The Lifshitz-Khalatnikov Kasner index parametrization and the Weyl Tensor

TL;DR

This paper explores the connection between the Petrov classification of the Weyl tensor and Kasner index parametrizations, revealing geometric insights into cosmological dynamics near singularities.

Contribution

It introduces a natural Kasner index parametrization aligned with the Petrov classification, clarifying the geometric meaning of the Kasner map in cosmological models.

Findings

Link between Petrov classification and Kasner indices

Geometric interpretation of the Kasner map

Insight into Mixmaster dynamics near singularities

Abstract

The scale invariant Petrov classification of the Weyl tensor is linked to the scale invariant combination of the Kasner index constraints, and the Lifshitz-Khalatnikov Kasner index parametrization scheme turns out to be a natural way of adapting to this symmetry, while hiding the permutation symmetry that is instead made manifest by the Misner parametrization scheme. While not so interesting for the Kasner spacetime by itself, it gives a geometrical meaning to the famous Kasner map transitioning between Kasner epochs and Kasner eras, equivalently bouncing between curvature walls, in the BLK-Mixmaster dynamics exhibited by spatially homogeneous cosmologies approaching the initial cosmological singularity and the inhomogeneous generalization of this dynamics.