Thermodynamic Interpretation of Field Equations at Horizon of BTZ Black Hole

TL;DR

This paper demonstrates that the field equations at the horizon of BTZ black holes can be interpreted as thermodynamic identities, extending from static to rotating cases, revealing intrinsic thermodynamic properties of the spacetime.

Contribution

It generalizes the thermodynamic interpretation of field equations at the horizon from static to rotating BTZ black holes, including angular momentum and velocity.

Findings

Field equations near horizon can be expressed as a thermodynamic identity.

Extension to non-static BTZ black holes includes angular momentum terms.

Field equations possess intrinsic thermodynamic properties at the horizon.

Abstract

A spacetime horizon comprising with a black hole singularity acts like a boundary of a thermal system associated with the notions of temperature and entropy. In case of static metric of BTZ black hole, the field equations near horizon boundary can be expressed as a thermal identity $dE = TdS + P_{r}dA$, where $E = M$ is the mass of BTZ black hole, $dA$ is the change in the area of the black hole horizon when the horizon is displaced infinitesimally small, $P_{r}$ is the radial pressure provided by the source of Einstein equations, $S= 4\pi a$ is the entropy and $T = \kappa / 2\pi$ is the Hawking temperature associated with the horizon. This approach is studied further to generalize it for non-static BTZ black hole and show that it is also possible to interpret the field equation near horizon as a thermodynamic identity $dE = TdS + P_{r}dA + \Omega_{+} dJ$, where $\Omega_{+}$ is the angular velocity and $J$ is the angular momentum of BTZ black hole. These results indicate that the field equations for BTZ black hole possess intrinsic thermodynamic properties near horizon.