Weyl-transverse gravity with boundaries

TL;DR

Weyl-transverse gravity (WTG) modifies GR by restricting gauge symmetry to Weyl transformations and transverse diffeomorphisms, with a fixed background volume form $\bm{\omega}$. The authors develop a covariant phase-space formulation for WTG in the presence of timelike and spacelike boundaries, derive the symplectic structure, and classify boundary conditions for differentiability, including Dirichlet, Neumann, and York-type data on both the auxiliary metric $\tilde{g}_{\mu\nu}$ and the dynamical metric $g_{\mu\nu}$. They construct the Noether current and Hamiltonian generators for transverse diffeomorphisms, showing that the Noether charge reduces to the GR charge plus a term involving the difference between the cosmological constant $\Lambda$ and the Lagrangian ambiguity $\lambda$, and they obtain a first-law relation on spacetimes with bifurcate Killing horizons that includes a $\delta\Lambda$ contribution. The results provide a clear covariant phase-space perspective on conserved quantities in WTG and reveal how boundary conditions and the fixed volume form influence black-hole thermodynamics and its connection to unimodular-like formulations of gravity.

Abstract

We develop the covariant phase space formulation of Weyl-transverse gravity (WTG) in the presence of general timelike and spacelike boundaries. WTG is classically equivalent to General Relativity (GR) but possesses a reduced gauge symmetry consisting of Weyl transformations and transverse diffeomorphisms, together with a fixed background volume form. This structure modifies the variational principle and the definition of conserved quantities relative to GR. We derive the symplectic potential, presymplectic current, and Hamiltonian generators associated with transverse diffeomorphisms, and we identify a set of boundary conditions under which the WTG action is differentiable. These include Dirichlet and Neumann conditions for both the auxiliary Weyl-invariant metric and the dynamical metric, as well as a natural implementation of York boundary conditions, for which WTG exhibits a particularly transparent geometric formulation. We obtain the Noether current and surface charge, clarify the role of the Lagrangian ambiguity related to the cosmological constant, and evaluate the Hamiltonian identity on spacetimes containing a bifurcate Killing horizon. The resulting first-law relation shows that variations of the cosmological constant can contribute nontrivially unless additional physical restrictions are imposed.