Information conservation in de Sitter tunneling
Baoyu Tan
 
Quantum reservoir computing on random regular graphs
    Tapio Ala-Nissila
    
====================================================





§ INTRODUCTION

It is well known that according to the unitary principle, a fundamental principle of quantum mechanics, pure states evolve over time as pure states and cannot become mixed states. But black holes seem to defy this. In 1975, Hawking used quantum field theory in curved spacetime to show that particles can escape black hole through quantum effects <cit.>. In 1976, Hawking proved that Hawking radiation is a pure thermal spectrum that carries no information <cit.>. That is, matter that enters the black hole in a pure state radiates out in a mixed state. The information is lost in the process of Hawking radiation, violating the principle of unitarity.

Page argues that Hawking's calculations did not take into account the effects of quantum entanglement. Although Hawking radiation does not carry information itself, it has been entangled with particles inside the black hole, thus transmitting information from the black hole to the outside. In 1993, Page proposed that in order to satisfy the unitary principle, the entanglement entropy of radiation should satisfy the Page curve <cit.>. In simple terms, although the radiation is a mixed state, the direct product of the radiation state and the black hole state is still a pure state. In recent years, there have been some breakthroughs in this approach <cit.>. But there is an essential difficulty with this approach. Since there is no complete theory of quantum gravity, it is impossible to know the concrete form of the density matrix. The entropy of entanglement is determined by the density matrix:

    S_EE=-Tr(ρlogρ).

Where ρ is the density matrix and S_EE is the entropy of entanglement. Much of the discussion of information conservation by quantum entanglement has focused on AdS space-time. The AdS-CFT duality has automatically implied the unitary principle, that is, the conformal field on the boundary must satisfy the unitary principle and must not lose information. And the discussion of general black holes is still difficult, even impossible. Because this requires the semi-classical theory of gravity to automatically include the unitary principle, which is obviously unreasonable, at least unfounded.

In 2000, Parikh and Wilczekr believed that Hawking radiation was a tunneling process. They started from the perspective of energy conservation and the change in spatiotemporal background caused by the contraction of black holes after radiation, that is, they considered the self gravity effect during the particle emission process, thus avoiding the difficulty of calculating the specific form of the density matrix. They also came to the conclusion that the radiation process satisfies the unitary principle and information conservation <cit.>. After the Parikh-Wilczek method was proposed, researchers successively studied various static and stationary rotating black holes and reached the same conclusion. Due to the effect of self gravity, the spatiotemporal background changes, and the radiation spectrum will deviate from the pure thermal spectrum, thus proving that Hawking radiation satisfies the unitary principle. <cit.>. But until now, apart from the Hawking radiation of the Schwarzschild de Sitter spacetime discussed in ref. <cit.>, there has been little discussion on the asymptotic de Sitter spacetime that is more similar to the real universe.

In Section <ref>, we separately calculated the radiation spectra near the event horizon and cosmic horizon of the magnetically charged Reissner-Nordström de Sitter black hole, and briefly discussed our results. In Section <ref> and Section <ref>, we performed similar calculations on the Kerr-Newman-Kasuya de Sitter black hole and Bardeen de Sitter black hole, respectively. Finally, in Section <ref>, we summarized the results and the discussion of the results obtained in the previous section. We have adopted the system of natural units in this paper (G≡ħ≡ c≡ 1).


§ MAGNETICALLY CHARGED REISSNER-NORDSTRÖM DE SITTER BLACK HOLE


According to ref. <cit.>, for the case of both electric and magnetic charges, the electromagnetic field tensor can be defined as:

    F_μν=∇_ν A_μ-∇_μ A_ν+G_μν^+.

Where G_μν^+ is the Dirac string term. The Maxwell equation can be rewritten as:

    ∇_ν F^μν=4πρ_eu^μ,


    ∇_ν F^+μν=4πρ_gu^μ.

Where F^+μν is the dual tensor of F^μν, ρ_e and ρ_g represent the charge density and magnetic charge density, respectively. For the convenience of discussion, we have defined a new antisymmetric tensor:

    F̃^μν=F^μνcosα+F^+μνsinα.

Where α is a fixed angle. From (<ref>) and (<ref>), we can deduce that:

    ∇_νF̃^μν=4π(ρ_ecosα+ρ_gsinα)u^μ=4πρ_hu^μ,


    ∇_νF̃^+μν=4π(-ρ_esinα+ρ_gcosα)u^μ=0.

According to the no-hair theorem, we can consider a black hole as a conductor sphere, and assume that the charge density and magnetic charge density satisfy the following relationship to ensure that (<ref>) and (<ref>) hold true:

    ρ_e/ρ_g=cotα.

We can conclude that:

    Q_h^2=Q_e^2+Q_g^2.

Where Q_e and Q_g are the total electric and magnetic charges of the system, respectively, and Q_h is the equivalent charge corresponding to density ρ_h. Finally, we obtain the Lagrangian density of the electromagnetic field, which can be expressed by the following equation:

    ℒ_h=-1/4F̃_μνF̃^μν.

The corresponding generalized coordinates for (<ref>) are:

    Ã_μ=(Ã_0,Ã_1,Ã_2,Ã_3)=(-Q_h/r,0,0,0).

Obviously, Ã_0 is a cyclic coordinate. The above discussion also applies to Section <ref> and Section <ref>, and will not be repeated in the following text.

The line element of Magnetically charged Reissner-Nordström de Sitter black hole is:

    ds^2=-(1-2M/r+Q_e^2+Q_g^2/r^2-Λ/3r^2)dt_s^2+(1-2M/r+Q_e^2+Q_g^2/r^2-Λ/3r^2)^-1dr^2+r^2dΩ^2.

Where M is the mass of the black hole, Q_e is the charge of the black hole, Q_g is the magnetic charge of the black hole, and Λ is the cosmological constant. According to (<ref>), the line element of Magnetically charged Reissner-Nordström de Sitter black hole can be rewritten as:

    ds^2   =-(1-2M/r+Q_h^2/r^2-Λ/3r^2)dt_s^2+(1-2M/r+Q_h^2/r^2-Λ/3r^2)^-1dr^2+r^2dΩ^2
       =-f(r)dt_s^2+1/f(r)dr^2+r^2dΩ^2.

Where f(r)=1-2M/r+Q_h^2/r^2-Λ/3r^2. The outer event horizon of black hole r_+ and cosmic horizon r_c satisfy the following equation:

    f(r)=1-2M/r+Q_h^2/r^2-Λ/3r^2=0.

Clearly, (<ref>) has four solutions: r_non, r_-, r_+, and r_c. Among these, r_non is a negative solution without physical significance, r_- represents the inner event horizon of black hole, r_+ denotes the outer event horizon of black hole, and r_c signifies the cosmic horizon.

The specific forms of these four solutions are:

    r_non=   -1/2η-1/2ζ,
    
    r_-=   -1/2η+1/2ζ,
    
    r_+=   1/2η-1/2ζ,
    
    r_c=   1/2η+1/2ζ.

Where:

    λ=   [-54+972Λ M^2-648Λ Q_h^2+√((-54+972Λ M^2-648Λ Q_h^2)^2-4(9-36Λ Q_h^2)^3)]^1/3,
    η=   √(2/Λ+3(2)^1/3(1-4Λ Q_h^2)/Λλ+λ/3(2)^1/3Λ),
    ζ=   √(4/Λ-3(2)^1/3(1-4Λ Q_h^2)/Λλ-λ/3(2)^1/3Λ-12M/η).

For the process of particles tunneling out of the event horizon of a black hole, the initial position of the particle is:

    r_i=1/2η_i-1/2ζ_i.

