Analytical Study of Gravitational Waves

TL;DR

This paper reviews the Blanchet-Damour analytical approach to modeling gravitational waves from localized perfect fluid sources using asymptotic expansions and analytic continuation to solve Einstein's equations.

Contribution

It explains the Blanchet-Damour method for solving post-Minkowskian and post-Newtonian equations with divergence issues via analytic continuation and matching procedures.

Findings

Derived general solutions for gravitational fields using analytic continuation.

Addressed divergence problems in post-Newtonian expansions.

Provided a framework for describing gravitational waves in $ ext{R}^3$.

Abstract

The aim of the present thesis is to review the Blanchet-Damour approach to analytical study of gravitational waves emitted by localized perfect fluid sources. It is assumed these perfect fluids are such that it is possible to define small parameters for asymptotic expansions. Asymptotic expansions in this approach are called post-Minkowskian and post-Newtonian expansions. By plugging these expansions into the Einstein field equation, post-Minkowskian and post-Newtonian equations are obtained. The usual methods for solving these equations are to use the retarded and Poisson integrals. However, they cannot provide solutions up to any arbitrary order in these cases because of their divergence. In fact, these divergences motivated Blanchet and Damour to employ a new approach to solve the post-Minkowskian and post-Newtonian equations. They obtained the general solutions by means of a specific process of analytic continuation. These solutions have some unknown terms that if one determines, the gravitational field is described everywhere in $\mathbb{R}^3$. A matching procedure, which depends deeply on analytic continuation, is used to determine these unknown terms.