\nu}g_{\mu\nu}=\frac{F_{r\theta}}{\sqrt{-g}}(g_{t\phi}+g_{\phi\phi}\omega),$
$\displaystyle h_{\sigma\sigma}$
$\displaystyle\equiv\lambda^{\mu}\lambda^{\nu}g_{\mu\nu}=\frac{1}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}|\nabla\psi|^{2}+g_{\phi\phi}\left(\frac{F_{r\theta}}{\sqrt{-g}}\right)^{2}.$

Since $\chi$ is the corotating vector, the light surface is characterized by
$\chi^{2}=h_{\tau\tau}=0$. In addition, $\chi$ is the Killing vector with
respect to the induced metric on the field sheet,555 Each component of the
induced metric is a function of only $r$ and $\theta$. Since
$\mathcal{L}_{\chi}r=\mathcal{L}_{\chi}\theta=0$, we have
$\mathcal{L}_{\chi}h_{ab}=0$ on the field sheet.  so that the light surface
corresponds to the Killing horizon on the field sheet. This Killing horizon
works as a causal boundary for various phenomena governed by the induced
metric Gralla:2014yja . We find the following correspondence between the
induced metric of the field sheet and that of the rigidly rotating string
(61):

$\frac{F_{r\theta}}{\sqrt{-g}}\leftrightarrow\varphi^{\prime},\quad\frac{1}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}|\nabla\psi|^{2}\leftrightarrow
g_{rr}r^{\prime 2}+g_{\theta\theta}\theta^{\prime 2}.$ (35)

Indeed, we can explicitly show

$\mathcal{L}_{\lambda}\varphi=\lambda^{\mu}\partial_{\mu}\varphi=\frac{1}{\sqrt{-g}}\left(\frac{\partial\psi}{\partial\theta}\frac{\partial\varphi}{\partial
r}-\frac{\partial\psi}{\partial
r}\frac{\partial\varphi}{\partial\theta}\right)=\frac{F_{r\theta}}{\sqrt{-g}},$
(36)

and

$\mathcal{L}_{\lambda}r=\lambda^{\mu}\partial_{\mu}r=\frac{\partial_{\theta}\psi}{\sqrt{-g}},\quad\mathcal{L}_{\lambda}\theta=\lambda^{\mu}\partial_{\mu}\theta=-\frac{\partial_{r}\psi}{\sqrt{-g}}.$
(37)

If we introduce the local coordinates as $\chi=\partial/\partial\tau$ and
$\lambda=\partial/\partial\sigma$ restricted on the field sheet, the
correspondence becomes more apparent. Thus, the intrinsic geometry on the
field sheet of the stationary, axisymmetric force-free electromagnetic field
and that on the world sheet of the rigidly rotating string have entirely
identical properties.

###  III.3 Angular-momentum flux and energy flux

Since the spacetime has time-translational symmetry and axisymmetry, we have
conservation laws for the energy–momentum tensor that are associated with each
symmetry. In particular, the energy–momentum tensor of the force-free
electromagnetic field is conserved independently of other matter. The angular-
momentum conservation $\nabla_{\mu}(T^{\mu}{}_{\nu}(\partial_{\phi})^{\nu})=0$
yields

$\displaystyle 0$
$\displaystyle=\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}T^{\mu}{}_{\nu}(\partial_{\phi})^{\nu})$
(38)
$\displaystyle=-\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}F^{\mu\nu}\partial_{\nu}\psi)$
$\displaystyle=-\frac{1}{\sqrt{-g}}\left(\partial_{\theta}\psi\frac{\partial}{\partial
r}-\partial_{r}\psi\frac{\partial}{\partial\theta}\right)\sqrt{-g}F^{r\theta}.$

This indicates that $\sqrt{-g}F^{r\theta}$ should depend only on $\psi$ as

$I(\psi)\equiv\sqrt{-g}F^{r\theta}.$ (39)

Note that this scalar function666 Since
$I(\psi)={}^{*}F_{\mu\nu}(\partial_{t})^{\mu}(\partial_{\phi})^{\nu}$, we can
see that $I(\psi)$ is scalar.  is constant on the field sheets, and it is one
of the characteristic quantities characterizing the stationary and
axisymmetric force-free electromagnetic field as well as $\psi$ and
$\omega(\psi)$. By using $\lambda$, this angular-momentum conservation can be
rewritten as

