On quasi-local Hamiltonians in General Relativity

TL;DR

This paper examines the definition of quasi-local energy in General Relativity through Hamiltonian analysis, emphasizing the importance of boundary constraints and the need for a well-posed initial boundary value problem framework.

Contribution

It highlights the critical role of boundary constraints, especially the Hamiltonian constraint, in defining quasi-local energy in GR, and discusses the necessity of a well-posed boundary value problem.

Findings

Boundary constraints are essential for quasi-local energy definitions.

Neglecting the Hamiltonian constraint leads to incomplete formulations.

A well-posed initial boundary value problem is crucial for consistent energy definitions.

Abstract

We analyse the definition of quasi-local energy in GR based on a Hamiltonian analysis of the Einstein-Hilbert action initiated by Brown-York. The role of the constraint equations, in particular the Hamiltonian constraint on the timelike boundary, neglected in previous studies, is emphasized here. We argue that a consistent definition of quasi-local energy in GR requires, at a minimum, a framework based on the (currently unknown) geometric well-posedness of the initial boundary value problem for the Einstein equations.