Researching on Hawking Effect in a Kerr Space Time via Open Quantum System Approach

Abstract

It has been proposed that Hawking radiation from a Schwarzschild or a de Sitter spacetime can be understood as the manifestation of thermalization phenomena in the framework of an open quantum system. Through examining the time evolution of a detector interacting with vacuum massless scalar fields, it is found that the detector would spontaneously excite with a probability the same as the thermal radiation at Hawking temperature. Following the proposals, the Hawking effect in a Kerr space time is investigated in the framework of an open quantum systems. It is shown that Hawking effect of the Kerr space time can also be understood as the the manifestation of thermalization phenomena via open quantum system approach. Furthermore, it is found that near horizon local conformal symmetry plays the key role in the quantum effect of the Kerr space time.

1. Introduction

Hawking radiation arising from the quantization of matter field in a curved background space-time with the event horizon is a prominent quantum effect. The research to understand Hawking radiation, which is related to general relativity, quantum theory, and thermodynamics, has attracted widespread interest in the physics community. Since Hawking’s original derivation of black hole thermal radiation [1–3], several alternative methods have been proposed, such as Damour-Ruffini method [4, 5], the tunneling method [6–11], and gravitational anomaly method [12, 13].

However, from a physical viewpoint, the black hole thermodynamics system should be more like a nonequilibrium system rather than an equilibrium system. Hawking effect should be investigated in the frame of nonequilibrium statistics physics. In quantum mechanics and nonequilibrium statistics physics, the open quantum theory system has gotten a lot of successful development [14]. The quantum dynamics of an open quantum system characterized by the effects of decoherence and dissipation cannot be represented in terms of a unitary time evolution. It has been applied to quantum information science, modern quantum optics, atomic and many-body systems, soft condensed matter physics, and biophysics. Recently, in the paradigm of open quantum system, based on [15], Yu and Zhang proposed a new insight to understand Hawking radiation in a Schwarzschild space time [16]. Through examining the time evolution of a detector interacting with vacuum massless scalar field, they got a conclusion that the detector in both Unruh and Hartle-Hawking vacua would spontaneously excite with a nonvanishing probability the same as Hawking thermal radiation from the black hole. This new approach has been extended to understand the Gibbons-Hawking effect of de Sitter space-time [17]. However, there remain some challenges to study Hawking radiation from a generic Kerr space time under the manifestation of thermalization phenomena in an open quantum system.

Our motivation comes from the fact that the near-horizon geometry plays the key role to the character of a black hole space time [18–29]. In 1998, Strominger [18] discussed the near-horizon asymptotic symmetry in a Kerr black hole and found that there was a holographic duality between extremal and near-extremal Kerr black hole and a 2-dimensional conformal field theory. In [23, 24], it was shown that around the horizon of a Kerr space time, the scalar field theory can be reduced to a 2-dimensional effective field theory. Now, the thermal radiation of scalar particles from a Kerr black hole can be derived on the basis of a conformal symmetry arising from the near-horizon geometry [29].

Using the near-horizon geometry and open quantum system approach, Hawking effect in a Kerr space time will be investigated. We will examine the time evolution of a static detector (modeled by a two-level atom) outside a Kerr space time immersed in a vacuum massless scalar field. The dynamics of the detector can be obtained from the complete time evolution describing the total system (detector plus external field) by integrating over the field degrees of freedom. Our results show that the detector would spontaneously excite in the Unruh vacuum state with a probability the same as the thermal radiation at Hawking temperature, indicating that Hawking radiation from a Kerr space time can be understood as the manifestation of thermalization phenomena in the framework of open quantum systems. The conformal invariance of the wave equation near the horizon is at the key point of Hawking quantum effect in the Kerr space time.

The organization of our paper is as follows. In Section 2, we will review the basic formulae, including the master equation describing the system of the detector plus external vacuum scalar field in the weak-coupling limit and the reduced dynamical equation for the finite time evolution of the detector. In Section 3, the dimensional reduction technique is used to investigate the massless scalar field in a Kerr space time, and the Wightman function is obtained. In Section 4, applying the method and results of the preceding sections to calculate the probability of a spontaneous transition of the detector from the ground state to the excited state outside a Kerr space time. Finally, some discussions and conclusions will be given in Section 5.

