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.

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

References in corpus (4)

• Asymptotic symmetries and subleading soft graviton theorem
• New symmetries for the Gravitational S-matrix
• On the Various Extensions of the BMS Group
• Lie Theory for Asymptotic Symmetries in General Relativity: The NU Group

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