Weyl-transverse gravity with boundaries

TL;DR

This paper formulates Weyl-transverse gravity with boundaries, clarifying its boundary conditions, conserved charges, and the role of the cosmological constant, revealing differences from General Relativity in a covariant phase space framework.

Contribution

It develops the covariant phase space formalism for Weyl-transverse gravity with boundaries, detailing boundary conditions, conserved quantities, and the impact of the cosmological constant.

Findings

Derived symplectic potential and Hamiltonian generators.

Identified boundary conditions for differentiability of the action.

Analyzed the first-law relation and cosmological constant variations.

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.