Where:

    λ_i=   [-54+972Λ M^2-648Λ Q_h^2+√((-54+972Λ M^2-648Λ Q_h^2)^2-4(9-36Λ Q_h^2)^3)]^1/3,
    η_i=   √(2/Λ+3(2)^1/3(1-4Λ Q_h^2)/Λλ_i+λ_i/3(2)^1/3Λ),
    ζ_i=   √(4/Λ-3(2)^1/3(1-4Λ Q_h^2)/Λλ_i-λ_i/3(2)^1/3Λ-12M/Λη_i).

The final position of the particle is:

    r_f=1/2η_f-1/2ζ_f.

Where:

    λ_f=   {-54+972Λ (M-ω)^2-648Λ (Q_h-q_h)^2+
       √([-54+972Λ (M-ω)^2-648Λ (Q_h-q_h)^2]^2-4[9-36Λ (Q_h-q_h)^2]^3)}^1/3,
    η_f=   √(2/Λ+3(2)^1/3[1-4Λ (Q_h-q_h)^2)]/Λλ_f+λ_f/3(2)^1/3Λ),
    ζ_f=   √(4/Λ-3(2)^1/3[1-4Λ (Q_h-q_h)^2]/Λλ_f-λ_f/3(2)^1/3Λ-12(M-ω)/Λη_f).

After the particle exits, the position of horizons change to:

    r_non^'=   -1/2η^'-1/2ζ^',
    
    r_-^'=   -1/2η^'+1/2ζ^',
    
    r_+^'=   1/2η^'-1/2ζ^',
    
    r_c^'=   1/2η^'+1/2ζ^'.

Where:

    λ^'=   {-54+972Λ (M-ω^')^2-648Λ (Q_h-q_h^')^2+
       √([-54+972Λ (M-ω^')^2-648Λ (Q_h-q_h^')^2]^2-4[9-36Λ (Q_h-q_h^')^2]^3)}^1/3,
    η^'=   √(2/Λ+3(2)^1/3[1-4Λ (Q_h-q_h^')^2]/Λλ^'+λ^'/3(2)^1/3Λ),
    ζ^'=   √(4/Λ-3(2)^1/3[1-4Λ (Q_h-q_h^')^2]/Λλ^'-λ^'/3(2)^1/3Λ-12(M-ω^')/Λη^').


For the process of particle tunneling into the cosmic horizon, the initial position of the particle is:

    r_i^'=1/2η_i^'+1/2ζ_i^'.

Where:

    λ_i^'=   [-54+972Λ M^2-648Λ Q_h^2+√((-54+972Λ M^2-648Λ Q_h^2)^2-4(9-36Λ Q_h^2)^3)]^1/3,
    η_i^'=   √(2/Λ+3(2)^1/3(1-4Λ Q_h^2)/Λλ_i^'+λ_i^'/3(2)^1/3Λ),
    ζ_i^'=   √(4/Λ-3(2)^1/3(1-4Λ Q_h^2)/Λλ_i^'-λ_i^'/3(2)^1/3Λ-12M/Λη_i^').

The final position of the particle is:

    r_f^'=1/2η_f^'+1/2ζ_f^'.

Where:

    λ_f^'=   {-54+972Λ (M+ω)^2-648Λ (Q_h+q_h)^2+
       √([-54+972Λ (M+ω)^2-648Λ (Q_h+q_h)^2]^2-4[9-36Λ (Q_h+q_h)^2]^3)}^1/3,
    η_f^'=   √(2/Λ+3(2)^1/3[1-4Λ (Q_h+q_h)^2)]/Λλ_f^'+λ_f^'/3(2)^1/3Λ),
    ζ_f^'=   √(4/Λ-3(2)^1/3[1-4Λ (Q_h+q_h)^2]/Λλ_f^'-λ_f^'/3(2)^1/3Λ-12(M+ω)/Λη_f^').

After the particle is incident, the position of horizons change to:

    r_non^''=   -1/2η^''-1/2ζ^'',
    
    r_-^''=   -1/2η^''+1/2ζ^'',
    
    r_+^''=   1/2η^''-1/2ζ^'',
    
    r_c^''=   1/2η^''+1/2ζ^''.

Where:

    λ^''=   {-54+972Λ (M+ω^')^2-648Λ (Q_h+q_h^')^2+
       √([-54+972Λ (M+ω^')^2-648Λ (Q_h+q_h^')^2]^2-4[9-36Λ (Q_h+q_h^')^2]^3)}^1/3,
    η^''=   √(2/Λ+3(2)^1/3[1-4Λ (Q_h+q_h^')^2]/Λλ^''+λ^''/3(2)^1/3Λ),
    ζ^''=   √(4/Λ-3(2)^1/3[1-4Λ (Q_h+q_h^')^2]/Λλ^''-λ^''/3(2)^1/3Λ-12(M+ω^')/Λη^'').



 §.§ Painlevé coordinate and time-like geodesic line equation

To describe the tunneling process of particles, Painlevé coordinates without event horizon singularity should be used. In order to obtain the Painlevé-Reissner-Nordström de Sitter coordinate, we performed the following coordinate transformation:

    t_s=t+F(r), dt_s=dt+F^'(r)dr.

Where t_s is the time coordinate before transformation. In order for the constant time slice of Painlevé line elements to be a flat Euclidean spacetime in the radial direction, F(r) must satisfy:

    1/1-2M/r+Q_h^2/r^2-Λ/3r^2-(1-2M/r+Q_h^2/r^2-Λ/3r^2)[F^'(r)]^2=1.

By substituting (<ref>) and (<ref>) into (<ref>), we obtain the Painlevé coordinate:

    ds^2=   -(1-2M/r+Q_h^2/r^2-Λ/3r^2)dt^2± 2√(2M/r-Q_h^2/r^2+Λ/3r^2)dtdr+dr^2+r^2dΩ^2
    
    =   g_00dt^2+2g_01dtdr+dr^2+r^2dΩ^2.

Obviously, in (<ref>), the coordinate singularity at the event horizon is eliminated. The selection of positive and negative signs in (<ref>) will have a clearer physical meaning in the equation of time-like geodesics.

In this article, we discuss particles that carry both electric and magnetic charges. So they are all massive particles, not following the optical geodesic equation. We consider massive particles as de Broglie s-waves, employing an approach analogous to that described in ref. <cit.>. The time-like geodesic equation, derived from the WKB approximation, is presented below:

    ṙ=v_p=1/2v_g=-1/2g_00/g_01.

Where v_p is the phase velocity and v_g is the group velocity, respectively represented by the following equations:

    v_p=dr/dt=ω/k.


    v_g=dr_c/dt=dω/dk.

Where k is de Broglie wave number. By substituting g_00 and g_01 in (<ref>) into (<ref>), we can obtain:

    ṙ=dr/dt=±1/2r√(Λ/3)(r-r_non)(r-r_-)(r-r_+)(r-r_c)/√(r^4+6M/Λr-3/ΛQ_h^2).

It can be clearly seen that when taking a positive sign, (<ref>) is the equation of motion for the outgoing particles near the outer event horizon of the black hole, while when taking a negative sign, (<ref>) is the equation of motion for the incoming particles near the cosmic horizon. Furthermore, if considering the influence of self gravity, M in (<ref>) and (<ref>) should be replaced by M∓ω, and Q_h in (<ref>) and (<ref>) should be replaced by Q_h∓ q_h. ω is the energy of the particle, q_h is the equivalent charge of the particle. The minus sign and plus sign represent the outgoing particles and incoming particles, respectively.


 §.§ Tunneling out of the black hole outer event horizon

Let's first calculate the particle emission rate near the event horizon of a black hole. We are considering the tunneling of particles that are both charged and magnetic, so we must take into account the influence of electromagnetic fields. We provide the Lagrangian density for the coupling of matter field and electromagnetic field:

    ℒ=ℒ_m+ℒ_h=ℒ_m-1/4F̃_μνF̃^μν.