2. Review of the Open Quantum System Approach

In this section, we will review the open quantum system approach to get the master equation describing the combined system 𝐵+𝑆, where a static detector (two-level atom) as an open system 𝑆 which is coupled to another quantum system 𝐵 of a vacuum massless scalar field in a Kerr space time. Our derivation mostly follows the works in [15–17]. Here, we will consider the evolution of the static detector in the proper time and assume the combined system (𝐵+𝑆) to be initially prepared in a factorized state, with the detector keeping static in the exterior region of the Kerr black hole and the field keeping in vacuum state. The static detector is a two-level simplest quantum system whose Hilbert space is spanned over just two states, an excited state |+⟩, and a ground state |−⟩. The Hilbert space of such a system is equivalent to that of a spin-(1/2) system. So the states of the detector can be represented by a 2×2 density matrix, which is Hermitian 𝜌^†=𝜌, and normalized Tr(𝜌)=1 with det(𝜌)⩾0. For simplicity, the Hamiltonian of the detector 𝐻_𝑆 may be taken as 𝐻_𝑆=𝜔₀2𝜎₃, (1) where 𝜎₃ is the Pauli matrix and 𝜔₀ is the energy level spacing. The standard Hamiltonian of massless, free scalar field in a Kerr space time can be denoted as 𝐻_𝐵, which will be discussed in detail in Section 3. The interaction Hamiltonian of the detector with the scalar field can be denoted as 𝐻^′_𝐼=𝜎₃𝜙(𝑥). (2) Therefore, the Hilbert space of the total system 𝑆+𝐵 is given by the tensor product space ℋ=ℋ_𝑆⊗ℋ_𝐵. The total Hamiltonian 𝐻(𝑡) can be taken as 𝐻=𝐻_𝑆⊗𝐼_𝐵+𝐼_𝑆⊗𝐻_𝐵+𝜆𝐻^′_𝐼, (3) where 𝜆 is the coupling constant and, 𝐼_𝑆 and 𝐼_𝐵 denote the identity operators in ℋ_𝑆 and ℋ_𝐵, respectively.

Now in order to get the reduced dynamics of the subsystem 𝑆, we assume that the interaction between the detector and the scalar field is weak as 𝜆 is small and the finite time evolution describing the dynamics of the detector takes the form of a one-parameter semigroup of completely positive map.

