Schr?dinger Symmetry in Gravitational Mini-Superspaces

We prove that the simplest gravitational symmetry-reduced models describing cosmology and black hole mechanics are invariant under the Schrodinger group. We consider the flat FRW cosmology filled with a massless scalar field and the Schwarzschild black hole mechanics and construct their conserved charges using the Eisenhart-Duval (ED) lift method in order to show that they form a Schrodinger algebra. Our method illustrates how the ED lift and the more standard approach analyzing the geometry of the field space are complementary in revealing different sets 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 hole models in terms of hydrodynamic equations in field space.