Horizon dynamics of a BTZ black hole

It has been suggested in the literature that, given a black-hole spacetime, a relativistic membrane can provide an effective description of the horizon dynamics. In this paper, we explore such a framework in the context of a (2 + 1)-dimensional BTZ black hole. Following this membrane prescription, we are able to translate the horizon dynamics (now described by a string) into the convenient form of a (1 + 1)-dimensional Klein-Gordon equation. We proceed to quantize the solutions and construct a thermodynamic partition function. Ultimately, we are able to extract the quantum-corrected entropy, which is shown to comply with the BTZ form of the Bekenstein-Hawking area law. We also substantiate that the leading-order correction is proportional to the logarithm of the area.