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A Covariant Phase Space Approach to Einstein-AEther Gravity

Walter Arata, Stefano Liberati, Giulio Neri

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.

A Covariant Phase Space Approach to Einstein-AEther 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 , 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 , 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.

Paper Structure

This paper contains 38 sections, 4 theorems, 219 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and Conventions
  3. Covariant Phase Space Formalism
  4. Field Space
  5. Dynamics
  6. Symmetries and Noether Charges
  7. Diffeomorphism Invariance and Covariant Lagrangians
  8. First Law of Black Hole Thermodynamics and Smarr Formula
  9. Scale Invariance, Extended Thermodynamics, and Smarr Formula
  10. Einstein--Æther Theory
  11. Relation with the Infrared Limit of Hořava--Lifshitz Gravity
  12. Universal Horizons
  13. Covariant Phase Space Analysis of Einstein--Æther theory
  14. Diffeomorphism Charges for Einstein--Æther
  15. Boundary Conditions for Einstein--Æther Gravity from Khronometric Theory
  16. ...and 23 more sections

Key Result

Proposition 2.1

The canonical generator defined in Eq. eq: Hamiltonian does not depend on the slice of integration $\Sigma$. Moreover the vertical variation of this quantity is related to the symplectic form contraction as in Eq. eq: vOmega = -dH. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 2.1: Representation of the cylindrical spacetime under consideration, with its different parts: the lateral boundary $\Gamma$, the two lids $\Sigma_i$, the two corners $\partial\Sigma_i$ and a generic slice $\Sigma$.
  • Figure 3.1: Spacetime in presence of an internal boundary $H$
  • Figure 4.1: Carter--Penrose diagram for a Schwarzschild-like solution featuring a universal horizon and three constant-Khronon slices.
  • Figure 6.1: Æther heat flux entering the horizon $H$ of the s-mode in the disformal frame.

Theorems & Definitions (7)

  • Proposition 2.1
  • Proposition
  • proof
  • Proposition A.1
  • proof
  • Proposition
  • proof