Backreaction of Mass and Angular Momentum Accretion on Black Holes: General Formulation of the Metric Perturbations and Application to the Blandford-Znajek Process

We study the metric backreaction of mass and angular momentum accretion on black holes. We first develop the formalism of monopole and dipole linear gravitational perturbations around the Schwarzschild black holes in the Eddington-Finkelstein coordinates against the generic time-dependent matters. We derive the relation between the time dependence of the mass and angular momentum of the black hole and the energy-momentum tensors of accreting matters. As a concrete example, we apply our formalism to the Blandford-Znajek process around the slowly rotating black holes. We find that the time dependence of the monopole and dipole perturbations can be interpreted as the slowly rotating Kerr metric with time-dependent mass and spin parameters, which are determined from the energy and angular momentum extraction rates of the Blandford-Znajek process. We also show that the Komar angular momentum and the area of the apparent horizon are decreasing and increasing in time, respectively, while they are consistent with the Blandford-Znajek argument of energy extraction in term of black hole mechanics if we regard the time-dependent mass parameter as the energy of the black hole.


I. INTRODUCTION
Black holes in astrophysical situations are usually assumed to be the Kerr black holes, and the matter fields are treated as test fields. This is because the effects of matter distribution on the curvature are typically small, and then the spacetime is determined from the vacuum Einstein equations which only admit the Kerr black holes as stationary regular black holes due to the uniqueness theorem in general relativity [1][2][3]. Nevertheless, if we take into account the effect of matter distribution on the spacetime, we can discuss the effect of energymomentum tensor on the metric by the gravitational perturbations around the background black holes. In particular, if matter accretions on black holes exist, we expect that the mass and angular momentum of black holes secularly change. In this paper, we would like to make clear this issue by explicitly studying the gravitational perturbations around black holes. As a first step, we focus on the case of the Schwarzschild black hole background.
The linear gravitational perturbations around the Schwarzschild black holes were studied by Regge and Wheeler [4], and Zerilli [5,6]. For the higher-order multipole perturbations, where the degrees of freedoms of the gravitational waves exist, the linearized Einstein equations reduce to the second order wave equations called the Regge and Wheeler, and Zerilli equations with the source terms [4][5][6][7]. Because now we are interested in the evolution of the black hole mass and angular momentum by the matter accretion, we need to study the monopole and dipole perturbations. In [8,9], the monopole perturbations against the generic stationary accreting matters around the Schwarzschild black holes were studied. Recently, in [10], it was shown that both monopole and dipole perturbations for the generic time-dependent matters around the Schwarzschild black holes can be solved in a static coordinate system. In this paper, we extend the formalism in [9], where the Eddington-Finkelstein coordinates are used, to the case of monopole and dipole perturbations for the generic time-dependent accreting matters. To study the evolution of the mass and angular momentum of black holes, the regularity of the accreting matters on the black hole horizon is required and the Eddington-Finkelstein coordinates are suitable for checking the regularity.
As shown in Sec. II, we derive the relation between the time dependence of the mass and angular momentum of the black hole and the energy-momentum tensors.
As interesting phenomena around the rotating black holes, we can consider the energy extraction from black holes, not only increasing the mass of the black holes. The energy extraction by test particles is known as the Penrose process [11,12], and that by the scattering waves is the superradiance [13][14][15][16][17][18]. The energy extraction process by the force-free electromagnetic fields is the Blandford-Znajek process [19], which is a candidate of the central engine for gamma-ray bursts and active galactic nucleus jets. The various aspects of the Blandford-Znajek process were studied in [20][21][22][23][24][25][26][27][28][29][30]. In this paper, we discuss the metric backreaction of the energy extraction from rotating black holes by the Blandford-Znajek process. Because the discussions by the Blandford and Znajek in [19] are based on the slow rotation approximation of the Kerr black holes, we discuss the backreaction using the non-linear gravitational perturbations around the Schwarzschild black holes, where both the effects of the slow rotation and the backreaction of the Blandford-Znajek process are taken into account. In the study of the non-linear gravitational perturbations, at each order, we need to solve equations whose forms are same as those of linear order but the non-linear effects appear in the source terms. For this reason, our formalism can be applied to this problem.
This paper is organized as follows. In Sec. II, we develop the formalism by extending the discussion in [9]. In Sec. III, we briefly review the force-free electromagnetic fields considered by the Blandford and Znajek [19]. Applying the formalism in Sec. II to the electromagnetic fields in Sec. III, we study the metric backreaction of the Blandford-Znajek process in Sec. IV.
The black hole mechanics in this situation is discussed in V. Sec. VI presents summary and discussions. We use the units in which c = G = 1.

