Novel logarithmic corrections to black hole entropy

Abstract

For a thermodynamic system, apart from thermal fluctuations, there are also fluctuations in thermodynamic volume when the system is in contact with a volume reservoir. For the case of black holes in anti de Sitter spacetime, the effect of thermal fluctuations on the entropy is well studied. The aim of this work is to compute novel logarithmic corrections to black hole entropy coming from simultaneous fluctuations in energy and volume. We work in the isothermal-isobaric ensemble and first obtain a general form of corrections to entropy which are valid for any thermodynamic system. Applying the formalism to Kerr black holes in AdS reveals that entropy gets corrected as: \(\mathcal{S} = S_0 - k \ln S_0 + \cdots\) where \(S_0\) is given by the Bekenstein-Hawking formula and \(k = - 1\). The same leading coefficient is also obtained in the canonical ensemble, i.e. by considering energy fluctuations alone. This coefficient is found to be unaltered in the slowly rotating and high temperature limits.

Statistical ensembles and logarithmic corrections to black hole entropy

“…This always gives the generic structure Ω(E) ≈ e S0 / √ D, where D can be interpreted as the 'one-loop determinant' in the statistical mechanics sense 2 . This leads to logarithmic corrections to the microcanonical entropy and such corrections have been reported in various works in the context of black hole thermodynamics [27,42,43,[47][48][49][50][51] (see also [53]). It should be noted that these corrections which stem from statistical thermodynamics arguments are distinct from those obtained as oneloop corrections in Euclidean quantum gravity.…”

“…The framework shall be applied to black holes in sections 3 and 4. We shall be quite brief and the reader is referred to [27,42,43,[47][48][49][50][51], where similar calculations for obtaining logarithmic corrections to black hole entropy have been discussed. We begin by discussing the canonical ensemble.…”

“…Examples include the grand canonical ensemble [65] where X can take the role of number of particles N for a gas, or electric charge Q/angular momentum J for black holes. For hydrostatic systems, and in the black hole chemistry framework [49][50][51], yet another ensemble of the type described above is the isothermal-isobaric ensemble [66,67] where X = V (thermodynamic volume) while x = βP. From equation (12), we can extract the density of states as…”

“…x S) 0 = (∆X) 2 , and (∂ β ∂ x S) 0 = ⟨∆E∆X⟩, meaning that D is the determinant of the covariance matrix [47,49,50]. Some straightforward computations give (see appendix A)…”

“…Some of these corrections are new while some of them were reported earlier but we include them anyway to make the presentation self-contained. We particularly emphasize upon the black hole chemistry framework, where the cosmological constant is treated as a thermodynamic variable [54][55][56][57][58][59][60] (see also [61][62][63][64] for some older works), for which some studies on logarithmic corrections have been reported recently [49][50][51]. The isoenthalpic-isobaric ensemble is introduced to describe black holes in this framework, which was not considered earlier.…”