Noncommutative black hole thermodynamics

Abstract

Abstract:We give a general derivation, for any static spherically symmetric metric, of the relation T h = K 2π connecting the black hole temperature (T h ) with the surface gravity (K), following the tunneling interpretation of Hawking radiation. This derivation is valid even beyond the semi classical regime i. e. when quantum effects are not negligible. The formalism is then applied to a spherically symmetric, stationary noncommutative Schwarzschild space time. The effects of back reaction are also included. For such a black hole the Hawking temperature is computed in a closed form. A graphical analysis reveals interesting features regarding the variation of the Hawking temperature (including corrections due to noncommutativity and back reaction) with the small radius of the black hole. The entropy and tunneling rate valid for the leading order in the noncommutative parameter are calculated. We also show that the noncommutative Bekenstein-Hawking area law has the same functional form as the usual one. IntroductionClassical general relativity gives the concept of black hole from which nothing can escape. This picture was changed dramatically when Hawking[1] incorporated the quantum nature into this classical problem. In fact he showed that black hole radiates a spectrum of particles which is quite analogous with a thermal black body radiation. Thus Hawking radiation emerges as a nontrivial consequence of combining gravity and quantum mechanics.After his original derivation which was based on the calculation of Bogoliubov coefficients in the asymptotic states, Hawking together with Hartle[2] gave a simpler, path integral derivation. Physically, black hole radiation can be interpreted as the quantum tunneling of vacuum fluctuations through the horizon. This picture was mathematically formulated in [3]. An important step of this method is the calculation of tunneling amplitude from which the Hawking temperature is obtained. This is done either by using the trajectory of a null geodesic [3] or by solving the Hamilton-Jacobi equation to calculate the imaginary part of the action variable [4].The tunneling approach was subsequently used to compute the Hawking temperature for black holes with different types of metric [5]. The results have agreed with the general *

Hawking radiation as quantum tunneling from a noncommutative Schwarzschild black hole

“…Note added: After we have completed this work, Banerjee et al have reported a similar treatment of the problem [35].…”

Quantum tunneling beyond semiclassical approximation

“…In our previous work [12] we found out the corrections to the temperature and entropy by including the effects of back reaction knowing the modified surface gravity of the black hole due to one loop back reaction for the Schwarzschild case by radial null geodesic method. As an extension we [13] also applied this method for a noncommutative Schwarzschild metric. Recently, a problem in this approach has been discussed in [14 -18] which corresponds to a factor two ambiguity in the original Hawking temperature.…”

Charged Particle’s Tunneling in a Modified Reissner-Nordstrom Black Hole

“…Thence, by the Hamilton-Jacobi method, the quantum correction to the tunneling program was discussed and the quantum corrected entropy of black holes was calculated [26][27][28][29][30][31][32][33]. Also, in the radial trajectory method, the quantum effects of spacetime was considered and the emission rate was related the quantum corrected entropy of black hole [34][35][36][37][38][39][40][41][42][43]. These results show that, considering the quantum corrections of black hole entropy and spacetime, information conservation of black hole is still possible.…”