$\frac{1}{\sqrt{-g}}\partial_{t}(\sqrt{-g}T^{t}{}_{\nu}(\partial_{\phi})^{\nu})=\lambda^{\mu}\partial_{\mu}I(\psi).$
(40)

Therefore, $I(\psi)$ is identified with the angular-momentum flux (per unit
magnetic flux $d\psi$) flowing on the field sheet, which should be conserved
for each field sheet. Note that $-I(\psi)$ gives the outward angular-momentum
flux for $\partial_{\theta}\psi>0$, while $I(\psi)$ is the outward flux for
$\partial_{\theta}\psi<0$. This is because the direction of $\lambda$ changes
depending on the sign of $\partial_{\theta}\psi$ (for instance, recall that
$\mathcal{L}_{\lambda}r=\partial_{\theta}\psi/\sqrt{-g}$ in Eq. (37)). It is
well known that $I(\psi)$ is connected with the electric current according to
Ampère’s law. However, we stress that this is electromagnetic angular-momentum
flux without involving matter such as charged particles, because the relevant
energy–momentum tensor consists only of the electromagnetic field and it is
individually conserved thanks to the force-free condition. Similarly, the
energy conservation $\nabla_{\mu}(T^{\mu}{}_{\nu}(\partial_{t})^{\nu})=0$
leads to

$\frac{1}{\sqrt{-g}}\partial_{t}(-\sqrt{-g}T^{t}{}_{\nu}(\partial_{t})^{\nu})=\lambda^{\mu}\partial_{\mu}[\omega(\psi)I(\psi)],$
(41)

where $\omega(\psi)I(\psi)$ is identified with the conserved energy flux
flowing on the field sheet. As in the case of the angular-momentum flux,
$-\omega(\psi)I(\psi)$ gives the outward energy flux for
$\partial_{\theta}\psi>0$, while $\omega(\psi)I(\psi)$ is the outward flux for
$\partial_{\theta}\psi<0$. Integrating Eqs. (40) and (41) over a 3D volume on
a $t$-constant hypersurface surrounding the black hole, we have total fluxes
of angular momentum $\mathcal{J}$ and energy $\mathcal{M}$, i.e., each
extraction rate from the black hole,777We denote the mass and angular momentum
of the black hole as $\mathcal{M}_{\mathrm{BH}}$ and
$\mathcal{J}_{\mathrm{BH}}$. Each conservation law yields
$\frac{d\mathcal{M}}{dt}+\frac{d\mathcal{M}_{\mathrm{BH}}}{dt}=0$ and
$\frac{d\mathcal{J}}{dt}+\frac{d\mathcal{J}_{\mathrm{BH}}}{dt}=0$.  as

$\frac{d\mathcal{J}}{dt}=-2\pi\int
I(\psi)d\psi,\quad\frac{d\mathcal{M}}{dt}=-2\pi\int\omega(\psi)I(\psi)d\psi.$
(42)

Now, we find from Eq. (36) that

$I(\psi)=\sqrt{-g}F^{r\theta}=(-g)g^{rr}g^{\theta\theta}\frac{F_{r\theta}}{\sqrt{-g}}=(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})\mathcal{L}_{\lambda}\varphi,$
(43)

and also we find from Eqs. (29) and (30) that

$\displaystyle I(\psi)^{2}$
$\displaystyle=(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})(B^{2}-\eta^{2}|\nabla\psi|^{2})$
(44)
$\displaystyle=(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})B^{2}+\chi^{2}|\nabla\psi|^{2}.$

Combining the above two results, we can obtain

$\hat{q}\equiv\mp\frac{I(\psi)}{B}=\mp\frac{(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})\mathcal{L}_{\lambda}\varphi}{\sqrt{(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})(\mathcal{L}_{\lambda}\varphi)^{2}-\chi^{2}[g_{rr}(\mathcal{L}_{\lambda}r)^{2}+g_{\theta\theta}(\mathcal{L}_{\lambda}\theta)^{2}]}},$
(45)

where we take the upper minus sign for $\partial_{\theta}\psi>0$ and the lower
plus sign for $\partial_{\theta}\psi<0$. In a unified manner, alternatively,
it can be rewritten as

