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Addendum: Hawking Radiation of Photons in a Variable-mass Kerr Black Hole

Hawking evaporation of photons in a variable-mass Kerr space-time is investigated by using a method of the generalized tortoise coordinate transformation. The blackbody radiant spectrum of photons displays a new spin-rotation coupling effect obviously dependent on different helicity states of photons.

Ever since Hawking [1] initiated the discussion on particle creation by a black hole event horizon, much work has been done on the Hawking evaporation of black holes in some spherically symmetric and non-static space-times [2][3][4]. In the case of a non-stationary axisymmetric black hole, though the Hawking radiation of scalar particles received a fairly extensive investigation [5], similar work to that of Dirac particles encounters great obstacles. The main difficulty lies in the non-separability of the radial and angular variables for Chandrasekhar-Dirac equations [6] in the non-stationary axisymmetry space-time. In a recent paper [7] (here refer to Paper I), this dilemma has been cast off by considering simultaneously the asymptotic behaviors of the first-order and second-order forms of Dirac equation near the event horizon. A new interaction due to the coupling of the spin of Dirac particles with the rotation of the evaporating Kerr black holes was observed in the thermal radiation spectrum of Dirac particles. The character of this spin-rotation coupling effect is its obvious dependence on different helicity states of particles with spin-1/2. This effect vanishes [8] when the space-time degenerates to a spherically symmetric black hole of Vaidya-type.
In this addendum, we apply the method presented in Paper I to deal with the thermal radiation of photons in a non-stationary Kerr space-time [9,10], that is, we consider the asymptotic behaviors of the first-order and second-order forms of Maxwell equations near the event horizon. By using the relations between the first-order derivatives of three complex Newman-Penrose [11] scalars of Maxwell fields, we eliminate the crossing-terms of the firstorder derivatives in the second-order equation and recast each second-order equation to a standard wave equation near the event horizon. The blackbody radiation spectrum of photons displays a spin-rotation coupling effect due to the interaction between the spin of photons and the angular momentum of radiating Kerr black holes.
As in Paper I, the line element of a variable-mass Kerr black hole [9,10] can be written in the advanced Eddington-Finkelstein system as where ∆ = r 2 − 2M(v)r + a 2 , Σ = r 2 + a 2 cos 2 θ = ρ * ρ, ρ * = r + ia cos θ, ρ = r − ia cos θ, and v is the standard advanced time. The mass M of the hole depends on the time v, but the specific angular momentum a is a constant. The metric (1) of an evaporating Kerr black hole is a natural non-stationary generalization of the stationary Kerr solution. The geometry of this Petrov type-II space-time is characterized by three kinds of surfaces of particular interest: the apparent horizons r ± AH = M ± (M 2 − a 2 ) 1/2 , the timelike limit surfaces r ± T LS = M ± (M 2 − a 2 cos 2 θ) 1/2 , and the event horizons r ± EH = r H . The event horizon is necessary a null-surface r = r(v, θ) that satisfies the null hypersurface conditions g ij ∂ i F ∂ j F = 0 and F (v, r, θ) = 0. The generalized tortoise coordinate transformation (GTCT) method is an effective one to determine the location of the event horizon of a dynamic black hole. To illuminate this method, we use it to derive the event horizon equation. Because the space-time under consideration is symmetric about ϕ-axis, we can introduce a GTCT [5,7] as follows where r H = r(v, θ) is the location of event horizon, and κ is an adjustable parameter. All parameters κ, v 0 and θ 0 characterize the initial state of the hole and are constant under the tortoise transformation.
