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

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.

arXiv comment29 pages, article; minor revisions; version to appear in Classical and Quantum Gravity

References in corpus (25)

• 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
• BMS charge algebra
• Asymptotic symmetries and subleading soft graviton theorem
• Gravitational waves and neutrino emission from the merger of binary neutron stars
• New symmetries for the Gravitational S-matrix
• Bondi-Sachs Formalism
• Asymptotic symmetries of gravity and soft theorems for massive particles
• Peeling or not peeling -- is that the question ?
• On the structure and applications of the Bondi-Metzner-Sachs group
• 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 Conformal BMS Group
• The diffeomorphism group of a non-compact orbifold
• Measurable regularity properties of infinite-dimensional Lie groups
• On the Various Extensions of the BMS Group
• Lie groupoids of mappings taking values in a Lie groupoid
• On the representation theory of the Bondi-Metzner-Sachs group and its variants in three space-time dimensions
• The Lie group of real analytic diffeomorphisms is not real analytic
• Lie Theory for Asymptotic Symmetries in General Relativity: The NU Group
• The Strong Trotter Property for Locally $\mu$-convex Lie Groups

Cited by in corpus (7)

• Lie Theory for Asymptotic Symmetries in General Relativity: The NU Group
• On the nature of Bondi-Metzner-Sachs transformations
• Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms
• On the singularities of the exponential function of a semidirect product
• Null infinity as a Killing horizon
• On the unit component of the Newman-Unti group
• Group Contractions via Infinite-Dimensional Lie Theory