The essence of the Blandford-Znajek process

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.


I. INTRODUCTION AND SUMMARY
Rotational energy of rotating black holes has been a promising energy source to form relativistic jets, which are ubiquitous in astrophysics. The Blandford-Znajek process [1], which is an energy-extraction mechanism by forcefree electromagnetic fields, can efficiently achieve powerful energy fluxes and then has been widely believed to be a viable mechanism to extract the rotational energy of a black hole. There are a number of analytical and numerical investigations from various aspects during four decades after this process proposed. Detailed analysis for the extraction mechanism was in terms of the membrane paradigm [2]. In recent years interpretations and explanations of this mechanism have been discussed (for example, [3][4][5]). Moreover, various numerical simulations are developed and demonstrated [6][7][8]. 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 is elucidated that rigidly rotating Nambu-Goto strings [9] twining around a rotating black hole can highly efficiently extract its rotational energy from the black hole [10]. The order of possible energy flux, namely energy-extraction rate, is given by a so-called Dyson luminosity 1 ∼ c 5 /G N 10 59 erg/s [12], consisting of only fundamental constants: the speed of light c and Newton's constant G N , and a dimensionless string tension G N µ/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 that the energy extraction can 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 ∼ B 2 r 2 h , we can reproduce energy extraction rate of the Blandford-Znajek process with 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, that is, the energy-extraction mechanism from rotating black holes via force-free electromagnetic fields, is the Penrose process for the magnetic field lines with angular momentum and energy transport mediated by their magnetic tension.
For this purpose, we will first reformulate dynamics of force-free electromagnetic fields in accordance with a "fieldsheet" formalism [13][14][15]. 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 two-dimensional extended object in a spacetime. We will call such objects "field sheets", the name of which was adopted in Ref. [15], 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 µν F µν > 0), particularly, the field sheets become two-dimensional timelike surfaces characterized by the electromagnetic field strength F µν . 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 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 its 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 outer-light cylinder, we can say that the Blandford-Znajek process is at work if the magnetic field lines extend to a sufficiently far region apart 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 neighborhood 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, it 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 magnetic field or electric current for extracting the energy are different questions. Furthermore, we should separately consider 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. [10].

II. FIELD-SHEET FORMALISM FOR FORCE-FREE ELECTROMAGNETIC FIELDS
In this section, in order to elucidate similarity between Nambu-Goto strings and magnetic field lines of forcefree fields, we will rewrite the equations of motion for force-free electromagnetic fields according to a "field-sheet" formalism [13][14][15]. Unless otherwise specified, Newton's constant G N and the speed of light c are set to unity hereafter.
Let F µν be the electromagnetic field strength and let j µ be the current density four-vector of electric charge. In terms of F µν , the Maxwell equations are given by In general, when an electromagnetic field interacts with charged matter such as plasma, the energy-momentum tensor for the electromagnetic field, satisfies where the right-hand side of the above equation means the Lorentz force. In the situation where the force density four-vector F να j α can be neglected, i.e., which is known as the force-free condition, the energy-momentum tensor of the electromagnetic field is individually conserved ∇ µ T µ ν = 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 The dynamics described by these equations is force-free electrodynamics (FFE). An important property for the force-free electromagnetic fields is satisfying where * F µν is the dual of F µν , defined by * F µν ≡ F αβ αβµν /2. These fields are called degenerate. Note that the force-free condition for nonzero j µ implies that F µν is degenerate, 2 but degeneracy does not always lead to forcefreeness. To clarify physical meanings, let t µ be a four-velocity of a timelike observer. Then, the electric and magnetic field measured by this observer are given by E µ = F µν t ν and B µ = − * F µν t ν , respectively. The degenerate condition physically means E α B α = 0, where this relation holds even for an arbitrary observer t µ because F αβ * F αβ is scalar. Now, the fact that F µν is closed together with the degeneracy implies that * F µν is tangent to a two-dimensional submanifold, S . Furthermore, we assume that F µν is magnetically dominated, i.e., Naively, this condition implies that the magnetic field should be stronger than the electric field. Because this condition is also described by scalar, there is a notion independent of observers. The degeneracy and magnetically dominated condition for F µν guarantee the existence of pure magnetic frame in which E µ = 0. Therefore, F αβ F αβ > 0 states that the electromagnetic field F µν is purely magnetic with its magnitude defined by Namely, we can take * F µν in the form * where σ µν denotes the two-dimensional volume element on S . It turns out that the magnetically dominated condition for F µν is equivalent to σ µν σ µν = −2, and then S becomes timelike. We call such 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 dynamics of field sheets as string world sheets. In terms of * F µν , the equations of FFE are rewritten as * Substituting Eq. (9) into the above, we obtain the equations of FFE in terms of B and σ µν given by where N µ ν is defined by Note that h µν is the induced metric on the field sheet and N µ ν 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, that is, magnetic pressure; the latter equation (12) does conservation of the magnetic flux. This expression tells us that dynamics of field sheets in FFE is similar to 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 where σ µν denotes the volume element of the world sheet in this case [27]. Comparing the above equation with Eqs. (11) and (12), we can easily notice that the only difference between dynamics of field sheets and world sheets is adding the extra scalar quantity B, which describes the magnetic tension and pressure. (In the flat spacetime such a correspondence was discussed in Refs. [28][29][30].) From a geometrical point of view, their meanings are so clear. In general, the extrinsic curvature of a submanifold is defined by where h µν is the induced metric on the submanifold. If the submanifold is two dimensional, we have Thus, dynamics of field sheets and world sheets belong to the same class of dynamics of two-dimensional surface (see Ref. [31] and references therein). Because every degenerate, closed two-form defines a foliation of spacetime, 3 the field sheets represented by F µν can define a two-dimensional timelike foliation with a coordinate transformation x µ = X µ (τ, σ, α, β), where (τ, σ) denote local coordinates on field sheets S defined by α = const. and β = const. Note that, once α and β are fixed, X µ give embedding functions of S . Let ∂ a be coordinate derivatives with respect to two-dimensional local coordinates on the field sheet. The latin indices denote intrinsic components on the field sheet. The volume element σ µν is related to X µ as where h a µ ≡ ∂ a X µ , and σ ab is the intrinsic volume element on S . Moreover, the induced metric can be intrinsically written as By using these intrinsic quantities, we can rewrite Eq. (11) in terms of X µ as where D a denotes the covariant derivative with respect to h ab and Γ µ αβ is the Christoffel symbol associated with g µν . Note that, intriguingly, this equation is derived from the following action S[ It is instructive to compare a perfect fluid in terms of conservation of the energy-momentum tensor. As well known, the energy-momentum tensor of a perfect fluid is given by where u µ is a four-velocity of the fluid, ρ and p are the energy density and pressure in its rest frame, respectively. Decomposing the conservation law ∇ µ T µν = 0 into tangential components and normal ones with respect to u µ yields If the fluid is pressure-less (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 where µ is a tension equal to its energy density andp is a normal pressure. Note that Nambu-Goto strings have a constant µ andp = 0, and magnetically dominated electromagnetic fields have µ =p = B 2 /2. Decomposing the energy-momentum conservation into components tangential and normal to the field sheet yields Thus, it turns out that the world line and its volume element u µ for a fluid element of perfect fluids correspond to the world sheet (field sheet) and its volume element σ µν for a Nambu-Goto string or a magnetic field line. Moreover, in each pressure-less 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 ξ µ , the conservation law of the energy-momentum tensor in terms of ξ µ yields where we have used ξ µ ∇ µp = 0 because of the symmetry. This conservation law is equivalent to that of a string with an effective tension µ +p. Thus, even though the pressure does not vanishp = 0, kinematic properties such as a conservation law can be identical thank to a symmetry.

III. ENERGY EXTRACTION BY FORCE-FREE ELECTROMAGNETIC FIELDS
A. 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 have been well known [19]. In the field-sheet approach, we will follow Ref. [15].
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 (∂ t ) µ and (∂ φ ) µ , which respectively represent time translation symmetry and axisymmetry of the spacetime as L ∂t g µν = L ∂ φ g µν = 0. Here, L denotes the Lie derivative. Its metric can be written as where we assume that g tt g φφ − g 2 tφ = 0 characterizes the event horizon and g tt g φφ − g 2 tφ < 0 implies the outside of the black hole. Here, we have taken a coordinate system in which the Killing vector fields (∂ t ) µ and (∂ φ ) µ are manifestly orthogonal to two-dimensional surfaces spanned by (r, θ) such as the Boyer-Lindquist coordinates for the Kerr spacetime. Moreover, we suppose that the electromagnetic fields share the same Killing symmetries such that It is known that stationary and axisymmetric force-free electromagnetic fields are given by the following field strength where η ≡ dφ − ω(ψ)dt, and both of ψ and ϕ are scalar functions independent of t and φ. This yields a common component representation of the field strength: which are related to an electric field, 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 = dψ ∧ dφ in terms of so-called Euler potentials given by the following scalar functions ψ andφ ≡ φ − ω(ψ)t − ϕ. (See, for example, Ref. [15].) This means that the field sheets are represented by the intersections of the hypersurfaces of constant ψ andφ. Each magnetic field line given by the field sheets is rotating with angular velocity ω(ψ). Obviously, the field sheets described by the same ψ have the same angular velocity because of the axisymmetry. Such ψ-constant surface is a so-called magnetic surface, and 2πψ = F gives the magnetic flux across the region enclosed by the magnetic surface.
