Causal production of the electromagnetic energy flux and role of the negative energies in Blandford-Znajek process
Blandford-Znajek process, the steady electromagnetic energy extraction from a rotating black hole (BH), is widely believed to work for driving relativistic jets in active galactic nuclei, gamma-ray bursts and Galactic microquasars, although it is still under debate how the Poynting flux is causally produced and how the rotational energy of the BH is reduced. We generically discuss the Kerr BH magnetosphere filled with a collisionless plasma screening the electric field along the magnetic field, extending the arguments of Komissarov and our previous paper, and propose a new picture for resolving the issues. For the magnetic field lines threading the equatorial plane in the ergosphere, we find that the inflow of particles with negative energy as measured in the coordinate basis is generated near that plane as a feedback from the Poynting flux production, which appears to be a similar process to the mechanical Penrose process. For the field lines threading the event horizon, we first show that the concept of the steady inflow of negative electromagnetic energy is not physically essential, partly because the sign of the electromagnetic energy density depends on the coordinates. Then we build an analytical toy model of a time-dependent process both in the Boyer-Lindquist and Kerr-Schild coordinate systems in which the force-free plasma injected continuously is filling a vacuum, and suggest that the structure of the steady outward Poynting flux is causally constructed by the displacement current and the cross-field current at the in-going boundary between the plasma and the vacuum. In the steady state, the Poynting flux is maintained without any electromagnetic source.
1,2]Kenji Toma 3]Fumio Takahara
The driving mechanism of collimated outflows or jets with relativistic speeds which are observed in active galactic nuclei (AGNs), gamma-ray bursts, and Galactic microquasars is one of the major problems in astrophysics. A most widely discussed model is based on Blandford-Znajek (BZ) process, the electromagnetic energy extraction from a rotating black hole (BH) along magnetic field lines threading it . This process produces Poynting-dominated outflows, which may be collimated by the pressure of the surrounding medium such as the accretion disk and the disk wind [e.g. 2, 3]. The particles in the outflow can be gradually accelerated depending on the geometrical structure [e.g. 4, 5, 6], which is consistent with the recent observational implications from the radio jet in the giant elliptical galaxy M87 [7, 8] [see also 9].
BZ process was proposed by a pioneering paper of , who found steady, axisymmetric, force-free solutions of Kerr BH magnetosphere in the slow rotation limit where the outward angular momentum (AM) and Poynting fluxes are non-zero along the field lines threading the event horizon. This was followed by demonstrations of analytical and numerical magnetohydrodynamic (MHD) solutions [e.g. 10, 11, 12, 2, 3, 13] and other force-free solutions [e.g. 14, 15, 16, 17]. However, the physical mechanism how the fluxes are created in the electromagnetically-dominated plasma has not been clearly explained. In contrast, the origin of pulsar winds is identified definitely with the rotation of the matter-dominated central star. The rotation velocity of the matters of the star and the magnetic field threading the star provide the electromotive force on charges, maintaining the electric field and the poloidal electric currents (with the toroidal magnetic field ) which form the outward Poynting flux in the magnetosphere. As its feedback, the rotation of the stellar matters slows down  [see also a review in 19, hereafter TT14]. BZ process, working in the electromagnetically-dominated BH magnetosphere, does not include any matter-dominated region in which the poloidal magnetic field is anchored. One should also note that the toroidal magnetic field cannot be produced in vacuum just by the rotation of the space-time [20, 14]. Then how the electric field and currents forming the AM and Poynting fluxes are created and how the BH rotational energy is reduced in BZ process are not simple problems and have been still matters of debate. See recent discussions in  (hereafter K09) and .
Among the numerous calculations, the force-free numerical simulations performed by  (hereafter K04) are most insightful for investigating the essential points on the origin of the fluxes. TT14 extended the arguments in K04 and K09 and analytically showed that for open magnetic field lines threading the ergosphere in the steady, axisymmetric Kerr BH magnetosphere, the situation of no electric potential difference with no poloidal electric current (i.e. no outward AM or Poynting flux) cannot be maintained. The origin of the electric potential differences is ascribed to the ergosphere. It was also shown that for the open field lines threading the equatorial plane in the ergosphere, the poloidal currents are driven by electric field (perpendicular to the magnetic field) stronger than the magnetic field in the ergosphere, where the force-free condition is violated (see also Section 3 below).
