International Journal of Theoretical Physics

, Volume 51, Issue 8, pp 2392–2397

Hawking Radiation of the Charged Particles via Tunneling from the (n+2)-Dimensional Topological Reissner-Nordström-de Sitter Black Hole

Authors

    • College of Mathematic and InformationChina West Normal University
Article

DOI: 10.1007/s10773-012-1118-6

Cite this article as:
Yan, H. Int J Theor Phys (2012) 51: 2392. doi:10.1007/s10773-012-1118-6
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Abstract

Extending Parikh-Wilczek’s semi-classical tunneling method, we discuss the Hawking radiation of the charged massive particles via tunneling from the cosmological horizon of (n+2)-dimensional Topological Reissner-Nordström-de Sitter black hole.The result shows that, when energy conservation and electric charge conservation are taken into account, the derived spectrum deviates from the pure thermal one, but satisfies the unitary theory, which provides a probability for the solution of the information loss paradox.

Keywords

Quantum tunnelingEnergy conservationElectric charge conservationHawking radiation

1 Introduction

In 1974, Stephen Hawking discovered the black hole radiate thermally [1, 2]. Hawking predicted that with the process of the thermal radiation, the black hole could lose energy, shrink, and finally evaporate completely, and owing to the pure thermality of the radiation, the information carried by the matter that initially formed the black hole would be lost after the black hole had disappeared. This implied, however, a pure quantum state would evolve into a mixed one, and violate the unitarity of evolution in Quantum Mechanics. Thereby, the famous paradox of information loss was produced. A great deal of work has been done in an attempt to overcome this paradox, but without any way to solve it. Also, it was claimed that this problem could be settled in favor of conservation of information by ADS/CFT which is a conjectured duality between string theory in anti-de Sitter space and a conformal field theory on the boundary of anti-de Sitter space at infinity [3]. But it still was not clear how information could get out of a black hole.

Hawking radiation can be attributed to the spontaneous creation of a pair of virtual particles at a point just inside of the black hole horizon. One of the particles with positive energy tunnels out to the opposite site of the horizon and materializes a real particle, while another one with negative energy remains behind the horizon and is absorbed by the black hole. Basing on the above tunneling picture, a new method utilized dynamic geometry to depict particle tunneling radiation was developed by Kraus and Wilczek and elaborated upon by Parikh and Wilczek [47]. In their methodology, two key points are necessary. First, energy conservation, which was often neglected in other former treatments of Hawking radiation, was taken into account. When this effect is included, Parikh regarded the potential barrier is created by the effect of the outgoing particle itself, and thereby the problem of no pre-existing barrier was overcome. Second, to remove the coordinate singularity at the black hole horizon, they introduced the Painlevé coordinate transformation. Finally, it is found that the derived spectrum deviates from the pure thermal one, but satisfies the unitary theory, which provides a probability for the solution of the information loss paradox. Later on, some attempts towards this direction have been broadly discussed in [827].

In this letter, we extend Parikh-Wilczek’s work to discuss the Hawking radiation of the charged and massive particle via tunneling from (n+2)-dimensional Topological Reissner- Nordström-de Sitter (TRNdS) black hole. Our result shows the derived spectrum deviates from the pure thermal one when the conservation of energy and charge are taken into account, but satisfies the unitary theory.

In the next section, we introduce the Painlevé coordinate to eliminate the coordinate singularity. Also, the geodesics of the charged massive particles is obtained. In Sect. 3, the Hawking radiation of the charged massive particles via tunneling from the cosmological horizon is discussed. In Sect. 4, some discussion and conclusion are included.

