Some properties of the Noether charge and a proposal for dynamical black hole entropy
Abstract
We consider a general, classical theory of gravity with arbitrary matter fields in n dimensions, arising from a diffeomorphisminvariant Lagrangian L. We first show that L alwasy can be written in a ``manifestly covariant'' form. We then show that the symplectic potential current (n1)form FTHETA and the symplectic current (n1)form ω for the theory always can be globally defined in a covariant manner. Associated with any infinitesimal diffeomorphism is a Noether current (n1)form J and corresponding Noether charge (n2)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 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 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 stressenergy pseudotensors.
 Publication:

Physical Review D
 Pub Date:
 July 1994
 DOI:
 10.1103/PhysRevD.50.846
 arXiv:
 arXiv:grqc/9403028
 Bibcode:
 1994PhRvD..50..846I
 Keywords:

 04.20.Fy;
 97.60.Lf;
 Canonical formalism Lagrangians and variational principles;
 Black holes;
 General Relativity and Quantum Cosmology
 EPrint:
 30 pages, LaTeX