Relativistic Celestial Mechanics with PPN Parameters

TL;DR

This paper develops a PPN-based framework for celestial mechanics, deriving local reference systems, equations of motion, and addressing violations of the equivalence principle, with applications to extended bodies and spin effects.

Contribution

It constructs explicit local PPN reference systems, derives equations of motion for extended bodies with multipole structures, and discusses violations of the equivalence principle in celestial mechanics.

Findings

Explicit local PPN reference systems are formulated.

Derived equations of motion for extended bodies with multipoles.

Reproduced known PPN equations for monopoles with spins.

Abstract

Starting from the global parametrized post-Newtonian (PPN) reference system with two PPN parameters $\gamma$ and $\beta$ we consider a space-bounded subsystem of matter and construct a local reference system for that subsystem in which the influence of external masses reduces to tidal effects. Both the metric tensor of the local PPN reference system in the first post-Newtonian approximation as well as the coordinate transformations between the global PPN reference system and the local one are constructed in explicit form. The terms proportional to $\eta=4\beta-\gamma-3$ reflecting a violation of the equivalence principle are discussed in detail. We suggest an empirical definition of multipole moments which are intended to play the same role in PPN celestial mechanics as the Blanchet-Damour moments in General Relativity. Starting with the metric tensor in the local PPN reference system we derive translational equations of motion of a test particle in that system. The translational and rotational equations of motion for center of mass and spin of each of $N$ extended massive bodies possessing arbitrary multipole structure are derived. As an application of the general equations of motion a monopole-spin dipole model is considered and the known PPN equations of motion of mass monopoles with spins are rederived.