Abstract
Black holes violate the third law of thermodynamics in its standard formulation. Schwarzschild black hole entropy is inverse proportional to the square of the temperature \(S=1/(16 \pi T^2)\) and tends to infinity rather than zero when the temperature goes to zero. We search for quantum statistical models with such exotic thermodynamic behaviour. It is shown that the Schwarzschild black hole in \(D=4\) spacetime dimensions corresponds to a Bose gas in a space with \(d=-4\) negative spatial dimensions. The Riemann zeta function is used to define the entropy of the Bose gas in negative dimension. The correspondence between black holes in higher dimensions and de Sitter space with Bose gas is considered. In particular case of \(D=5\), the corresponding Bose gas model lives in a space with \(d=3\) spatial dimensions and contains a nonlocal kinetic term.
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Notes
In particular, for the Reissner–Nordstrom case the entropy at the extremal regime is equal to \(S=\pi Q^2\), i.e. it dependents on parameter of the black hole. Note that one can get the zero entropy for the extremal configuration of the Reissner–Nordstrom black hole properly using an integration constant and get zero entropy for the extremal case.
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Acknowledgements
We would like to thank D. Ageev, V. Berezin, V. Frolov, M. Khramtsov, K. Rannu, P. Slepov, A. Teretenkov, A. Trushechkin and V. Zagrebnov for fruitful discussions. This work is supported by the Russian Science Foundation (project19-11-00320, V.A. Steklov Mathematical Institute).
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Aref’eva, I., Volovich, I. Violation of the third law of thermodynamics by black holes, Riemann zeta function and Bose gas in negative dimensions. Eur. Phys. J. Plus 139, 300 (2024). https://doi.org/10.1140/epjp/s13360-024-05049-7
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DOI: https://doi.org/10.1140/epjp/s13360-024-05049-7