Twisting asymptotically-flat spacetimes

TL;DR

This paper extends the Bondi formalism to asymptotically-flat spacetimes with non-zero twist, revealing new symmetries, solution structures, and applications to algebraically special solutions like Kerr-Taub-NUT.

Contribution

It introduces a generalized Bondi hierarchy for twisted spacetimes, explores their asymptotic symmetries, and connects the twist potential to Carrollian geometry, broadening the understanding of asymptotic structures.

Findings

Extended Bondi formalism to include twist in null congruences

Identified Carrollian interpretation of the twist potential

Applied results to algebraically special solutions like Kerr-Taub-NUT

Abstract

We extend the Bondi formalism to describe asymptotically-flat spacetimes where the outgoing null geodesic congruence is not hypersurface-orthogonal, i.e. has a non-vanishing twist. In the Newman-Penrose formulation, the twist $\text{Im}(\rho)$ is sourced by a twist potential sitting in the transverse null dyad $(m,\bar{m})$, while in the metric formulation this potential arises from $g_{ra}\neq0$. We explain how to arrange and solve the Einstein equations for such generalized line elements, thereby providing an extension of the Bondi hierarchy to asymptotically-flat spacetimes with non-vanishing 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. One of the advantages of the Bondi gauge with non-vanishing twist is that it allows to write algebraically special solutions in a manifestly finite radial expansion, and with a repeated principal null direction such that $\Psi_0=\Psi_1=0$. 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 to study the supertranslated Schwarzschild solution and its charges quite straightforwardly. We expect that these results will find applications in the development of flat holography for algebraically special solutions and in the study of their perturbations. 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.