$\hat{q}=-\frac{(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})d\varphi/dr}{\sqrt{(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})(d\varphi/dr)^{2}-\chi^{2}[g_{rr}+g_{\theta\theta}(d\theta/dr)^{2}]}},$
(46)

where $d\varphi/dr\equiv\mathcal{L}_{\lambda}\varphi/\mathcal{L}_{\lambda}r$
and $d\theta/dr\equiv\mathcal{L}_{\lambda}\theta/\mathcal{L}_{\lambda}r$. This
quantity $\hat{q}$ means the specific angular-momentum flux per unit tension,
and its expression is identical to that of the specific angular-momentum flux
for the rigidly rotating strings Kinoshita:2016lqd  (see also Eq. (65)).
Similarly, $\omega(\psi)\hat{q}$ is the specific energy flux per unit tension.
Since the sign of $\hat{q}$ has been defined such that $\hat{q}$ should be
positive if a radially outward flux, $\hat{q}>0$ and $\omega\hat{q}>0$ mean
angular momentum extracting and energy extracting, respectively. It is worth
noting that the sign of $\hat{q}$ is irrelevant to the direction of $\lambda$
as is clear from Eq. (46). In other words, whether it is an extracting process
or an injecting one does not depend on the direction of the magnetic field but
the configuration of the magnetic field line. One thing to keep in mind as a
difference from the cases of the rigidly rotating strings is that the specific
angular-momentum flux $\hat{q}$ is not conserved while $I(\psi)$ is conserved
on the field sheet. The tension of Nambu–Goto strings is constant, while the
magnetic tension associated with $B$ can vary even on the field sheet.
Therefore, the specific angular-momentum flux per tension should be not
necessarily conserved, or more accurately the scalar function $B$ can work as
an effective tension for the field sheet.888 As can be seen from the
energy–momentum tensor, $B^{2}$ describes the magnetic tension per unit area,
which has the same dimension as pressure. Now, $B$ describes the magnetic
tension per unit magnetic flux.

So far we have shown that the intrinsic structures on the field sheet are
identical to those on the world sheet of the rigidly rotating string. This
does not mean that the global, extrinsic structures of both objects will be
identical. The equations of motion for each are indeed similar, but they are
not identical, as seen in the previous section. In general, the global
configurations of the Nambu–Goto strings and the magnetic field lines are
different, and such global configurations should be determined by each
dynamics by solving the equations of motion. However, it is noteworthy that
kinetic properties can be locally determined without solving the equations of
motion. In what follows we will see that the specific angular-momentum flux is
in fact governed by local relations on the light surface. On the light
surface, which is characterized by

$\chi^{2}=g_{tt}+2g_{t\phi}\omega(\psi)+g_{\phi\phi}\omega(\psi)^{2}=0,$ (47)

we have the specific angular-momentum flux

$\hat{q}^{2}=(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})|_{\chi^{2}=0},$ (48)

by substituting $\chi^{2}=0$ into Eq. (44). It turns out that this expression
is identical to that of the specific angular-momentum flux for the rigidly
rotating strings (64). Since Eqs. (47) and (48) are equations in terms of $r$
and $\theta$ essentially, they give relations among the angular velocity
$\omega$, the specific angular-momentum flux $\hat{q}$, and the locus of the
light surface $(r_{\mathrm{LS}},\theta_{\mathrm{LS}})$. Thus, we conclude that
both the stationary, axisymmetric force-free electromagnetic fields and the
rigidly rotating strings have identical relations. It is worth noting that
these relations are determined by the background spacetime metric irrelevant
to the dynamics of the electromagnetic field. Even though the dynamics of
electromagnetic fields and rigidly rotating strings are different, i.e., their
global configurations are different, the same locus of the light surface
provides the same angular velocity and specific angular-momentum flux. This
fact implies that the specific angular-momentum flux is kinematically
determined and both extracting mechanisms of energy and angular momentum via
the force-free electromagnetic fields and the rigidly rotating strings are
essentially identical.

###  III.4 Energy extraction in the Kerr spacetime

Now, to discuss the energy-extraction process from a rotating black hole
explicitly, let us focus on the Kerr spacetime. Since the major properties of
the force-free electromagnetic fields and the rigidly rotating strings are
identical, as seen, most of the following argument and its results are the
same as those shown in Ref. Kinoshita:2016lqd . For more details, refer to it.

