Schr\"{o}dinger symmetry in cosmology and black hole mechanics

We show that the (symmetry-reduced) gravitational sectors describing i) the flat FRLW cosmology filled with a massless scalar field and ii) the Schwarzschild black hole mechanics both admit the two dimensional centrally extended Schr\"{o}dinger group as a dynamical symmetry, a symmetry shared by the compressible Navier-Stokes equation. To this end, we use the Eisenhart-Duval (ED) lift method to identify the Schr\"{o}dinger observables. The Casimirs of the observables algebra coincide respectively with the conserved kinetic energy of the scalar field, for cosmology, and with the mass of the black hole, for the Schwarzschild mechanics. Moreover, in both models, the central extension is found to encode the ratio between the IR and UV scales of the gravitational systems. We pursue this analysis by comparing the ED method to the superspace approach and demonstrate their complementarity. We use the superspace approach to show that these two models possess also an infinite dimensional symmetry whose conserved charges organize in two copies of a Witt algebra. Finally, we consider the anisotropic Bianchi I dynamics and show that it admits a dynamical symmetry under the SO$(4,2)$ conformal group, within which one can identify another Schr\"{o}dinger algebra of observables. These new symmetries provide a new way to algebraically characterize these homogeneous gravitational sectors, to guide their quantization or emergence from quantum gravity models, and they suggest new dictionaries with non-linear Schr\"{o}dinger systems and fluid mechanics.Comment: 41+8 page