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

TL;DR

This paper develops a systematic 3+1 formulation of the 1/c expansion of General Relativity, extending previous work and enabling more efficient calculations of the effective Lagrangian with new all-order insights.

Contribution

It introduces a 3+1 formulation based on the Kol and Smolkin approach for the 1/c expansion, improving computational efficiency and extending the effective Lagrangian analysis.

Findings

Extended the effective Lagrangian beyond previous limits.

Made new all-order observations in the 1/c expansion.

Provided a more systematic approach to nonrelativistic gravity theories.

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.