Initially, the complete system is described by the total density matrix 𝜌_tot=𝜌(0)⊗|0⟩⟨0|, where 𝜌(0) is the initial reduced density matrix of the detector and |0⟩ is the Kerr space-time vacuum state of field 𝜙(𝑥). In the frame of the atom, the evolution in the proper time 𝜏 of the total density 𝜌_tot of the complete system satisfies 𝜕𝜌_tot(𝜏)𝜕𝜏=−𝑖𝐿_𝐻𝜌_tot(𝜏), (4) which is often referred to the von Neumann or Liouville-von Neumann equation, where 𝐿_𝐻 represents the Liouville operator associated with 𝐻 as follows: 𝐿_𝐻[𝑆]≡[𝐻,𝑆]. (5) The dynamics of the detector can be obtained by summing over the degrees of freedom of the field 𝜙; that is, by applying to 𝜌_tot(𝜏) with the trace projection operator 𝑃 as follows: 𝜌(𝜏)=𝑃𝜌_tot(𝜏)≡Tr_𝜙𝜌_tot(𝜏). (6) In the limit of weak coupling, we can find that the reduced density obeys an equation in the Kossakowski-Lindblad form [30–32] as follows: 𝜕𝜌(𝜏)𝐻𝜕𝜏=−𝑖_eff,𝜌(𝜏)+ℒ𝜌(𝜏), (7) where ℒ𝜌=12^3𝑖,𝑗=1𝑎_𝑖𝑗2𝜎_𝑗𝜌𝜎_𝑖−𝜎_𝑖𝜎_𝑗𝜌−𝜌𝜎_𝑖𝜎_𝑗. (8) The matrix 𝑎_𝑖𝑗 and the effective Hamiltonian 𝐻_eff are determined by the Fourier transform 𝒢(𝜆) and Hilbert transform 𝒦(𝜆) of the vacuum field correlation function (the Wightman function) as follows: 𝐺⁺𝑥−𝑦=⟨0|𝜙(𝑥)𝜙𝑦|0⟩, (9) and they are defined as 𝒢(𝜆)=𝑑𝜏𝑒^𝑖𝜆𝜏𝐺⁺(𝑥(𝜏,𝒦(𝜆)=𝑃))𝒢(𝜔)𝜋𝑖𝑑𝜔.𝜔−𝜆 (10) The coefficients of the Kossakowski matrix 𝑎_𝑖𝑗 can be written as 𝑎_𝑖𝑗=𝐴𝛿_𝑖𝑗−𝑖𝐵𝜖_𝑖𝑗𝑘𝛿_𝑘3+𝐶𝛿_𝑖3𝛿_𝑗3, (11) with 1𝐴=2𝒢𝜔₀+𝒢−𝜔₀,1𝐵=2𝒢𝜔₀−𝒢−𝜔₀,𝐶=𝒢(0)−𝐴. (12) The effective Hamiltonian 𝐻_eff contains a correction term, the so-called Lamb shift, and one can find that it can be obtained by replacing 𝜔₀ in 𝐻_𝑠 with a renormalized energy level spacing Ω as follows: 𝐻_eff=Ω2𝜎₃=𝜔₀+𝑖𝒦−𝜔₀−𝒦𝜔₀𝜎₃, (13) where a suitable subtraction is assumed in the definition of 𝒦(−𝜔₀)−𝒦(𝜔₀) to remove the logarithmic divergence which would otherwise be presented.

To facilitate the discussion of the properties of solutions for (7) and (8), let us express the density matrix in terms of the Pauli matrices as follows: 𝜌(𝜏)=121+³_𝑖=1𝜌_𝑖(𝜏)𝜎_𝑖. (14) Substituting (14) into (8), the Bloch vector |𝜌(𝜏)⟩ of components 𝜌₁(𝜏),𝜌₂(𝜏),𝜌₃(𝜏) satisfies 𝜕𝜌(𝜏𝜌(𝜏+𝜂,𝜕𝜏)=−2ℋ) (15) where |𝜂⟩ denotes a constant vector {0,0,−4𝐵}. The exact form of the matrix ℋ reads Ωℋ=2𝐴+𝐶20−2Ω2𝐴+𝐶000𝐴. (16) Equation (15) can be solved exactly and its solution is 𝜌(𝜏)=𝑒^−2ℋ𝜏𝜌(0)+1−𝑒^−2ℋ𝜏𝜌_∞, (17) where 𝜌_∞=12ℋ⁻¹𝜂𝐵=−𝐴001, (18) the matrix 𝑒^−2ℋ𝜏 is defined by series expansion as usual. However, ℋ obeys a cubic eigenvalue equation, so powers of ℋ higher than 2 can always be written in terms of combinations of ℋ², ℋ, and 𝐼. Actually, three eigenvalues of ℋ are 𝜆₁=2𝐴, 𝜆_±=(2𝐴+𝐶)±𝑖Ω/2. We can write 𝑒^−2ℋ𝜏=4Ω²+4𝐶²𝑒^−2𝐴𝜏Λ₁+2𝑒^{−2(2𝐴+𝐶)𝜏}×Λ₂cos(Ω𝜏)+Λ₃sin(Ω𝜏)Ω, (19) where Λ₁=(2𝐴+𝐶)²+(Ω)²4𝐼−2(2𝐴+𝐶)ℋ+ℋ²,Λ₂()()1=−2𝐴𝐴+𝐶𝐼+2𝐴+𝐶ℋ−2ℋ²,Λ₃Ω=2𝐴²4(𝐼+𝐶()−Ω−𝐶2𝐴+𝐶)4𝐴+𝐶²4ℋ−𝐶ℋ². (20) Equation (19) reveals that a freely falling atom in a Kerr space time is subjected to the effects of decoherence and dissipation by the exponentially decaying factors including the real parts of the eigenvalues of ℋ and oscillating terms associated with the imaginary part. These nonunitary effects can be analyzed by examining the evolution behavior in time of suitable atom observable. For any observable of the atom represented by a Hermitian operator 𝒪, the behavior of its mean value is determined by ⟨𝒪⟩=Tr𝒪𝜌(𝜏). (21) Let the observable 𝒪 be an admissible atom state 𝜌_𝑓, the probability 𝒫_𝑖→𝑓, that the atom evolves to the expected state represented by density matrix 𝜌_𝑓(𝜏) from an initial one 𝜌_𝑖≡𝜌(0), should be 𝒫_𝑖→𝑓(𝜏)=Tr𝜌_𝑓𝜌(𝜏). (22) If initially the atom is in the ground state, its Bloch vector |𝜌(0)⟩ is {0,0,−1}, and the final state 𝜌_𝑓 is the excited state given by the Bloch vector |𝜌_𝑓⟩={0,0,1}, according to (17)–(22), we have 𝒫_𝑖→𝑓=121−𝑒^−4𝐴𝜏𝐵1−𝐴. (23)