In the Boyer–Lindquist coordinates the metric of the Kerr spacetime with mass
$M$ and angular momentum $aM$ is given by

$g_{\mu\nu}dx^{\mu}dx^{\nu}=-dt^{2}+\frac{2Mr}{\Sigma}(dt-a\sin^{2}\theta
d\phi)^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2}+(r^{2}+a^{2})\sin^{2}\theta d\phi^{2},$ (49)

where

$\Sigma(r,\theta)=r^{2}+a^{2}\cos^{2}\theta,\quad\Delta(r)=r^{2}+a^{2}-2Mr.$
(50)

For simplicity and without loss of generality, we assume $a>0$. This metric
gives $g_{t\phi}^{2}-g_{tt}g_{\phi\phi}=\Delta\sin^{2}\theta$, and the event
horizon lies at $r=r_{\mathrm{h}}\equiv M+\sqrt{M^{2}-a^{2}}$ defined by
$\Delta(r_{\mathrm{h}})=0$. Note that we have $\Delta>0$ outside the black
hole $r>r_{\mathrm{h}}$. The angular velocity of the black hole is given by
$\Omega_{\mathrm{h}}\equiv a/(r_{\mathrm{h}}^{2}+a^{2})$. The ergosphere is
characterized by the locus where the stationary Killing vector
$(\partial_{t})^{\mu}$ becomes null, i.e. $g_{tt}=0$, and its radius is
$r_{\mathrm{ergo}}(\theta)\equiv M+\sqrt{M^{2}-a^{2}\cos^{2}\theta}$.

In this spacetime, the conditions (47) and (48) at the light surface
($r_{\mathrm{LS}},\theta_{\mathrm{LS}}$) become

$\chi^{2}=-\left[1-\frac{2Mr}{\Sigma}(1-\omega
a\sin^{2}\theta)^{2}-\omega^{2}(r^{2}+a^{2})\sin^{2}\theta\right]=0,$ (51)

and

$\hat{q}^{2}=\Delta\sin^{2}\theta.$ (52)

Solving these equations in terms of $r$ and $\theta$ yields the locus of the
light surface as999 As shown in Ref. Kinoshita:2016lqd , these equations have
another branch of solutions, $r^{-}(\omega,\hat{q})\equiv
r_{\mathrm{LS}}(\omega,-\hat{q})$. This branch describes time-reversal
processes rather than physically natural processes.

$r_{\mathrm{LS}}(\omega,\hat{q})=\frac{M}{1+\hat{q}\omega}+\sqrt{\left(\frac{M}{1+\hat{q}\omega}\right)^{2}-a(a-\hat{q})},$
(53)

and

$\sin\theta_{\mathrm{LS}}=\frac{|\hat{q}|}{\sqrt{\Delta(r_{\mathrm{LS}})}}.$
(54)

Here, we focus on the northern hemisphere and should take
$\pi-\theta_{\mathrm{LS}}$ in the southern hemisphere. In order to satisfy
$\Delta(r_{\mathrm{LS}})>0$, namely, the light surface being located outside
the horizon, and $0\leq\sin^{2}\theta_{\mathrm{LS}}\leq 1$, we obtain an
allowed parameter region in terms of $(\omega,\hat{q})$. The allowed region
can be described by the intervals in which $\omega$ lies as

$\displaystyle\omega_{\mathrm{axis}}(\hat{q})<\omega\leq\omega_{\mathrm{eq}}(\hat{q})\quad\text{for}\quad\hat{q}>0,$
(55)
$\displaystyle\omega_{\mathrm{eq}}(\hat{q})\leq\omega<\omega_{\mathrm{axis}}(\hat{q})\quad\text{for}\quad\hat{q}<0,$

where

$\omega_{\mathrm{axis}}(\hat{q})\equiv-\frac{1}{\hat{q}},\quad\omega_{\mathrm{eq}}(\hat{q})\equiv\frac{a-\hat{q}}{2M^{2}-\hat{q}(a-\hat{q})+2M\sqrt{M^{2}-(a^{2}-\hat{q}^{2})}}.$
(56)

