Twisting asymptotically-flat spacetimes

Abstract

We extend the Bondi formalism to describe asymptotically-flat spacetimes where the outgoing null geodesic congruence is not hypersurface-orthogonal, i.e. has non-vanishing twist. In the Newman-Penrose formulation, the twist $\text{Im}(ρ)$ is sourced by a twist potential sitting in the transverse null dyad $(m,\bar{m})$, while in the metric formulation this potential enters $g_{ra}\neq0$. We explain how to solve the Einstein equations for such generalized line elements, thereby providing an extension of the Bondi hierarchy to asymptotically-flat spacetimes with twist. We work out the twisting generalizations of all the well-known features pertaining to asymptotically-flat spacetimes in Bondi gauge, such as the solution space, the flux-balance laws, the asymptotic symmetries, and the transformation laws. The twist potential has a natural Carrollian interpretation as an Ehresmann connection, and gives rise to Carroll boosts as extra asymptotic symmetries. The supertranslation generators in this gauge with twist have no extension in the bulk and therefore only act at null infinity. This enables us to study finite supertranslations of any given solution. One of the advantages of the Bondi gauge with twist is also that it allows to write algebraically special solutions in a manifestly finite radial expansion, and with a repeated principal null direction such that $Ψ_0=Ψ_1=0$. This is in particular the case for the Kerr-Taub-NUT solution. The asymptotic symmetries of algebraically special solutions also have a finite radial expansion, which enables us to study, for example, the supertranslated Schwarzschild solution and its charges quite straightforwardly. We also study an analogue of the twist in three-dimensional spacetimes with non-vanishing cosmological constant, and find an 8-dimensional solution space which encompasses and generalizes the existing results in the literature.