Abstract
We examine the thermodynamics of a near-extremal Kerr black hole, and demonstrate that the geometry behaves as an ordinary quantum system with a vanishingly small degeneracy at low temperatures. This is in contrast with the classical analysis, which instead predicts a macroscopic entropy for the extremal Kerr black hole. Our results follow from a careful analysis of the gravitational path integral. Specifically, the low temperature canonical partition function behaves as \( Z\sim {T}^{\frac{3}{2}}\ {e}^{S_0+c\log {S}_0} \), with S0 the classical degeneracy and c a numerical coefficient we compute. This is in line with the general expectations for non-supersymmetric near-extremal black hole thermodynamics, as has been clarified in the recent past, although cases without spherical symmetry have not yet been fully analyzed until now. We also point out some curious features relating to the rotational zero modes of the near-extremal Kerr black hole background that affects the coefficient c. This raises a puzzle when considering similar black holes in string theory. Our results generalize to other rotating black holes, as we briefly exemplify.
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Acknowledgments
It is a pleasure to thank R. Emparan, C. Ferko, M. Heydeman, V. Hubeny, L. Iliesiu, R. Loganayagam, J. Maldacena, S. Murthy, L. Pando Zayas, A. Porfyriadis, W. Song, and S. Trivedi for useful discussions. We would specifically like to acknowledge important insightful discussions with D. Marolf and A. Sen; we are very grateful to them for sharing their thoughts. MR was supported by U.S. Department of Energy grant DE-SC0009999 and funds from the University of California. MR is grateful to the long-term workshop YITP-T-23-01 held at YITP, Kyoto University, for hospitality during the concluding stages of this project. GJT was supported by the Institute for Advanced Study and the NSF under Grant No. PHY-2207584, and by the Sivian Fund, and currently by the University of Washington and the DOE award DE-SC0024363.
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Rakic, I., Rangamani, M. & Turiaci, G.J. Thermodynamics of the near-extremal Kerr spacetime. J. High Energ. Phys. 2024, 11 (2024). https://doi.org/10.1007/JHEP06(2024)011
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DOI: https://doi.org/10.1007/JHEP06(2024)011