Symmetries and conformal bridge in Schwarzschild-(A)dS black hole mechanics

We show that the Schwarzschild-(A)dS black hole mechanics possesses a hidden SL$(2,\mathbb{R})\ltimes \mathbb{R}^3$ symmetry which fully dictates the black hole geometry. This symmetry shows up after having gauge-fixed the diffeomorphism invariance in the symmetry-reduced homogeneous Einstein-$\Lambda$ model and stands as a physical symmetry of the system. It follows that one can associate a set of non-trivial conserved charges to the Schwarzschild-(A)dS black hole which act in each causally disconnected regions. In $T$-region, they act on fields living on spacelike hypersurface of constant time, while in $R$-regions, they act on time-like hypersurface of constant radius. We find that while the expression of the charges depend explicitly on the location of the hypersurface, the sl$(2,\mathbb{R})\ltimes \mathbb{R}^3$ charge algebra remains the same at any radius in R-regions (or time in T-regions). The conserved charges represent the evolving constants of motion of the system and are built from the Hamiltonian, the trace of the extrinsic curvature of the considered hypersurface, the 3d volume and the area of the 2-sphere. Finally, the analysis of the Casimirs of the charge algebra reveals a new solution-generating map. The sl$(2,\mathbb{R})$ Casimir is shown to generate flow along the cosmological constant. This gives rise to a new conformal bridge allowing one to continuously deform the Schwarzschild-AdS geometry to the Schwarzschild and the Schwarzschild-dS solutions.