A lower bound for the eigenvalues of the Sen-Witten operator on closed spacelike hypersurfaces

TL;DR

This paper establishes a precise lower bound for the eigenvalues of the Sen-Witten operator on closed spacelike hypersurfaces, linking it to the energy-momentum densities of matter and gravity in general relativity.

Contribution

It provides the first sharp lower bound for the eigenvalues of the Sen-Witten operator based on the Einstein tensor's constraint components.

Findings

Eigenvalues relate exactly to the total energy-momentum integral.

A sharp lower bound is derived in terms of matter energy and momentum densities.

The results connect spectral properties to physical energy conditions in spacetime.

Abstract

The eigenvalue problem for the Sen--Witten operator on closed spacelike hypersurfaces is investigated. The (square of its) eigenvalues are shown to be given exactly by the 3-surface integral appearing in the expression of the total energy-momentum of the matter+gravity systems in Witten's energy positivity proof. A sharp lower bound for the eigenvalues, given in terms of the constraint parts of the spacetime Einstein tensor, i.e. the energy and momentum densities of the matter fields, is given.