Quasi-local Energy for Spherically Symmetric Spacetimes

TL;DR

This paper introduces two methods for calculating quasi-local energy in spherically symmetric spacetimes, comparing their effectiveness and properties, including positivity and reference matching techniques.

Contribution

It proposes and tests two complementary approaches for defining quasi-local energy, enhancing understanding of reference choices and energy positivity in spherically symmetric spacetimes.

Findings

Program I can yield positive, zero, or negative energies depending on observer choice.

Program II ensures non-negative energies that vanish only for Minkowski or anti-de Sitter spacetime.

Matching orthonormal frames and extremizing energy are effective strategies for reference selection.

Abstract

We present two complementary approaches for determining the reference for the covariant Hamiltonian boundary term quasi-local energy and test them on spherically symmetric spacetimes. On the one hand, we isometrically match the 2-surface and extremize the energy. This can be done in two ways, which we call programs I (without constraint) and II (with additional constraints). On the other hand, we match the orthonormal 4-frames of the dynamic and the reference spacetimes. Then, if we further specify the observer by requiring the reference displacement to be the timelike Killing vector of the reference, the result is the same as program I, and the energy can be positive, zero, or even negative. If, instead, we require that the Lie derivatives of the two-area along the displacement vector in both the dynamic and reference spacetimes to be the same, the result is the same as program II, and it satisfies the usual criteria: the energies are non-negative and vanish only for Minkowski (or anti-de Sitter) spacetime.