CRETION ON SCHWARZSCHILD BLACK HOLES
Let us consider the situation where the effect of matter distribution on curvature is weak.
Then, we need to solve the Einstein equations with the small parameter ǫ. At the lowest order O(ǫ 0 ), the metic is given by a vacuum solution of the Einstein equations. For the later convenience, in this section, we choose the lowest-order vacuum solution as the Schwarzschild metric g Sch with f = 1 − r 0 /r and r 0 = 2M, where M denotes the background black hole mass. When we consider the effect of ǫT µν , the spacetime will be described by the metric with the small deviation from the Schwarzschild metric The Einstein tensor of this metric becomes where ∇ µ denotes the the covariant derivative of the Schwarzschild metric g Sch µν , and we raise or lower indices by g Sch µν . At the order of O(ǫ), we need to solve the equations The energy-momentum tensor satisfies due to the Bianchi identity. We should note that the following discussion holds if only the basic equations formally take the form of Eq. (5). In particular, when we discuss the effect of the backreaction for the Blandford-Znajek process in Sec. IV, we will solve Eq. (5) with the effective energy-momentum tensor.
For the spherically symmetric spacetime background, we can decompose tensor quantities by the tensor spherical harmonics characterized by ℓ, m (ℓ = 0, 1, 2 · · · , m = 0, ±1, · · · ± ℓ), and we can separately discuss even and odd parities and different ℓ, m modes when we solve Eq. (5) [4][5][6]. In this section, we study ℓ = 0 and ℓ = 1 odd-parity time-dependent gravitational perturbations for generic time-dependent matter distribution because those modes are important for the study of the backreaction of accreting matters on the mass and angular momentum of black holes. In [9], the case of stationary energy-momentum tensor was discussed, and recently, in [10], the generic time-dependent case was discussed in the static coordinate system. In this paper, we work in the Eddington-Finkelstein coordinates (V, r, θ, Φ) with dV = dt + f −1 dr, dΦ = dφ, and the line element becomes g Sch µν dx µ dx ν = −f dV 2 + 2dV dr + r 2 (dθ 2 + sin 2 θdΦ 2 ).
In this coordinate system, it is easy to discuss the regularity of tensor quantities at r = r 0 because the finiteness of the tensor components at r = r 0 coincides with the regularity condition at the horizon.

