The energy spectrum of the membrane effective model for quantum black holes

We study first order fluctuations of a relativistic membrane in the curved background of a black hole. The zeroth-order solution corresponds to a spherical membrane tightly covering the event horizon. We obtain a massive Klein-Gordon equation for the fluctuations of the membraneʹs radial coordinate on the 2+1 dimensional world-volume. We finally suggest that quantization of the fluctuations can be related to black holeʹs mass quantization and the corresponding entropy is computed. This entropy is proportional to the membrane area and is related to the one-loop corretion to the thermodynamical entropy AH4. With regards to the membrane model for describing effectively a quantum black hole, we connect these results with previous work on critical phenomena in black hole thermodynamics.