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

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.

arXiv comment37 pages, article; minor revisions; version to appear in Classical and Quantum Gravity

References in corpus (22)

• Observation of Gravitational Waves from a Binary Black Hole Merger
• On BMS Invariance of Gravitational Scattering
• Aspects of the BMS/CFT correspondence
• BMS supertranslations and Weinberg's soft graviton theorem
• Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited
• Asymptotic symmetries and subleading soft graviton theorem
• Peeling or not peeling -- is that the question ?
• On the structure and applications of the Bondi-Metzner-Sachs group
• A note on the Newman-Unti group and the BMS charge algebra in terms of Newman-Penrose coefficients
• Differentiable mappings on products with different degrees of differentiability in the two factors
• Gravity-Matter Feynman Rules for any Valence
• Regularity properties of infinite-dimensional Lie groups, and semiregularity
• The diffeomorphism group of a non-compact orbifold
• Measurable regularity properties of infinite-dimensional Lie groups
• An introduction to infinite-dimensional differential geometry
• On the Various Extensions of the BMS Group
• Lie groupoids of mappings taking values in a Lie groupoid
• Lie Theory for Asymptotic Symmetries in General Relativity: The BMS Group
• Strong topologies for spaces of smooth maps with infinite-dimensional target
• The Lie group of real analytic diffeomorphisms is not real analytic
• Differentiable vectors and unitary representations of Frechet-Lie supergroups
• The Strong Trotter Property for Locally $\mu$-convex Lie Groups

Cited by in corpus (6)

• Lie Theory for Asymptotic Symmetries in General Relativity: The BMS Group
• Using thermodynamics to learn gravitational wave physics
• Null infinity as a Killing horizon
• Group Contractions via Infinite-Dimensional Lie Theory
• Lectures on the Bondi--Metzner--Sachs group and related topics in infrared physics
• On the unit component of the Newman-Unti group