The probability per unit time of the transition from the ground state to the excited state, in the limit of infinitely slow switching on and off the atom-field interaction, that is, the spontaneous excitation rate, can be calculated by taking the time derivative of 𝒫_𝑖→𝑓(𝜏) at 𝜏=0 as Γ_𝑖→𝑓=𝜕𝒫𝜕𝜏^𝑖→𝑓(𝜏)_𝜏=0=2𝐴−2𝐵=2𝒢−𝜔₀. (24)

3. Scalar Wave Equation Near the Event Horizon in a Kerr Space-Time

3.1. Dimensional Reduction Near the Horizon

In order to find out how the reduced density evolves with proper time from (7), we will investigate the scalar wave equation of the Kerr space time. In Boyer-Lindquist coordinates, the stationary Kerr space time can be written as 𝑑𝑠²Δ=−𝜌²𝑑𝑡−𝑎sin²𝜃𝑑𝜑²+sin²𝜃𝜌²𝑟²+𝑎²𝑑𝜑−𝑎𝑑𝑡²+𝜌²Δ𝑑𝑟²+𝜌²𝑑𝜃², (25) where Δ=(𝑟−𝑟₊)(𝑟−𝑟₋), 𝜌²=𝑟²+𝑎²cos²𝜃 and 𝑟_±=𝑀±(𝑀²−𝑎²)^1/2. The parameters 𝑀 and 𝑎 represent the mass and the angular momentum per unit mass of the black hole, respectively. The event horizon of the Kerr black hole is located at 𝑟=𝑟₊. The line element in (25) is stationary and axisymmetric, with 𝜕^𝜇_𝑡 and 𝜕^𝜇_𝜑 as the corresponding Killing vector fields.

And then, we will show that the scalar field theory in the background (25) can be reduced to a 2-dimensional field theory in the near-horizon region with the dimensional reduction technique. This technique firstly has been employed for the Kerr black hole by Murata and Soda [24] and developed with a more general technique by Iso et al. [23].

The action for the scalar field in a Kerr space time is 1𝑆=2𝑑𝑥⁴√−𝑔𝑔^𝜇𝜈𝜕_𝜇𝜙𝜕_𝜈𝜙+𝑆_int, (26) where the first term is the kinetic term and the second term 𝑆_int represents the mass, potential, and interaction terms.

