Bondi mass cannot become negative in higher dimensions

TL;DR

The work proves that the Bondi mass cannot become negative in any even dimension d ≥ 4 for asymptotically flat vacuum spacetimes under broad geometric and spin assumptions. It introduces conformal Gaussian null coordinates to obtain precise asymptotic expansions of the metric and Bondi quantities, derives an explicit Bondi-mass formula and a mass-loss law, and then uses a Witten spinor construction to establish positivity. The results extend classical d = 4 positivity proofs to higher dimensions and to general null infinities admitting a real Killing spinor on Σ, highlighting a deep link between geometry, spinors, and gravitational radiation. The paper also outlines open directions, including odd dimensions, non-spherical infinities, and connections to asymptotic symmetries.

Abstract

We prove that the Bondi mass of an asymptotically flat, vacuum, spacetime cannot become negative in any even dimension $d \ge 4$. The notion of Bondi mass is more subtle in $d > 4$ dimensions because radiating metrics have a slower decay than stationary ones, and those subtleties are reflected by a considerably more difficult proof of positivity. Our proof holds for the standard spherical infinities, but also extends to infinities of more general type which are $(d-2)$-dimensional manifolds admitting a real Killing spinor. Such manifolds typically have special holonomy and Sasakian structures. The main technical advance of the paper is an expansion technique based on "conformal Gaussian null coordinates". This expansion helps us to understand the consequences imposed by Einstein's equations on the asymptotic tail of the metric field. As a by-product, we derive a coordinate expression for the geometrically invariant formula for the Bondi mass originally given by Hollands and Ishibashi.

Theorems & Definitions (10)

• Theorem 1
• Remark 2
• Theorem 3
• Remark 4
• Remark 5
• Theorem 6
• Remark 7
• Lemma 8
• Lemma 9
• Lemma 10