In this paper, we mainly discuss the field lines threading the event horizon. Some theorists consider that the membrane paradigm  is useful for effectively understanding the production mechanism of the AM and Poynting fluxes for such field lines [e.g. 25, 24]. This interprets the condition at the horizon as a boundary condition [26, 1] and the horizon as a rotating conductor which creates the potential differences and drives the electric currents in an analogy with the unipolar induction for pulsar winds explained above. However, the horizon does not actively affect its exterior, but just passively absorbs particles and waves . The condition at the horizon should be interpreted as a regularity condition [10, K04]. The mechanism of producing the steady AM and Poynting fluxes has to work outside the horizon, making the physical quantities consistent with the regularity condition.
For such a causal flux production, some other theorists proposed a scenario that certain types of negative energies (as measured in the coordinate basis, i.e. as measured at infinity) created outside the horizon flows towards the horizon, resulting in the positive outward energy flux. This is an analogy with the mechanical Penrose process, in which the rotational energy of a BH is extracted by making it absorb negative-energy particles [28, 29, 30]. However, MHD simulations demonstrate that no regions of negative particle energy are seen in the steady state , although a transient inflow of negative particle energy is possible as a feedback from generation of an outward MHD wave . The role of negative electromagnetic energy density in the steady state has been discussed recently [K09; 31, 32], although the concept of ‘advection of the steady field’ is ambiguous. Below we show that the sign of the electromagnetic energy density depends on the coordinates, and thus the negative field energy is not physically essential (see Section 4.4 below).
We argue that the causal production mechanism of the electromagnetic AM and Poynting fluxes cannot be fully understood by investigating only the steady-state structure. We examine a time-dependent process evolving towards the steady state with an analytical toy model to clarify how the steady outward fluxes are created. In order to find the essential physics, our analysis is performed both in the Boyer-Lindquist (BL) and the Kerr-Schild (KS) coordinate systems. Most of the previous analytical studies used the BL coordinates [e.g. 33, 10, 34, 35, 17, 36] [but see 37], most of the recent numerical simulations used the KS coordinates [e.g. K04; 12, 2, 3, 13] [but see e.g. 38, 16], and both of them focused on the steady-state structure.222The force-free electrodynamics without decomposition of tensors into spatial and temporal components has also been developed [39, 40, 41, 42]. Our new analytical studies of time-dependent process in the BL and KS coordinates will be highly helpful for understanding physics in BZ process.
This paper is organized as follows. In Section 2, we explain our formulation of general relativistic electrodynamics, set generic assumptions for Kerr BH magnetosphere, and review the recent analytical understandings given by K04 and TT14. Section 3 concentrates on the field lines threading the equatorial plane, for which we show the flux production mechanism and the role of the negative energy of particles. In Section 4, we explain differences between the equatorial plane and the horizon, and then we focus on the field lines threading the horizon, discussing differences of the electromagnetic structures as seen in the BL and KS coordinates and the role of the negative electromagnetic energy density. In Section 5, we discuss the time-dependent process towards the steady state. Section 6 is devoted to conclusion.
2 Formulation and Assumptions
2.1 The decomposition of space-time
The space-time metric can be generally written as
where is called the lapse function, the shift vector and the three-dimensional metric tensor of the space-like hypersurfaces. Those hypersurfaces are regarded as the absolute space at different instants of time [cf. 23]. We focus on Kerr space-time with fixed BH mass and angular momentum . (The electromagnetic field with outward fluxes which we consider below is a test field for Kerr space-time.) We adopt the units of and , for which the gravitational radius . We use the dimensionless spin parameter .
Kerr space-time has two symmetries, i.e. . These correspond to the existence of the Killing vector fields and . In the coordinates ,
The event horizon, where , is located at . The ergosphere is the region , where the Killing vector is space-like, . At infinity, this space-time asymptotes to the flat one.