A. monopole perturbations
The perturbed metric for ℓ = 0 is given by where we choose the gauge condition h rr = h θθ (= h ΦΦ / sin 2 θ) = 0 (see Appendix. A). In this is an arbitrary function of V . We note that this residual gauge mode corresponds to the rescale of the coordinate V . The generic energy-momentum tensor for ℓ = 0 becomes The equation ∇ µ T µV = 0 shows that the quantity satisfies The quantity A is interpreted as the accretion rate of the energy which is related to the flux associated with the conservation law ∇ µ (T µν (∂ V ) ν ) = 0. We note that (∂ V ) µ := ∂x µ /∂V and (dr) ν := ∂r/∂x µ . If the energy-momentum tensor is stationary, A becomes constant. Introducing a quantity E as we can write Eq. (11) as In the static coordinates (t, r, θ, φ), Eq. (14) becomes f ∂ r A = ∂ t E, then we can easily see that this corresponds to the local energy conservation law. 1 The other components of the equations ∇ µ T µν = 0 shows the relation among T V r , T rr and T Ω 4rT In the same manner as [9], introducing new variables δM(V, r) and λ(V, r) as the (V, V ), (V, r) and (r, r) components of the Einstein equations give These equations can be solved as where δm and V 0 are constants and χ(V ) is an arbitrary function of V . The function χ(V ) corresponds to the residual gauge mode, i.e., the rescaling of V . We note that the other components of the Einstein equations are automatically satisfied. To summarize, the perturbed metric for ℓ = 0 is described by where δM and λ are given by Eqs. (21) and (22).
If there do not exist ℓ ≥ 1 perturbations, due to the spherical symmetry of the spacetime, we can calculate the Misner-Sharp mass 2 for the metric g Sch µν + ǫh µν at (V, r) as where δM is given by Eq. (21). We can see that the constant δm denotes the deviation of the Misner-Sharp mass from the background mass parameter M at V = V 0 and r = r 0 . We note that the degrees of freedom in choosing δm and V 0 are degenerate because if we change V 0 , δm is shifted. Also, the quantity A determines the time dependence of the mass B. odd-parity dipole perturbations The perturbed metric for the ℓ = 1 odd-parity modes is given by where Y 1,0 = 2 −1 3/π cos θ, 3 and we choose the gauge condition h rΦ = 0 (see Appendix. A).
In this gauge choice, there is a residual gauge mode, is an arbitrary function of V . Note that this residual gauge mode corresponds to the shift of the coordinate Φ by the function of V . The generic energy-momentum tensor for the ℓ = 1 odd-parity modes becomes The non-trivial component of ∇ µ T µν = 0 shows that the quantity satisfies The quantity B is interpreted as the accretion rate of the angular momentum where M = r 0 /2, and (∂ Φ ) µ := ∂x µ /∂Φ. This is related to the flux associated with the conservation law ∇ µ (T µν (∂ Φ ) ν ) = 0. When the energy-momentum tensor is stationary, B becomes constant. Introducing a quantity J as we can write Eq. (29) as In the static coordinates (t, r, θ, φ), Eq. (32) becomes f ∂ r B = ∂ t J , then we can easily see that this corresponds to the local angular momentum conservation law. 4 The (r, Φ) component of the Einstein equations becomes The general solutions of this equation are given by where C 1 and C 2 are arbitrary functions of V , and h IH 0 is an inhomogeneous solution The other components of the Einstein equations give The general solution of this equation is given by where δa is a constant. To summarize, the perturbed metric becomes where the function C 2 (V ) corresponds to the residual gauge mode. 4 In our definition, B is positive when a positive angular momentum accretion onto the black hole exists.
The equation can be written in the conventional conservation form If there do not exist m = 0 perturbations, we can calculate the Komar angular momentum associated with the Killing vector ∂ Φ for the metric g Sch µν + ǫh µν at the radius r as We note that h IH 0 = ∂ r h IH 0 = 0 at r = r 0 . The time dependence of J Komar at the radius r becomes We can see that δa corresponds to a constant shift in the Kerr parameter for slowly rotating cases, and B determines the time dependence of the angular momentum at the radius r.
C. Remarks

Uniqueness of the Kerr metric if
Let us consider that the energy-momentum tensor T µν in r ≥ r 0 vanishes for V ≥ V 1 (> V 0 ). This corresponds to the situation that the matter fields are electrically neutral and they completely fall into the black hole at V = V 1 . In that case, according to our formalism, the perturbed metric for ℓ = 0 and odd-parity ℓ = 1 modes becomes that for the slowly rotating Kerr black holes for r ≥ r 0 and V ≥ V 1 with Note that we can evaluate δM in Eq. (21) at r = r 0 for V ≥ V 1 because of the relation Thus, the integrals of A and B at r = r 0 give the changes of the mass and the angular momentum of the black hole, respectively.

Vaidya metric
The Vaidya metric [31] is the exact spherically symmetric solution with the radiating matter On the other hand, using our formalism with Eq. (44), we obtain Thus, we can see that our linear perturbation formalism reproduces the exact Vaidya metric [31]. 5

The conservation laws and fluxes
The quantities A in Eq. (11) and B in Eq. (29) are related to the energy and angular momentum fluxes associated with the conservation laws ∇ µ (T µν (∂ V ) ν ) = 0 and ∇ µ (T µν (∂ Φ ) ν ) = 0, respectively. We should note that this discussion holds if only the basic equations formally take the form of Eq. (5). In particular, as is discussed later, the equations whose forms are same as Eq. (5), but T µν is replaced by the effective energy momentum tensors T eff µν , appear in the context of the non-linear perturbations around the Schwarzschild metric. In that case, equations ∇ µ (T eff µν (∂ V ) ν ) = 0 and ∇ µ (T eff µν (∂ Φ ) ν ) = 0 hold for the covariant derivative with respect to the Schwarzschild metric, and the corresponding global conservation laws exist.

