Entropy relations and the application of black holes with the cosmological constant and Gauss–Bonnet term

Abstract

Keywords:(A)dS black hole First law of thermodynamics Smarr relation Thermodynamic bound Thermodynamic relation Based on entropy relations, we derive the thermodynamic bound for entropy and the area of horizons for a Schwarzschild-dS black hole, including the event horizon, Cauchy horizon, and negative horizon (i.e., the horizon with negative value), which are all geometrically bound and comprised by the cosmological radius. We consider the first derivative of the entropy relations to obtain the first law of thermodynamics for all horizons. We also obtain the Smarr relation for the horizons using the scaling discussion. For the thermodynamics of all horizons, the cosmological constant is treated as a thermodynamic variable. In particular, the thermodynamics of the negative horizon are defined well in the r < 0 side of space-time.This formula appears to be valid for three-horizon black holes. We also generalize the discussion to thermodynamics for the event horizon and Cauchy horizon of Gauss-Bonnet charged flat black holes because the Gauss-Bonnet coupling constant is also considered to be thermodynamic variable. These results provide further insights into the crucial role played by the entropy relations of multi-horizons in black hole thermodynamics as well as improving our understanding of entropy at the microscopic level.

Entropy relations and bounds of horizons in modified gravity

“…We introduce the entropy bounds of four dimensional Schwarzschild-dS black hole here [47]. The area entropy S i = A i 4 = πr 2 i , (i = E, C, V ) lead to the mass-dependence entropy product…”

“…These geometrical area (entropy) bounds are also studied in Gauss-Bonnet gravity [47], where the Gauss-Bonnet coupling constant should also be treated as a thermodynamical variable (see, e.g. [42,[48][49][50][51][52]).…”

“…Follow the same procedure as the discussion for Schwarzschild-dS black hole [47], we find the entropy bounds for all horizons…”

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Entropy of higher-dimensional topological dS black holes with nonlinear source

“…For constructing the whole history of evolution of the Universe, it's necessary to understand both the classical and quantum properties of the de Sitter spacetime. People usually define the entropy of de Sitter spacetime as the sum of two horizon's entropies [34,[37][38][39][40]. However, there is no theoretical proof supports this consequence.…”