Higher Dimensional Black Hole Corrected Tunneling Radiation

Corrected tunneling radiation of a higher dimensional black hole and generalized second law

S. S. Mortazavi[*]1, A. Farmany1, H. Noorizadeh2, V. Fayaz1, H. Hosseinkhani1

Abstract

Study the quantum gravitational effects on a higher dimensional horizon, the semiclassical s-wave tunneling radiation of black holes are calculated. It is shown that quantum gravitational effects correct the semiclassical radiation of the horizon space time. Within this background, the generalized second law of thermodynamics is applied to the black hole entropy.

1. Introduction

It is interesting that that radiation of black holes can be viewed as simple tunneling phenomena. In this view, a particle-antiparticle pair may form close to a black hole event horizon. The ingoing mode is trapped inside the horizon while the outgoing mode can tunnel through the event horizon. It is interesting that this effect is a quantum mechanically and the present of an event horizon is essential (Hawking, 1975). Recently, the semiclassical analysis of this phenomenon carried out by Parikh and Wilczek (Parikh, Wilczek, 2000; Parikh, 2002; Parikh, 2004; Parikh, 2004). Parikh-Wilczek proposal of black hole tunneling radiation is based on the computation of incoming part of action for classically forbidden of s-wave emission across the horizon (Parikh, Wilczek, 2000; Parikh, 2002; Parikh, 2004; Parikh, 2004; Kraus, Wilczek, 1994; Kraus, Wilczek, 1995; Kraus, Wilczek, 1995; Kraus, Keski-Vakkuri, 1997; Berezin, Boyarsky, Neronov, 1999; Volovik, 1999;1999; Calogeracos, Volovik,1999). As a comparison between Hawking original calculation and tunneling method, it is easy to see that the hawking method is a direct method but its complication to generalization to all other space times is failed while the Parikh-Wilczeck proposal, the tunneling approaches have been successfully applied to a wide range of both the black hole space time horizon and cosmological horizon. For example, 3- dimensional BTZ black holes (Agheben, Nadalini, Vanzo, Zerbini, 2005; Wu, Jiang, 2006), Vaidya space time(Ren, Zhang, Zhao, 2006), dynamical black holes(Di Criscienzo, Nadalini, Vanzo, Zerbini, Zoccatelli, 2007), black rings(Zhao, 2006), Kerr and Kerr-Newman black holes(Jiang, Wu, Cai, 2006; Zhang, Zhao, 2006; Hu, Zhang, Zhao, 2006; Kerner, Mann, 2006), Taub-NUT space time(Kerner, Mann, 2006), Godel space time (Kerner, Mann, 2007), dynamical horizons(Di Criscienzo, Nadalini, Vanzo, Zerbini, Zoccatelli, 2007), cosmological horizons(Parikh, 2002; Medved,2002; Sekiwa, 2008), Rindler space time (Medved, 2002), de Sitter space time. Of course in all of these approaches the Unruh temperature is recovered successfully (Unruh, 1976; Akhmedova, Pilling, Gill, Singleton, 2008; Banerjee, Kulkarni, 2008; Banerjee, Majhi, 2008).

This model is applied to not only the black hole event horizon, but also to the cosmological horizon (Parikh, 2002; Medved, 2002; Sekiwa, 2008). The black hole tunneling method was studied in different space-times and different frames and the time contribution to the black hole radiation is developed in (Chowdhury, 2008; Akhmedov, Akhmedova, Pilling, Singleton, 2007; Zhang, Cai, Zhan, 2009; Banerjee, Majhi, 2009; Akhmedov, et al, 2006; Akhmedov, Pilling, Singleton, 2008). In continue, the spectrum form of the tunneling mechanism is analyzed using the density matrix technique (Banerjee, Majhi, 2009). However the Parikh-Wilczek method is based on the classical analysis, when it comes into the high-energy regime, for example a small black hole whose size can be compared with Planck scale, the effect of quantum gravity should not be forbidden. In this case, the conventional semiclassical approaches are not proper and the complete quantum gravity analysis is required. To study the quantum gravitational effects on the tunneling mechanism it is interesting to relate the analysis under a minimal length quantum gravity scale ( Adler, Chen, Santiago, 2001; Han, Li, Ling, 2008; Farmany, et al, 2008; Shu, Shen, 2008; Wang, Gui, Ma, 2008; Setare, 2004; Kim, Park, 2007; Nouicer, 2007; Zhao, Zhang, 2006; Xiang, 2006; Dehghani, Farmany, 2009). In this paper, the black hole tunneling radiation is studied based on the generalized uncertainty principle. It is shown that the generalized second law of thermodynamics applie a bound on the tunneling radiation.