By substituting (25) into (26), we obtain 1𝑆=−2−𝑟𝑑𝑟𝑑𝑡𝑑𝜃𝑑𝜑sin𝜃𝜙²+𝑎²²Δ−𝑎²sin²𝜃𝜕²_𝑡−2𝑎𝑟²+𝑎²−ΔΔ𝜕_𝑡𝜕_𝜑+1sin²𝜃−𝑎²Δ𝜕²_𝜑+𝜕_𝑟Δ𝜕_𝑟+1𝜕sin𝜃^𝜃sin𝜃𝜕_𝜃𝜙+𝑆_int. (27) Now, we transform the radial coordinate 𝑟 into the tortoise coordinate 𝑟_∗ defined by 𝑑𝑟_∗=1𝑑𝑟𝐹(𝑟)≡𝑟²+𝑎²Δ. (28) After the transformation, the action (27) can be written as 1𝑆=−2𝑑𝑟_∗×−𝑟𝑑𝑡𝑑𝜃𝑑𝜑sin𝜃𝜙₂+𝑎²−𝐹(𝑟)𝑎²sin²𝜃𝜕²_𝑡−2𝑎(1−𝐹(𝑟))𝜕_𝑡𝜕_𝜑+𝐹(𝑟)sin²𝜃−𝑎²𝑟²+𝑎²𝜕²_𝜑+𝜕_𝑟∗𝑟²+𝑎²𝜕_𝑟∗+𝐹(𝑟)𝜕sin𝜃^𝜃sin𝜃𝜕_𝜃𝜙+𝑆_int. (29) Now we consider this action in the region near the horizon. Since 𝐹(𝑟₊)=0 at 𝑟→𝑟₊, we only retain dominant terms in (29). We have 1𝑆=−2𝑑𝑟_∗×−𝑟𝑑𝑡𝑑𝜃𝑑𝜑sin𝜃𝜙₂+𝑎²𝜕²_𝑡−2𝑎𝜕_𝑡𝜕_𝜑−𝑎²𝑟²+𝑎²𝜕²_𝜑+𝜕_𝑟∗𝑟²+𝑎²𝜕_𝑟∗𝜙, (30) where we have ignored 𝑆_int by using 𝐹(𝑟₊)=0 at 𝑟→𝑟₊. Because the theory becomes high-energy case near the horizon and the kinetic term dominates, we can ignore all the terms in 𝑆_int. After this analysis, we return to the expression written in terms of 𝑟. So, we have 1𝑆=−2𝑑𝑡𝑑𝑟𝑑𝜃𝑑𝜑sin𝜃𝑟²+𝑎²𝜙×−1𝐹(𝑟)𝜕²_𝑡−2𝑎Δ𝜕_𝑡𝜕_𝜑−𝑎²Δ𝑟²+𝑎²𝜕²_𝜑+𝜕_𝑟𝐹(𝑟)𝜕_𝑟𝜙. (31)

Following Murata and Soda’s method [24], we transform the coordinates to the corotating coordinate system. They employed a locally corotating coordinate system, and we will use a globally corotating coordinate system as 𝑎𝜓=𝜑−𝑟²+𝑎²𝑡,𝜉=𝑡. (32) Under these new coordinates, we can rewrite the action (31) as 𝑆𝜙1=−2𝑑𝜉𝑑𝑟𝑑𝜓𝑑𝜃𝑟²+𝑎²−1×sin𝜃𝜙𝐹(𝑟)𝜕²_𝜉+𝜕_𝑟𝐹(𝑟)𝜕_𝑟𝜙, (33) so the angular terms disappear completely. Using the spherical harmonics expansion 𝜙=Σ_𝑙,𝑚𝜙_𝑙𝑚(𝜉,𝑟)𝑌_𝑙𝑚(𝜃,𝜓), we obtain the effective 2-dimensional action 𝑆𝜙=_𝑙,𝑚12𝑟²+𝑎²𝑑𝜉𝑑𝑟𝜙_𝑙𝑚×−1𝐹(𝑟)𝜕²_𝜉+𝜕_𝑟𝐹(𝑟)𝜕_𝑟𝜙_𝑙𝑚, (34) where we have used the orthonormal condition for the spherical harmonics as follows: 𝑑𝜓𝑑𝜃sin𝜃𝑌^∗_{𝑙′𝑚′}𝑌_𝑙𝑚=𝛿_𝑙′,𝑙𝛿_𝑚′,𝑚. (35)

From the action (34), it is obvious to find that 𝜙 can be considered as a (1+1)-dimensional massless scalar field in the backgrounds of the dilaton Φ. The effective 2-dimensional metric and the dilaton Φ can be written as 𝑑𝑠²=−𝐹(𝑟)𝑑𝜉²+1𝐹(𝑟)𝑑𝑟²,Φ=𝑟²+𝑎². (36)(37)