The "proper" magnetic field is given by Here, we have written the contractions for vector fields u and v as u · v = u µ v µ and u 2 = u µ u µ in the abbreviated notation. In addition, we have 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, χ is a corotating vector defined by which satisfies χ µ η µ = 0 and χ 2 = g tt + 2g tφ ω + g φφ ω 2 = −(g 2 tφ − g tt g φφ )η 2 . Since the corotating vector χ satisfies F µν χ ν = 0, χ is tangential to the field sheet. It turns out that χ represents a corotating frame with the angular velocity ω(ψ) for each field sheet labeled by constant ψ. If χ (also, η) becomes null, the velocity of such corotating frame has reached to the speed of light. The locus characterized by χ 2 = 0 is called light surface. 4 Meanwhile, because a field sheet is a two-dimensional surface, there is another tangent vector linearly independent of χ. We will introduce the other tangential vector as By construction, this vector is tangential to the field sheet and normal to ∇ µ t, so that λ becomes a generator of the intersection of the field sheet and a constant-t surface. This means that the integral curves of λ are magnetic field lines at a time t. The direction of λ is radially outward for F θφ = ∂ θ ψ > 0, while its direction is radially inward for ∂ θ ψ < 0. It turns out that the direction of λ corresponds to that of the magnetic field. Note that, since [λ, χ] = L λ χ = 0 are satisfied, χ and λ can be coordinate bases on the field sheet as χ = ∂/∂τ and λ = ∂/∂σ, where τ and σ denote local coordinates on the field sheet. The dual of F µν is expressed as * This expression manifestly shows that it is a two-form tangential to the field sheet, that is, proportional to the volume form of the field sheet as * F µν = Bσ µν .
B. Induced geometry on the field sheet As we mentioned, χ and λ 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 (τ, σ) are given by h τ τ ≡ χ µ χ ν g µν = g tt + 2g tφ ω + g φφ ω 2 , Since χ is the corotating vector, the light surface is characterized by χ 2 = h τ τ = 0. In addition, χ is the Killing vector with respect to the induced metric on the field sheet, 5 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 [15]. We find the following correspondence between the induced metric of the field sheet and that of the rigidly rotating string (A3): 4 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) at far region. Since the inner-light surface is important for extracting the rotational energy from the black hole [10], we will focus only on the inner-light surface hereafter. 5 Each component of the induced metric is a function of only r and θ. Since Lχr = Lχθ = 0, we have Lχh ab = 0 on the field sheet.
Indeed, we can explicitly show and If we introduce the local coordinates as χ = ∂/∂τ and λ = ∂/∂σ 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.

C. Angular momentum flux and energy flux
Since the spacetime has time-translational symmetry and axisymmetry, we have the conservation laws for the energy-momentum tensor which 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 This indicates that √ −gF rθ should depend only on ψ as Note that this scalar function 6 is constant on the field sheets, and it is one of characteristic quantities to characterize the stationary and axisymmetric force-free electromagnetic field as well as ψ and ω(ψ). By using λ, this angular momentum conservation can be rewritten as Therefore, I(ψ) is identified with the angular momentum flux (per unit magnetic flux dψ) flowing on the field sheet, which should be conserved for each field sheet. Note that −I(ψ) gives the outward angular momentum flux for ∂ θ ψ > 0, while I(ψ) is the outward flux for ∂ θ ψ < 0. This is because the direction of λ changes depending on the sign of ∂ θ ψ (for instance, recall that L λ r = ∂ θ ψ/ √ −g in Eq. (37)). It is well known that I(ψ) 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 ∇ µ (T µ ν (∂ t ) ν ) = 0 leads to where ω(ψ)I(ψ) is identified with the conserved energy flux flowing on the field sheet. As in the case of the angular momentum flux, −ω(ψ)I(ψ) gives the outward energy flux for ∂ θ ψ > 0, while ω(ψ)I(ψ) is the outward flux for ∂ θ ψ < 0. Integrating Eqs. (40) and (41) over a three-dimensional volume on a t-constant hypersurface surrounding the black hole, we have total fluxes of angular momentum J and energy M, that is, each extraction rate from the black hole, 7 as Now, we find from Eq. (36) that and also we find from Eqs. (29) and (30) that Combining the above two results, we can obtain where we take the upper minus sign for ∂ θ ψ > 0 and the lower plus sign for ∂ θ ψ < 0. In an unified manner, alternatively, it can be rewritten aŝ where dϕ/dr ≡ L λ ϕ/L λ r and dθ/dr ≡ L λ θ/L λ r. This quantityq 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 [10] (see also Eq. (A7)). Similarly, ω(ψ)q is the specific energy flux per unit tension. Since the sign ofq has been defined such thatq should be positive if a radially outward flux,q > 0 and ωq > 0 mean angular momentum extracting and energy extracting, respectively. It is worth noting that the sign ofq is irrelevant to the direction of λ as is clear from Eq. (46). In other words whether extracting process or injecting one does not depend on a direction of the magnetic field but a configuration of the magnetic field line. One thing to keep in mind as a difference with the cases of the rigidly rotating strings is that the specific angular momentum fluxq is not conserved while I(ψ) 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. 