Dynamical symmetries of homogeneous minisuperspace models

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 nontrivial 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 eight-dimensional symmetry algebra $\mathcal{A}=(\mathfrak{sl}(2,\mathbb{R})\ensuremath{\bigoplus}\mathbb{R})⨭{\mathfrak{h}}_{2}$, based on the two-dimensional Heisenberg algebra ${\mathfrak{h}}_{2}\ensuremath{\simeq}{\mathbb{R}}^{4}$. We apply this to the systematic study of the Bianchi models for homogeneous cosmology. This extends the previous results on the $\mathfrak{sl}(2,\mathbb{R})$ algebra for Friedmann-Lema\^{\i}tre-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 toward quantization and the study of solution generating mechanisms.