III. THE ENERGY-MOMENTUM TENSOR OF THE BLANDFORD-ZNAJEK PROCESS
A. The force-free electromagnetic fields around the Kerr spacetime We consider the test electromagnetic field F µν = ∂ µ A ν − ∂ ν A µ with the electric current density vector j µ on the Kerr spacetime. In this section, g µν denotes the Kerr metric, and ∇ µ denotes the corresponding covariant derivative. The metric of the Kerr spacetime in the Boyer-Lindquist coordinates (t, r, θ, φ) is given by where Σ and ∆ are The constants M and a denote the mass and spin parameters. The black hole horizon locates The Maxwell equations on this spacetime are given by We note that the equations The energy-momentum tensor of the electromagnetic field satisfies If the right hand side of Eq. (52), i.e., the Lorentz force term, is neglected, the force-free condition is satisfied. Then, the energy-momentum tensor of the electromagnetic field satisfies We should note that under the condition T particle µν ≪ T EM µν , where T particle µν is the particle energy density, the total energy momentum conservation equations ∇ µ (T particle µν + T EM µν ) = 0 reduce to Eq. (54). This implies that T particle µν ≪ T EM µν is the sufficient condition for Eq. (53). To summarize, the force-free electromagnetic fields F µν can be obtained by solving Eq. (54) with Eq. (51), and the electric current density vector j µ can be calculated from Eq (49).
Because the Boyer-Lindquist coordinates do not cover the black hole horizon, the location r = r + becomes a coordinate singularity and tensors have apparently singular behavior there. In order to solve this problem, we introduce the Kerr-Schild coordinates (T, r, θ, Φ)

B. The Blandford-Znajek solutions in the the Kerr-Schild coordinates
In [19], Blandford and Znajek studied the stationary and axisymmetric force-free electromagnetic fields around the slowly rotating Kerr metric, and the energy and angular momentum extraction though the magnetic fields, called the Blandford-Znajek process. In this paper, we focus on the so-called split-monopole solution, and the solution in the Kerr-Schild coordinates is given by [19,20] where the explicit forms of F µν are and The functions A Φ0 , A Φ2 , ω 1 and B Φ1 are given by where C is implicitly assumed to be of different signs for different signs of cos θ, and the function F (r) is a regular solution of the differential equation The explicit form of F (r) is where Li 2 is the second polylogarithm function The asymptotic behaviors of F at r = 2M and r = ∞ are respectively. We can confirm that T BZ µν satisfies the equations for the force-free electromagnetic fields (54). 6 Also F µν satisfies the degenerate condition where ⋆F µν = F αβ ǫ αβµν /2 and ǫ αβµν is the Levi-Civita tensor. 7 As shown in [19,20], the energy and angular momentum extraction rate are given bẏ We can see that the relationĖ BZ = ωJ BZ holds at this order.

A. Perturbation scheme
The discussions in Sec. III are based on the test field approximation around the Kerr black holes. When we consider the backreaction of the Blandford-Znajek process on the spacetime, we regard the parameter C 2 as a small parameter so that the effect of the energy-momentum tensor T BZ µν (∝ C 2 ) on the spacetime is weak. Introducing dimensionless small parameters α, β as 8 6 We note that F µν satisfies Eq. (50), and this implies that A µ with F µν = ∂ µ A ν − ∂ ν A µ exists. 7 The degenerate condition Eq. (75) can be derived from the force-free condition Eq. (53) for non-zero j µ (see e.g. [28]). We also note that Eq. (75) is compatible with the ideal magnetohydrodynamic (MHD) condition [20]. 8 The shape of the letter α is similar to a, and β reminds us of the magnetic fields B.
the energy-momentum tensor T BZ µν can be written by the Taylor series around (α, β) = (0, 0) as To discuss the backreaction of the Blandford-Znajek process, we need to solve the Einstein We expand the metric tensor as with The Einstein tensor becomes We note that the Einstein tensor at O(β 0 ) vanishes because O(β 0 ) metric is the Kerr metric.
At each order, we need to solve the following equations: If we regard α n β (n = 0, 1, 2) as small parameters, we can apply the formalism in Sec. II to these equations at each order.

