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

TL;DR

This paper develops a new method to perform a systematic $1/c$ expansion of general relativity in a 3+1 formulation, applicable to different decompositions, and explores the duality and all-order properties of the expansion.

Contribution

It introduces a novel approach to expand the Einstein-Hilbert action without fixing a slicing, up to third order in $1/c$, and demonstrates duality between ADM and Kol-Smolkin decompositions.

Findings

Expansion up to $c^{-3}$ in ADM decomposition explicitly performed.

Expansion up to $c^{-1}$ in Kol-Smolkin decomposition shown.

All-order observations about the $1/c$ expansion made.

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.