The Hamiltonian boundary term and quasi-local energy flux

TL;DR

This paper explores the Hamiltonian boundary terms in gravitating regions, deriving quasi-local energy flux expressions, distinguishing gauge-invariant electromagnetic energy, and connecting boundary conditions to well-known energy measures in general relativity.

Contribution

It introduces four covariant Hamiltonian boundary term expressions, linking them to boundary conditions and deriving associated energy fluxes in electromagnetism and general relativity.

Findings

Identifies four quasi-local energy-momentum boundary terms with distinct boundary conditions.

Derives energy flux expressions from Hamiltonian identities, including electromagnetic Poynting flux.

Connects boundary conditions to ADM and Trautman-Bondi energies in asymptotic regimes.

Abstract

The Hamiltonian for a gravitating region includes a boundary term which determines not only the quasi-local values but also, via the boundary variation principle, the boundary conditions. Using our covariant Hamiltonian formalism, we found four particular quasi-local energy-momentum boundary term expressions; each corresponds to a physically distinct and geometrically clear boundary condition. Here, from a consideration of the asymptotics, we show how a fundamental Hamiltonian identity naturally leads to the associated quasi-local energy flux expressions. For electromagnetism one of the four is distinguished: the only one which is gauge invariant; it gives the familiar energy density and Poynting flux. For Einstein's general relativity two different boundary condition choices correspond to quasi-local expressions which asymptotically give the ADM energy, the Trautman-Bondi energy and, moreover, an associated energy flux (both outgoing and incoming). Again there is a distinguished expression: the one which is covariant.