Schr\"{o}dinger Symmetry in Gravitational Mini-Superspaces

TL;DR

This paper demonstrates that simplified gravitational models for cosmology and black holes exhibit Schr"{o}dinger symmetry, revealing new algebraic structures that could influence their quantization and extensions in quantum gravity.

Contribution

It shows that key gravitational mini-superspace models possess Schr"{o}dinger symmetry and identifies an infinite-dimensional Witt algebra extension, providing new insights into their algebraic structure.

Findings

Models are invariant under Schr"{o}dinger group.

Constructed conserved charges form Schr"{o}dinger algebra.

Discovered infinite-dimensional Witt algebra symmetry.

Abstract

We prove that the simplest gravitational symmetry reduced models describing cosmology and black holes mechanics are invariant under the Schr\"{o}dinger group. We consider the flat FRW cosmology filled with a massless scalar field and the Schwarzschild black hole mechanics, construct their conserved charges using the Eisenhart-Duval (ED) lift method and show that they form a Schr\"{o}dinger algebra. Our method illustrates how the ED lift and the more standard approach analyzing the geometry of the field space are complementarity in revealing different set of symmetries of these systems. We further identify an infinite-dimensional symmetry for those two models, generated by conserved charges organized in two copies of a Witt algebra. These extended charge algebras provide a new algebraic characterization of these homogeneous gravitational sectors. They guide the path to their quantization and open the road to non-linear extensions of quantum cosmology and quantum black holes models in terms of hydrodynamic equations in field space.