Positivity Bounds for the Y-ADM Mass Density

TL;DR

This paper extends spinorial techniques to establish positivity bounds for Y-ADM mass density in p-brane spacetimes, revealing conditions under which the mass density is positive, thus generalizing the positive energy theorem.

Contribution

It introduces a formalism for proving positivity of Y-ADM mass density using spinorial methods, including new conditions on the Weyl tensor and energy conditions for specific spacetime classes.

Findings

Positivity bounds are established for Y-ADM mass density.

Conditions on the Weyl tensor are necessary for positivity.

The formalism applies to conformally flat, algebraically special, and Killing vector spacetimes.

Abstract

Killing-Yano tensors are natural generalizations of Killing vectors to arbitrary rank anti-symmetric tensor fields. It was recently shown that Killing-Yano tensors lead to conserved gravitational charges, called Y-ADM charges. These new charges are interesting because they measure, for example, the mass density of a p-brane, rather than the total ADM mass which may be infinite. In this paper, we show that the spinorial techniques used by Witten, in his proof of the positive energy theorem, may be straightforwardly extended to study the positivity properties of the Y-ADM mass density for p-brane spacetimes. Although the resulting formalism is quite similar to the ADM case, we show that establishing a positivity bound in the higher rank Y-ADM case requires imposing a condition on the Weyl tensor in addition to an energy condition. We find appropriate energy conditions for spacetimes that are conformally flat or algebraically special, and for spacetimes that have an exact Killing vector along the brane. Finally we discuss our expression for the Y-ADM mass density from the Hamiltonian point of view.