Black hole thermodynamics at 4 derivatives, natural variables and BPS limits

TL;DR

This work extends Einstein–Maxwell theory by including all parity-even four-derivative corrections and analyzes static and rotating black holes in $D\ge3$ to first order in the HD expansion. It develops a natural-variables formalism that splits thermodynamics into left- and right-moving sectors, simplifying the computation and yielding a smooth BPS limit, while also clarifying the correct application of the Reall–Santos method in multi-horizon spacetimes. The paper provides explicit results in $D=4$ and $D=5$, including SUSY-coupled reductions (supergravity) and detailed BPS/almost-BPS limits, and uncovers a subtle discrepancy between higher-derivative localization predictions and the $D=4$/$D=5$ connection under dimensional reduction. These findings sharpen the understanding of HD corrections to black hole thermodynamics and highlight the need for careful cross-dimension checks in localization-based approaches.

Abstract

We study Einstein-Maxwell theory in $D \geq 3$ spacetime dimensions including all Lorentz-invariant parity-even four-derivative couplings. Building on the results of arXiv:2312.11610, we consider static, charged, asymptotically flat black hole solutions to first order in the higher-derivative expansion. In $D=4$ and $D=5$, we compute the corrected black hole thermodynamics and compare with the Reall-Santos prescription based on the two-derivative background, highlighting a subtlety when both inner and outer horizons are involved. By introducing natural variables, as in arXiv:2304.07320, we recast the on-shell actions in terms of left- and right-moving chemical potentials, which significantly simplifies the analysis. We also compute first-order thermodynamic corrections for the most general rotating black holes in $D=4$ and $D=5$, without modifying the background solutions. We identify a novel BPS-like limit in $D=4$, extending known supergravity results beyond their traditional domain of validity. Finally, in $D=5$, the analysis of BPS and almost BPS limits enables an independent verification of the five-dimensional BPS thermodynamics. We clarify the origin of a discrepancy in the literature concerning higher-derivative supergravity localization, sharpening the tension between direct computations and predictions based on the $D=4$/$D=5$ connection.