A generalization of the ADM mass for asymptotically Euclidean manifolds of weak regularity

TL;DR

This paper introduces a new, more flexible definition of the ADM mass for asymptotically Euclidean manifolds with weak regularity, extending its applicability to less smooth metrics while maintaining key properties.

Contribution

It proposes a generalized ADM mass definition for metrics with Sobolev regularity, ensuring finiteness, invariance, and consistency with classical definitions.

Findings

Mass is finite under suitable asymptotic conditions.

Mass is invariant under coordinate changes at infinity.

New expression in terms of Ricci tensor matches existing Ricci-based ADM mass.

Abstract

We propose a new definition of the ADM mass for asymptotically Euclidean manifolds inspired by the definition of mass for weakly regular asymptotically hyperbolic manifolds by Gicquaud and Sakovich. This version of the mass allows one to work with metrics of local Sobolev regularity $ W^{1,2}_\text{loc} \cap L^\infty $ and we show, under suitable asymptotic assumptions, that the mass is finite, invariant under a change of coordinates at infinity and that it agrees with the classical ADM mass in the smooth setting. We also provide an expression in terms of the Ricci tensor that agrees with the Ricci version of the ADM mass studied by Herzlich.