The boundaries represented by $\omega_{\mathrm{eq}}$ and
$\omega_{\mathrm{axis}}$ indicate when the light surface is located at the
equatorial plane ($\theta_{\mathrm{LS}}=\pi/2$) and when it approaches the
rotation axis ($\theta_{\mathrm{LS}}\to 0$), respectively. Moreover, the
radius of the light surface coincides with that of the ergosphere
($r_{\mathrm{LS}}=r_{\mathrm{ergo}}$) when $\omega=0$ and that of the event
horizon ($r_{\mathrm{LS}}=r_{\mathrm{h}}$) when $\hat{q}=0$ including
$\omega=\Omega_{\mathrm{h}}$. The typical shape of the allowed region is shown
in Fig. 1. Since the region where the energy extraction occurs should be
located in $\omega\hat{q}>0$, a necessary condition for the energy extraction
is that a magnetic field line intersects the light surface inside the
ergoregion of the Kerr black hole
($r_{\mathrm{h}}<r_{\mathrm{LS}}<r_{\mathrm{ergo}}$) and the angular velocity
of the magnetic field line is less than that of the black hole
($\omega<\Omega_{\mathrm{h}}$). It turns out that the curve of the upper
boundary of the energy-extraction region, represented by
$\omega=\omega_{\mathrm{eq}}(\hat{q})$, should pass through
$(\omega,\hat{q})=(0,a),(\Omega_{\mathrm{h}},0)$. Therefore, for an arbitrary
$a$ the extraction rate of the specific energy can be bounded by $\sim
a\Omega_{\mathrm{h}}/4$, and such a maximum extraction rate will be achieved
when the angular velocity of the magnetic field line is approximately half of
the black hole angular velocity near the equatorial plane
($\omega\simeq\Omega_{\mathrm{h}}/2$ and $\theta_{\mathrm{LS}}\simeq\pi/2$).
It is worth noting that these extraction rates are irrelevant to the mass
scale of the central black hole.

Figure 1: Typical shape of the allowed parameter region ($\omega,\hat{q}$)
focusing on the energy-extraction region (an example of $a/M=0.9$). The shaded
region is the allowed region, and in particular the triangular region
$\omega\hat{q}>0$ is where the energy extraction occurs. The solid and dotted
curves denote contour lines for the locus of the light surface
$r_{\mathrm{LS}}$ and $\theta_{\mathrm{LS}}$, respectively. The radius of the
light surface will coincide with that of the ergosphere
$r_{\mathrm{LS}}=r_{\mathrm{ergo}}$ on the vertical axis $\omega=0$ and that
of the event horizon $r_{\mathrm{LS}}=r_{\mathrm{h}}$ on the horizontal axis
$\hat{q}=0$. The curve represented by $\omega=\omega_{\mathrm{eq}}(\hat{q})$
indicates that the light surface is located on the equatorial plane
$\theta_{\mathrm{LS}}=\pi/2$; the horizontal axis that it is on the rotation
axis $\theta_{\mathrm{LS}}=0$.

This necessary condition applies to each field sheet, i.e., each magnetic
field line. It follows that each local configuration of the magnetic field
line at the light surface governs whether energy and angular-momentum
extraction can occur or not, i.e., the sign of $\omega\hat{q}$ and $\hat{q}$.
Figure 2 shows relations among local configurations of the magnetic field line
and directions of the specific angular-momentum flux $\hat{q}$. From Eq. (46)
we find that the sign of $\hat{q}$ is directly connected with the sign of
$d\varphi/dr\equiv\mathcal{L}_{\lambda}\varphi/\mathcal{L}_{\lambda}r$
regardless of the direction of $\lambda$. When the black hole is rotating
faster than a magnetic field line, $\omega(\psi)<\Omega_{\mathrm{h}}$, the
magnetic field line is braking the rotation of the black hole, namely,
extracting the angular momentum from the black hole, $\hat{q}>0$. When a
magnetic field line is rotating faster than the black hole,
$\omega(\psi)>\Omega_{\mathrm{h}}$, the magnetic field line is accelerating
the rotation of the black hole, namely, injecting the angular momentum into
the black hole, $\hat{q}<0$. Furthermore, on the premise that the angular-
momentum extraction has occurred $\hat{q}>0$, the magnetic field line can
extract the rotational energy of the black hole $\omega(\psi)\hat{q}>0$ if the
light surface enters into the ergoregion
$r_{\mathrm{h}}<r_{\mathrm{LS}}<r_{\mathrm{ergo}}$.

|
---|---
Figure 2: Relations among local configurations of the magnetic field line and
directions of the specific angular-momentum flux $\hat{q}$. The left panel
shows that the black hole is rotating faster than the magnetic field line
($\omega<\Omega_{\mathrm{h}}$) in the corotating frame of the field line; the
right panel shows that the magnetic field line is rotating faster than the
black hole ($\omega>\Omega_{\mathrm{h}}$) in the corotating frame of the black
hole. The directions of the magnetic field are irrelevant. The dashed circle
and the black disk depict, respectively, a light surface and a black hole
viewed from the top along the rotation axis. The magnetic field line is
stretching inside the light surface.

