Entropy corrections for Schwarzschild and Reissner-Nordstroem black holes

A Schwarzschild black hole being thermodynamically unstable, corrections to its entropy due to small thermal fluctuations cannot be computed. However, a thermodynamically stable Schwarzschild solution can be obtained within a cavity of any finite radius by immersing it in an isothermal bath. For these boundary conditions, classically there are either two black-hole solutions or no solution. In the former case, the larger mass solution has a positive specific heat and hence is locally thermodynamically stable. We find that the entropy of this black hole, including first-order fluctuation corrections, is given by: S=SBH-1n[(3/R)(SBH/4π)1/2-2]-1+1/21n(4π, where SBH = A/4 is its Bekenstein-Hawking entropy and R is the radius of the cavity. We extend our results to four-dimensional Reissner-Nordstroem black holes, for which the corresponding expression is: S=SBH-1/21n[(SBH/πR2) (3SBH/πR2-2√SBH/πR2-α2) (√SBH/πR2-α2)/SBH/πR2-α2)2]-1+1/21n(4π). Finally, we generalize the stability analysis to Reissner-Nordstroem black holes in arbitrary spacetime dimensions, and compute their leading order entropy corrections. In contrast to previously studied examples, we find that the entropy corrections in these cases have a different character