The AM of this observer is , and thus FIDO is also a zero AM observer [ZAMO; 23]. Note that the FIDO frame is not inertial, but it can be used as a convenient orthonormal basis to investigate the local physics [43, 23, 44]. It should also be confirmed that the FIDOs are time-like, physical observers (i.e. ).
In the BL coordinates, one has the following non-zero metric components:
BL FIDOs rotate in the same direction as the BH with the coordinate angular velocity
The BL coordinates have the well-known coordinate singularity ( and , where ) at the horizon. The BL FIDOs are physical observers only outside the horizon.
The KS coordinates have no coordinate singularity at the event horizon. The coordinates and are different from those in the BL coordinates. The non-zero metric components in this coordinate system are:
where [K04; 37]. The KS spatial coordinates are no longer orthogonal (). From the space-time symmetries,
are the same in the BL and KS coordinates. We should note that the KS FIDOs are different from the BL FIDOs (K04).
2.2 The electrodynamics
We follow the definitions and formulations of K04 for electrodynamics in Kerr space-time (except for keeping in Maxwell equations), in a similar way to TT14 [see also K09, references therein, and 45]. The covariant Maxwell equations and are reduced to
where and denote and , respectively, and is the Levi-Civita pseudo-tensor of the absolute space. The condition of zero electric and magnetic susceptibilities for general fully-ionized plasmas leads to the following constitutive equations,
where denotes . At infinity, and , so that and . Here and are the electric and magnetic fields as measured by the FIDOs, while and are the electric and magnetic fields in the coordinate basis, where . The current is related to the current as measured by the FIDOs, , as
See Appendix A on the relation between convective current and particle velocity.
The covariant energy-momentum equation of the electromagnetic field gives us the AM equation as
and the energy equation as
where denotes , ,
is the AM density,
is the AM flux (),
is the energy density, and
is the Poynting flux ().
2.3 Kerr BH magnetosphere
2.3.1 Electromagnetic fields
We study the axisymmetric electromagnetic field in Kerr space-time which is filled with a plasma. (The steadiness of the field is partly discarded in Section 5.) We put the additional assumptions similarly to TT14: (1) The poloidal field produced by the external currents (whose distribution is symmetric with respect to the equatorial plane) is threading the ergosphere. We call the field lines threading the ergosphere ‘ergospheric field lines’. (2) The plasma in the BH magnetosphere is dilute and collisionless, but its number density is high enough to screen the electric field along the field lines, i.e.
The energy density of the particles is much smaller than that of the electromagnetic fields. (3) The gravitational force is negligible compared with the Lorentz force. (The gravitational force overwhelms the Lorentz force in a region very close to the event horizon , but the physical condition in that region hardly affects its exterior.)
The condition and equation (11) lead to . In the steady state, we have , which means that is a potential field, and the axisymmetry leads to . Then one can write
Substituting this equation into , one obtains
That is, is constant along each field line. The field is also described by in terms of the magnetic flux function , so that each field line is equipotential and corresponds to the potential difference between the field lines.
where the subscript p denotes the poloidal component. Here one sees that the poloidal AM and Poynting fluxes are described by
respectively. It should be noted that and are the same in the BL and KS coordinates (K04).
TT14 shows that the condition is inevitable for the ergospheric field lines in the steady, axisymmetric state (see also K04; K09). Furthermore, for the ergospheric field lines crossing the outer light surface (see Section 2.3.2), the condition
has to be maintained, i.e. the poloidal AM and Poynting fluxes are steadily non-zero (TT14). The following discussion in this paper focuses on how their values are causally determined and the role of the negative energies as measured in the coordinate basis.
