A 3+1 formulation of the 1/c expansion of General Relativity

Abstract

Expanding General Relativity in the inverse speed of light, 1/c, leads to a nonrelativistic gravitational theory that extends the Post-Newtonian expansion by the inclusion of additional strong gravitational potentials. This theory has a fully covariant formulation in the language of Newton-Cartan geometry but we revisit it here in a 3+1 formulation. The appropriate 3+1 formulation of General Relativity is one first described by Kol and Smolkin (KS), rather than the better known Arnowitt-Deser-Misner (ADM) formalism. As we review, the KS formulation is dual to the ADM formulation in that the role of tangent and co-tangent spaces get interchanged. In this 3+1 formulation the 1/c expansion can be performed in a more systematic and efficient fashion, something we use to extend the computation of the effective Lagrangian beyond what was previously achieved and to make a number of new all order observations.

45 pages + 3 appendices. Version accepted for publication in JHEP with corrected typos in the appendix

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• Memory Effect and Carroll Symmetry: 50 years later
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