As in the case of the rigidly rotating strings, this energy-extraction
mechanism via the force-free electromagnetic fields can be simply interpreted
as an analogy of the Penrose process. Outside the light surface, each field
sheet is stationary with respect to each corotating vector
$\chi=\partial_{t}+\omega(\psi)\partial_{\phi}$, which is tangential to the
field sheet. In other words, the configuration of the magnetic field line does
not change in the corotating frame with angular velocity $\omega(\psi)$.
However, inside the light surface the field sheet cannot be stationary because
the corotating vector is spacelike. This means that the proper motion of each
line element of the magnetic field line cannot follow the corotating angular
velocity, and therefore the magnetic field line is stretching while its
configuration remains unchanged. As a result, angular momentum associated with
each line element is transferred toward the central black hole. When the black
hole is rotating faster than the magnetic field line, this angular-momentum
transfer will make the black hole spin down; when the magnetic field line is
rotating faster than the black hole, it will make the black hole spin up. If
angular-momentum transfer of the magnetic field line such that the black hole
will be spun down occurs in the ergoregion, the magnetic field line can gain
energy as a reaction to the angular-momentum transfer. The energy gain will be
transferred away from the black hole by the magnetic tension of the magnetic
field line. This process is quite similar to the Penrose process for
particles. Roughly speaking, the stretching part of the magnetic field lines
plays the role of the infalling “object” in the Penrose process.

The total amount of energy and angular-momentum flux is given by multiplying
the specific quantities by an effective magnetic tension and integrating the
contributions of every magnetic line. In fact, because $B\omega\hat{q}$ and
$B\hat{q}$ are conserved on each field sheet, once values of $B$ on the light
surface are given, we can obtain the total energy and angular-momentum flux by
integrating $B\omega\hat{q}$ and $B\hat{q}$ on the whole light surface as in
Eq. (42). To know the details of $B$, we have to solve the equations of motion
and need global information on configurations of the magnetic field. However,
the order of possible fluxes together with whether they are injecting or
extracting has been locally determined by the specific quantities per unit
tension, so that $B$ plays the role of a weight function of the magnetic
tension.

An average magnetic tension surrounding the black hole is defined by

$4\pi r_{\mathrm{h}}^{2}\langle B^{2}\rangle\equiv
2\pi\int_{\mathrm{LS}}B|d\psi|.$ (57)

The energy-extraction rate can be estimated as

$\frac{d\mathcal{M}}{dt}=\frac{\pi}{2}M^{2}\langle B^{2}\rangle
u(\alpha)\int_{\mathrm{LS}}\frac{B|d\psi|}{2r_{\mathrm{h}}^{2}\langle
B^{2}\rangle}\frac{\omega\hat{q}}{\Omega_{\mathrm{h}}a/4},$ (58)

where $u(\alpha)\equiv\alpha^{2}(1+\sqrt{1-\alpha^{2}})$ and $\alpha\equiv
a/M$.

The whole of the Blandford–Znajek process can be classified into what is
governed by local kinematics and what is governed by dynamics by solving the
equations of motion. The former is the energy-extraction mechanism
characterized by the relations between the specific energy and angular-
momentum fluxes, $\omega\hat{q}$ and $\hat{q}$, and the locus of the light
surface, which we have shown in this paper; the latter is global
configurations of the magnetic field lines and the electric current, the value
of the magnetic tension $B$, and so on. This fact suggests the following whole
picture of the Blandford–Znajek process. Depending on the environments in an
outer region, the global configurations of the magnetic field lines associated
with plasmas are dynamically determined according to the equations of motion.
If only appropriate configurations of the magnetic field lines can be locally
realized in an inner region around the ergoregion, the magnetic field lines
start to extract the energy and angular momentum of the black hole by a
kinematical mechanism irrelevant to situations in the outer region. For
example, to obtain explicitly the directional dependence of the energy and
angular-momentum fluxes in the far region, we have to solve globally the
equations of motion for the electromagnetic fields and to know the entire
configurations of the magnetic field lines. This implies that we should
examine the details of the surrounding system, i.e., the boundary conditions
in the outer region, in order to explore the directional dependence in the far
region where astrophysical jets are generated. However, in order to explain
and understand the energy-extraction mechanism in the Blandford–Znajek
process, we should consider just local kinematics in the inner region between
the light surface and the ergosphere. The directional dependence of the
specific energy and angular-momentum fluxes at the light surface has been
shown in the contour of, e.g., Fig. 1. This provides the potential to generate
the energy and angular-momentum fluxes in the Blandford–Znajek process. We can
see that the closer to the equatorial plane $\theta_{\mathrm{LS}}=\pi/2$ the
locus where the magnetic field lines intersect the light surface is, the
larger specific energy flux $\omega\hat{q}$ the magnetic field lines can
generate in general.

