Asymptotic symmetries in Carrollian theories of gravity

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.

Asymptotic symmetries in Carrollian theories of gravity with a negative cosmological constant

“…As in the case with Λ = 0 [10], the asymptotic structure of the magnetic Carrollian contraction of General Relativity with a negative cosmological constant is richer and with a better behavior than its electric counterpart.…”

“…Note that, as in the case with Λ = 0 discussed in [10], the canonical variables, as well as the lapse and the shift in eqs. (2.32), (2.33) exactly coincide with those of the Schwarzschild-AdS solution in General Relativity.…”

“…Recently, an analysis of the asymptotic structure of electric Carroll gravity was performed in the case of vanishing cosmological constant [10], where it was shown that, from the point of view of the asymptotic symmetries, this theory has some unwanted properties. In particular, the asymptotic symmetry algebra does not contain a generator associated with time translations at the boundary, i.e., there is no a notion of energy in this theory 3 .…”

“…For both contractions, magnetic and electric, the asymptotic symmetries were recently studied in ref. [10] in 3+1 space-time dimensions, with a vanishing cosmological constant. It is then natural to wonder if in the presence of a negative cosmological constant there exists a similar interpretation as in the case of Einstein gravity, i.e., that the asymptotic symmetries in the Carrollian gravitational theories correspond to conformal Carrollian algebras in one lower dimension.…”

“…The presence of a negative cosmological constant turns out to be incompatible with a consistent set of asymptotic conditions. Indeed, the naive ground state configuration analogous to the AdS spacetime in General Relativity is not a solution of the constraints of the electric theory [10,50]. As a consequence, it does not seems to be possible to construct a well-defined set of asymptotic conditions with non-trivial conserved charges in this case.…”