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 *

A planar phase space having both position and momentum noncommutativity is defined in a more inclusive setting than that considered elsewhere. The dynamics of a particle in a gravitational quantum well in this space is studied. The use of the WKB approximation and the virial theorem enable analytic discussions on the effect of noncommutativity. Consistent results are obtained following either commutative space or noncommutative space descriptions. Comparison with recent experimental data with cold neutrons at Grenoble imposes an upper bound on the noncommutative parameter. Also, our results are compared with a recent numerical analysis of a similar problem. Finally, we provide a noncommutative version of the virial theorem for the case at hand.

We discuss a scheme based on Ehrenfest like equations to exhibit and classify transitions between two phases (with "smaller" and "larger" masses) of Kerr AdS black holes. We show that for fixed angular velocity this phase transition is of second order as both Ehrenfest's equations are satisfied. Finally we make a close connection of the results found from this analysis with those obtained from the thermodynamic state space geometry approach. *

We describe the deformed Poincare-conformal symmetries implying the covariance of the noncommutative space obeying Snyder's algebra. Relativistic particle models invariant under these deformed symmetries are presented. A gauge (reparametrisation) independent derivation of Snyder's algebra from such models is given. The algebraic transformations relating the deformed symmetries with the usual (undeformed) ones are provided. Finally, an alternative form of an action yielding Snyder's algebra is discussed where the mass of a relativistic particle gets identified with the inverse of the noncommutativity parameter. *

We exploit the Seiberg-Witten map technique to formulate the theory of gravity defined on a Lie algebraic noncommutative space-time. Detailed expressions of the Seiberg-Witten maps for the gauge parameters, gauge potentials, and the field strengths have been worked out. Our results demonstrate that notwithstanding the introduction of more general noncommutative structure there is no first order correction, exactly as happens for a canonical (i.e. constant) noncommutativity.