B. The Eddington-Finkelstein like coordinates
We discuss the backreaction of the Blandford-Znajek process using the formalism developed in Sec. II which is written in the Eddington-Finkelstein coordinates. Then, it is convenient to introduce the Eddington-Finkelstein like coordinates (V, r, θ, Φ) by dV = dT + dr.

C. O(β) corrections
We can read A (0,1) , T From Eqs. (16), (17), (21) and (22), we obtain the perturbed metric as where we set the residual gauge mode as χ (0,1) (V ) = 0. For the later convenience, we choose δm (0,1) = −πM, then the total metric at this order is This is the Reissner-Nordström metric with a magnetic charge parameter Q = 2 √ πβM = 2 √ πC. We can see that the mass of the spacetime is M and the location of the event horizon is r = r H with One may think that it is strange for the spacetime to be the magnetic Reissner-Nordström metric because the Blandford-Znajek solution is globally different from the magnetic monopole, but it describes the split monopole [19]. The answer to this question is because the Birkhoff's theorem for the specially symmetric spacetime holds locally, and the energy-momentum tensor of the Blandford-Znajek solution at O(β) is locally same as that for the global magnetic monopole.

D. O(αβ) corrections: time dependence of the angular momentum
After some calculations, we obtain 8παβT (1,1) µν dx µ dx ν = 16παβ sin 2 θdΦ Thus, the effective energy-momentum tensor becomes We can read  (29)). From Eq. (38), the perturbed metric becomes At this order, the Komar angular momentum at the radius r is We find that the time dependence of J Komar coincides with the prediction from the angular momentum extraction rate of the Blandford-Znajek process in Eq. (77) We set δa (1,1) = 0, then J Komar = αM 2 at V = V 0 and r = r 0 . We also choose the gauge mode as C In a similar way, we obtainT with and The effective energy-momentum tensor T eff(2,1) µν is given by We can see that A eff(2,1) is not a constant but does not depend on V . The value of A eff(2,1) From Eqs. (16), (17), (21) and (22), we obtain the perturbed metric as where δm (2,1) is a constant and the function χ (2,1) (V ) corresponds to the residual gauge mode. We can see that "the mass term" δM (2,1) depends on time. However, because the spacetime is not spherically symmetric at this order, the appropriate definition of the mass is not clear. We discuss this topic in the next section.
We should note that ℓ = 2 even-parity metric perturbations also exist at O(α 2 β) The perturbed metric can be obtained by solving the Zerilli equation [5][6][7]. As shown in the next section, ℓ = 2 metric perturbations do not affect the area of the apparent horizon, and thus these modes are not relevant for the discussion of the black hole mechanics.

V. BLACK HOLE MECHANICS
In this section, we discuss the relation among the area, mass and angular momentum of the black hole. In the standard derivation of the black hole mechanics [36,37], assuming the time-translational and rotational Killing vectors in a vacuum spacetime before and after the dynamical process, the differences in the Bondi-Sachs energy and angular momentum and therefore the energy and angular momentum of the whole system are discussed. In the present case, however, we would like to determine the energy and angular momentum extraction in the presence of the force-free electromagnetic fields without a time-translational Killing vector. We do not assume the stationary stages before and after the energy extraction. Moreover, to isolate the energy and angular momentum of the black hole from the ambient electromagnetic fields, we need to discuss them in terms of quasi-local quantities.
In the present situation, we show that the apparent horizon is a good candidate for the black hole horizon for this purpose and that the first law of black hole mechanics holds if we take the appropriate time-dependent mass parameter of the apparent horizon.

