Entropy corrections for Schwarzschild and Reissner–Nordström black holes

doi:10.1088/0264-9381/21/6/007

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Classical and Quantum Gravity

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Classical and Quantum Gravity Volume 21 Number 6 Create an alert RSS this journal

M M Akbar and Saurya Das 2004 Class. Quantum Grav. 21 1383 doi:10.1088/0264-9381/21/6/007

Entropy corrections for Schwarzschild and Reissner–Nordström black holes

M M Akbar¹ and Saurya Das^2,3

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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: ${\cal S} = S_{BH} - \ln\big[\frac{3}{R} (S_{BH}/4{\pi})^{1/2} -2\big]^{-1} + \case{1}{2} \ln(4{\pi})$ , where S_BH = A/4 is its Bekenstein–Hawking entropy and R is the radius of the cavity. We extend our results to four-dimensional Reissner–Nordström black holes, for which the corresponding expression is: ${\cal S} = S_{BH} - \frac{1}{2} \ln [(S_{BH}/ {\pi} R^2) \big({3S_{BH}}/ {{\pi} R^2} - 2\sqrt{{S_{BH}}/{{\pi} R^2 - {\alpha}^2}}\big)$ $\big(\sqrt{{S_{BH}}/{{\pi} R^2}} - {\alpha}^2\big)\big/ ({S_{BH}}/ {{\pi} R^2} -{\alpha}^2)^2 ]^{-1} +\case{1}{2} \ln(4{\pi})$ . Finally, we generalize the stability analysis to Reissner–Nordström 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.