Fast and Slow solutions in General Relativity: The Initialization Procedure

TL;DR

This paper justifies the use of Postnewtonian approximations in General Relativity by analyzing initial data procedures for problems with multiple time scales, ensuring accurate modeling of slow and fast dynamics.

Contribution

It introduces a method to determine the necessary order of initialization in Einstein's equations to accurately capture slow motion effects in Postnewtonian approximations.

Findings

Initialization order affects quadrupole moment calculations.

Compatibility of initialization with constraint equations is established.

Green function solutions for Postnewtonian equations are presented.

Abstract

We apply recent results in the theory of PDE, specifically in problems with two different time scales, on Einstein's equations near their Newtonian limit. The results imply a justification to Postnewtonian approximations when initialization procedures to different orders are made on the initial data. We determine up to what order initialization is needed in order to detect the contribution to the quadrupole moment due to the slow motion of a massive body as distinct from initial data contributions to fast solutions and prove that such initialization is compatible with the constraint equations. Using the results mentioned the first Postnewtonian equations and their solutions in terms of Green functions are presented in order to indicate how to proceed in calculations with this approach.