Hamiltonian and Noether charges in first order gravity

TL;DR

The paper analyzes gravity in four dimensions using the first-order vielbein formulation with boundaries, employing the covariant phase space formalism to study how boundary and topological terms affect conserved quantities. It proves that adding topological/boundary terms does not modify the symplectic structure (hence the Hamiltonian charges remain unchanged), but Noether charges do change with these terms, revealing a nontrivial relation between Hamiltonian and Noether charges in bounded spacetimes. The authors provide explicit calculations for asymptotically flat spacetimes with isolated horizons, showing that ADM energy emerges from Palatini boundary contributions, while Noether charges for Holst and topological terms can differ, particularly at horizons, without altering the Hamiltonian energy. These findings clarify longstanding ambiguities about the correspondence between Hamiltonian generators and Noether charges in gravity and highlight the subtle role boundary conditions play in the covariant Hamiltonian framework.

Abstract

We consider gravity in four dimensions in the vielbein formulation, where the fundamental variables are a tetrad $e$ and a SO(3,1) connection $ω$. We start with the most general action principle compatible with diffeomorphism invariance which includes, besides the standard Palatini term, other terms that either do not change the equations of motion, or are topological in nature. For our analysis we employ the covariant Hamiltonian formalism where the phase space $Γ$ is given by solutions to the equations of motion. We consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. For this extended action we study the effect of the topological terms on the Hamiltonian formulation. We prove two results. The first one is rather generic, applicable to any field theory with boundaries: The addition of topological terms (and any other boundary term) does not modify the symplectic structure of the theory. The second result pertains to the conserved Hamiltonian and Noether charges, whose properties we analyze in detail, including their relationship. While the Hamiltonian charges are unaffected by the addition of topological and boundary terms, we show in detail that the Noether charges {\em do} change. Thus, a non-trivial relation between these two sets of charges arises when the boundary and topological terms needed for a consistent formulation are included.

Figures (1)

• Figure 1: The region $\mathcal{M}$.