| Summary: | 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.
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