Absolute Parallelism: Spherical Symmetry and Singularities

TL;DR

This paper explores spherically symmetric solutions in Absolute Parallelism equations, identifying conditions under which non-static, potentially singular or non-singular solutions exist, with implications for cosmological models.

Contribution

It introduces a specific class of second order AP equations with spherical symmetry and analyzes their solutions, revealing conditions for singularities and non-singular cosmological-like behaviors.

Findings

Non-static solutions dominate under certain gauges.

Singularities can arise in specific coordinate choices.

Covariant gauge reduces equations to a Chaplygin gas-like system.

Abstract

Observing the list of compatible second order equations of Absolute Parallelism (AP) found by Einstein and Mayer (they used D=4), we choose the one-parameter class of equations which take on a 3-linear form (when contra-frame density of some weight is in use). Spherically symmetric solutions to these equations are considered, and we try not to add any delta-sources (ie, $\delta(x)$-sources of unknown nature) during integrations allowed due to this high symmetry. Using two different ways to fix the radius and time, we have found that only non-static solutions (except for trivial one, of course) are possible. If D=5, such solutions, looking like a single wave moving along the radius, could serve as an expanding cosmological model (with a simple Hubble diagram). With one coordinate choice (gauge), a single second order equation remains and there exist spherically symmetric solutions with arising singularities. On the other hand, a more reasonable (covariant) choice of the radius and time reduces the problem to a system of two first order equations looking like Chaplygin gas dynamics, where solutions are seemingly free of emerging singularities and gradient catastrophe.