Comparing space+time decompositions in the post-Newtonian limit

TL;DR

This paper compares ADM and NRG space+time decompositions in the post-Newtonian limit, highlighting their differences and computational complexities at higher orders within the effective field theory approach.

Contribution

It demonstrates the equivalence of ADM and NRG at 1PN and analyzes the increased computational complexity of ADM at 2PN and beyond.

Findings

ADM and NRG are identical at 1PN order.

ADM requires an extra Feynman diagram at 2PN.

Computational complexity of ADM increases at higher PN orders.

Abstract

The relationship between the Arnowitt-Deser-Misner (ADM) field decomposition and the non-relativistic gravitational (NRG) fields attracted considerable interest recently. This paper compares the two, especially with respect to computing the two-body post-Newtonian (PN) effective action within the effective field theory (EFT) approach. Both are space+time decompositions and hence do better than using the standard metric. However, ADM is essentially a reduction over space whereas NRG is essentially a reduction over time. We use a variant of ADM which is linearly equivalent to NRG and the two are identical at order 1PN. We compare the two at order 2PN and find that ADM requires the computation of an additional Feynman diagram. We argue that the computational excess will further increase at higher orders.