Lie Theory for Asymptotic Symmetries in General Relativity: The BMS Group

TL;DR

This paper explores the infinite-dimensional Lie group structure of the BMS group in General Relativity, revealing its regularity properties and laying groundwork for understanding asymptotic symmetries.

Contribution

It constructs and analyzes the Lie group structure of the BMS group, showing its regularity and non-analyticity, and discusses related asymptotic symmetry groups.

Findings

BMS group is regular in Milnor's sense

BMS group is not real analytic

BMS group likely not locally exponential

Abstract

We study the Lie group structure of asymptotic symmetry groups in General Relativity from the viewpoint of infinite-dimensional geometry. To this end, we review the geometric definition of asymptotic simplicity and emptiness due to Penrose and the coordinate-wise definition of asymptotic flatness due to Bondi et al. Then we construct the Lie group structure of the Bondi--Metzner--Sachs (BMS) group and discuss its Lie theoretic properties. We find that the BMS group is regular in the sense of Milnor, but not real analytic. This motivates us to conjecture that it is not locally exponential. Finally, we verify the Trotter property as well as the commutator property. As an outlook, we comment on the situation of related asymptotic symmetry groups. In particular, the much more involved situation of the Newman--Unti group is highlighted, which will be studied in future work.