The generalized coordinates corresponding to ℒ_h are Ã_μ=(Ã_0,0,0,0). Obviously, Ã_0 is a cyclic coordinate. To eliminate the degrees of freedom corresponding to Ã_0, the action can be written as:

    S=∫_t_i^t_f(L-P_Ã_0Ã̇_0)dt.

Due to the fact that the constant time slice of the Painlevé line element is a flat Euclidean spacetime in the radial direction, and the Reissner-Nordström black hole is a steady-state black hole, the WKB approximation can be applied to it. Particle emission rate and the imaginary part of action has the following relationship:

    Γ∼ e^-2ImS.

According to (<ref>), the imaginary part of the action can be written as:

    ImS=Im{∫_r_i^r_f[P_r-P_Ã_0Ã̇_0/ṙ]dr}=Im{∫_r_i^r_f[∫_(0,0)^(P_r,P_Ã_0)dP^'_r-Ã̇_0/ṙdP^'_Ã_0]dr}.

Where p_r is the generalized momentum conjugate with r, and P_Ã_0 is the generalized momentum conjugate with Ã_0. r_i is the initial position of the particle radiation process, slightly within the event horizon of the black hole. r_f is the final position of the particle radiation process, slightly outside the event horizon of the black hole.

To proceed with our calculations, we substitute the Hamilton's equation into (<ref>):

    ṙ=dH/dP_r|_(r;Ã_0,P_Ã_0).


    Ã̇_0=dH/dP_Ã_0|_(Ã_0;r,P_r).

By substituting (<ref>) and (<ref>) into (<ref>), we can obtain:

    ImS=Im{∫_r_i^r_f[∫_(M,E_Q_h)^(M-ω,E_Q_h-q_h)1/ṙ(dH)_r;Ã_0,P_Ã_0-1/ṙ(dH)_Ã_0;r,P_r]dr}.

Where E_Q_h is the energy corresponding to the electromagnetic field. M, E_Q_h is a fixed and unchanging quantity. M-ω, E_Q_h-q_h is the residual gravitational and electromagnetic energy of the black hole after radiation. It can be clearly seen that in our formula, energy conservation, charge conservation, and magnetic charge conservation are all reflected. It is worth emphasizing that in order to simplify the equivalent charge of a black hole after radiation to Q_h-q_h, we adopt the following assumption:

    q_e/q_g=Q_e/Q_g.

Where q_e and q_g are the charges and magnetic charges of the emitted particles, respectively. The loss of gravitational energy and electromagnetic energy caused by particle tunneling in a black hole is represented by the following equations:

    (dH)_r;Ã_0,P_Ã_0=d(M-ω^')=-dω^'.


    (dH)_Ã_0;r,P_r=-Q_h-q_h^'/rdq^'_h.

By substituting (<ref>) and (<ref>) into (<ref>), we can obtain:

    ImS=-Im{∫_r_i^r_f[∫_(0,0)^(ω,q_h)1/ṙ(dω^'-Q_h-q_h^'/rdq^'_h)]dr}.

For the process of particles tunneling out of the event horizon of a black hole, ṙ takes a positive sign, and accounting for the self-gravitational force, we make the following substitution: M→ M-ω^', Q_h→ Q_h-q^'_h, r_non→ r_non^', r_-→ r_-^', r_+→ r_+^', r_c→ r_c^', that is:

    ṙ=1/2r√(Λ/3)(r-r_non^')(r-r_-^')(r-r_+^')(r-r_c^')/√(r^4+6(M-ω^')/Λr-3/Λ(Q_h-q_h^')^2).

Substituting (<ref>) into (<ref>), we have:

    ImS=-Im{∫_r_i^r_f[∫_(0,0)^(ω,q_h)2r√(r^4+6(M-ω^')/Λr-3/Λ(Q_h-q_h^')^2)/√(Λ/3)(r-r_non^')(r-r_-^')(r-r_+^')(r-r_c^')(dω^'-Q_h-q_h^'/rdq^'_h)]dr}.

To proceed with the calculation, switch the order of integration and integrate r first. It can be clearly observed that r=r_+^' is a pole. By selecting a new integral path to bypass the poles and applying the residue theorem, we can obtain:

    ImS=-6π/Λ∫_(0,0)^(ω,q_h)r_+^' 2/(r_+^'-r_non^')(r_+^'-r_-^')(r_+^'-r_c^')(dω^'-Q_h-q_h^'/r_+^'dq^'_h).


According to the relationship between r_non^', r_-^', r_+^', r_c^' and ω^', q_h^', we can obtain:

    6/Λ[r_+^'dω^'-(Q_h-q_h^')dq^'_h]=(r_+^'-r_non^')(r_+^'-r_-^')(r_+^'-r_c^')dr_+^'.

Continuing the integration operation, through simple calculations, we can obtain the final result as:

    ImS=-π∫_r_i^r_fr_+^'dr_+^'=-π/2(r_f^2-r_i^2)=-1/2Δ S_BH.

Where Δ S_BH=S_BH(M-ω,Q_e-q_e,Q_g-q_g)-S_BH(M,Q_e,Q_g) is the change in entropy before and after the black hole radiates particles. So the emission rate of particles can be expressed by the following equation:

    Γ∼ e^-2ImS=e^Δ S_BH.

This result conforms to the unitary principle and supports the conservation of information.


 §.§ Tunneling into the cosmic horizon

Next, we will discuss the particle tunneling process near the de Sitter cosmic horizon. Obviously, for the process of particles tunneling from outside the cosmic horizon to inside the cosmic horizon, the final position is smaller than the initial position, r_f^'<r_i^'. The total energy, total charge, and total magnetic charge of space-time will all increase, M→ M+ω^', Q_e→ Q_e+q^'_e, Q_h→ Q_h+q^'_h, Q_h→ Q_h+q^'_h. And the position of the horizon will also change, r_non→ r_non^'', r_-→ r_-^'', r_+→ r_+^'', r_c→ r_c^''. What we are considering now is the tunneling of the cosmic horizon, so ṙ should take the negative sign. Considering the influence of self gravity, the geodesic equation becomes:

    ṙ=-1/2r√(Λ/3)(r-r_non^'')(r-r_-^'')(r-r_+^'')(r-r_c^'')/√(r^4+6(M+ω^')/Λr-3/Λ(Q_h+q_h^')^2).

Similarly, we consider the incident particle as a de Broglie s-wave and subtract the degrees of freedom of the cyclic coordinate Ã_0 to obtain the imaginary part of the action as:

    ImS=Im{∫_r_i^'^r_f^'[P_r-P_Ã_0Ã̇_0/ṙ]dr}=Im{∫_r_i^'^r_f^'[∫_(0,0)^(P_r,P_Ã_0)dP^'_r-Ã̇_0/ṙdP^'_Ã_t]dr}.

r_i^' is the initial position of particle radiation, slightly outside the cosmic horizon. And r_f^' is the final position of particle radiation, slightly within the cosmic horizon.

The Hamilton's equation is:

    ṙ=dH/dP_r|_(r;Ã_0,P_Ã_0)=d(M+ω^')/dP_r=dω^'/dP_r.


    Ã̇_0=dH/dP_Ã_0|_(Ã_0;r,P_r)=Q_h+q_h^'/rdq^'_h/dP_Ã_0.

(<ref>) and (<ref>) respectively represent the changes in gravitational energy and electromagnetic energy after particles enter the cosmic horizon, from which it can be clearly seen that energy conservation, charge conservation, and magnetic charge conservation occur. Substituting (<ref>), (<ref>) and (<ref>) into (<ref>) yields:

    ImS=-Im{∫_r_i^'^r_f^'[∫_(0,0)^(ω,q_h)2r√(r^4+6(M+ω^')/Λr-3/Λ(Q_h+q_h^')^2)/√(Λ/3)(r-r_non^'')(r-r_-^'')(r-r_+^'')(r-r_c^'')(dω^'-Q_h+q_h^'/rdq^'_h)]dr}.

