Abstract
We present some entropy and temperature relations of multi-horizons, even including the "virtual" horizon. These relations are related to product, division and sum of entropy and temperature of multi-horizons. We obtain the additional thermodynamic relations of both static and rotating black holes in three and four dimensional (A)dS spacetime. Especially, a new dimensionless, charges-independence and $T_+S_+=T_-S_-$ like relation is presented. This relation does not depend on the mass, electric charge, angular momentum and cosmological constant, as it is always a constant. These relations lead us to get some interesting thermodynamic bound of entropy and temperature, including the Penrose inequality which is the first geometrical inequality of black holes. Besides, based on these new relations, one can obtain the first law of thermodynamics and Smarr relation for all horizons of black hole.
Highlights
One of the central issues in quantum gravity is to understand the entropy of a black hole microscopically.Significant insights have been achieved for four- and five- dimensional, supersymmetric, asymptotically flat, multi-charged black holes [1], where the microscopic degrees of freedom can be explained in terms of a two-dimensional conformal field theory
We study the additional thermodynamic relations of black holes with multi-horizons, in order further study the understanding of the origin of black hole entropy at the microscopic level
Based on the thermodynamic relations presented in this paper, we obtain some thermodynamic bounds of the thermodynamic quantities, including the entropy and temperature
Summary
One of the central issues in quantum gravity is to understand the entropy of a black hole microscopically. When the discussion generalizes to some cases of black holes with more than two horizons, the thermodynamics relations T+ S+ = T− S− (equivalently, entropy product S+ S− being mass independent) break down [12,15,17,21,25], and there are even only two physical horizons. This does not depend on the mass, electric charge, angular momentum and cosmological constant, as it is always a constant for black holes, as shown in the present paper This relation is expected to be helpful of constructing the BH/CFT correspondence for more than two horizons, in order to understand the black hole entropy microscopically.
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