Schrödinger Symmetry in Gravitational Mini-Superspaces

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.