A generalization of the ADM mass for asymptotically Euclidean manifolds of weak regularity
Stig Lundgren, Benjamin Meco
TL;DR
This work extends the notion of ADM mass to asymptotically Euclidean manifolds with very low metric regularity by defining a weak mass $m_W$ via bulk integrals with a family of cutoff functions, mirroring the hyperbolic weak mass construction of Gicquaud–Sakovich. It proves the weak mass is finite, coordinate-invariant, and agrees with the classical ADM mass when the metric is smooth enough, while also providing a Ricci-tensor expression that aligns with the Ricci ADM definitions of Herzlich and Miao–Tam. The paper further develops a Ricci-based weak mass $m_{RW}$ and shows $m_W = m_{RW}$ under appropriate $W^{2,2}$-type regularity, unifying scalar- and Ricci-driven notions of mass in low regularity. Overall, the results enable robust mass concepts for impulsive gravitational phenomena and other low-regularity geometries, while preserving consistency with classical definitions in the smooth limit.
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.