Dynamical symmetries of homogeneous minisuperspace models

TL;DR

This paper explores the phase space symmetries of homogeneous minisuperspace models in general relativity, revealing a universal 8-dimensional symmetry algebra that aids in understanding cosmological solutions and their quantization.

Contribution

It identifies a universal symmetry algebra for two-dimensional minisuperspaces and applies it systematically to Bianchi cosmological models, extending previous symmetry results.

Findings

Discovered a universal 8-dimensional symmetry algebra for minisuperspaces.

Extended known symmetries to Bianchi models in cosmology.

Facilitates new approaches to quantization and solution generation.

Abstract

We investigate the phase space symmetries and conserved charges of homogeneous gravitational minisuperspaces. These (0+1)-dimensional reductions of general relativity are defined by spacetime metrics in which the dynamical variables depend only on a time coordinate, and are formulated as mechanical systems with a non-trivial field space metric (or supermetric) and effective potential. We show how to extract conserved charges for those minisuperspaces from the homothetic Killing vectors of the field space metric. In the case of two-dimensional field spaces, we exhibit a universal 8-dimensional symmetry algebra given by the semi-direct sum of $\mathfrak{sl}(2,\mathbb{R})\oplus\mathbb{R}$ with the two-dimensional Heisenberg algebra $\mathfrak{h}_2\simeq\mathbb{R}^4$. We apply this to the systematic study of the Bianchi models for homogeneous cosmology. This extends previous results on the $\mathfrak{sl}(2,\mathbb{R})$ algebra for Friedmann-Lemaitre-Robertson-Walker cosmology, and the Poincar\'e symmetry for Kantowski-Sachs metrics describing the black hole interior. The presence of this rich symmetry structure already in minisuperspace models opens new doors towards quantization and the study of solution generating mechanisms.