The Trautman-Bondi mass of initial data sets

TL;DR

This paper introduces a geometric definition of mass for initial data sets in general relativity that accommodates gravitational radiation and polyhomogeneous compactifications, proving positivity under certain conditions.

Contribution

It provides a new, invariant mass definition for conformally compactifiable initial data sets compatible with gravitational radiation and establishes positivity in specific cases.

Findings

Mass is a geometric invariant.

Positivity of mass is proven for spherical conformal infinity.

Mass can be expressed in terms of lower-dimensional objects when certain curvature conditions hold.

Abstract

We give a definition of mass for conformally compactifiable initial data sets. The asymptotic conditions are compatible with existence of gravitational radiation, and the compactifications are allowed to be polyhomogeneous. We show that the resulting mass is a geometric invariant, and we prove positivity thereof in the case of a spherical conformal infinity. When R(g) - or, equivalently, the trace of the extrinsic curvature tensor - tends to a negative constant to order one at infinity, the definition is expressed purely in terms of three-dimensional or two-dimensional objects.