2.3.2 Particle motions and light surfaces
Under the assumption (2) for the magnetospheric plasma stated in Section 2.3.1, the force-free condition (or ) is satisfied when (e.g. K04; TT14; see also Appendix A).333Finite particle mass may cause some inertial drift currents to flow across the field lines, which transfer the AM and energy between the particles and the electromagnetic fields . We assume that this effect is negligible for the flux production. Then equation (23) indicates
In this case, in the BL coordinates, the particles drift in the azimuthal direction with angular velocity when (TT14). The light surfaces are thus defined as where the four-velocity of a particle with the coordinate angular velocity becomes null, i.e. , where
There can be two light surfaces; the outer light surface (outside which ) and the inner light surface (inside which ). In the case of , at the outer light surface, and at the inner light surface, which is located in the ergosphere [33, K04; TT14]. The condition is satisfied when the outward Poynting flux is non-zero either for the field lines threading the horizon  or the ergospheric field lines threading the equatorial plane (TT14). Note that is a scalar, so that the location of each light surface is the same in the BL and KS coordinates.
If is realized, the cross-field current flows, i.e. (TT14). The force-free condition is violated, and varies along a field line.
3 Field lines threading the equatorial plane
3.1 Production of AM and Poynting fluxes
The mechanism of and being determined along the ergospheric field lines threading the equatorial plane has been already clarified (K04; TT14). Generally in the BL coordinates, one has from equations (11), (12), and (21)
The key point is that on the equatorial plane because of the symmetry. Therefore, the region of (i.e. inside the inner light surface, which is within the ergosphere) can satisfy the condition around the equatorial plane, driving the poloidal current to flow across the field lines. (Note that is a scalar, so that one has also in the KS coordinates.) This leads to outside the region where , which we call ‘current crossing region’. The value of will be regulated so that the current crossing region is finite (see Figure 4 of TT14), and thus it is expected to depend on the microphysics in the ergosphere. The values of and will be determined by the conditions around the equatorial plane and at infinity.
In the current crossing region, is in the opposite direction of , i.e. , as seen in the BL coordinates (see Figure 3 of TT14). This leads to and , which generate the poloidal electromagnetic AM and Poynting fluxes (see equations 23 and 24). (We confirm that also in the KS coordinates in Appendix B.)
All the ergospheric field lines crossing the outer light surface have and for the northern hemisphere ( for the southern hemisphere), while ‘the last ergospheric field line’, which passes the equatorial plane at , has . This means that the current flows inward along the ergospheric field lines and outward along the last ergospheric field line (see Figure 3 below). Correspondingly, the current crossing region extends over . Such a poloidal current structure prevents the BH from charging up.
3.2 Production of particle negative energy
Equations (23) and (24) imply that the particles in the current crossing region lose their AMs and energies by the feedback, and , from the production of the electromagnetic AM and Poynting fluxes. We find that this feedback can make the particles have negative energy as measured in the coordinate basis.
The production of the particle negative energy can be explained by showing the particle motions in the local orthonormal basis carried with the BL FIDOs, in which the equation of a particle motion with four-velocity , three-velocity , charge , and mass is written as
where denotes the vector component in respect of the FIDO’s orthonormal basis [23, TT14]. In this basis one can investigate local, instantaneous particle motions under the Lorentz force as special relativistic dynamics. (The FIDO frame is not inertial and a particle feels the gravitational force, although we neglect it compared with the Lorentz force, based on the assumption (3) set in Section 2.3.1.) The AM and energy per mass of a particle as measured in the coordinate basis are
where we have used [cf. 44].
Near the equatorial plane, the field is approximately perpendicular to that plane, because at that plane, and then the field is radial in that plane (see Figure 1). The motion of a test particle can be easily solved in such fields . For the case of which we focus on, the positively (negatively) charged particles are accelerated in the directions of () and . In Figure 2, we show the calculation results for and , where we fix the basis and assume that the electromagnetic fields are uniform. For (i.e. ) in particular, the particles are strongly accelerated in the direction of , and then one obtains
in the ergosphere, where . For , does not approach , so that near the boundary of the ergosphere where . However, for implies that can be realized near the horizon. Here we emphasize that and are scalars, and thus and also in the KS coordinates.
