Bondi Mass Cannot Become Negative in Higher Dimensions

Abstract

We prove that the Bondi mass of an asymptotically flat, vacuum spacetime cannot become negative in any even dimension \({d \geq 4}\) . 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 it 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 as well as auxiliary spinorial fields. As a by-product, we derive a coordinate expression for the geometrically invariant formula for the Bondi mass originally given by Hollands and Ishibashi.