Mass, gauge conditions and spectral properties of the Sen-Witten and 3-surface twistor operators in closed universes

TL;DR

This paper introduces a positive definite measure related to the Sen-Witten and 3-surface twistor operators that quantifies gravitational field strength and total mass in closed universes, linking spectral properties to spacetime geometry.

Contribution

It establishes a new non-negative expression connecting spectral properties of operators with mass bounds and gauge conditions in closed universes, extending previous asymptotic results.

Findings

Provides a lower bound for ADM/Bondi--Sachs mass in asymptotic cases.

Shows the expression's vanishing characterizes flat spacetime with toroidal topology.

Identifies the first eigenvalue of the Sen--Witten operator as a measure of gravitational strength.

Abstract

A non-negative expression, built from the norm of the 3-surface twistor operator and the energy-momentum tensor of the matter fields on a spacelike hypersurface, is found which, in the asymptotically flat/hyperboloidal case, provides a lower bound for the ADM/Bondi--Sachs mass, while on closed hypersurfaces it gives the first eigenvalue of the Sen--Witten operator. Also in the closed case, its vanishing is equivalent to the existence of non-trivial solutions of Witten's gauge condition. Moreover, it is vanishing if and only if the closed data set is in a flat spacetime with toroidal spatial topology. Thus it provides a positive definite measure of the strength of the gravitational field (with physical dimension mass) on closed hypersurfaces, i.e. some sort of the total mass of closed universes.