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 the case of static metric of Banados--Teitelboim--Zanelli (BTZ) black hole, the field equations near the horizon boundary can be expressed as a thermal identity dE = TdS + P_rdA, 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= 4πa is the entropy and T =k/2π is the Hawking temperature associated with the horizon. This approach is studied further to generalize it for non-static BTZ black hole, showing that it is also possible to interpret the field equation near horizon as a thermodynamic identity dE = TdS + P_rdA +Ω₊dJ, where Ω₊ 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 the horizon.