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

TL;DR

This paper investigates the infinite-dimensional geometric structure of the Newman-Unti (NU) group in general relativity, revealing it cannot be a Lie group in the natural topology but becomes one under a finer topology, with implications for its subgroup BMS.

Contribution

It introduces a Whitney-type topology to give the NU group an infinite-dimensional Lie group structure and analyzes its properties and relation to the BMS group.

Findings

NU group is a topological group but not a Lie group in natural topology.

Finer topology makes the NU group into an infinite-dimensional Lie group.

The full NU group's operations are discontinuous, preventing a Lie group structure.

Abstract

We study the Newman--Unti (NU) group from the viewpoint of infinite-dimensional geometry. The NU group is a topological group in a natural coarse topology, but it does not become a manifold and hence a Lie group in this topology. To obtain a manifold structure we consider a finer Whitney-type topology. This turns the unit component of the NU group into an infinite-dimensional Lie group. We then study the Lie theoretic properties of this group. Surprisingly, the group operations of the full NU group become discontinuous, whence the NU group does not support a Lie group structure. The NU group contains the Bondi--Metzner--Sachs (BMS) group as a subgroup, whose Lie group structure was constructed in a previous article. It is well known that the NU Lie algebra splits into a direct sum of Lie ideals of the Lie algebras of the BMS group and conformal rescalings of scri. However, the lack of a Lie group structure on the NU group implies that the BMS group cannot be embedded as a Lie subgroup therein.