On a class of 2-surface observables in general relativity

TL;DR

This paper explores boundary conditions in vacuum general relativity at the quasi-local level, showing that fixing the area element on a 2-surface yields well-defined algebraic structures and gauge-invariant observables.

Contribution

It introduces a boundary condition fixing the area element on 2-surfaces, leading to well-defined constraint and Poisson algebras with covariant, gauge-invariant observables.

Findings

Fixing the area element ensures a well-defined constraint algebra.

The basic Hamiltonian yields covariant, gauge-invariant 2-surface observables.

Evolution equations preserve the boundary conditions.

Abstract

The boundary conditions for canonical vacuum general relativity is investigated at the quasi-local level. It is shown that fixing the area element on the 2- surface S (rather than the induced 2-metric) is enough to have a well defined constraint algebra, and a well defined Poisson algebra of basic Hamiltonians parameterized by shifts that are tangent to and divergence-free on $. The evolution equations preserve these boundary conditions and the value of the basic Hamiltonian gives 2+2 covariant, gauge-invariant 2-surface observables. The meaning of these observables is also discussed.