Anisotropic Homogeneous Cosmologies in the Post-Newtonian Approximation

TL;DR

This paper investigates the capabilities of post-Newtonian theory in modeling anisotropic homogeneous cosmologies, showing it aligns more closely with general relativity than Newtonian theory, especially regarding singularities and pressure effects.

Contribution

It demonstrates that post-Newtonian approximation improves the modeling of anisotropic cosmologies, eliminating arbitrary functions needed in Newtonian theory and better capturing pressure effects.

Findings

Post-Newtonian theory aligns more with general relativity in Bianchi cosmologies.

In post-Newtonian approximation, no arbitrary functions are needed for initial conditions.

Post-Newtonian approximation captures pressure effects in perfect fluid cosmologies.

Abstract

In this paper we explore how far the post-Newtonian theory goes in overcoming the difficulties associated with anisotropic homogeneous cosmologies in the Newtonian approximation. It will be shown that, unlike in the Newtonian case, the cosmological equations of the post-Newtonian approximation are much more in the spirit of general relativity with regard to the nine Bianchi types and issues of singularities. The situations of vanishing rotation and vanishing shear are treated separately. The homogeneous Bianchi I model is considered as an example of a rotation-free cosmology with anisotropy. It is found in the Newtonian approximation that there are arbitrary functions that need to be given for all time if the initial value problem is to be well-posed, while in the post-Newtonian case there is no such need. For the general case of a perfect fluid only the post-Newtonian theory can satisfactorily describe the effects of pressure. This is in accordance with findings in an earlier paper where the post-Newtonian approximation was applied to homogeneous cosmologies. For a shear-free anisotropic homogeneous cosmology the Newtonian theory of Heckmann and Sch\"ucking is explored. Comparisons with its relativistic and post-Newtonian counterparts are made. In the Newtonian theory solutions exist to which there are no analogues in general relativity. The post-Newtonian approximation may provide a way out.