Comparing space+time decompositions in the post-Newtonian limit

TL;DR

This work compares ADM and non-relativistic gravitational (NRG) field decompositions within the EFT framework for two-body post-Newtonian dynamics. It shows a linear equivalence between a modified ADM and NRG, identical at 1PN, and analyzes their 2PN equivalence, where ADM requires an extra diagram from additional A^2 couplings. The authors reproduce the 2PN action and demonstrate that ADM and NRG yield the same effective action, though ADM incurs extra diagrammatic work that increases at higher PN orders due to ADM-specific vertices. The results inform method choice for PN EFT computations, highlighting potential efficiency advantages of NRG over ADM at higher 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.

Figures (3)

• Figure 1: 2PN Feynman diagrams of two-field exchange including differences from diagrams with NRG fields. (b) An additional diagram that is eliminated when NRG fields are used. These diagrams should be included together with their mirror images.
• Figure 2: 2PN Feynman diagrams with a 3-field vertex including differences from diagrams with NRG fields. These diagrams should be included together with their mirror images.
• Figure 3: Additional Feynman diagrams including the extra 2-field $A^2$ worldline vertex, which appear at 3PN if the modified ADM fields are used. In NRG fields, an $A^2$ vertex does not appear altogether at 3PN, but rather it appears first at 4PN through the term $(\vec{A} \cdot \vec{v})^2\, v^2$. Diagrams (a1)-(a2) contribute at order $G^2$, while diagrams (b1)-(c5) contribute at order $G^3$. Note that while diagram (a1) appears already at 2PN, the velocity dependence of its vertices contributes also to 3PN.