Obviously, there is a pole at the cosmic horizon r=r_+^''. In order to proceed with the calculation, exchange the order of integration, first integrate r and apply the residue theorem, which yields:

    ImS=-6π/Λ∫_(0,0)^(ω,q_h)r_c^'' 2/(r_c^''-r_non^'')(r_c^''-r_-^'')(r_c^''-r_+^'')(dω^'-Q_h+q_h^'/r_c^''dq^'_h).


According to the relationship between r_non^'', r_-^'', r_+^'', r_c^'' and ω^', q_h^', we can obtain:

    6/Λ[r_c^''dω^'-(Q_h+q_h^')dq^'_h]=(r_c^''-r_non^'')(r_c^''-r_-^'')(r_c^''-r_+^'')dr_c^''.

By completing the integration operation through simple calculations, the final result obtained is:

    ImS=-π∫_r_i^'^r_f^'r_c^''dr_c^''=-π/2(r_f^' 2-r_i^' 2)=-1/2Δ S_CH.

Where Δ S_CH=S_CH(M+ω,Q_e+q_e,Q_g+q_g)-S_CH(M,Q_e,Q_g) is the change in entropy of particles before and after passing through the cosmic horizon. The emission rate of particles can be expressed by the following equation:

    Γ∼ e^-2ImS=e^Δ S_CH.

Obviously, such a result follows the unitary principle and supports the conservation of information.


 §.§ Discussion

The above results can be easily obtained from the first law of black hole thermodynamics, and we can obtain the temperature at the surface of the black hole event horizon and the cosmic horizon from it. For the case of magnetic charge, the first law of black hole thermodynamics should be written as:

    dS^'=1/T^'dM^'-V_e^'/T^'dQ_e^'-V_g^'/T^'dQ_g^'=1/T^'(dM^'-V_h^'dQ_h^').

Among them, for the process of particles passing through the event horizon of a black hole, dM^'=-dω^', dQ_h^'=-dq_h^', V_h^'=Q_h-q_h^'/r_+^'. For the process of particles crossing the cosmic horizon, dM^'=dω^', dQ_h^'=dq_h^', V_h^'=Q_h+q_h^'/r_c^''. So:

    dS_BH^'=-1/T^'(dω^'-Q_h-q_h^'/r_+^'dq_h^').


    dS_CH^'=1/T^''(dω^'-Q_h+q_h^'/r_c^''dq_h^').

Comparing (<ref>) and (<ref>) with (<ref>) and (<ref>) respectively, the temperature T^' at the event horizon surface of the black hole and the temperature T^'' at the cosmic horizon surface are obtained:

    β^'=1/T^'=-12π/Λr_+^' 2/(r_+^'-r_non^')(r_+^'-r_-^')(r_+^'-r_c^')=lim_r→ r_+^'4π/|f^'(r)|.


    β^''=1/T^''=12π/Λr_c^'' 2/(r_c^''-r_non^'')(r_c^''-r_-^'')(r_c^''-r_+^'')=lim_r→ r_c^''4π/|f^'(r)|.

Where f(r)=-Λ/3r^2(r-r_non)(r-r_-)(r-r_+)(r-r_c). This is consistent with the conclusion drawn from general black hole thermodynamics.


§ KERR-NEWMAN-KASUYA DE SITTER BLACK HOLE


The line element of Kerr Newman Kasuya de Sitter black hole is:

    ds^2=g_00dt_k^2+g_11dr^2+g_22dθ^2+g_33dϕ^2+2g_03dt_kdϕ.

Where:

    g_00=   -Δ_r-a^2Δ_θsin^2θ/Ξ^2Σ,
    
    g_11=   Σ/Δ_r,
    
    g_22=   Σ/Δ_θ,
    
    g_33=   -h^2Δ_r-ξ^2Δ_θsin^2θ/Ξ^2Σ,
    
    g_03=   hΔ_r-aξΔ_θsin^2θ/Ξ^2Σ.

Where Ξ=1+Λ/3a^2, h=asin^2θ, Σ=r^2+a^2cos^2θ, ξ=a^2+r^2, Δ_θ=1+Λ/3a^2cos^2θ, Δ_r=ξ(1-Λ/3r^2)-2Mr+Q_h^2, Q_h^2=Q_e^2+Q_g^2. M is the mass of the black hole. Q_e and Q_g are the charges and magnetic charges of the black hole, respectively. a=J/M is the angular momentum per unit mass of a black hole.

The position of the horizon are given by equation Δ_r=0, that is:

    Δ_r=-Λ/3r^4+(1-Λ/3a^2)r^2-2Mr+Q_h^2+a^2=0.

Let A=3/Λ(a^2+Q_h^2), B=3/Λ-a^2. The equation (<ref>) has four solutions, r_non, r_-, r_+, r_c. Among them, r_non is a negative value solution with no physical meaning, r_- is the inner event horizon of black hole, r_+ is the outer event horizon of black hole, and r_c is the cosmic horizon. The specific forms of these four solutions are:

    r_non=   -1/2η̃-1/2ζ̃,
    
    r_-=   -1/2η̃+1/2ζ̃,
    
    r_+=   1/2η̃-1/2ζ̃,
    
    r_c=   1/2η̃+1/2ζ̃.

Where:

    λ̃=   [972Λ M^2-72Λ^3AB-2Λ^3B^3+
       √((72Λ^3AB+2Λ^3B^3-972Λ M^2)^2-4(Λ^2B^2-12Λ^2A)^3)]^1/3,
    η̃=   √(2B/3+(2)^1/3(Λ^2B^2-12Λ^2A)/3Λλ̃+λ̃/3(2)^1/3Λ),
    ζ̃=   √(4B/3-(2)^1/3(Λ^2B^2-12Λ^2A)/3Λλ̃-λ̃/3(2)^1/3Λ-12M/Λη̃).

For the process of particles tunneling out of the event horizon of a black hole, the initial position of the particle is:

    r_i=1/2η̃_i-1/2ζ̃_i.

Where:

    λ̃_i=   [972Λ M^2-72Λ^3AB-2Λ^3B^3+
       √((72Λ^3AB+2Λ^3B^3-972Λ M^2)^2-4(Λ^2B^2-12Λ^2A)^3)]^1/3,
    η̃_i=   √(2B/3+(2)^1/3(Λ^2B^2-12Λ^2A)/3Λλ̃_i+λ̃_i/3(2)^1/3Λ),
    ζ̃_i=   √(4B/3-(2)^1/3(Λ^2B^2-12Λ^2A)/3Λλ̃_i-λ̃_i/3(2)^1/3Λ-12M/Λη̃_i).

The final position of the particle is:

    r_f=1/2η̃_f-1/2ζ̃_f.

Where:

    λ̃_f=   {972Λ (M-ω)^2-72Λ^3A_fB_f-2Λ^3B_f^3+
       √([72Λ^3A_fB_f+2Λ^3B_f^3-972Λ (M-ω)^2]^2-4(Λ^2B_f^2-12Λ^2A_f)^3)}^1/3,
    η̃_f=   √(2B_f/3+(2)^1/3(Λ^2B_f^2-12Λ^2A_f)/3Λλ̃_f+λ̃_f/3(2)^1/3Λ),
    ζ̃_f=   √(4B_f/3-(2)^1/3(Λ^2B_f^2-12Λ^2A_f)/3Λλ̃_f-λ̃_f/3(2)^1/3Λ-12(M-ω)/Λη̃_f).

