Thermodynamic Interpretation of Field Equations at Horizon of BTZ Black Hole

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.