## Carroll Expansion of General Relativity

Dennis Hansen, Niels A. Obers, Gerben Oling, Benjamin T. Søgaard

SciPost Phys. 13, 055 (2022) · published 8 September 2022

- doi: 10.21468/SciPostPhys.13.3.055
- Submissions/Reports

### Abstract

We study the small speed of light expansion of general relativity, utilizing the modern perspective on non-Lorentzian geometry. This is an expansion around the ultra-local Carroll limit, in which light cones close up. To this end, we first rewrite the Einstein-Hilbert action in pre-ultra-local variables, which is closely related to the 3+1 decomposition of general relativity. At leading order in the expansion, these pre-ultra-local variables yield Carroll geometry and the resulting action describes the electric Carroll limit of general relativity. We also obtain the next-to-leading order action in terms of Carroll geometry and next-to-leading order geometric fields. The leading order theory yields constraint and evolution equations, and we can solve the evolution analytically. We furthermore construct a Carroll version of Bowen-York initial data, which has associated conserved boundary linear and angular momentum charges. The notion of mass is not present at leading order and only enters at next-to-leading order. This is illustrated by considering a particular truncation of the next-to-leading order action, corresponding to the magnetic Carroll limit, where we find a solution that describes the Carroll limit of a Schwarzschild black hole. Finally, we comment on how a cosmological constant can be incorporated in our analysis.

### Cited by 42

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1}Dennis Hansen, -
^{2}^{3}Niels Obers, -
^{2}^{3}Gerben Oling, -
^{2}^{4}Benjamin T. Søgaard

^{1}Eidgenössische Technische Hochschule Zürich / Swiss Federal Institute of Technology in Zurich (ETH) [ETH Zurich]^{2}Niels Bohr Institute [NBI]^{3}Nordisk Institut for Teoretisk Fysik / Nordic Institute for Theoretical Physics [NORDITA]^{4}Princeton University