Noether charges and black hole mechanics in Einstein-aether theory

TL;DR

This paper applies the Noether charge method to Einstein-aether theory to derive expressions for conserved quantities and analyze black hole mechanics, revealing limitations in deriving the first law of thermodynamics due to singularities at the horizon.

Contribution

It extends the Noether charge approach to Einstein-aether theory and investigates black hole thermodynamics within this framework, highlighting challenges in defining entropy.

Findings

Expressions for energy, momentum, and angular momentum in Einstein-aether spacetime.

The first law of black hole thermodynamics cannot be derived due to horizon singularities.

A general identity relating energy and angular momentum variations to horizon surface integrals is established.

Abstract

The Noether charge method for defining the Hamiltonian of a diffeomorphism-invariant field theory is applied to "Einstein-aether" theory, in which gravity couples to a dynamical, timelike, unit-norm vector field. Using the method, expressions are obtained for the total energy, momentum, and angular momentum of an Einstein-aether space-time. The method is also used to discuss the mechanics of Einstein-aether black holes. The derivation of Wald, and Iyer and Wald, of the first law of black hole thermodynamics fails for this theory, because the unit vector is necessarily singular at the bifurcation surface of the Killing horizon. A general identity relating variations of energy and angular momentum to a surface integral at the horizon is obtained, but a thermodynamic interpretation, including a definitive expression for the black hole entropy, is not found.