Static post-Newtonian equivalence of GR and gravity with a dynamical preferred frame

TL;DR

This paper studies a generally covariant gravity theory with a dynamical unit timelike vector (the aether) in the asymptotic weak-field limit of static, spherically symmetric spacetimes. Using isotropic coordinates and a series expansion in $x=1/r$, it derives the field equations and solves for the metric and aether components, identifying how the ERS post-Newtonian parameters depend on the Lagrangian coefficients $c_i$. It shows that for generic choices with $c_1+c_2+c_3 eq0$, the ERS parameters reproduce GR values ($eta= abla eta=1$ and $oldsymbol{oldsymbol{eta}=1}$), with deviations arising only in special cases (notably $c_1+c_2+c_3=0$) or from higher-order/preferred-frame effects. The results imply the aether model closely mimics GR at leading order in the solar system, while indicating routes to constrain the theory via full PPN analyses, gravitational waves, and strong-field tests.

Abstract

A generally covariant extension of general relativity (GR) in which a dynamical unit timelike vector field is coupled to the metric is studied in the asymptotic weak field limit of spherically symmetric static solutions. The two post-Newtonian parameters known as the Eddington-Robertson-Schiff parameters are found to be identical to those in the case of pure GR, except for some non-generic values of the coefficients in the Lagrangian.