Carroll Expansion of General Relativity

TL;DR

This paper explores the ultra-local Carroll limit of general relativity through a small speed of light expansion, revealing new geometric structures, equations, and initial data relevant to black holes and cosmology.

Contribution

It introduces a novel expansion around the Carroll limit, reformulates Einstein-Hilbert action in Carroll geometry, and constructs initial data for black holes within this framework.

Findings

Derivation of leading and next-to-leading order Carroll gravity actions

Analytical solutions for Carroll evolution equations

Construction of Carroll black hole initial data with conserved charges

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.