So far, we have reduced the 4-dimensional field theory to a 2-dimensional case. This is consistent with [23]. This 2-dimensional metric tells us that, near the horizon, the geometry of a Kerr space time can be regarded as a Rindler space time when 𝑟₊>𝑟₋. In the extremal case 𝑟₊=𝑟₋, the near horizon geometry reduces to 𝐴𝑑𝑆₂ which is consistent with [19, 23]. The same as the Schwarzschild space time [33], we will define two vacuum states by using the two natural notions of time translation of this effective 2-dimensional metric, namely, the Killing time and the proper time as measured by a congruence of freely falling observers.

3.2. The Boulware Vacuum

Using the tortoise coordinate in (28), the effective 2-dimensional metric (36) can be changed into 𝑑𝑠²_𝐼=−𝐹(𝑟)𝑑𝜉²+𝑑𝑟²_∗. (38)

We can see that the (𝜉,𝑟_∗) part of the metric has the form of Minkowski metric. Now in this 2-dimensional space time, the wave equation of 𝜙(𝜉,𝑟_∗) can be written as 𝜕²_𝜉−𝜕²_𝑟∗𝜙𝜉,𝑟_∗≡𝜕_𝑢𝜕_𝑣𝜙(𝑢,𝑣)=0. (39)

Its standard ingoing and outgoing orthonormal mode solutions are 𝜙𝜉,𝑟_∗∼𝑒^{−𝑖𝜎(𝜉+𝑟∗)},𝑒^{−𝑖𝜎(𝜉−𝑟∗)}∼𝑒^{−𝑖𝜎𝑣},𝑒^{−𝑖𝜎𝑢}, (40) where 𝑣=𝜉+𝑟_∗,𝑢=𝜉−𝑟_∗ are null coordinates. These modes are positive frequency modes with respect to the killing vector field 𝜕/𝜕𝜉 for 𝜎>0, and they satisfy 𝐿_{𝜕/𝜕𝜉}𝜙=−𝑖𝜎𝜙. (41)

It is obvious that the wave equation (39) is manifestly invariant under the infinite-dimensional group of conformal transformation in two dimensions 𝑢→𝑢^′(𝑢), 𝑣→𝑣^′(𝑣). In the following, we will show how this conformal symmetry does play the key role in the quantum effect of a Kerr space time.

Near the event horizon, we only consider the outgoing modes 𝜙^out𝜉,𝑟_∗=1√𝑒4𝜋𝜔^{−𝑖𝜎(𝜉−𝑟∗)}=1√𝑒4𝜋𝜔^{−𝑖𝜎𝑢}, (42) along the rays 𝑢= constant. Quantizing the field 𝜙^out in the exterior of the black hole, we can expand it as follows 𝜙^𝐼=_𝜎𝑎^𝐼_𝜎𝜙^out_𝜎𝜉,𝑟_∗+𝑎_𝜎^𝐼†𝜙^out†_𝜎𝜉,𝑟_∗, (43) where 𝑎^𝐼_𝜎 and 𝑎_𝜎^𝐼† are the annihilation and creation operators acting on the 𝐼 vacuum state, which corresponds to the Boulware vacuum. The Fock vacuum state can be defined as 𝑎^𝐼_𝜎|0⟩=0. So, with the proper 𝑖𝜖 prescription, the Wightman function of 𝐼 state can be written as 𝐺^𝐼+_Kerr𝑥,𝑥^′1=−4𝜋²1𝑥⁰−𝑖𝜖²−𝑥¹²1=−4𝜋²1(Δ𝜉−𝑖𝜖)², (44) where 𝑥⁰=𝜉, 𝑥¹=𝑟_∗.

