A Positivity Theorem for Gravitational Tension in Brane Spacetimes

TL;DR

The paper addresses the positivity of the gravitational tension $\mu$ in transverse-asymptotically flat $p$-brane spacetimes by extending the Witten–Nester positive energy framework to a co-dimension two setting. It introduces a Dirac–Witten spinor equation on $V_{tx}$ slices and a Nester form to express $\mu$ as a boundary/volume integral, proving the purely gravitational contribution is positive definite and clarifying how stress-energy affects the total tension. The results establish an analogue of the positive mass theorem for brane tensions and illuminate the relationship between mass and tension under static or boost-invariant conditions, with horizons and nonuniform static configurations identified as avenues for future work.

Abstract

We study transverse asymptotically flat spacetimes without horizons that arise from brane matter sources. We assume that asymptotically there is a spatial translation Killing vector that is tangent to the brane. Such spacetimes are characterized by a tension, analogous to the ADM mass, which is a gravitational charge associated with the asymptotic spatial translation Killing vector. Using spinor techniques, we prove that the purely gravitational contribution to the spacetime tension is positive definite.