Non-Relativistic Gravitation: From Newton to Einstein and Back

TL;DR

The paper develops an improved Classical Effective Field Theory approach to non-relativistic gravity by performing a temporal Kaluza-Klein reduction to NR gravity (NRG) fields, enabling a diagonal scalar propagator and a compact, diagrammatic organization of PN corrections. The method maps Newtonian gravity to GR through the fields φ (Newtonian potential), A_i (gravito-magnetic potential), and γ_ij, and applies this framework to derive the Einstein-Infeld-Hoffmann Lagrangian with a minimal set of diagrams, plus a higher-dimensional generalization. It also clarifies spin interactions, showing the leading spin-spin force arises from gravito-magnetic exchange at 2PN and outlining 3PN and 2.5PN corrections, with spin-orbit terms emerging at 1.5PN. Extending to arbitrary dimensions, the authors generalize the EIH action and compare with prior results, finding broad agreement except for one coefficient, thereby providing a transparent, field-theoretic lens on non-relativistic gravity and its PN expansion.

Abstract

We present an improvement to the Classical Effective Theory approach to the non-relativistic or Post-Newtonian approximation of General Relativity. The "potential metric field" is decomposed through a temporal Kaluza-Klein ansatz into three NRG-fields: a scalar identified with the Newtonian potential, a 3-vector corresponding to the gravito-magnetic vector potential and a 3-tensor. The derivation of the Einstein-Infeld-Hoffmann Lagrangian simplifies such that each term corresponds to a single Feynman diagram providing a clear physical interpretation. Spin interactions are dominated by the exchange of the gravito-magnetic field. Leading correction diagrams corresponding to the 3PN correction to the spin-spin interaction and the 2.5PN correction to the spin-orbit interaction are presented.

Figures (3)

• Figure 1: Feynman diagrams contributing to the Newtonian two body Lagrangian (\ref{['Npart']}) and the 1PN Einstein-Infeld-Hoffmann Lagrangian (\ref{['EIH']}). Double lines represent the masses and do not carry propagators. Solid lines represent $\phi$, the Newtonian potential, dashed lines represent $A_i$, the vector potential, while spring-like lines are non-discriminating notation for all the polarizations of the original graviton $g$. A power of $v$ near a vertex denotes that the vertex was expanded in $v$, while a circled cross denoted $v$-dependent correction to the (static) propagator. A dot vertex for a dashed line represent the $\vec{v} \cdot \vec{A}$ vertex of the world-line action (\ref{['pp']}). Diagram (a) is the Newtonian potential. Each diagram in (b,c) matches a specific terms in the EIH Lagrangian (\ref{['EIH']}) as described in the text. Diagrams in (b) represent the $v$ dependent EIH terms, while $(c)$ are $v$-independent. Note that the relatively complicated triple vertex diagram, fig. 5(a) of GoldbergerRothstein1, disappeared.
• Figure 2: Diagrams representing spin-spin interactions. The cross vertex represents the $\vec{J} \cdot \vec{B}$ vertex (\ref{['LOspin']}). Unlike the previous figure, here the spring-like line represents the 3-tensor $\gamma$. Other notation remains the same. Diagram (a) represents the leading order at 2PN, in terms of the gravito-magnetic field. The other diagrams represent the next to leading contributions at 3PN: (b) represent $v^2$ corrections, while $(c)$ represent $Gm/r$ corrections.
• Figure 3: Diagrams representing contributions to the spin-orbit interaction. Diagrams (a) represent the leading order at 1.5PN. The diagrams in (b) and (c) are a sample of those representing the first correction at 2.5PN. The notation is the same as in the previous figures.