2. The corrected Bekenstein-Hawking entropy

A d-dimensional spherical symmetric black hole background is defined by

(1)

where . The uncertainty in the position of a particle, during the emission,

(2)

where applying the uncertainty principle, we obtain the energy of radiated particle,

(3)

Where and Mpl is Planck mass. Temperature of black hole in a d-dimension space time may be obtained by setting the radiated particle mass m to. The d-dimensional black hole temperature may be obtained as,

(4)

where d3. Eqs. (4) shows the temperature of a d-dimensional black hole with . The Bekenstein-Hawking entropy is usually derived from the Hawking temperature. The entropy S may be found from the well known thermodynamics relation,

(5)

From (3-5) we obtain,

(6)

Quantum gravitational effects of horizon may affect on the thermodynamics of black hole and modifies its usual thermodynamical behavior. Study of black hole thermodynamics in the quantum gravity theory was made using a generalized uncertainty principle (Adler, 1999; Hossenfelder et al, 2004; Maggiore, 1994; Kempf, Managano, 1997; Farmany, Abbasi, Naghipour, 2007)

(7)

Where lpl is the Planck length. Setting 2rh as , we obtain,

(8)

Solving for minimum and expanding around lpl²=0, eq. (8) reads,

(9)

Comparing (9) with (7) we obtain,

(10)

inserting (4) into (10), the d-dimensional black hole temperature me be obtained,

(11)

The corrected entropy S’ may be obtained from the thermodynamics relation (5),

(12)

3. The corrected black hole radiation

As shown by Parick and Wilczek (2000) the WKB approximation relate the tunneling probability to the imaginary part of the action

(13)

Where I is the classical action of trajectory. The difference between all approaches of tunneling method is in how the action is calculated. As shown by Arzano et al (Arzano, Medved, Vagenas, 2005),

(14)

in terms of black hole mass M and energy E, which is correspond to

(15)

provided the Bekenstein-Hawking entropy/area relation.

Consider the above relation, eq.(13) can be written in the following general form,

(16)

The quantum gravity-corrected black hole entropy is given by eq.(12), so,

(17)

Substituting (17) into (16) we obtain,

(18)

which shows the corrected tunneling probability and

.

4. Generalized second law of thermodynamics and modified black hole tunneling radiation

Bekenstein (1981) has conjectured that the entropy S and energy E of any thermodynamic system must obey,

(19)

where R is defined as the circumferential radius. This bound is universal in the sense that it is supposed to hold in any matter system. The Bekenstein bound has been confirmed in wide classes of systems. However, as pointed by Bekenstein, the bound is valid for systems with finite size and limited self-gravity. Counterexamples can be easily found in systems undergoing gravitational collapse (Bousso, 1999). Another entropy bound is related to the holographic principle, which says that the entropy in a spherical volume satisfies

(20)

where A is the area of the system. It was shown that this bound is violated for sufficiently large volumes (Fischler and Susskind, 1998). As shown by eqs.(19-20), there is a bound on the entropy of the black hole when it related to the black hole area. While the black hole entropy bound applied to eq. (7), we obtain,

(21)

So, in the presence of entropy bound, eq. (16) may be,

(22)

Combining eq.(22) and (18) we obtain the corrected tunneling probability of black hole radiation.

(23)

Conclusion

The semiclassical black hole tunneling radiation is calculated by the Parikh-Wilczek tunneling proposal of black hole radiation based on the generalized uncertainty principle. It is shown that the Bekenstein-Hawking entropy of black holes receives a correction that affects on the radiation tunneling probability. In continue applying the generalized second law of thermodynamics to the modified black hole tunneling radiation is obtained.

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