An Optimal Choice of Reference for the Quasi-Local Gravitational Energy and Angular Momentum

TL;DR

This paper develops a method to determine the optimal reference and quasi-Killing vectors for calculating quasi-local gravitational energy and angular momentum, providing explicit solutions for the Kerr spacetime.

Contribution

It introduces a four-dimensional isometric matching and energy extremization approach to find optimal references and vectors in axisymmetric spacetimes, especially Kerr.

Findings

Explicit solutions for Kerr metric reference and vectors.

Exact expressions for quasi-local energy and angular momentum.

Validation of the method in axisymmetric cases.

Abstract

The boundary term of the gravitational Hamiltonian can be used to give the values of the quasi-local quantities as long as one can provide a suitable evolution vector field and an appropriate reference. On the two-surface boundary of a region we have proposed using {\em four dimensional isometric matching} between the dynamic spacetime and the reference geometry along with energy extremization to find both the optimal reference matching and the appropriate quasi-Killing vectors. Here we consider the axisymmetric spacetime case. For the Kerr metric in particular we can explicitly solve the equations to find the best matched reference and quasi-Killing vectors. This leads to the exact expression for the quasi-local boundary term and the values of our optimal quasi-local energy and angular momentum.