For typical AGN jets, is expected to be orders of magnitude smaller than [cf. 47], so that the distance which a particle travels until it achieves the asymptotic azimuthal velocity is tiny compared to the size of the ergosphere. This justifies our calculations of the particle motion in the fixed orthonormal basis with the uniform electromagnetic fields, and the asymptotic velocities can be interpreted as the local velocities of the test particles.
Since the current crossing region is bounded at , the positively charged particles do not cross the last ergospheric field line and will gyrate around this field line. When they emerge out of the ergosphere, they contribute to the current flowing outward along the last ergospheric field line (see Figure 3). The particles outside the ergosphere generally have positive energies.
3.3 Comparison to the mechanical Penrose process
We argue that BZ process for the ergospheric field lines threading the equatorial plane is similar to the mechanical Penrose process, in which the rotational energy of a BH is extracted as mechanical energy by making the BH absorb negative-energy particles [28, 29]. For simplicity, let us consider the positively and negatively charged particles in the geometrically thin current crossing region as a one-fluid. The energy equation for this fluid in the steady state is written as
where is the energy-momentum tensor of the fluid. The boundary condition at is . Therefore, the solutions of equation (36) should be in the current crossing region. Then one has , where and are the comoving mass density and the four-velocity of the fluid, respectively. Since all the particles may have negative energy, it is reasonable to estimate
This means that the current crossing region generates the inflow of the negative-energy fluid and the outward Poynting flux, which appears to be a similar process to the mechanical Penrose process.
However, it is too simple to treat the charged particles in the current crossing region as a one-fluid, since the average velocities of the positively and negatively charged particles should be different. Furthermore, Figure 2 is just the result of the test particle calculations. More detailed studies of the plasma particle motions are required to confirm whether the condition of equation (37) is realistic in the current crossing region.
3.4 Comparison to MHD numerical simulation results
MHD numerical simulations treat the energy of particles (while force-free simulations not), so that they can observe the negative particle energy in principle. However, the MHD simulation results of the dilute Kerr BH magnetosphere with cylindrical magnetic field at the far zone in  do not show any negative particle energy in the steady state. This is just due to the disappearance of the ergospheric field lines threading the equatorial plane, although the reason of this disappearance has not been identified. Such behavior is also seen in the three dimensional MHD simulations including the dense accretion flow [3, 13][but see 48].
4 Field lines threading the event horizon
4.1 Force-free condition is satisfied
In contrast to the equatorial plane where from the symmetry, one generally has at the horizon. Thus the above argument on the field lines threading the equatorial plane is not applicable for the field lines threading the horizon. At the horizon, the regularity condition should be satisfied [26, 23]:
For the BZ split-monopole solution  as an example, in which , , and (to the zeroth order of ), so that one has
Therefore the force-free condition is satisfied at the horizon.
We confirm this fact more generally in the KS coordinates. From the calculation shown in Appendix B, we obtain
where is generally satisfied (see equation B2). For the BZ split-monopole solution as an example, in which and at the northern hemisphere (see Section 4.2 below), , and , one has in the region where . We see that is generally satisfied where is weak, , and . (Note that for the field lines threading the equatorial plane, is the dominant field component near that plane, where can be realized.)
Therefore, for the field lines threading the horizon, the force-free condition can be satisfied, and then no poloidal current is driven to flow across the field lines in the steady state. No AM or energy is transferred from the particles to the electromagnetic fields. These properties clearly indicate that the flux production mechanism for the field lines threading the horizon is different from that for the field lines threading the equatorial plane.
4.2 Electromagnetic structure
Here we focus on the electromagnetic structure of the BZ split-monopole solution, and show that some properties are measured differently in the BL and KS coordinates. This analysis is useful for finding the essential physics in BZ process for the field lines threading the horizon, which should be independent of the adopted coordinate systems.
The split-monopole field is given by
which satisfies . (Note that for .) In the BL coordinates, one has
As is well known, changes its sign at the point where (see Figure 4, left). We can see that and diverge as , while is finite, and one has
near the horizon.
