Some properties of the Noether charge and a proposal for dynamical black hole entropy
1994/03/15 by Vivek Iyer, Robert M. Wald · 70 citations
Physics and Astronomy · #Black Holes and Theoretical Physics #Black hole (networking) #Charge (physics) #Classical mechanics #Computer science #Cosmology and Gravitation Theories #Entropy (arrow of time) #Lagrangian #Mathematical physics #Noether's theorem #Noncommutative and Quantum Gravity Theories #Physics #Quantum mechanics #Statistical physics #Theoretical physics #gr-qc
paper · pdf · doi:10.1103/physrevd.50.846
published as Phys.Rev. D50 (1994) 846-864 · 30 pages, LaTeX
arxiv created 1994/03/15 · openalex publication_date 1994/07/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract
We consider a general, classical theory of gravity with arbitrary matter fields in n dimensions, arising from a diffeomorphism-invariant Lagrangian L. We first show that L alwasy can be written in a ``manifestly covariant'' form. We then show that the symplectic potential current (n-1)-form FTHETA and the symplectic current (n-1)-form \ensuremathω for the theory always can be globally defined in a covariant manner. Associated with any infinitesimal diffeomorphism is a Noether current (n-1)-form J and corresponding Noether charge (n-2)-form Q. We derive a general ``decomposition formula'' for Q. 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 Sdyn 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 L, FTHETA, and Q. 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.
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