2 The Painlevé Coordinate Transformation and the Geodesics of the Charged Particles

The line element of the (n+2)-dimensional TRNdS black hole can be written as follows
$$ds^{2} = - f(r)dt_{\mathit{TRN}}^{2} + f^{ - 1}(r)dr^{2}+ r^{2}\gamma_{ij}dx^{i}dx^{j}, $$
(1)
where
$$f( r ) = k - \frac{\omega_{n}M}{r^{n - 1}} + \frac{n\omega_{n}Q^{2}}{8( n - 1 )r^{2n - 2}} - \frac{r^{2}}{l^{2}},\quad \omega_{n} = \frac{16\pi G_{n + 2}}{n \operatorname{Vol}( \sum )},$$
γij denotes the line element of an n-dimensional hypersurface ∑ with constant curvature n(n−1)k and volume \(\operatorname{Vol}( \sum), G_{n + 2}\) is the (n+2)-dimensional Newtonian gravity constant. When k=1, the metric (1) is just the Reissner-Nordström-de Sitter solution. For general M and Q, the equation f(r)=0 may have four real root. Three of them are real, the largest one is the cosmological horizon (CH) rc, the smallest is the inner horizon (IH) r, the one in between is the event horizon (EH) r+ of the black hole, and the fourth is negative and has no physical meaning. When k=0 or k<0, there is only one positive real root of the equation f(r)=0, which locates the position of the cosmological horizon rc.
In this paper, we mainly discuss the Hawking radiation of the charged massive particles via tunneling from the cosmological horizon. The electromagnetic potential of the TRNdS black hole at the cosmological horizon is Aμ=(At,0,0,0), where \(A_{t} = - \frac{n\omega_{n}Q}{4(n - 1)r^{n - 1}}\), and the corresponding entropy is given by
$$S_{c} = \frac{r_{c}^{n}\operatorname{Vol}(\sum )}{4G_{n + 2}} = \frac{4\pi r_{c}^{n}}{n\omega_{n}}. $$
(2)
In the tunneling picture, it is more convenient to adopt the stationary coordinates that are manifestly asymmetric under time reversal. In addition, there exists coordinate singularity in metric (1). Now, introducing the following Painlevé coordinate transformation [28], dtTRN=dt+Δ(r)dr, the metric (1) can be rewritten as
$$ds^{2} = - f(r)dt^{2} \pm 2\sqrt{1 - f(r)} dtdr +dr^{2} + r^{2}\gamma_{ij}dx^{i}dx^{j}, $$
(3)
where +(−) sign denotes the line element of the outgoing (ingoing) particle from the event (cosmological) horizon of the black hole, respectively. In (3), the Painlevé-like coordinate system has many attractive features. First, the metric is well behaved at the EH and CH; Secondly, it satisfies Landau’s condition of the coordinate clock synchronization; Thirdly, the new form of the line element is stationary, but not static. These characters are useful to investigate the tunneling radiation of the charged massive particles across the horizons.
It is well-known that the TRNdS black hole is charged, so the particle emitted from it should be charged or massive. According to Ref. [29], the geodesics of the charged particle is
$$\dot{r} = v_{p} = - \frac{1}{2}\frac{g_{00}}{g_{01}} = \pm \frac{1}{2}\frac{f( r )}{\sqrt{1 - f( r )}} , $$
(4)
where the +(−) sign identifies with the outgoing (ingoing) radial motion.