3.3. The Unruh Vacuum

In order to define the Unruh vacuum state, one can write the Kerr line element in the near-horizon region in terms of Kruskal-like coordinates defined as 𝑈=𝑇−𝑅=−𝜅⁻¹𝑒^−𝜅𝑢,𝑉=𝑇+𝑅=𝜅⁻¹𝑒^𝜅𝑣, (45) where 𝑇=𝜅⁻¹𝑒^𝜅𝑟∗sinh𝜅𝜉, 𝑅=𝜅⁻¹𝑒^𝜅𝑟∗cosh𝜅𝜉 and 𝜅=(𝑟₊−𝑟₋)/2(𝑟²₊+𝑎²) is the surface gravity of the event horizon. The effective 2-dimensional space time (36) becomes 𝑑𝑠²_𝐼𝐼=𝐶(𝑟)−𝑑𝑇²+𝑑𝑅², (46) where 𝐶(𝑟)=𝑒^{−2𝜅𝑟∗}𝐹(𝑟), which is a finite constant near the event horizon 𝑟=𝑟₊. The (𝑇,𝑅) part of the metric also has the form of Minkowski metric. The interval of time Δ𝑇 then corresponds to the interval of proper time of a radial freely falling observer crossing the horizon. The 2-dimensional space time in the (𝑇,𝑅) coordinates is well-behaved near the event horizon. One can obtain the wave equation of 𝜙(𝑇,𝑅) as 𝜕²_𝑇−𝜕²_𝑅𝜙(𝑇,𝑅)≡𝜕_𝑈𝜕_𝑉𝜙(𝑈,𝑉)=0. (47)

Similar to previous proceeding, we can obtain the out-going wave solution as 𝜙^out(𝑇,𝑅)∼𝑒^{−𝑖𝜔(𝑇−𝑅)}=𝑒^{−𝑖𝜔𝑈}. These modes are positive frequency modes with respect to the freely falling observer for 𝜔>0, satisfying 𝐿_{𝜕/𝜕𝑇}𝜙=−𝑖𝜔𝜙. (48) As pointed in previous subsection, there are conformal transformations from 𝜙(𝑈,𝑉) to 𝜙(𝑢,𝑣), as (45).

Subsequently, quantizing the field 𝜙^out(𝑇,𝑅) in the exterior of the black hole, we can expand it as follows: 𝜙^𝐼𝐼=_𝜔𝑎_𝜔^𝐼𝐼𝜙^out_𝜔(𝑇,𝑅)+𝑎_𝜔^𝐼𝐼†𝜙^out†_𝜔(𝑇,𝑅), (49) where 𝑎_𝜔^𝐼𝐼 and 𝑎_𝜔^𝐼𝐼† are the annihilation and creation operators acting on the 𝐼𝐼 vacuum state, which can be defined as 𝑎_𝜔^𝐼𝐼|0⟩=0. This vacuum state is just the so-called Unruh vacuum defined in the maximally extended geometry. So, with the proper 𝑖𝜖 prescription, the Wightman function of the 𝐼𝐼 state can be written as 𝐺^𝐼𝐼+_Kerr𝑥,𝑥^′1=−4𝜋²1𝑥⁰−𝑖𝜖²−𝑥¹²1=−16𝜋²𝜅⁻²sinh²𝜉−𝜉^′,𝜅/2−𝑖𝜖 (50) where 𝑥⁰=𝑇, 𝑥¹=𝑅.

4. Probability of Spontaneous Transition of the Detector in a Kerr Space-Time

In what follows, we will calculate the spontaneous excitation rate in the two vacuum states with the open quantum system approach.

4.1. The 𝐼 State-Boulware Vacuum

Firstly, let us turn to the 𝐼 state-Boulware vacuum case. Thinking of the relation between the proper time and the coordinate time, 𝑑𝜏=𝐹(𝑟)𝑑𝜉, (51) the Fourier transform of the Wightman function (44) with respect to the proper time can be expressed as 𝒢_Kerr(𝜆)=^+∞_−∞𝑑𝜏𝑒^𝑖𝜆𝜏𝐺^𝐼+_Kerr𝑥,𝑥^′=−^+∞_−∞𝑑𝜉𝐹(𝑟)𝑒^{𝑖𝜆𝐹(𝑟)𝜉}14𝜋²1(Δ𝜉−𝑖𝜖)²=0. (52)

According to (12), we have 𝐴=𝐵=0. (53)

So the spontaneous excitation rate can be obtained as Γ_𝑖→𝑓=2(𝐴−𝐵)=0. (54)

Therefore, no spontaneous excitation would ever occur in the 𝐼 state-Boulware vacuum. In fact, the Boulware vacuum corresponds to our familiar notion of a vacuum state. This result is consistent with the conclusion in [16].