A. Apparent Horizon
In this subsection, we discuss the apparent horizon for the metric g µν = g Kerr µν + g BZ µν . Because V = const. surface of this spacetime is timelike at O(α 2 β), we work in the Kerr Schild coordinates (T, r, θ, Φ). We set the relation between T and V as V = T + r − 2M.
The unit normal to T = const. surface is given by where the function F n is chosen so that g µν n µ n ν = −1 and n µ is future directed. The induced metric on T = const. surface is given by and the projection operator γ µ ν becomes Because of the facts • Y 1,0 perturbations come from O(α) and O(αβ), • Y 2,0 perturbations come from O(α 2 ) and O(α 2 β), in the metric g µν = g Kerr µν + g BZ µν , we can assume that the location of the apparent horizon at each hypersurface T = const. is where the coefficients only depend on T [38]. From the results for the Kerr metric and O(β) perturbations, where the location of the apparent horizon coincides with that of the event horizon at this order, discussed in Sec. IV C, we obtain Thus, we need to fix R (2,1) ℓ=2 . The unit normal to r = R(θ; T ) at each T = const surface is where the function F s is chosen so that g µν s µ s ν = 1 and s µ is an outward vector. The induced metric on the T = const. and r = R(θ; T ) surface is The location of the apparent horizon is determined by the equation After some calculations, we obtain as solutions of Eq. (141), where T 0 = V 0 . Using our results in the previous section, we have a relation The area of the apparent horizon is given by We should note that ℓ = 2 terms in Eq. (133) do not affect the area because of the orthogonality of the spherical harmonics. Thus, the time dependence of the apparent horizon area is

B. Angular Momentum
The Komar angular momentum at the apparent horizon is The time dependence of J Komar | AH is whereJ BZ is given by Eq. (77). Thus, this reproduces the angular momentum extraction rate of the Blandford-Znajek process in Eq. (77). This explicitly shows that the total angular momentum conservation law holds, i.e., the decreasing rate of the angular momentum of the black hole is balanced with the angular momentum extraction rate of the Blandford-Znajek process.

C. Implication from the Black Hole Mechanics
If we assume the relation of the first law of black hole mechanics [36] we can obtain implication of the time dependence of the black hole mass. Setting dA and dJ as ∂ T A AH and ∂ T J Komar | AH in Eqs. (147) and (149), the time dependence of the mass is suggested by If we assume we obtain whereĖ BZ is given by Eq. (76). This reproduces the energy extraction rate of the Blandford- Znajek process in Eq. (77), although the quantity M in the first law is yet undefined as a quasi-local quantity of the apparent horizon.

D. The Hawking Mass
Because the spacetime at O(α 2 β) is not stationary nor spherically symmetric, it is not obvious how to define the mass of the black hole. In that case, a possible choice of the quasi local mass is the Hawking mass. The Hawking mass at the apparent horizon is given by [39,41] The time dependence of M Hawking | AH is While the absolute value is the desired value, this is positive. This is because the Hawking mass at the apparent horizon is the square root of the apparent horizon area which is increasing in time. Thus, we consider that the Hawking mass is not suitable for the description of energy extraction by the Blandford-Znajek process. We note that even if we use the Hayward mass [40,41], the mass is increasing in time.

