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Covariant phase space approach to Einstein-æther gravity

Walter Arata*, Stefano Liberati†, and Giulio Neri‡

  • SISSA, via Bonomea 265, 34136 Trieste, Italy, INFN (Sez. Trieste), via Valerio 2, 34127 Trieste, Italy, and IFPU, via Beirut 2, 34014 Trieste, Italy

  • *Contact author: warata@sissa.it
  • †Contact author: liberati@sissa.it
  • ‡Contact author: gneri@sissa.it
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Phys. Rev. D 114, 084071 – Published 22 October, 2026

DOI: https://doi.org/10.1103/gwmt-ywwy

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–æther 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 s, we introduce a disformal frame in which the corresponding s-horizon is a Killing horizon, so that a standard Wald-type derivation can be carried out. We then map the result back to the original (horizon) frame, showing that the first law retains the same formal structure. The associated horizon charge contains, besides the usual geometric (Komar/Wald) contribution, an irreducible æther term that admits an interpretation as an additional entropy sourced by the æther flux across the horizon; accordingly, the total entropy splits into a gravitational part and an æther part. We further develop an extended-thermodynamics framework in which the couplings (and Λ) are allowed to vary, obtaining generalized Smarr relations. Finally, we analyze the probe-mode limit cs→+∞, 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 a nonarea entropy. Once the independent æther entropy contribution is properly taken into account, the two prescriptions are reconciled.

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Phys. Rev. D 114, 084071– Published 22 October, 2026
  • Received 14 April 2026
  • Accepted 17 August 2026
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