##  IV Discussion

In this paper we have shown that the essence of the energy extraction by the
Blandford–Znajek process is local kinematics inside the ergoregion. Therefore,
global configurations of the magnetic field line, such as whether the magnetic
field lines can thread the event horizon or not, should not be the essence of
this process. The Znajek condition Znajek1977 , which is well known as a
condition to be satisfied at the event horizon, cannot be a necessary
condition for the energy extraction. This condition is a consistency condition
for the force-free magnetic field lines to regularly cross the event horizon
(an identical condition (69) can be derived for the rigidly rotating strings).
The main process for the energy extraction occurs between the inner-light
surface and the ergosphere, and, besides, the light surface is the causal
boundary for this system. Therefore, the energy extraction can occur even if
this condition is not necessarily satisfied. For instance, it does not matter
if the force-free condition for plasma has been violated at the event horizon.
For the same reason, moreover, the so-called Meissner-like effect, in which
the extremal Kerr black hole tends to expel magnetic fields (see, e.g., Ref.
Komissarov:2007rc  and references therein), should not have a direct
connection with the Blandford–Znajek process as long as we do not require an
extra assumption at the horizon for a different reason.

Throughout this paper we have discussed stationary and axisymmetric force-free
electromagnetic fields for the energy extraction. At the least, we need these
assumptions between the inner-light surface and the ergosphere. If such
assumptions are violated, it is expected that the energy extraction by the
Blandford–Znajek process will become less efficient. The reason why the
Blandford–Znajek process can efficiently extract the rotational energy from
the black hole and produce a highly powerful energy flux is that a
relativistic tension, which is equal to its energy density, carries the
angular-momentum and energy flux along the magnetic field lines in the same
way as Nambu–Goto strings. If the axisymmetry, stationariness, or force-
freeness are violated, the magnetic pressure or other matters begin to affect
the angular-momentum and energy transfer and then alter the correspondence to
the Nambu–Goto strings. This seems to be a disadvantage for the energy
extraction.

What is necessary for the Blandford–Znajek process to occur is toroidal
magnetic fields winding the black hole at the light surface inside the
ergoregion. As we showed in Eq. (46), the angular-momentum and energy fluxes
depend on toroidal configurations of the magnetic field lines, namely, the
toroidal component of the magnetic field. This is because toroidal magnetic
fields take the central role in the angular-momentum transfer of the magnetic
field lines. The poloidal components of the magnetic fields determine the
directions of the angular-momentum and energy flows.

For the Blandford–Znajek process, since magnetic field lines with magnetic
tension are essential and the transferred energy and angular momentum are
purely electromagnetic, it seems that the electric current or plasma has only
an auxiliary role to sustain the magnetic fields. In fact, Ref.
Jacobson:2017xam  recently showed that magnetic fields without plasma can
extract the rotational energy of a black hole in lower spacetime dimensions.

###### Acknowledgements.

We would like to thank Kenji Toma and Yudai Suwa for useful comments. We would
like to thank the participants of the workshop “FuwakuBZ77” for helpful
discussions. This work was supported by JSPS KAKENHI Grant Numbers JP16K17704
(S.K.) and the MEXT-Supported Program for the Strategic Research Foundation at
Private Universities, 2014-2017, S1411024 (T.I.).

##  Appendix A RIGIDLY ROTATING STRINGS

In this appendix we summarize useful results for rigidly rotating strings
around a rotating black hole, some of which were obtained in Ref.
Kinoshita:2016lqd .

