Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy
Vivek Iyer, Robert M. Wald
TL;DR
The paper establishes a fully covariant Noether-charge framework for diffeomorphism-invariant gravity, showing Lagrangians can be written in manifestly covariant form and that the symplectic potential and current admit global covariant definitions. It derives a universal decomposition of the Noether charge and proves the first law of black hole mechanics holds for arbitrary nonstationary perturbations. Building on this, it proposes a local, geometric prescription for the entropy of dynamical black holes, S_dyn, which recovers Wald’s Noether-charge entropy for stationary cases and is independent of ambiguities in L, Θ, and Q. The proposal is instantiated in GR, 2D dilaton gravity, and Lovelock gravity, with explicit expressions and consistency checks, though a general second-law proof for S_dyn remains open. The work lays the groundwork for a covariant, horizon-local notion of entropy in a broad class of gravitational theories and motivates further study of dynamical-law constraints.
Abstract
We consider a general, classical theory of gravity with arbitrary matter fields in $n$ dimensions, arising from a diffeomorphism invariant Lagrangian, $\bL$. We first show that $\bL$ always can be written in a ``manifestly covariant" form. We then show that the symplectic potential current $(n-1)$-form, $þ$, and the symplectic current $(n-1)$-form, $\om$, for the theory always can be globally defined in a covariant manner. Associated with any infinitesimal diffeomorphism is a Noether current $(n-1)$-form, $\bJ$, and corresponding Noether charge $(n-2)$-form, $\bQ$. We derive a general ``decomposition formula" for $\bQ$. Using this formula for the Noether charge, we prove that the first law of black hole mechanics holds for arbitrary perturbations of a stationary black hole. (For higher derivative theories, previous arguments had established this law only for stationary perturbations.) Finally, we propose a local, geometrical prescription for the entropy, $S_{dyn}$, of a dynamical black hole. This prescription agrees with the Noether charge formula for stationary black holes and their perturbations, and is independent of all ambiguities associated with the choices of $\bL$, $þ$, and $\bQ$. However, the issue of whether this dynamical entropy in general obeys a ``second law" of black hole mechanics remains open. In an appendix, we apply some of our results to theories with a nondynamical metric and also briefly develop the theory of stress-energy pseudotensors.