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

TL;DR

This paper introduces a method to define the energy of isolated systems at retarded times using the null limit of quasilocal energy, aligning with the Bondi-Sachs mass, and relates it to Hamiltonian formulations and supertranslations.

Contribution

It establishes a new Hamiltonian-based approach to define system energy at null infinity, connecting quasilocal energy, Bondi-Sachs mass, and supertranslations.

Findings

The null limit of the Hamiltonian yields the Bondi-Sachs mass.

The approach links Hamiltonian boundary values to the full Bondi-Sachs four-momentum.

The method applies to isolated systems at retarded times using quasilocal energy.

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 $\Sigma$ contained within a finite topologically spherical boundary $B = \partial \Sigma$. 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.