Where A_f=3/Λ[a_f^2+(Q_h-q_h)^2], B_f=3/Λ-a_f^2, a_f=J-j/M-ω. After the particle exits, the position of horizons change to:

    r_non^'=   -1/2η̃^'-1/2ζ̃^',
    
    r_-^'=   -1/2η̃^'+1/2ζ̃^',
    
    r_+^'=   1/2η̃^'-1/2ζ̃^',
    
    r_c^'=   1/2η̃^'+1/2ζ̃^'.

Where:

    λ̃^'=   {972Λ (M-ω^')^2-72Λ^3A^' B^'-2Λ^3B^' 3+
       √([72Λ^3A^' B^'+2Λ^3B^' 3-972Λ (M-ω^')^2]^2-4(Λ^2B^' 2-12Λ^2A^')^3)}^1/3,
    η̃^'=   √(2B^'/3+(2)^1/3(Λ^2B^' 2-12Λ^2A^')/3Λλ̃^'+λ̃^'/3(2)^1/3Λ),
    ζ̃^'=   √(4B^'/3-(2)^1/3(Λ^2B^' 2-12Λ^2A^')/3Λλ̃^'-λ̃^'/3(2)^1/3Λ-12(M-ω^')/Λη̃^').

Where A^'=3/Λ[a^' 2+(Q_h-q_h^')^2], B^'=3/Λ-a^' 2, a^'=J-j^'/M-ω^'.

For the process of particle tunneling into the cosmic horizon, the initial position of the particle is:

    r_i^'=1/2η̃_i^'+1/2ζ̃_i^'.

Where:

    λ̃_i^'=   [972Λ M^2-72Λ^3AB-2Λ^3B^3+
       √((72Λ^3AB+2Λ^3B^3-972Λ M^2)^2-4(Λ^2B^2-12Λ^2A)^3)]^1/3,
    η̃_i^'=   √(2B/3+(2)^1/3(Λ^2B^2-12Λ^2A)/3Λλ̃_i^'+λ̃_i^'/3(2)^1/3Λ),
    ζ̃_i^'=   √(4B/3-(2)^1/3(Λ^2B^2-12Λ^2A)/3Λλ̃_i^'-λ̃_i^'/3(2)^1/3Λ-12M/Λη̃_i^').

The final position of the particle is:

    r_f^'=1/2η̃_f^'+1/2ζ̃_f^'.

Where:

    λ̃_f^'=   {972Λ (M+ω)^2-72Λ^3A_f^' B_f^'-2Λ^3B_f^' 3+
       √([72Λ^3A_f^' B_f^'+2Λ^3B_f^' 3-972Λ (M+ω)^2]^2-4(Λ^2B_f^' 2-12Λ^2A_f^')^3)}^1/3,
    η̃_f^'=   √(2B_f^'/3+(2)^1/3(Λ^2B_f^' 2-12Λ^2A_f^')/3Λλ̃_f^'+λ̃_f^'/3(2)^1/3Λ),
    ζ̃_f^'=   √(4B_f^'/3-(2)^1/3(Λ^2B_f^' 2-12Λ^2A_f^')/3Λλ̃_f^'-λ̃_f^'/3(2)^1/3Λ-12(M+ω)/Λη̃_f^').

Where A_f^'=3/Λ[a_f^' 2+(Q_h+q_h)^2], B_f^'=3/Λ-a_f^' 2, a_f^'=J+j/M+ω. After the particle is incident, the position of horizons change to:

    r_non^''=   -1/2η̃^''-1/2ζ̃^'',
    
    r_-^''=   -1/2η̃^''+1/2ζ̃^'',
    
    r_+^''=   1/2η̃^''-1/2ζ̃^'',
    
    r_c^''=   1/2η̃^''+1/2ζ̃^''.

Where:

    λ̃^''=   {972Λ (M+ω^')^2-72Λ^3A^'' B^''-2Λ^3B^'' 3+
       √([72Λ^3A^'' B^''+2Λ^3B^'' 3-972Λ (M+ω^')^2]^2-4(Λ^2B^'' 2-12Λ^2A^'')^3)}^1/3,
    η̃^''=   √(2B^''/3+(2)^1/3(Λ^2B^'' 2-12Λ^2A^'')/3Λλ̃^''+λ̃^''/3(2)^1/3Λ),
    ζ̃^''=   √(4B^''/3-(2)^1/3(Λ^2B^'' 2-12Λ^2A^'')/3Λλ̃^''-λ̃^''/3(2)^1/3Λ-12(M+ω^')/Λη̃^'').

Where A^''=3/Λ[a^'' 2+(Q_h+q_h^')^2], B^''=3/Λ-a^'' 2, a^''=J+j^'/M+ω^'.


 §.§ Painlevé coordinate and time-like geodesic line equation

To eliminate coordinate singularity at the horizon, we first introduce a dragged coordinate system for the Kerr Newman Kasuya de Sitter spacetime. Let:

    Ω=dϕ/dt_k=-g_03/g_33=hΔ_r-aξΔ_θsin^2θ/h^2Δ_r-ξ^2Δ_θsin^2θ.


    ĝ_00=g_00-g_03^2/g_33=ΣΔ_θsin^2θΔ_r/Ξ^2(h^2Δ_r-ξ^2Δ_θsin^2θ).

The line elements of Kerr Newman Kasuya de Sitter black hole can be rewritten as:

    ds^2=ĝ_00dt_k^2+g_11dr^2+g_22dθ^2=ΣΔ_θsin^2θΔ_r/Ξ^2(h^2Δ_r-ξ^2Δ_θsin^2θ)dt_k^2+Σ/Δ_rdr^2+Σ/Δ_θdθ^2.

Due to the fact that the constant time slice of the dragging coordinate system is not a flat Euclidean space in the radial direction, it is not the coordinate system we want. We adopt a method similar to ref. <cit.> to perform coordinate transformation again:

    dt_k=dt+F(r,θ)dr+G(r,θ)dθ.


    t=t_k-∫[F(r,θ)dr+G(r,θ)dθ].

F(r,θ) and G(r,θ) satisfy the following relationship:

    ∂ F(r,θ)/∂θ=∂ G(r,θ)/∂ r.

In order to make the constant time slice of the new coordinates a flat Euclidean space in the radial direction, we require F(r,θ) to satisfy the following relationship:

    g_11+ĝ_00F^2(r,θ)=1.

So we obtained the coordinates of Painlevé-Kerr-Newman-Kasuya de Sitter:

    ds^2=   ĝ_00dt^2± 2√(ĝ_00(1-g_11))dtdr+dr^2+[ĝ_00G^2(r,θ)+g_22]dθ^2
       +2ĝ_00G(r,θ)dtdθ+2√(ĝ_00(1-g_11))G(r,θ)drdθ.


Similar to Section <ref>, we treat particles as de Broglie s-waves and derive the time-like geodesic equation for particle motion using the WKB approximate:

    ṙ=-1/2ĝ_00/ĝ_01=±ΣΔ_r/2Ξ√(Δ_θsin^2θ/(Δ_r-Σ)Σ(h^2Δ_r-ξ^2Δ_θsin^2θ)).

Where ĝ_01=√(ĝ_00(1-g_11)). When a particle exits the event horizon of a black hole, (<ref>) takes a positive sign. When a particle is incident from the cosmic horizon, (<ref>) takes a negative sign. When considering self gravity, replace (<ref>) with the following, M→ M∓ω, Q_h→ Q_h-q_h, a=J/M→J∓ j/M∓ω=ã. The minus sign and plus sign represent the outgoing particles and incoming particles, respectively.


 §.§ Tunneling out of the black hole outer event horizon

Similar to Section <ref>, it can be inferred from the Lagrangian density of matter-electromagnetic field that:

    ℒ=ℒ_m+ℒ_h=ℒ_m-1/4F̃_μνF̃^μν.

