Optimal Choices of Reference for a Quasi-local Energy: Spherically Symmetric Spacetimes

TL;DR

This paper introduces a method for selecting an optimal reference spacetime for calculating quasi-local energy in spherically symmetric spacetimes by embedding boundary surfaces into Minkowski space and extremizing the energy, ensuring unique and physically meaningful energy values.

Contribution

The authors propose a novel embedding and extremization approach to determine the reference spacetime for quasi-local energy, providing unique and physically consistent energy measures in spherically symmetric cases.

Findings

Unique energy values for Schwarzschild spacetime for each displacement vector.

Maximum energy measured by static observers.

Nonnegative maximum energy in FLRW spacetime with specific displacement vector.

Abstract

For a given timelike displacement vector the covariant Hamiltonian quasi-local energy expression requires a proper choice of reference spacetime. We propose a program for determining the reference by embedding a neighborhood of the two-sphere boundary in the dynamic spacetime into a Minkowski reference, so that the two sphere is embedded isometrically, and then extremizing the energy to determine the embedding variables. Applying this idea to Schwarzschild spacetime, we found that for each given future timelike displacement vector our program gives a unique energy value. The static observer measures the maximal energy. Applied to the Friedmann-Lemaitre-Robertson-Walker spacetime, we find that the maximum energy value is nonnegative; the associated displacement vector is the unit dual mean curvature vector, and the expansion of the two-sphere boundary matches that of its reference image. For these spherically symmetric cases the reference determined by our program is equivalent to isometrically matching the geometry at the two-sphere boundary and taking the displacement vector to be orthogonal to the spacelike constant coordinate time hypersurface, like the timelike Killing vector of the Minkowski reference.