4.2. The 𝐼𝐼 State-Unruh Vacuum

Using the Wightman function (50) and the relation between the proper time and coordinate time (51), the Fourier transform can be given as 𝒢_Kerr(𝜆)=^+∞_−∞𝑑𝜏𝑒^𝑖𝜆𝜏𝐺^𝐼𝐼+_Kerr𝑥,𝑥^′=𝜆𝑒2𝜋𝑒^{2𝜋𝜅𝑟−1𝜆2𝜋𝜅𝑟−1𝜆},−1 (55) where 𝜅_𝑟=𝜅/𝐹(𝑟)=𝜅(𝑟²+𝑎²)/Δ.

According to (12), we have 1𝐴=2𝒢_Kerr𝜔₀+𝒢_Kerr−𝜔₀=𝜔₀coth𝜋𝜔₀/𝜅_𝑟,14𝜋𝐵=2𝒢_Kerr𝜔₀−𝒢_Kerr−𝜔₀=𝜔₀,4𝜋𝐶=𝒢_Kerr(0)−𝜔₀coth𝜋𝜔₀/𝜅_𝑟.4𝜋 (56)

Using (23) and (24), we have 𝒫_𝑖→𝑓=121−𝑒^{−(𝜔0coth(𝜋𝜔0/𝜅𝑟)/𝜋)𝑡}×11−coth𝜋𝜔₀/𝜅_𝑟, (57) and the spontaneous excitation rate is Γ_𝑖→𝑓=𝜕𝒫𝜕𝑡^𝑖→𝑓(𝑡)_𝑡=0=𝜔₀𝜋𝑒^{2𝜋𝜔0/𝜅𝑟},−1 (58) which reveals that, the ground state detector in the 𝐼𝐼 vacuum would spontaneously excite with an excitation rate that one would expect in the case of a flux of thermal radiation at the temperature 𝜅𝑇=^𝑟.2𝜋 (59) It is obvious that there is a near horizon conformal symmetry from 𝜙(𝑈,𝑉) to 𝜙(𝑢,𝑣) as (45), which is just the reason of the nonvanishing spontaneously excitation. This proposal is similar to [29]. In fact, the effective temperature 𝑇 in (59) approaches to Hawking temperature 𝑇=𝜅/2𝜋=(𝑟₊−𝑟₋)/4𝜋(𝑟²₊+𝑎²) as 𝑟→∞. This suggests that the thermal radiation emanating from the horizon of a Kerr black hole is just Hawking radiation, which is in agreement with [16, 17].

5. Conclusions and Discussions

In brief, we have investigated the Hawking radiation from a Kerr space time through examining the evolution of a detector (modeled by a two-level atom) interacting with the vacuum massless scalar field in the framework of open quantum systems.

First of all, using the dimensional reduction technique, the 4-dimensional spherically nonsymmetric Kerr metric can be regarded as a 2-dimensional effective spherically symmetric metric near the event horizon. So we can construct two conformal vacuum states in this 2-dimensional effective space time: one is the 𝐼 state-Boulware vacuum, the other is the 𝐼𝐼 state-Unruh vacuum. Then we give the Wightman functions of the two vacuum states, respectively.

On the basis of these, we have calculated the time evolution of the detector in the two vacuum states. It is found that the detector in the 𝐼𝐼 state-Unruh vacuum would spontaneously excite with a nonvanishing probability the same as thermal radiation at Hawking temperature from a Kerr black hole. Hawking-Unruh effect of a Kerr space time can be understood as a manifestation of thermalization phenomena in an open quantum system. Meanwhile, it is also found that the probability of spontaneous transition of the detector would be vanishing in the 𝐼 state-Boulware vacuum. It suggests that near horizon conformal symmetry plays the key role in the full quantum phenomena in the Kerr space time.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank Dr. Kui Xiao and Dr. Shi-Wei Zhou for their helpful discussions and suggestions. This research is supported by the National Natural Science Foundation of China under Grant no. 11365008.