Black hole entropy and Lorentz-diffeomorphism Noether charge

TL;DR

This work resolves a long-standing puzzle about black hole entropy in frame formalisms by introducing a Lorentz–covariant Lorentz-Lie derivative ${\mathcal K}^e_\xi$ that implements a combined Lorentz–diffeomorphism symmetry. The entropy becomes a horizon Noether charge $S=\frac{2\pi}{\hbar}\oint_{\cal B}\widehat Q^{\mathcal K}_\xi$, valid without singular limits, and reproduces the Bekenstein–Hawking result $S_{BH}=\frac{A}{4\hbar G}$ in GR, while generalizing to Lovelock gravity and to four-dimensional topological terms where Euler contributes a horizon-topology term and Pontryagin vanishes. The formalism yields a coefficient structure where Lovelock terms contribute with factors $2\kappa m c_m$, tying entropy to intrinsic horizon geometry, and shows Holst, Euler, and Pontryagin terms have well-defined, calculable influences on $S$. The approach also suggests applicability to nonparallelizable manifolds and internal gauge symmetries, broadening the covariance and robustness of black hole thermodynamics.

Abstract

We show that, in the first or second order orthonormal frame formalism, black hole entropy is the horizon Noether charge for a combination of diffeomorphism and local Lorentz symmetry involving the Lie derivative of the frame. The Noether charge for diffeomorphisms alone is unsuitable, since a regular frame cannot be invariant under the flow of the Killing field at the bifurcation surface. We apply this formalism to Lagrangians polynomial in wedge products of the frame field 1-form and curvature 2-form, including general relativity, Lovelock gravity, and "topological" terms in four dimensions.