Quasi-local gravitational angular momentum and centre of mass from generalised Witten equations

TL;DR

This paper extends Witten's spinor-based approach to define quasi-local gravitational angular momentum and centre of mass, providing new integral formulas and conservation laws at spacetime boundaries.

Contribution

It generalizes Witten's equations to include angular momentum and centre of mass, leading to quasi-local expressions and conservation laws for gravitational charges.

Findings

Derived integral formulas for angular momentum and centre of mass.

Extended differential equations to arbitrary boundary signatures.

Established conservation laws for matter and gravitational radiation flux.

Abstract

Witten's proof for the positivity of the ADM mass gives a definition of energy in terms of three-surface spinors. In this paper, we give a generalisation for the remaining six Poincar\'e charges at spacelike infinity, which are the angular momentum and centre of mass. The construction improves on certain three-surface spinor equations introduced by Shaw. We solve these equations asymptotically obtaining the ten Poincar\'e charges as integrals over the Nester--Witten two-form. We point out that the defining differential equations can be extended to three-surfaces of arbitrary signature and we study them on the entire boundary of a compact four-dimensional region of spacetime. The resulting quasi-local expressions for energy and angular momentum are integrals over a two-dimensional cross-section of the boundary. For any two consecutive such cross-sections, conservation laws are derived that determine the influx (outflow) of matter and gravitational radiation.