The Poincaré group as the symmetry group of canonical general relativity
type: Item
instance of: scholarly article
title: The Poincaré group as the symmetry group of canonical general relativity
zbMATH Open document ID: 0617.70021
DOI: 10.1016/0003-4916(87)90037-6
author: Niall Ó. Murchadha
author: Robert Beig
published in: Annals of Physics
publication date: 1987-01-01T00:00:00Z
review text: \textit{T. Regge} and \textit{C. Teitelboim} have in a well known paper [ibid. 88, 286-318 (1974; Zbl 0328.70016)] that the Poincaré group acts as a symmetry group on the phase space of general relativity in the asymptotically flat case. In the present paper the authors further elaborate on this important result and on the exact form of the boundary conditions at spatial infinity needed to obtain it. They use the language of symplectic geometry in index notation. The paper is organized as follows: Section 1 is of introductory character including a review on the historical development of the subject. Section 2 presents some material concerning the reduction of phase space in the case of a theory with first class constraints, and with symmetries including gauge freedom. Section 3 applies these concepts to the dynamics of asymptotically flat general relativistic spacetimes thereby reproducing the Regge-Teitelboim result mentioned above. The authors emphasize the importance of so called ''parity conditions'' (i.e., behavior of the canonical variables \(g_{ij}\) and \(\pi^{ij}\) under reflection to the leading order in 1/r) in order to get finite integrals. The last two sections are concerned with the relationship of this work to the ''spi formulation'' (''spi'' stands for ''spatial infinity'') due to \textit{A. Ashtekar} and \textit{R. Hansen} [J. Math. Phys. 19, 1542-1566 (1978)].
reviewed by: Volker Perlick
Mathematics Subject Classification ID: 70H40
Mathematics Subject Classification ID: 83C05
Mathematics Subject Classification ID: 70G10
Mathematics Subject Classification ID: 83C30
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 22E70
zbMATH DE Number: 4001548
zbMATH Keywords: Poincaré group
zbMATH Keywords: symmetry group
zbMATH Keywords: phase space
zbMATH Keywords: general relativity
zbMATH Keywords: asymptotically flat
zbMATH Keywords: boundary conditions at spatial infinity
zbMATH Keywords: symplectic geometry
zbMATH Keywords: index notation
zbMATH Keywords: asymptotically flat general relativistic spacetimes
zbMATH Keywords: parity conditions
MaRDI profile type: Publication
full work available at URL: 0003-4916(87)90037-6
OpenAlex ID: W1984537216
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description: scientific article; zbMATH DE number 4001548