A Covariant Phase Space Approach to Einstein-AEther Gravity

TL;DR

This paper develops a covariant phase space formalism for Einstein-Aether gravity, deriving a consistent black hole first law that includes both gravitational and Aether contributions, and clarifies thermodynamic relations in Lorentz-violating contexts.

Contribution

It introduces a disformal frame approach to handle different mode speeds, deriving a generalized first law and thermodynamics for black holes in Einstein-Aether gravity.

Findings

Derived a Wald-type first law including Aether entropy contribution.

Established a connection between universal horizon thermodynamics and mode propagation speeds.

Reconciled different approaches to black hole entropy and temperature in Lorentz-violating gravity.

Abstract

Black hole thermodynamics in Lorentz-violating gravity is subtle because different excitations propagate at different speeds and hence identify different causal horizons. We revisit Einstein--AEther gravity using the covariant phase space formalism with boundaries and derive a consistent first law for stationary black holes. For a mode of propagation speed $c_s$, we introduce a disformal frame in which the corresponding causal horizon is a Killing horizon, so that the standard Wald-type derivation can be carried out. The result is then mapped back to the original frame, where it mantains the same structure. The associated horizon charge contains, besides the usual Komar term, an irreducible entropic AEther contribution that can be interpreted as heat due to the AEther flux across the horizon; accordingly, the total entropy splits into a gravitational part and an AEther part. We further develop an extended-thermodynamics framework in which the couplings of the theory are allowed to vary, obtaining generalized Smarr relations. Finally, we analyze the probe-mode limit $c_s \to +\infty$, clarifying its connection to universal-horizon thermodynamics and resolving the apparent tension in the literature between approaches that (i) fix the entropy to be proportional to the area and infer a corresponding temperature, and (ii) impose the Hawking temperature associated with modes peeling from the universal horizon and infer the entropy. Once the independent AEther contribution is properly taken into account, the two prescriptions are reconciled.