Evaluating quasilocal energy and solving optimal embedding equation at null infinity

TL;DR

This paper investigates the behavior of quasilocal energy near null infinity in asymptotically flat spacetimes, demonstrating Lorentzian symmetry recovery and deriving an energy-momentum vector consistent with Bondi-Sachs, while analyzing the solvability of the embedding equation.

Contribution

It introduces a method to analyze quasilocal energy limits at null infinity and solves the optimal embedding equation, establishing a connection with Bondi-Sachs energy-momentum.

Findings

Lorentzian symmetry is recovered at null infinity.

An energy-momentum 4-vector consistent with Bondi-Sachs is obtained.

The optimal embedding equation is solvable under analyticity assumptions.

Abstract

We study the limit of quasilocal energy defined in [7] and [8] for a family of spacelike 2-surfaces approaching null infinity of an asymptotically flat spacetime. It is shown that Lorentzian symmetry is recovered and an energy-momentum 4-vector is obtained. In particular, the result is consistent with the Bondi-Sachs energy-momentum at a retarded time. The quasilocal mass in [7] and [8] is defined by minimizing quasilocal energy among admissible isometric embeddings and observers. The solvability of the Euler-Lagrange equation for this variational problem is also discussed in both the asymptotically flat and asymptotically null cases. Assuming analyticity, the equation can be solved and the solution is locally minimizing in all orders. In particular, this produces an optimal reference hypersurface in the Minkowski space for the spatial or null exterior region of an asymptotically flat spacetime.