3 The Hawking Radiation of the Charged Particle via Tunneling from the Cosmological Horizon

When a particle with energy ω and charge q tunnels into the cosmological horizon of the black hole, the mass M and electric charge Q of the black hole will be changed into M+ω and Q+q. Now, the ingoing radial geodesics and non-null electromagnetic potential are, respectively, given by.
https://static-content.springer.com/image/art%3A10.1007%2Fs10773-012-1118-6/MediaObjects/10773_2012_1118_Equ5_HTML.gif
(5)
https://static-content.springer.com/image/art%3A10.1007%2Fs10773-012-1118-6/MediaObjects/10773_2012_1118_Equ6_HTML.gif
(6)
where
$$f'( r ) = k - \frac{\omega_{n}( M + \omega )}{r^{n - 1}} + \frac{n\omega_{n}( Q + q )^{2}}{8( n - 1 )r^{2n - 2}} -\frac{r^{2}}{l^{2}}.$$
Applying WKB approximation, the tunneling probability of the particle is expressed by
$$\Gamma\propto\exp( - 2{\mathop{\mathrm{Im}}\nolimits} S), $$
(7)
where S is the action of the ingoing particle, given by
$$S = \int_{t_{i}}^{t_{f}} L( r,\dot{r};A_{t},\dot{A}_{t} ) dt. $$
(8)
When the charged particle tunnels across the cosmological horizon, the effect of the electromagnetic field should be taken into account and the matter-gravity system must consist of the black hole and the electromagnetic field outside the black hole. As −1/4FμνFμν is the Lagrangian function of the electromagnetic field corresponding to the generalized coordinateAt, we find that At is an ignorable coordinate. For eliminating the freedom, the imaginary part of the particle action should be written as
$$\operatorname{Im}S = \operatorname{Im}\int_{t_{i}}^{t_{f}} ( L - P_{A}\dot{A}_{t} )dt = \operatorname{Im}\biggl[ \int_{r_{i}}^{r_{f}}\int_{0}^{P_{r}} dP'_{r}dr -\int_{A_{ti}}^{A_{tf}} \int_{0}^{P_{A}}dP'_{A}dA_{t} \biggr], $$
(9)
where ri and rf denote the location of the cosmological horizon before and after the charged particle emitted respectively, and the distance between them depends on the energy and charge of the emitted particles. Also, (Pr,PA) are the canonical momentum conjugate to(r,At). To proceed with explicit calculation, it is convenient to utilize Hamilton’s canonical equation
https://static-content.springer.com/image/art%3A10.1007%2Fs10773-012-1118-6/MediaObjects/10773_2012_1118_Equ10_HTML.gif
(10)
Substituting (7), (8) and (12) into (11) and switching the order of integration yields
https://static-content.springer.com/image/art%3A10.1007%2Fs10773-012-1118-6/MediaObjects/10773_2012_1118_Equ11_HTML.gif
(11)
On the other hand, the Bekenstein-Hawking entropy of the black hole before and after the charged particle emission can be written as
$$S_{\mathit{CH}}( M,Q ) = \frac{4\pi r_{i}^{n}}{n\omega_{n}},\quad S_{\mathit{CH}}( M +\omega,Q + q ) = \frac{4\pi r_{f}^{n}}{n\omega_{n}}. $$
(12)
So the tunneling probability from the cosmological horizon is
$$\Gamma\propto\exp( - 2\operatorname{Im}S) = \exp \biggl[\frac{4\pi}{n\omega_{n}}\bigl(r_{f}^{n} - r_{i}^{n}\bigr)\biggr]= \exp(\Delta S_{CH}), $$
(13)
where ΔSCH=SCH(M+ω,Q+q)−SCH(M,Q) is the change of the Bekenstein-Hawking entropy. Our result shows the derived radiation spectrum deviates from the pure thermal one, but is related to the change of Bekenstein-Hawking entropy. This realization satisfies the unitary theory, and provides a probability for the solution of the information loss paradox.

4 Discussion and Conclusions

When k=1, the line element of TRNdS black hole is reduced into the Reissner-Nordström-de Sitter (RNdS) black hole in higher dimensions, and which also exists the cosmological horizon and the event horizon. The Hawking radiation via tunneling from the cosmological horizon has already been discussed. Now, we will focus our attention on the Hawking radiation from the event horizon of the higher-dimensional Reissner-Nordström-de Sitter (RNdS) black hole. At the event horizon, the outgoing geodesics is obtained from (4). When the particle with energy ω and electric charge q tunnels across the event horizon, the mass M and electric charge Q in (4) is replaced by Mω and Qq. Thus, we have
https://static-content.springer.com/image/art%3A10.1007%2Fs10773-012-1118-6/MediaObjects/10773_2012_1118_Equ14_HTML.gif
(14)
where rin and rout represent the event horizon before and after the particle with energy ω and charge q tunnels out. On the other hand, the change of the Bekenstein-Hawking entropy is given by
$$\Delta S_{BH} = \frac{4\pi}{n\omega_{n}}\bigl( r_{out}^{n}- r_{in}^{n} \bigr). $$
(15)
So the tunneling probability from the event horizon of the higher-dimensional Reissner- Nordström-de Sitter black hole can be expressed naturally as
$$\Gamma\sim\exp( - 2\operatorname{Im}S_{ +} ) = \exp\biggl[ \frac{4\pi}{n\omega_{n}}\bigl(r_{out}^{n} - r_{in}^{n} \bigr) \biggr] =\exp( \Delta S_{BH} ), $$
(16)
Also, the derived spectrum deviates from the precise thermal one, satisfies the unitary theory, and provides a probability for the solution of the information loss paradox.

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© Springer Science+Business Media, LLC 2012