Entropy corresponding to the interior of a Schwarzschild black hole

Interior volume within the horizon of a black hole is a non-trivial concept which turns out to be very important to explain several issues in the context of quantum nature of black hole. Here we show that the entropy, contained by the {\it maximum} interior volume for massless modes, is proportional to the Bekenstein-Hawking expression. The proportionality constant is less than unity implying the horizon bears maximum entropy than that by the interior. The derivation is very systematic and free of any ambiguity. To do so the precise value of the energy of the modes, living in the interior, is derived by constraint analysis. Finally, the implications of the result are discussed.Comment: Two new references and additional discussions added, to appear in Phys. Lett.

P-V criticality of AdS black holes in a general framework

In black hole thermodynamics, it has been observed that AdS black holes behave as van der Waals system if one interprets the cosmological constant as a pressure term. Also the critical exponents for the phase transition of AdS black holes and the van der Waals systems are same. Till now this type of analysis is done by two steps. In the first step one shows that a particular metric allows phase transition and in the second step, using this information, one calculates the exponents. Here, we present a different approach based on two universal inputs (the general forms of the Smarr formula and the first law of thermodynamics) and one assumption regarding the existence of van der Waal like critical point for a metric. We find that the same values of the critical exponents can be obtained by this approach. Thus we demonstrate that, though the existence of van der Waal like phase transition depends on specific metrics, the values of critical exponents are then fixed for that set of metrics.Comment: Extensively modified version, to appear in Phys. Lett.

Gauge Generators, Transformations and Identities on a Noncommutative Space

By abstracting a connection between gauge symmetry and gauge identity on a noncommutative space, we analyse star (deformed) gauge transformations with usual Leibnitz rule as well as undeformed gauge transformations with a twisted Leibnitz rule. Explicit structures of the gauge generators in either case are computed. It is shown that, in the former case, the relation mapping the generator with the gauge identity is a star deformation of the commutative space result. In the latter case, on the other hand, this result gets twisted to yield the desired map.Comment: LaTex, 21 pages, Section 2 has been considerably expanded, 3 new references added, To appear in EPJ

Second Order Phase Transition and Thermodynamic Geometry in Kerr-AdS Black Hole

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.Comment: LaTex, v3: 12 pages, 3 figures; minor modifications to match the journal version; to appear in Phys. Rev.

Gravitational surface Hamiltonian and entropy quantization

The surface Hamiltonian corresponding to the surface part of a gravitational action has $xp$ structure where $p$ is conjugate momentum of $x$. Moreover, it leads to $TS$ on the horizon of a black hole. Here $T$ and $S$ are temperature and entropy of the horizon. Imposing the hermiticity condition we quantize this Hamiltonian. This leads to an equidistant spectrum of its eigenvalues. Using this we show that the entropy of the horizon is quantized. This analysis holds for any order of Lanczos-Lovelock gravity. For general relativity, the area spectrum is consistent with Bekenstein's observation. This provides a more robust confirmation of this earlier result as the calculation is based on the direct quantization of the Hamiltonian in the sense of usual quantum mechanics.Comment: Revised version, accepted in Phys. Lett.

Killing Symmetries and Smarr Formula for Black Holes in Arbitrary Dimensions

We calculate the effective Komar conserved quantities for the $N+1$ dimensional charged Myers-Perry spacetime. At the event horizon we derive a new identity $K_{\chi^{\mu}}=2ST$ where the left hand side is the Komar conserved quantity corresponding to the null Killing vector $\chi^{\mu}$ while in the right hand side $S,~T$ are the black hole entropy and Hawking temperature. From this identity we also derive the generalized Smarr formula connecting the macroscopic parameters $M,~J,~Q$ of the black hole with its surface gravity and horizon area. The consistency of this new formula is established by an independent algebraic approach.Comment: v3, minor changes over v2, LaTex, 16 pages, no figures, to appear in Phys. Rev.