8 So far we have shown that the intrinsic structures on the field sheet are identical to that on the world sheet of the rigidly rotating string. This does not mean that 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, global configurations of the Nambu-Goto strings and the magnetic field lines are different, and such global configurations should be determined by each dynamics with 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 governed by local relations on the light surface in fact. On the light surface, which is characterized by we have the specific angular momentum fluxq 7 We denote mass and angular momentum of the black hole as M BH and J BH . Each conservation law yields dM dt + dM BH dt = 0 and dJ dt + dJ BH dt = 0. 8 As can be seen from the energy-momentum tensor, B 2 describes a magnetic tension per unit area, which has the same dimension as pressure. Now, B does a magnetic tension per unit magnetic flux.
by substituting χ 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 (A6). Since Eqs. (47) and (48) are equations in terms of r and θ essentially, they give relations among the angular velocity ω, the specific angular momentum fluxq, and the locus of the light surface (r LS , θ LS ). Thus, we conclude that both the stationary, axisymmetric force-free electromagnetic fields and the rigidly rotating strings have the identical relations. It is worth noting that these relations are determined by the background spacetime metric irrelevant to dynamics of the electromagnetic field. Even though each dynamics of electromagnetic fields and rigidly rotating strings are different, that is, 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 mechanism of energy and angular momentum via the force-free electromagnetic fields and the rigidly rotating strings are essentially identical.

D. 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 major properties are identical between the force-free electromagnetic fields and the rigidly rotating strings as seen, most of the following argument and its results are the same as those shown in Ref. [10]. For more details, refer to it.
In the Boyer-Lindquist coordinates the metric of the Kerr spacetime with mass M and angular momentum aM are given by where Σ(r, θ) = r 2 + a 2 cos 2 θ, ∆(r) = r 2 + a 2 − 2M r.
For simplicity and without loss of generality, we assume a > 0. This metric gives g 2 tφ − g tt g φφ = ∆ sin 2 θ, and the event horizon lies at r = r h ≡ M + √ M 2 − a 2 defined by ∆(r h ) = 0. Note that we have ∆ > 0 outside the black hole r > r h . The angular velocity of the black hole is given by Ω h ≡ a/(r 2 h + a 2 ). The ergosphere is characterized by the locus where the stationary Killing vector (∂ t ) µ becomes null, i.e. g tt = 0, and its radius is r ergo (θ) ≡ M + √ M 2 − a 2 cos 2 θ. In this spacetime, the conditions (47) and (48) at the light surface (r LS , θ LS ) become Solving these equations in terms of r and θ yields the locus of the light surface as 9 and Here, we focus on the northern hemisphere and should take π − θ LS in the southern hemisphere. In order to satisfy ∆(r LS ) > 0, namely the light surface to be located outside the horizon, and 0 ≤ sin 2 θ LS ≤ 1, we obtain an allowed parameter region in terms of (ω,q). The allowed region can be described by the intervals in which ω lies as ω axis (q) < ω ≤ ω eq (q) forq > 0, ω eq (q) ≤ ω < ω axis (q) forq < 0, where The boundaries represented by ω eq and ω axis indicate when the light surface is located at the equatorial plane (θ LS = π/2) and approaches the rotation axis (θ LS → 0), respectively. Moreover, the radius of the light surface coincides with that of the ergosphere (r LS = r ergo ) when ω = 0 and that of the event horizon (r LS = r h ) when q = 0 including ω = Ω h . Typical shape of the allowed region is shown in Fig. 1. Since the region where the energy extraction occurs should be located in ωq > 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 h < r LS < r ergo ) and the angular velocity of the magnetic field line is less than that of the black hole (ω < Ω h ). It turns out that the curve of the upper boundary of the energy extraction region, represented by ω = ω eq (q), should pass through (ω,q) = (0, a), (Ω h , 0). Therefore, for an arbitrary a the extraction rate of the specific energy can be bounded by ∼ aΩ h /4, and such maximum extraction rate will achieve when the angular velocity of the magnetic field line is approximately half of the black hole angular velocity near the equatorial plane (ω Ω h /2 and θ LS π/2). It is worth noting that these extraction rates are irrelevant to a mass scale of the central black hole.