E. Comparison with the Kerr Metric with Time-Dependent Parameters
As shown in Appendix B, the Kerr metric with the small parameter shifts of the mass and angular momentum takes the form of Eq. (B13). In this subsection, we show that the time dependence of g µν = g Kerr µν + g BZ µν can be understood in terms of the Kerr metric Eq. (B13) but with time-decreasing mass and angular momentum.
Let us consider the Kerr metric in the form of Eq. (B13), but we replace the constants δM (phys) and δJ (phys) by δM (V ) and δJ(V ) which are the functions of V . We denote by g Kerr+(δM ,δJ) µν this metric. We would like to compare g µν = g Kerr µν + g BZ µν with g Kerr+(δM ,δJ) µν . We set δM and δJ as We also choose χ(V ) in g Kerr+(δM ,δJ) µν as (see Eq. (B15)) Then, after some calculations, we obtain where g other µν does not depend on time. We note that g other µν contains the O(β) effect, i.e., the perturbation corresponding to the magnetic Reissner-Nordström metric duscussed in Sec. IV C. Eq. (160) shows that the ℓ = 0, 1 time-dependent terms of g µν = g Kerr µν + g BZ µν can be expressed as the Kerr metric with time-dependent parameters, g Kerr+(δM ,δJ) µν , whose time dependence is determined from the energy and angular momentum extraction rates of the Blandford-Znajek process. If we regard M + δM as a black hole mass, its time dependence coincides with Eq. (154). Therefore, this gives an appropriate time-dependent mass for the energy extraction in the present setting.
We have seen that the ℓ = 0, 1 time-dependent terms of our results can be fit by the Kerr metric with time-dependent parameters in the Eddington-Finkelstein like coordinates.
We should note that it is essential on which time coordinate we let the mass and angular momentum parameters depend. For example, if we let them depend on time in the Boyer-Lindquist coordinates, our results cannot be fit by the corresponding spacetime. Finding an appropriate coordinate system is not such a trivial problem, and what we showed is that the Eddington-Finkelstein like coordinates is the appropriate choice. 9 Finally, we comment on A eff(2,1) in Eq. (122). While A eff(2,1) is related to the flux associated with ∇ µ (T eff µν (∂ V ) ν ) = 0 (see Sec. II C 3), at this stage, the physical meaning of A eff(2,1) is not clear. Because δM (2,1) in Eq. (127) is written by A eff(2,1) , it is useful to consider the meaning of "the mass term" δM (2,1) . If we compare the situation with the Kerr black hole case, δM (2,1) corresponds to δM  Kerr , the second term of the right-hand side of Eq. (B21). This suggests that δM (2,1) also contains information of both the mass and angular momentum of black holes, and this is the reason why A eff(2,1) | r=r 0 in Eq. (125) does not coincide with −Ė BZ . In fact, 9 In [42], as an extension of the Vaidya metric [31], the Kerr metric but the time-dependent mass and angular momentum parameters are discussed in a different coordinate system from our paper. It is interesting to discuss the relation with our perturbative solution, but we leave this problem for future work.
A eff(2,1) can be written in terms ofĖ BZ andJ BZ as Applying our formalism to the Blandford-Znajek process [19], we studied the metric backreaction. While we need to study the non-linear gravitational perturbations to discuss the backreaction of the Blandford-Znajek process, our formalism can be applied to this problem because the forms of equations at each order are same as those of linear order with the source terms which contain the non-linear effects. We calculated the time-dependent Komar angular momentum and area of the apparent horizon. The decreasing rate of the former coincides with the angular momentum loss rate estimated in terms of the stressenergy tensor of the force-free electromagnetic fields at infinity.
According to the test-field calculation of the energy and angular momentum extraction rates of the Blandford-Znajek process [19], there is no doubt that energy and angular momentum are transfered to asymptotic regions. However, it is not clear how to describe the local metric behavior of the backreaction. In this paper, we showed that the time dependence of ℓ = 0, 1 modes are expressed by the Kerr metric but with time-decreasing mass and angular momentum parameters, which depend only on the ingoing null coordinate V . This suggests that the corresponding outgoing fluxes come directly from the vicinity of the event horizon. If we regard the corresponding mass parameter as the black hole mass, we saw that its decreasing rate coincides with the energy extraction rate of the Blandford-Znajek process and that the first law of black hole mechanics holds for the apparent horizon in terms of this mass parameter but not the Hawking mass.
Finally, we comment on future works. It is interesting to extend our analysis to the higher-order solutions of the Blandford-Znajek process [21][22][23]. The applications to other situations, e.g., the Penrose process or the superradiance phenomena, is possible. It is also interesting to consider the applications to modified gravity theories. If we consider some modified gravity theories and they admit solutions close to the Schwarzschild black holes, we expect that the field equations for the monopole and dipole gravitational perturbations take the same form as Eq. (5), then our formalism can be applied.
Under this gauge transformation, the components of the perturbed metric change as If we choose the gauge with H 2 = K = 0, the residual gauge modes become ξ V = −fη(V ), ξ r =η(V ), whereη(V ) is an arbitrary function of V , and the components of the perturbed metric transform as H 0 → H 0 − 2f ∂ Vη and H 1 → H 1 + ∂ Vη .
Under this gauge transformation, the components of the perturbed metric change as If we choose the gauge with h 1 = 0, the residual gauge modes become ξ (−) = r 2ζ (V ), wherẽ ζ(V ) is an arbitrary function of V , and h 0 transforms as h 0 → h 0 + r 2 ∂ Vζ .
the metric becomes g