In a stationary and axisymmetric spacetime (26), a stationary rotating string
with angular velocity $\omega$ is embedded as

$t=\tau,\quad\phi=\omega\tau+\varphi(\sigma),\quad
r=r(\sigma),\quad\theta=\theta(\sigma).$ (59)

The corotating (Killing) vector $\chi$ that is tangential to the world sheet
becomes

$\chi=\frac{\partial}{\partial t}+\omega\frac{\partial}{\partial\phi}.$ (60)

The induced metric of the world sheet is given by

$h_{ab}d\sigma^{a}d\sigma^{b}=(g_{tt}+2g_{t\phi}\omega+g_{\phi\phi}\omega^{2})d\tau^{2}+2\varphi^{\prime}(g_{t\phi}+g_{\phi\phi}\omega)d\tau
d\sigma+(g_{rr}r^{\prime 2}+g_{\theta\theta}\theta^{\prime
2}+g_{\phi\phi}\varphi^{\prime 2})d\sigma^{2},$ (61)

where the prime denotes a derivative with respect to $\sigma$. From this
induced metric the Lagrangian density for the Nambu–Goto string with unit
tension is given by

$\mathscr{L}=-\sqrt{(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})\varphi^{\prime
2}-\chi^{2}(g_{rr}r^{\prime 2}+g_{\theta\theta}\theta^{\prime 2})},$ (62)

where $\chi^{2}=\chi^{\mu}\chi^{\nu}g_{\mu\nu}=h_{\tau\tau}$. The light
surface is characterized by

$\chi^{2}=g_{tt}+2g_{t\phi}\omega+g_{\phi\phi}\omega^{2}=0,$ (63)

and a consistency condition that the string can pass through the light surface
regularly is given by

$q^{2}=(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})|_{\chi^{2}=0}.$ (64)

Here, $q$ is a conserved quantity characterizing the configuration of the
rigidly rotating strings, which means the specific angular-momentum flux per
unit tension flowing on the string world sheet. Its expression is written as

$q\equiv\frac{\partial\mathscr{L}}{\partial\varphi^{\prime}}=-\frac{(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})\varphi^{\prime}}{\sqrt{(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})\varphi^{\prime
2}-\chi^{2}(g_{rr}r^{\prime 2}+g_{\theta\theta}\theta^{\prime 2})}},$ (65)

where the sign of $q$ has been defined so that $q>0$ describes a radially
outward flux when $r^{\prime}>0$.

Using Eqs. (62) and (65), we have the following identity:

$q^{2}\mathscr{L}^{2}=(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})[\mathscr{L}^{2}+\chi^{2}(g_{rr}r^{\prime
2}+g_{\theta\theta}\theta^{\prime 2})].$ (66)

Suppose that $(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})|_{r=r_{\mathrm{h}}}=0$ and
$(g_{rr})^{-1}|_{r=r_{\mathrm{h}}}=0$ should be satisfied at the event horizon
$r=r_{\mathrm{h}}$. The determinant of the metric components $g$ should not be
degenerate, so that the equation

$(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})g_{rr}=-g/g_{\theta\theta}$ (67)

is satisfied even at $r=r_{\mathrm{h}}$. If the string can regularly pass
through the event horizon, the volume element of the string world sheet,
namely, the Lagrangian density $\mathscr{L}$, should be finite and nonzero at
the event horizon. Moreover, at $r=r_{\mathrm{h}}$ we find

$\chi^{2}|_{r=r_{\mathrm{h}}}=g_{\phi\phi}(\omega-\Omega_{\mathrm{h}})^{2},$
(68)

where we have used $g_{t\phi}^{2}=g_{tt}g_{\phi\phi}$ and
$\Omega_{\mathrm{h}}=-g_{t\phi}/g_{\phi\phi}$ at $r=r_{\mathrm{h}}$. Combining
the above results (66), (67), and (68), we obtain the following condition at
the event horizon:

$q^{2}=\left.(\omega-\Omega_{\mathrm{h}})^{2}\frac{(-g)}{\mathscr{L}^{2}}\left(\frac{dr}{d\sigma}\right)^{2}\frac{g_{\phi\phi}}{g_{\theta\theta}}\right|_{r=r_{\mathrm{h}}}.$
(69)

Note that, by definition, the combination $\mathscr{L}d\sigma$ is invariant
under reparameterization of the world-sheet coordinate $\sigma$. This
condition is identical to the Znajek condition for force-free electromagnetic
fields:

$I(\psi)=\left.(\omega-\Omega_{\mathrm{h}})\partial_{\theta}\psi\sqrt{\frac{g_{\phi\phi}}{g_{\theta\theta}}}\right|_{r=r_{\mathrm{h}}},$
(70)

which we can also obtain by evaluating Eq. (44) at the event horizon.

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