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(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 infinite small small, P-r is the radial pressure provided by the source of Einstein equations, S = 4 pi alpha 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, showing 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 the horizon.