Ã_0 is the cyclic coordinate. Under the condition of rotating with electricity, Ã_0=-Q_hr/Σ(1-aΩsin^2θ). The electromagnetic potential at the event horizon of a black hole is Ã_0|_r_+=-Q_hr_+/r_+^2+a^2 and it at the cosmic horizon is Ã_0|_r_c=-Q_hr_c/r_c^2+a^2. When the line element of a rotating black hole is written in a dragged coordinate system, the line element does not contain ϕ, so ϕ is also a cyclic coordinate. To eliminate the degrees of freedom corresponding to ϕ and Ã_0, the action can be written as:

    S=∫^t_f_t_i(L-p_Ã_0Ã̇_0-p_ϕϕ̇)dt.

For the same reason as in Section <ref>, the WKB approximation can be applied. The emission rate of particles has the same relationship as (<ref>):

    Γ∼ e^-2ImS.

We obtain the imaginary part of the action as:

    ImS=   Im{∫_r_i^r_f[P_r-P_Ã_0Ã̇_0/ṙ-p_ϕϕ̇/ṙ]dr}
    
    =   Im{∫_r_i^r_f[∫_(0,0)^(P_r,P_Ã_0)dP^'_r-Ã̇_0/ṙdP^'_Ã_0-ϕ̇/ṙdP^'_ϕ]dr}.


To proceed with the calculation, we write down the Hamilton's equation:

    ṙ=dH/dP_r|_(r;Ã_0,P_Ã_0;ϕ,P_ϕ)=d(M-ω^')/dP_r=-dω^'/dP_r.


    Ã̇_0=dH/dP_Ã_0|_(Ã_0;r,P_r;ϕ,P_ϕ)=-(Q_h-q_h^')r/r^2+a^' 2dq_h^'/dP_Ã_0.

Additionally, we have the following relationship:

    ϕ̇=dϕ/dt=Ω^'.


    p_ϕ^'=J^'=J-j^'=(M-ω^')a^'.

In (<ref>), (<ref>) and (<ref>), conservation of energy, conservation of angular momentum, conservation of electric charge and conservation of magnetic charge are manifested. Substitute (<ref>), (<ref>), (<ref>), (<ref>) into (<ref>), we can obtain:

    ImS=-Im{∫_r_i^r_f[∫_(0,0,0)^(ω,q_h,j)1/ṙ(dω^'-(Q_h-q_h^')r/r^2+a^' 2dq^'_h-Ω^'dj^')]dr}.

When particles tunnel out of the event horizon of a black hole, ṙ takes a positive sign. And considering the self gravity effect, make the following substitution, M→ M-ω^', Q_h=Q_h-q_h^', a=J/M→J-j^'/M-ω^'=a^', Ξ=1+Λ/3a^2→ 1+Λ/3a^' 2=Ξ^', h=asin^2θ→ a^'sin^2θ=h^', Σ=r^2+a^2cos^2θ→ r^2+a^' 2cos^2θ=Σ^', ξ=a^2+r^2→ a^' 2+r^2=ξ^', Δ_θ=1+Λ/3a^2cos^2θ→ 1+Λ/3a^' 2cos^2θ=Δ_θ^', Δ_r=ξ(1-Λ/3r^2)-2Mr+Q_h^2→ξ^'(1-Λ/3r^2)-2(M-ω^')r+(Q_h-q_h^')^2=Δ_r^', r_non→ r_non^', r_-→ r_-^', r_+→ r_+^', r_c→ r_c^', that is:

    ṙ=Σ^'Δ_r^'/2Ξ^'√(Δ_θ^'sin^2θ/(Δ_r^'-Σ^')Σ^'(h^' 2Δ_r^'-ξ^' 2Δ_θ^'sin^2θ)).

Where Δ_r=-Λ/3(r-r_non)(r-r_-)(r-r_+)(r-r_c). Substitute (<ref>) into (<ref>), we can obtain:

    ImS=   -Im{∫_r_i^r_f[∫_(0,0,0)^(ω,q_h,j)6Ξ^'/ΛΣ^'1/(r-r_non^')(r-r_-^')(r-r_+^')(r-r_c^')×
       √((Δ_r^'-Σ^')Σ^'(h^' 2Δ_r^'-ξ^' 2Δ_θ^'sin^2θ)/Δ_θ^'sin^2θ)(dω^'-(Q_h-q_h^')r/r^2+a^' 2dq^'_h-Ω^'dj^')]dr}.

Switch the order of integration, integrating r first. It is obvious that r=r_+^' is a pole. It can be obtained by re-selecting the integral circumference and applying the residue theorem:

    ImS=-2π∫_(0,0,0)^(ω,q_h,j)(Λ/3+a^' 2)(a^' 2+r_+^' 2)/(r_+^'-r_non^')(r_+^'-r_-^')(r_+^'-r_c^')(dω^'-(Q_h-q_h^')r_+^'/r_+^' 2+a^' 2dq^'_h-Ω^'dj^').

Where Ω^'|_r_+^'=a^'/r_+^' 2+a^' 2. Based on the relationship between r_non^', r_-^', r_+^', r_c^' and ω^', q_h^', j^', we can get:

    (6/Λ+2a^' 2)[(r_+^' 2+a^' 2)dω^'-(Q_h-q_h^')r_+^'dq^'_h-a^'dj^']=(r_+^'-r_non^')(r_+^'-r_-^')(r_+^'-r_c^')dr_+^'.

By substituting (<ref>) into (<ref>), we get our final result:

    ImS=-π∫_r_i^r_fr_+^'dr_+^'=-π/2(r_f^2-r_i^2)=-1/2Δ S_BH.

Where Δ S_BH=S_BH(M-ω,Q_e-q_e,Q_g-q_g,J-j)-S_BH(M,Q_e,Q_g,J) is the change in entropy before and after the black hole radiates particles. So the emission rate of particles can be expressed by the following equation:

    Γ∼ e^-2ImS=e^Δ S_BH.

(<ref>) indicates that our results conform to the unitary principle and support the conservation of information.


 §.§ Tunneling into the cosmic horizon

The process of Kerr-Newman-Kasuya de Sitter space-time particles tunneling into the cosmic horizon increases the total energy, total charge, total magnetic charge, and total angular momentum of space-time, M→ M+ω^', Q_h→ Q_h+q_h^', a=J/M→J+j^'/M+ω^'=a^''. Other quantities that depend on M, Q_h, J also change, Ξ=1+Λ/3a^2→ 1+Λ/3a^'' 2=Ξ^'', h=asin^2θ→ a^''sin^2θ=h^'', Σ=r^2+a^2cos^2θ→ r^2+a^' 2cos^2θ=Σ^'', ξ=a^2+r^2→ a^'' 2+r^2=ξ^'', Δ_θ=1+Λ/3a^2cos^2θ→ 1+Λ/3a^'' 2cos^2θ=Δ_θ^'', Δ_r=ξ(1-Λ/3r^2)-2Mr+Q_h^2→ξ^''(1-Λ/3r^2)-2(M+ω^')r+(Q_h+q_h^')^2=Δ_r^'', r_non→ r_non^'', r_-→ r_-^'', r_+→ r_+^'', r_c→ r_c^''. For particles tunneling into the cosmic horizon, ṙ is a negative sign. Taking into account the effect of self-gravitation, the geodesic equation becomes:

    ṙ=-Σ^''Δ_r^''/2Ξ^''√(Δ_θ^''sin^2θ/(Δ_r^''-Σ^'')Σ^''(h^'' 2Δ_r^''-ξ^'' 2Δ_θ^''sin^2θ)).

We treat the incident particle as a de Broglie s-wave and subtract the degrees of freedom of the cyclic coordinates Ã_0 and ϕ to get the imaginary part of the action:

    ImS=   Im{∫_r_i^'^r_f^'[P_r-P_Ã_0Ã̇_0/ṙ-p_ϕϕ̇/ṙ]dr}
    
    =   Im{∫_r_i^'^r_f^'[∫_(0,0)^(P_r,P_Ã_0)dP^'_r-Ã̇_0/ṙdP^'_Ã_0-ϕ̇/ṙdP^'_ϕ]dr}.