Now applying the GTCT of Eq. (2) to the null surface equation in which the vanishing of the coefficient in the square bracket can give the following equation to determine the location of the event horizon of an evaporating Kerr black hole where we denote ∆ H = r 2 H − 2Mr H + a 2 . The quantitiesṙ H = ∂r H /∂v is the rate of the event horizon varying in time, r ′ H = ∂r H /∂θ is its rate changing with the angle θ. They describe the evolution of the black hole event horizon in the time and the change in the direction, which reflect the presence of quantum ergosphere near the event horizon. Eq. (4) is exactly Eq. (11) derived in Paper I. Now we consider the sourceless Maxwell equations in the spacetime (1). When the back reaction of the massless spin-1 field on this geometry is neglected, the field equation is given by the Maxwell equations on the background spacetime (1). The sourceless Maxwell equations in the Newman-Penrose formalism [6,11] read To write out their explicit form in the spacetime (1), we assume the complex null tetrad system established in Paper I and insert for the appropriate spin-coefficients in Eq. (A4) given in Paper I and then after substituting here we have defined operators Eq. (6) can not be decoupled except in the case of a stationary Kerr black hole [6] (M = const). However, to deal with the problem of Hawking radiation, one should be concerned about the asymptotic behavior of Eq.  (6) can be reduced to the following forms after being taken limits r → r H (v 0 , θ 0 ), v → v 0 and θ → θ 0 . If the derivatives ∂ ∂r * Φ 0 , ∂ ∂r * Φ 1 and ∂ ∂r * Φ 2 in Eq. (7) are not equal to zero, the existence condition of nontrial solutions for Φ 0 , Φ 1 and Φ 2 is that the determinant of two pairs of Eq. (7) vanishes, which gives exactly the event horizon equation (4). It will be seen that the relations (7) play a crucial role to eliminate the crossing-term of the first-order derivatives in the second-order equation.
Next we turn to the second-order form of Maxwell equations. A tedious but straightforward calculation gives Given the GTCT in Eq. (2), the limiting form of Eq. (8), when r approaches r H (v 0 , θ 0 ), v goes to v 0 and θ goes to θ 0 , reads and and where we have replaced the first-order derivative term ∂ ∂r * Φ 0 in Eqs. (10,11) by using the relations (7). In the calculations, the L'Hôsptial's rule has been used to treat an infinite form of 0/0-type.
In order to reduce Eqs. (9), (10) and (11) to a standard form of wave equation near the event horizon, we adjust the parameter κ as did in Paper I, then these wave equations can be recast into an united form as follows where the angular velocity of the event horizon of the evaporating Kerr black hole, Ω, and the coefficient C 3 are presented in Paper I, while both C 1 and C 2 are real, , in which the correspondence should be Ψ p = Φ 0 , Φ 1 , Φ 2 when p = −1, 0, 1 for (s = 1), respectively. It should be pointed out that Eq. (12) includes, as a special case, the wave equation (19) of Dirac particles discussed in Paper I, that is, Ψ p = P 2 , P 1 , when p = −1/2, 1/2 for (s = 1/2).
The subsequent analysis parallels the treatment of Paper I, and one can obtain the Hawking radiation spectrum of photons from the black hole by using the method of Damour-Ruffini-Sannan's [12], where ω is the energy of photons, m is the azimuthal quantum number. The explicit expression of the surface gravity κ is referred to Paper I. The thermal radiation spectrum (13) due to the Bose-Einstein statistics of photons shows that the black hole emits radiation just like a black body. It demonstrate that the energy spectrum of photons in an evaporating Kerr space-time is composed of two parts: the rotational energy mΩ arising from the coupling of the orbital angular momentum of photons with the rotation of the black hole and C 1 due to the coupling of the intrinsic spin of photons and the angular momentum of the black hole. From the explicit expression of the "spin-dependent" term C 1 one can easily find that it vanishes in the case of a stationary Kerr black hole (M = const, r H = r ′ H = 0) or a Vaidya-type black hole (a = 0, r ′ H = 0,ṙ H = 0). The term C 1 is obviously related to the helicity of photons in different spin states, it characterizes a new effect arising from the interaction between the spin of photons and the rotation of an evaporating black hole.
In this addendum, we have dealt with Hawking radiation of photons in a variable-mass Kerr black hole. Eq. (13) shows the thermal radiation spectrum of photons in the nonstationary Kerr space-time, in which an extra term C 1 represents a new spin-rotation coupling effect probably arising from the interaction between the spin of photons and the angular momentum of the evaporating Kerr black hole. The feature of this spin-rotation coupling effect is its dependence on different helicity states of photons. As is pointed out in Paper I, this effect vanishes when a black hole is stationary (ṙ H = r ′ H = 0) or it has a zero angular momentum (a = 0).
In summary, this study confirms that the thermal radiation spectra of particles with higher spins in the non-stationary Kerr black hole displays a new effect due to the coupling of the intrinsic spin of particles with the rotation of the black holes. The spin-rotation coupling effect shows that a non-stationary Kerr space-time has some distinct effects different from that of a stationary Kerr black hole. This effect is not shared by a stationary Kerr black hole or a Vaidya-type spherically symmetric black hole.
This work is supported in part by the NSFC in China.