Whereas the usual understanding is that the entropy of only a non-extremal black hole is given by the area of the horizon, there are derivations of an area law for extremal black holes in some model calculations. It is explained here how such results can arise in an approach where one sums over topologies and imposes the extremality condition after quantization.It has been known for quite some time now that a black hole can be assigned a temperature, which is a quantum effect, and is proportional to Planck's constant. Correspondingly, there is also an entropy [1, 2], with inverse dependence on Planck's constant and proportional to the area of the horizon. This entropy can be understood in a euclidean functional integral approach [3] where the integral is evaluated in the semiclassical approximation, i.e., replaced by the exponential of the negative of the minimum classical action, which is essentially a quarter of the area.All this is about what are now called non-extremal black holes. One is now more often interested in a different class of black holes -the extremal ones. These are characterized by coinciding horizons and have qualitatively different features. Thus, the euclidean topology of an extremal black hole is different from that of the related non-extremal black holes. Again, the classical action of an extremal black hole vanishes. This results in an entropy which vanishes [4] or behaves like the mass [5], but certainly does not behave like the area.

The formulation of quasi-local conformal Killling horizons(CKH) is extended to include rotation. This necessitates that the horizon be foliated by 2-spheres which may be distorted. Matter degrees of freedom which fall through the horizon is taken to be a real scalar field. We show that these rotating CKHs also admit a first law in differential form.Comment: 15 pages. sequel to arXiv:1412.5115. Definition slightly modified, an appendix adde

The change in holographic entanglement entropy (HEE) for small fluctuations about pure anti-de Sitter (AdS) is obtained by a perturbative expansion of the area functional in terms of the change in the bulk metric and the embedded extremal surface. However it is known that change in the embedding appears at second order or higher. It was shown that these changes in the embedding can be calculated in the 2 þ 1 dimensional case by solving a "generalized geodesic deviation equation." We generalize this result to arbitrary dimensions by deriving an inhomogeneous form of the Jacobi equation for minimal surfaces. The solutions of this equation map a minimal surface in a given space time to a minimal surface in a space time which is a perturbation over the initial space time. Using this we perturbatively calculate the changes in HEE up to second order for boosted black brane like perturbations over AdS 4 .

Physical process version of the first law of black hole mechanics relates the change in entropy of a perturbed Killing horizon, between two asymptotic cross sections, to the matter flow into the horizon. Here, we study the mathematical structure of the physical process first law for a general diffeomorphism invariant theory of gravity. We analyze the effect of ambiguities in the Wald's definition of entropy on the physical process first law. We show that for linearized perturbations, the integrated version of the physical process law, which determines the change of entropy between two asymptotic cross-sections, is independent of these ambiguities. In case of entropy change between two intermediate cross sections of the horizon, we show that it inherits additional contributions, which coincide with the membrane energy associated with the horizon fluid. Using this interpretation, we write down a physical process first law for entropy change between two arbitrary non-stationary cross sections of the horizon for both general relativity and Lanczos-Lovelock gravity. * akash.mishra@iitgn.ac.in † sumantac.physics@gmail.com ‡ avirup.ghosh@iitgn.ac.in § sudiptas@iitgn.ac.in 1 The requirement that a regular bifurcation surface be present can however be avoided in some quasi-local definitions of horizons viz. Isolated horizons [11,12].

Hawking radiation is nowadays being understood as tunnelling through black hole horizons. Here, the extension of the Hamilton-Jacobi approach to tunnelling for non-rotating and rotating black holes in different non-singular coordinate systems not only confirms this quantum emission from black holes but also reveals the new phenomenon of absorption into white holes by quantum mechanical tunnelling. The role of a boundary condition of total absorption or emission is also clarified.Comment: REVTeX, 6 pages; changed interpretation for white hole spacetimes; to appear in PL