The Hamilton's equation is:

    ṙ=dH/dP_r|_(r;Ã_0,P_Ã_0;ϕ,P_ϕ)=d(M+ω^')/dP_r=dω^'/dP_r.


    Ã̇_0=dH/dP_Ã_0|_(Ã_0;r,P_r;ϕ,P_ϕ)=(Q_h+q_h^')r/r^2+a^'' 2dq_h^'/dP_Ã_0.

In addition, we have the following relationship:

    ϕ̇=dϕ/dt=Ω^'.


    p_ϕ^'=J^'=J+j^'=(M+ω^')a^''.

(<ref>), (<ref>) and (<ref>) clearly represent conservation of energy, conservation of angular momentum, conservation of charge and conservation of magnetic charge. Substitute (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) into (<ref>), we can obtain:

    ImS=   -Im{∫_r_i^'^r_f^'[∫_(0,0,0)^(ω,q_h,j)6Ξ^''/ΛΣ^''1/(r-r_non^'')(r-r_-^'')(r-r_+^'')(r-r_c^'')×
       √((Δ_r^''-Σ^'')Σ^''(h^'' 2Δ_r^''-ξ^'' 2Δ_θ^''sin^2θ)/Δ_θ^''sin^2θ)(dω^'-(Q_h+q_h^')r/r^2+a^'' 2dq^'_h-Ω^'dj^')]dr}.

Switch the order of integration, integrating r first. Clearly, r=r_c^'' is a pole. It can be obtained by re-selecting the integral circumference and applying the residue theorem:

    ImS=-2π∫_(0,0,0)^(ω,q_h,j)(Λ/3+a^'' 2)(a^'' 2+r_c^'' 2)/(r_c^''-r_non^'')(r_c^''-r_-^'')(r_c^''-r_+^'')(dω^'-(Q_h+q_h^')r_c^''/r_c^'' 2+a^'' 2dq^'_h-Ω^'dj^').

Where Ω^'|_r_c^''=a^''/r_c^'' 2+a^'' 2. Based on the relationship between r_non^'', r_-^'', r_+^'', r_c^'' and ω^', q_h^', j^', we can get:

    (6/Λ+2a^'' 2)[(r_c^'' 2+a^'' 2)dω^'-(Q_h+q_h^')r_c^''dq^'_h-a^''dj^']=(r_c^''-r_non^'')(r_c^''-r_-^'')(r_c^''-r_+^'')dr_c^''.

By substituting (<ref>) into (<ref>), we get our final result:

    ImS=-π∫_r_i^'^r_f^'r_c^''dr_c^''=-π/2(r_f^' 2-r_i^' 2)=-1/2Δ S_CH.

Where Δ S_CH=S_CH(M+ω,Q_e+q_e,Q_g+q_g,J+j)-S_CH(M,Q_e,Q_g,J) is the change in entropy before and after particle enters the cosmic horizon. So the emission rate of particles can be expressed by the following equation:

    Γ∼ e^-2ImS=e^Δ S_CH.

Clearly, (<ref>) indicates that our results conform to the unitary principle and support the conservation of information.


 §.§ Discussion

Similarly, the results in this section can be easily obtained from the first law of black hole thermodynamics, and the temperature at the surface of black hole event horizon and cosmic horizon can be read from it. For the most general case of a rotating black hole that is both charged and magnetic, the first law of black hole thermodynamics can be written:

    dS^'=1/T^'dM^'-V_e^'/T^'dQ_e^'-V_g^'/T^'dQ_g^'-Ω^'/T^'dJ^'=1/T^'(dM^'-V_h^'dQ_h^'-Ω^'dJ^').

For the process by which particles tunnel out of a black hole's event horizon, dM^'=-dω^', dQ_h^'=-dq_h^', V_h^'=(Q_h-q_h^')r_+^'/r_+^' 2+a^' 2, dJ^'=-dj^', Ω^'=a^'/r_+^' 2+a^' 2. For the process by which particles tunnel into the cosmic horizon, dM^'=dω^', dQ_h^'=dq_h^', V_h^'=(Q_h+q_h^')r_+^'/r_c^'' 2+a^'' 2, dJ^'=dj^', Ω^'=a^''/r_+^'' 2+a^'' 2. So:

    dS_BH^'=-1/T^'(dω^'-(Q_h-q_h^')r_+^'/r_+^' 2+a^' 2dq_h^'-a^'/r_+^' 2+a^' 2dj^').


    dS_CH^'=1/T^''(dω^'-(Q_h+q_h^')r_+^'/r_c^'' 2+a^'' 2dq_h^'-a^''/r_+^'' 2+a^'' 2dj^').

Comparing (<ref>) and (<ref>) with (<ref>) and (<ref>) respectively, the temperature T^' at the event horizon surface of the black hole and the temperature T^'' at the cosmic horizon surface are obtained:

    β^'=1/T^'=-4π(Λ/3+a^' 2)(a^' 2+r_+^' 2)/(r_+^'-r_non^')(r_+^'-r_-^')(r_+^'-r_c^')=lim_r→ r_+^'4π√(-ĝ_00g_11)/|ĝ_00,1|.


    β^''=1/T^''=4π(Λ/3+a^'' 2)(a^'' 2+r_c^'' 2)/(r_c^''-r_non^'')(r_c^''-r_-^'')(r_c^''-r_+^'')=lim_r→ r_c^''4π√(-ĝ_00g_11)/|ĝ_00,1|.

The Hawking temperature obtained by this method is consistent with the results obtained by traditional black hole thermodynamics.


§ BARDEEN DE SITTER BLACK HOLE


The line element of Bardeen de Sitter black hole is:

    ds^2=-f(r)dt_s^2+1/f(r)dr^2+r^2dΩ^2.

Where

    f(r)=1-2Mr^2/(r^2+Q_h^2)^3/2-Λ/3r^2.

It is evident from (<ref>) that the electromagnetic field in Bardeen de Sitter spacetime is nonlinear. Therefore, it is impossible to prove the conservation of information by the concrete form of the horizon r_h(M,Q_h). In this section, we use the first law of thermodynamics of black holes to prove the conservation of information. According to the basic knowledge of black hole thermodynamics, the Hawking temperature for spherically symmetric black holes has the following expression:

    T=lim_r→ r_hκ/2π=lim_r→ r_h|f^'(r)|/4π.

Where r=r_h is the position of the horizon.


 §.§ Painlevé coordinate and time-like geodesic line equation

Similar to Section <ref>, to eliminate coordinate singularity at the event horizon, we use Painlevé coordinates. To get the Painlevé-Bardeen de Sitter coordinates, we do the following coordinate transformation:

    t_s=t+F(r), dt=dt_s-F^'(r)dr.

F(r) must satisfy:

    1/1-2Mr^2/(r^2+Q_h^2)^3/2-Λ/3r^2-[1-2Mr^2/(r^2+Q_h^2)^3/2-Λ/3r^2][F^'(r)]^2=1.

So we have Painlevé coordinates:

    ds^2=   -[1-2Mr^2/(r^2+Q_h^2)^3/2-Λ/3r^2]dt^2± 2√(2Mr^2/(r^2+Q_h^2)^3/2+Λ/3r^2)dtdr+dr^2+r^2dΩ^2
    
    =   g_00dt^2+2g_01dtdr+dr^2+r^2dΩ^2.

