Energy of Isolated Systems at Retarded Times as the Null Limit of Quasilocal Energy

TL;DR

The paper defines the energy of a perfectly isolated system at a retarded time as the null limit of the quasilocal energy, showing it equals the Bondi-Sachs mass. It then analyzes smeared Hamiltonians with vanishing shift to identify a dual-space element to supertranslations, i.e., the supermomentum, and demonstrates how the full Bondi-Sachs four-momentum emerges as a Hamiltonian charge. The results unify quasilocal energy with standard asymptotic energy-momentum notions in general relativity and clarify the role of the BMS group in a Hamiltonian framework. This provides a coherent Hamiltonian basis for Bondi-Sachs quantities and the Geroch–Dray–Streubel supermomentum concept.

Abstract

We define the energy of a perfectly isolated system at a given retarded time as the suitable null limit of the quasilocal energy $E$. The result coincides with the Bondi-Sachs mass. Our $E$ is the lapse-unity shift-zero boundary value of the gravitational Hamiltonian appropriate for the partial system $Σ$ contained within a finite topologically spherical boundary $B = \partial Σ$. Moreover, we show that with an arbitrary lapse and zero shift the same null limit of the Hamiltonian defines a physically meaningful element in the space dual to supertranslations. This result is specialized to yield an expression for the full Bondi-Sachs four-momentum in terms of Hamiltonian values.

Figures (1)

• Figure 1: Geometry of the asymptotic slice $\Sigma_{\infty}$