Abstract
AbstractWe investigate the properties of inner and outer horizon thermodynamics of Sen black hole (BH) both in Einstein frame (EF) and string frame (SF). We also compute area (or entropy) product, area (or entropy) sum of the said BH in EF as well as SF. In the EF, we observe that the area (or entropy) product is universal, whereas area (or entropy) sum is not universal. On the other hand, in the SF, area (or entropy) product and area (or entropy) sum don’t have any universal behaviour because they all are depends on Arnowitt–Deser–Misner (ADM) mass parameter. We also verify that the first law is satisfied at the Cauchy horizon as well as event horizon (EH). In addition, we also compute other thermodynamic products and sums in the EF as well as in the SF. We further compute the Smarr mass formula and Christodoulou’s irreducible mass formula for Sen BH. Moreover, we compute the area bound and entropy bound for both the horizons. The upper area bound for EH is actually the Penrose like inequality, which is the first geometric inequality in BHs. Furthermore, we compute the central charges of the left and right moving sectors of the dual CFT in Sen/CFT correspondence using thermodynamic relations. These thermodynamic relations on the multi-horizons give us further understanding the microscopic nature of BH entropy (both interior and exterior).
Highlights
In an un-quantized general relativity theory any black hole (BH) in thermal equilibrium has an entropy and a temperature
In the string frame (SF), area product, entropy product, area sum and entropy sum formula don’t have any universal nature because they all are depends on Arnowitt–Deser– Misner (ADM) mass parameter
We have examined various thermodynamic products for rotating charged black hole solution in four dimensional heterotic string theory
Summary
In an un-quantized (classical) general relativity theory any BH in thermal equilibrium has an entropy and a temperature. We observe that every BH thermodynamic quantities (e.g. area, entropy, temperature, surface gravity etc.), other than the mass (M), the angular momentum (J ) and the charge (Q), can form a quadratic equation whose roots are contained the three basic parameter M, J, Q. We shall see that every BH thermodynamic quantities (e.g. area, entropy, temperature, surface gravity etc.), other than the mass (M), the angular momentum (J ) and the charge (Q), which is defined on H± can form a quadratic equation of thermodynamic quantities like horizon radii (r±). We derive the dimensionless temperature of microscopic CFT, which is perfect agreement with the ones derived from hidden conformal symmetry in the low frequency scattering off the BH [19] Based on these above relations, we would like to compute the entropy bound of H± which is exactly Penrose-like inequality for event horizon. We are going to derive the Smarr formula for Sen BH
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