The $1/c$ expansion of general relativity in a $3+1$ formulation, revisited

TL;DR

This work develops a decomposition-agnostic, covariant $1/c$ expansion framework for GR by leveraging a shared parent Einstein–Hilbert action valid for both ADM and KS 3+1 decompositions. The method proceeds via a Matryoshka-style, stepwise expansion that yields compact, order-by-order expressions up to $c^{-3}$ for the ADM case and up to $c^{-1}$ for KS, while preserving duality at the level of the expansion. Central to the approach are generic identities for expanding the inverse metric, determinant, covariant derivatives, and Ricci tensor, which allow all-order control and cross-decomposition applicability. A concrete all-orders demonstration shows a universal linear-in-order piece, and the ADM computation up to $c^{-3}$ is presented in full detail, illustrating the practical utility and potential for higher PN-order extensions. The results reinforce the link between Newton–Cartan-type structures and large-$c$ GR, with potential applications to strong-field systems and beyond.

Abstract

We study the $1/c$ expansion of general relativity within a formulation that is compatible with both the Arnowitt-Deser-Misner and the Kol-Smolkin decompositions. The Einstein-Hilbert action takes a common form for those decompositions as they are dual to each other. We first develop a method to expand this generic form without choosing a particular slicing and then push the expansion up to $c^{-3}$ order within this novel approach. Next, we apply our technique to the Arnowitt-Deser-Misner decomposition and expand it up to $c^{-3}$ order explicitly. In order to demonstrate the applicability of our method and to highlight the duality at the level of expansion, we also perform the expansion in the Kol-Smolkin decomposition up to $c^{-1}$ order. Lastly, we make some all-order observations.