Carroll Expansion of General Relativity

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