On the other hand, in the KS coordinates, one has
Equation (45) is derived by rewriting in equation (12) with equation (11) and (B3) (K04). The regularity condition at the horizon () for the steady flow to pass with no diverging physical quantities is given by
near the horizon. That is, the field in the KS coordinates is not only so weak that it cannot drive the cross-field current but also it does not change its direction, i.e. in the whole region. This situation is in stark contrast to the field lines threading the equatorial plane, for which the cross-field current is driven by the strong field with .
In the BL coordinates the point where and appears special, and it was considered as a key in some previous analytical discussions [e.g. 35, K09]. However, the electromagnetic quantities are clearly continuous or seamless in the KS coordinates, as shown in Figure 4.
Below we generically consider the cases in which the force-free condition is satisfied along the field lines threading the horizon (see Section 4.1). In those cases, an essential point is that the outward AM and Poynting fluxes, and , are seamless along each field line from the event horizon to infinity in the steady state (from equation 27), with no transfer of AM and energy from the particles. This situation is described in Figure 3.
4.3 The issue
Now we discuss how and are created along the field lines threading the horizon. Blandford & Znajek  show that and in the steady state are determined mathematically from equations (21) and (27) with the conditions at the horizon and at infinity (see also K04). This mathematics and the seamless property shown above may lead to an incorrect consideration that the fluxes are created at the horizon. The conditions at the horizon and at infinity are not boundary conditions but regularity conditions [10, K04], as stated in Section 1. The place where the fluxes are created must not be the horizon, but outside the horizon.
We note that the non-zero outward AM and Poynting fluxes at the horizon in the KS coordinates do not violate causality, because the steady fluxes carry no information. It should be also noted that the steady Poynting flux is not a product of a certain energy density and its advection speed like steady particle energy flux (see Section 4.4 for a related discussion). The Poynting flux is just a result of the currents flowing in the plasma with the potential differences.
Consequently, the issue on the field lines threading the horizon is well defined as “How is the steady current structure causally built?” We consider that this issue may not be resolved by investigating only the steady-state structure. The phenomena at the horizon should be a result from those having occurred outside the horizon in the prior times . In Section 5, we address this issue by discussing a time-dependent state evolving towards the steady state.
4.4 Negative electromagnetic energy?
Lasota et al.  and Koide & Baba  argue that the outward Poynting flux is mediated by ‘inflow of the negative electromagnetic energy’ (see also K09). Although this interpretation analogous to the mechanical Penrose process looks attractive for causal production of the Poynting flux, it is difficult to consider the flow of the steady field (rather than waves). Furthermore, we find that the sign of the electromagnetic energy density depends on the coordinates.
In the BL coordinates, the electromagnetic AM and energy densities can be written down by (K09)
Thus and is negative (and diverges) near the horizon when . This condition is satisfied in the BZ split-monopole solution.
On the other hand, in the KS coordinates, the calculations shown in Appendix B lead to
In the BZ split-monopole solution as an example, in which and in the northern hemisphere, one has
This condition is generally valid when is weak and .
depend on the coordinates, while and are scalars. The concept of the negative electromagnetic energy density depends on the coordinates, and thus it is not physically essential.444Lasota et al.  argue that the electromagnetic energy density calculated in the KS coordinates is negative near the horizon, but they define the electromagnetic energy density as where and is the four-velocity of the BL FIDO.
5 Process towards the Steady State
As stated in Section 4.3, we address the issue how the steady poloidal current structure is built causally, by discussing a time-dependent state evolving towards the steady state.
In the steady state, the plasma has the inner and outer light surfaces (see Section 2.3.2). The particles flow in across the inner light surface and flow out across the outer light surface. Therefore, new particles have to keep injected between the two light surfaces, as discussed in many literatures [e.g. 33, 10, 2, 49]. In this paper we have assumed that the plasma particles keep injected from outside the magnetosphere through electron-positron pair creation by collisions of two photons [2, 50, 51] and/or diffusion of high-energy hadrons 555In the geometrically thick accretion disk the particles can be non-thermally accelerated and diffused out of the disk. The amount of those high-energy hadrons does not appear to be sufficient for the total mass loading of AGN jets which provides the observationally inferred Lorentz factor , but sufficient for satisfying [52, 53]., and that those particles maintain and carry the currents.
