Entropy relations and bounds of BTZ black hole in gravity's rainbow

In this paper, we present the entropy relations and bounds of Banados–Teitelboim–Zanelli (BTZ) black hole in two models of gravity's rainbow. Because of the effect of gravity's rainbow, one can find that the entropy product and sum both lost their universality and become mass-dependent. On the other hand, comparing the entropy bound of event horizon to the BTZ case, it is shown that the angular momentum [Formula: see text] enlarges the entropy bound while the gravity's rainbow parameter [Formula: see text] diminishes it. For the entropy bound of Cauchy horizon, the gravity's rainbow parameter [Formula: see text] enlarges it at the large [Formula: see text] limit, while [Formula: see text] diminishes it at the small [Formula: see text] limit. These suggest some clues on the geometrical origin of black hole entropy bounds.

Horizon Areas and Logarithmic Correction to the Charged Accelerating Black Hole Entropy

It has been shown by explicit and exact calculation that the geometric product formula i.e., area (or entropy) product formula of outer horizon ( H + ) and inner horizon ( H - ) for charged accelerating black hole (BH) should neither be mass-independent nor quantized. This implies that the area (or entropy ) product is mass-independent conjecture has been broken down for charged accelerating BH. This also further implies that the mass-independent feature of the area product of H ± is not a generic feature at all. We also compute the Cosmic-Censorship-Inequality for this BH. Moreover, we compute the specific heat for this BH to determine the local thermodynamic stability. Under certain criterion, the BH shows the second order phase transition. Furthermore, we compute logarithmic corrections to the entropy for the said BH due to small statistical fluctuations around the thermal equilibrium.

Thermodynamic products in extended phase-space

We have examined the thermodynamic properties for a variety of spherically symmetric charged-AdS black hole (BH) solutions, including the charged AdS BH surrounded by quintessence dark energy and charged AdS BH in [Formula: see text] gravity in extended phase-space. This framework involves treating the cosmological constant as thermodynamic variable (for example: thermodynamic pressure and thermodynamic volume). Then they should behave as an analog of Van-der-Waal (VdW) like systems. In the extended phase-space we have calculated the entropy product and thermodynamic volume product of all horizons. The mass (or enthalpy) independent nature of the said product signals they are universal quantities. The divergence of the specific heat indicates that the second-order phase transition occurs under certain condition. In Appendix A, we have studied the thermodynamic volume products for axisymmetric spacetime and it is shown to be not universal in nature. Finally, in Appendix B, we have studied the [Formula: see text] criticality of Cauchy horizon for charged-AdS BH and found to be an universal relation of critical values between two horizons as [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]. The symbols are defined in the main work.

Entropy Product Formula for Gravitational Instanton

We investigate the entropy product formula for various gravitational instantons. We speculate that due to the mass-independent features of the said instatons they are universal as well as quantized. For isolated Euclidean Schwarzschild black hole, these properties simply fail.

CFT and Logarithmic Corrections to the Black Hole Entropy Product Formula

We examine the logarithmic corrections to the black hole (BH) entropy product formula of outer horizon and inner horizon by taking into account the effects of statistical quantum fluctuations around the thermal equilibrium and via conformal field theory (CFT). We argue that, in logarithmic corrections to the BH entropy product formula when calculated using CFT and taking into account the effects of quantum fluctuations around the thermal equilibrium, the formula should not be universal and it also should not be quantized. These results have been explicitly checked by giving several examples.