Similar to Section <ref> and Section <ref>, we consider the particle as a de Broglie s-wave and get the subsequent time-like geodesic equation using the WKB approximation:

    ṙ=-1/2g_00/g_01=±√(Λ/3)/2r3/Λ(r^2+Q_h^2)^3/2-6M/Λr^2-r^2(r^2+Q_h^2)^3/2/√(6M/Λ(r^2+Q_h^2)^3/2+(r^2+Q_h^2)^3)=±f(r)/2r1/√(2M/(r^2+Q_h^2)^3/2+Λ/3).

The positive sign represents particles exiting from the black hole event horizon, and the negative sign represents particles entering from the cosmic horizon. Considering the effect of self-gravitation, (<ref>) and (<ref>) should be replaced as follows, M→ M∓ω, Q_h→ Q_h∓ q_h. The minus sign and plus sign represent outgoing particles and incident particles, respectively.


 §.§ Tunneling out of the black hole outer event horizon

Similar to Section <ref> and Section <ref>, we give the Lagrangian density of the matter-electromagnetic field:

    ℒ=ℒ_m+ℒ_h=ℒ_m-1/4F̃_μνF̃^μν.

Obviously, Ã_0 is the circular coordinate. To eliminate the degree of freedom of Ã_0, the action can be written as:

    S=∫_t_i^t_f(L-P_Ã_0Ã̇_0)dt.

For Painlevé line elements, the WKB approximation is valid. The relationship between emission rate of a particle and imaginary part of action is as follows:

    Γ∼ e^-2ImS.

The imaginary part of action can be written as:

    ImS=Im{∫_r_i^r_f[P_r-P_Ã_0Ã̇_0/ṙ]dr}=Im{∫_r_i^r_f[∫_(0,0)^(P_r,P_Ã_0)dP^'_r-Ã̇_0/ṙdP^'_Ã_0]dr}.

The Hamilton's equation is:

    ṙ=dH/dP_r|_(r;Ã_0,P_Ã_0)=d(M-ω^')/dP_r=-dω^'/dP_r.


    Ã̇_0=dH/dP_Ã_0|_(Ã_0;r,P_r)=-Q_h-q_h^'/rdq_h^'/dP_Ã_0.

In (<ref>) and (<ref>), energy conservation, charge conservation, and magnetic charge conservation are all reflected. By substituting (<ref>) and (<ref>) into (<ref>), we can obtain:

    ImS=-Im{∫_r_i^r_f[∫_(0,0)^(ω,q_h)1/ṙ(dω^'-Q_h-q_h^'/rdq_h^')]}.

For the process of particles tunneling out of the event horizon of a black hole, ṙ takes a positive sign, that is:

    ṙ=f(r)/2r1/√(2(M-ω^')/[r^2+(Q_h-q_h^')^2]^3/2+Λ/3).

Substituting (<ref>) into (<ref>) and considering the influence of self gravity, we can make the following substitution, M→ M-ω, Q_h→ q_h^'. We can obtain:

    ImS=-Im{∫_r_i^r_f[∫_(0,0)^(ω,q_h)2r/f(r)√(2(M-ω^')/[r^2+(Q_h-q_h^')^2]^3/2+Λ/3)(dω^'-Q_h-q_h^'/rdq_h^')]}.


Exchange the order of points, first integrate r. Obviously, r=r_+^' is a pole. Re-select the integral contour and apply the residue theorem to obtain:

    ImS=   -∫_(0,0)^(ω,q_h)2π/f^'(r_+^')r_+^'√(1-f^(r_+^')/r_+^' 2)(dω^'-Q_h-q_h^'/r_+^'dq^'_h)
    
    =   -1/2∫_(M,E_Q_h)^(M-ω,E_Q_h-q_h)1/T^'(dM^'-Q_h^'/r_+^'dQ^'_h)=-1/2∫_r_i^r_fdS_BH^'=-1/2Δ S_BH.

Where Δ S_BH=S_BH(M-ω,Q_e-q_e,Q_g-q_g)-S_BH(M,Q_e,Q_g) is the change in entropy before and after the black hole radiates particles. So the emission rate of particles can be expressed by the following equation:

    Γ∼ e^-2ImS=e^Δ S_BH.

Clearly, our results conform to the unitary principle and support the conservation of information.


 §.§ Tunneling into the cosmic horizon

During the process of particle tunneling into the cosmic horizon, the total energy, total charge, and total magnetic charge of spacetime all increase, M→ M+ω^', Q_h→ Q_h+q_h^'. And the symbol of ṙ should be negative:

    ṙ=-f(r)/2r1/√(2(M+ω^')/[r^2+(Q_h+q_h^')^2]^3/2+Λ/3).

We consider the incident particle as a de Broglie s-wave and subtract the degrees of freedom of the cyclic coordinate Ã_0 to obtain the imaginary part of action:

    ImS=Im{∫_r_i^'^r_f^'[P_r-P_Ã_0Ã̇_0/ṙ]dr}=Im{∫_r_i^'^r_f^'[∫_(0,0)^(P_r,P_Ã_0)dP^'_r-Ã̇_0/ṙdP^'_Ã_0]dr}.

The Hamilton's equation is:

    ṙ=dH/dP_r|_(r;Ã_0,P_Ã_0)=d(M+ω^')/dP_r=dω^'/dP_r.


    Ã̇_0=dH/dP_Ã_0|_(Ã_0;r,P_r)=Q_h+q_h^'/rdq_h^'/dP_Ã_0.

In (<ref>) and (<ref>), energy conservation, charge conservation, and magnetic charge conservation are all reflected. By substituting (<ref>), (<ref>) and (<ref>) into (<ref>), we can obtain:

    ImS=-Im{∫_r_i^r_f[∫_(0,0)^(ω,q_h)2r/f(r)√(2(M+ω^')/[r^2+(Q_h+q_h^')^2]^3/2+Λ/3)(dω^'-Q_h+q_h^'/rdq_h^')]}.

Switch the order of integration, integrating r first. Clearly, r=r_c^'' is a pole. It can be obtained by re-selecting the integral circumference and applying the residue theorem:

    ImS=   -∫_(0,0)^(ω,q_h)2π/f^'(r_c^'')(dω^'-Q_h+q_h^'/r_c^''dq_h^')
    
    =   -1/2∫_(M,E_Q_h)^(M+ω,E_Q_h+q_h)1/T^''(dM^'-Q_h^'/r_c^''dQ_h^')=-1/2∫_r_i^'^r_f^'dS_CH^'=-1/2Δ S_CH.

Where Δ S_CH=S_CH(M+ω,Q_e+q_e,Q_g+q_g)-S_CH(M,Q_e,Q_g) is the change in entropy before and after particle enters the cosmic horizon. So the emission rate of particles can be expressed by the following equation:

    Γ∼ e^-2ImS=e^Δ S_CH.

(<ref>) indicates that our results conform to the unitary principle and support the conservation of information.


§ CONCLUSION AND DISCUSSION


In the above calculation, we discussed the three most general cases of asymptotic de Sitter spacetime. The charged and magnetic particles tunnel into the magnetically charged Reissner-Nordström de Sitter black hole (the most general case of a static black hole), the Kerr-Newman-Kasuya de Sitter black hole (the most general case of a rotating black hole), and Bardeen de Sitter black hole (black hole without singularities). We calculated the radiation spectra of particles exiting event horizon of a black hole and entering cosmic horizon using the Parikh-Wilczek method. Our results indicate that for these three most general cases, the particle radiation spectrum deviates from the pure thermal spectrum, in accordance with the unitary principle and supporting information conservation. That is, information conservation is generally valid for general asymptotic de Sitter space-times. The fundamental reason for information conservation in the Parikh-Wilczek framework is the requirement that radiation process of particles is a reversible quasi-static process. For reversible processes, entropy is conserved, and information is naturally conserved as well.
The authors would like to thank Prof. Jian Jing and his student LiuBiao Ma and Zheng Wang from the Department of Physics , Beijing University of Chemical Technology for their valuable comments and suggestions during the completion of this manuscript.








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