Abstract
Asymptotic symmetries in Carrollian gravitational theories in 3+1 space and time dimensions obtained from “magnetic” and “electric” ultrarelativistic contractions of General Relativity are analyzed. In both cases, parity conditions are needed to guarantee a finite symplectic term, in analogy with Einstein gravity. For the magnetic contraction, when Regge-Teitelboim parity conditions are imposed, the asymptotic symmetries are described by the Carroll group. With Henneaux-Troessaert parity conditions, the asymptotic symmetry algebra corresponds to a BMS-like extension of the Carroll algebra. For the electric contraction, because the lapse function does not appear in the boundary term needed to ensure a well-defined action principle, the asymptotic symmetry algebra is truncated, for Regge-Teitelboim parity conditions, to the semidirect sum of spatial rotations and spatial translations. Similarly, with Henneaux-Troessaert parity conditions, the asymptotic symmetries are given by the semidirect sum of spatial rotations and an infinite number of parity odd supertranslations. Thus, from the point of view of the asymptotic symmetries, the magnetic contraction can be seen as a smooth limit of General Relativity, in contrast to its electric counterpart.
Highlights
The Carroll symmetry was introduced by Levy-Leblond in 1965 as an “ultrarelativistic” limit of the Poincaré symmetry, i.e., the limit in which the speed of light vanishes (c → 0) [1]
With Henneaux-Troessaert parity conditions, the asymptotic symmetry algebra corresponds to a BMS-like extension of the Carroll algebra
The asymptotic symmetries of the magnetic and electric Carrollian contractions of General Relativity were analyzed in 3+1 space and time dimensions
Summary
The Carroll symmetry was introduced by Levy-Leblond in 1965 as an “ultrarelativistic” limit of the Poincaré symmetry, i.e., the limit in which the speed of light vanishes (c → 0) [1]. In the case of the electric contraction, and because the lapse function does not appear in the boundary term needed to ensure a well-defined action principle, the asymptotic symmetry algebra is truncated to the semi-direct sum of spatial rotations and spatial translations when Regge-Teitelboim parity conditions are used. The Carroll group is not present in the electric contraction, and there is no generator associated with time translations that allow defining energy in this theory In this sense, from the point of view of the asymptotic symmetries, the magnetic contraction can be seen as a smooth limit of General Relativity, in contrast to its electric counterpart. Regge-Teitelboim and Henneaux-Troessaert parity conditions are considered, and the asymptotic symmetry algebra together with their canonical generators are determined for each case.
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