Now let us first consider a vacuum in Kerr space-time, and then begin the continuous injection of force-free plasma particles between the two light surfaces as a gedankenexperiment. The inflow (outflow) will fill the vacuum near the horizon (at infinity). Simultaneously we will see a process building the poloidal current structure. Hereafter we will call the (inflow + outflow) region filled with the force-free plasma ‘force-free region’. Figure 5 is a schematic picture of this process focusing on the inflow.
We show the space-time diagrams of the inner and outer boundaries of the force-free region in the BL and KS coordinates in Figure 6, in which the radial light signals are represented by the small arrows. The outflow continues to propagate into the vacuum, i.e. the radius of the outer boundary for . In the BL coordinates, the inflow also continues to propagate towards the horizon, for . In the KS coordinates, the inflow can pass the horizon in a finite time of . In both of the coordinates, when the inner boundary approaches the horizon, the outward signal from it becomes slower and slower and it can hardly affect the force-free region. This will lead to the steady state.666In some MHD simulations, a static plasma (not a vacuum) is initially given and then a central star starts rotating  or a BH starts rotating . They show that a switching-on wave propagates outward and that the outflow region settles down to the steady state after it passes the outer fast magnetosonic point .
Although such a time-dependent state should be analyzed numerically, we use a toy model to qualitatively illustrate the process of building the poloidal current structure. This model assumes that (1) is fixed to be split-monopole
in the whole region, and that (2) the Kerr BH magnetosphere is separated into the force-free region and the vacuum by geometrically thin boundaries moving radially. For further simplicity, (3) we assume that the force-free region and the vacuum have their steady-state structures, but the values of the physical quantities, particularly and , keep updated as determined by the varying conditions of the inner and outer boundaries.
Some of these assumptions would be violated in realistic experiments. Nevertheless we consider that our toy model is useful to suggest the key points for resolving the issue on the causality in the coordinate basis (Section 5.1.4), which also allows us to understand how the steady state is maintained (Section 5.3).
5.1 Analysis in the BL coordinates
5.1.1 The force-free and vacuum regions
The electromagnetic quantities in the force-free region are given as follows. The condition and lead to
Equation and the force-free condition lead to
These two equations imply that . We focus on the northern hemisphere, where and . The current flowing outward , which prevents the BH from charging up, is assumed to be concentrated on the equatorial plane. The poloidal AM and Poynting fluxes are
which satisfy and .
In the vacuum region, one has . Equations and lead to
Hereafter we will put the subscript and superscript ‘vac’ on the quantities in the vacuum region.
5.1.2 The inner boundary of the force-free region
Let us focus on the inner boundary of the force-free (inflow) region, and derive the conditions on the boundary, i.e. the junction conditions between the force-free and vacuum regions. The similar analysis can be done for the outer boundary. For equation
where and are the Heaviside step function and the Dirac delta function, respectively, and
where and are the initial radius and the velocity of the boundary. The location of the boundary is represented by . We have introduced in equation (68), i.e. possible contribution to from moving surface charges at the boundary. The assumption (3) stated in the first part of this section implies that the timescale for the quantities in the force-free and vacuum regions becoming adjusted for steady-state structure is much smaller than the timescale of the boundary propagation. We focus on the latter timescale, considering that only depends on in equation (65). Then we have
Taking account of equation (61), we obtain
which implies that the surface charge density on the boundary . This can be confirmed by integrating over the infinitesimally thin (in the direction) region enclosing the small area on the boundary and taking account of .
and equation (67). We have introduced , possible contribution to from the surface current flowing on the boundary. Then we have
which leads to
The last one of Maxwell equations nontrivial for the present problem is
for which we substitute
Then we have
Integrating equation (81) over and take a limit of , the last term vanishes, and we obtain
Here we take the minus sign, since we have assumed that the inner boundary keeps moving inward. In Section 5.1.3, we will confirm that this assumption is consistent with the electromagnetic structure which we found.
Let us consider the case of . Then we have