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, that is, the sign of ωq andq. Figure 2 shows relations among local configurations of the magnetic field line and directions of the specific angular momentum fluxq. From Eq. (46) we find that the sign ofq is directly connected with the sign of dϕ/dr ≡ L λ ϕ/L λ r regardless of the direction of λ. When the black hole is rotating faster than a magnetic field line, ω(ψ) < Ω h , the magnetic field line is braking the rotation of the black hole, namely extracting the angular momentum from the black hole,q > 0. When a magnetic field line is rotating faster than the black hole, ω(ψ) > Ω h , the magnetic field line is accelerating the rotation of the black hole, namely injecting the angular momentum into the black hole,q < 0. Furthermore, on the premise that the angular momentum extraction has occurredq > 0, the magnetic field line can extract the rotational energy of the black hole ω(ψ)q > 0 if the light surface enters into the ergoregion r h < r LS < r ergo .
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 χ = ∂ t + ω(ψ)∂ φ , 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 ω(ψ). However, inside the light surface the field sheet cannot be stationary because the corotating vector has been 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 The left panel shows that the black hole is rotating faster than a magnetic field line (ω < Ω h ) in the corotating frame of the field line; the right panel shows that a magnetic field line is rotating faster than the black hole (ω > Ω h ) in the corotating frame of the black hole. The directions of the magnetic field is 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.
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 an energy as a reaction of the angular-momentum transfer. The energy gain will be transferred apart 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, a stretching part of the magnetic field lines plays a role of infalling "object" in the Penrose process.
Total amount of energy and angular momentum flux is given by multiplying the specific quantities by an effective magnetic tension and integrating contributions of every magnetic line. In fact, because Bωq and Bq 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 Bωq and Bq on the whole light surface as in Eq. (42). To know 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 a role of a weight function of the magnetic tension.
An average magnetic tension surrounding the black hole is defined by The energy-extraction rate can be estimated as where u(α) ≡ α 2 (1 + √ 1 − α 2 ) and α ≡ a/M .

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 for this process. The Znajek condition [32], 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 across the event horizon (an identical condition (A11) can be derived for the rigidly rotating strings). The main process for the energy extraction has done 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. In the same reason, moreover, the so-called Meissner-like effect that the extremal Kerr black hole tends to expel magnetic fields (for example, Ref. [33] 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 under a different reason.
Throughout this paper we have argued about stationary and axisymmetric force-free electromagnetic fields for the energy extraction. At least, we need these assumptions between the inner-light surface and the ergosphere. If such assumptions have been 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 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 axisymmetry, stationary, or force-free have been 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. It 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 shown in Eq. (46), the angular-momentum and energy fluxes depend on toroidal configurations of the magnetic field lines, namely 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 directions the angular momentum and energy flows.
For the Blandford-Znajek process, since magnetic field lines with the magnetic tension are essential and 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. [34] shows that magnetic fields without plasma can extract the rotational energy of a black hole in lower spacetime dimensions, recently.
Suppose that (g 2 tφ − g tt g φφ )| r=r h = 0 and (g rr ) −1 | r=r h = 0 should be satisfied at the event horizon r = r h . The determinant of the metric components g should not be degenerate, so that the equation, (g 2 tφ − g tt g φφ )g rr = −g/g θθ , is satisfied even at r = r h . If the string can regularly pass through the event horizon, the volume element of the string world sheet, namely the Lagrangian density L should be finite and nonzero at the event horizon. Moreover, at r = r h we find where we have used g 2 tφ = g tt g φφ and Ω h = −g tφ /g φφ at r = r h . Combining the above results (A8), (A9), and (A10), we obtain the following condition at the event horizon, Note that, by definition, the combination L dσ is invariant under reparameterization of the world-sheet coordinate σ. This condition is identical to the Znajek condition for force-free electromagnetic fields: which we can also obtain by evaluating Eq. (44) at the event horizon.