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

Abstract

We show that the Schwarzschild-(A)dS black hole mechanics possesses a hidden SL(2, R)⋉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-Λ 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, R) ⋉ 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, 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.

Hidden symmetry of the static response of black holes: Applications to Love numbers

“…It has been shown recently in Ref. [41] that the Schwarzschild-(A)dS mechanics possesses a set of dynamical symmetries under the Poincaré group SL(2, R) ⋉ R 3 , extending previous results in Ref. [42].…”

“…Once more, finding the more general symmetry acting on the test field ψ(r, θ, φ) and not only on each multipole would be desirable, as it might provide the key to understand if this observation hides a deeper connection with the symmetry of black hole mechanics recently discussed in Ref. [41].…”

“…For metric perturbations, a covariant boundary term should be common not only for all multipoles of linear perturbations but also for the background and nonlinear perturbations 7. See Eq (4.2), Eq (4.4) and Eq (4.6) in Section 4 in Ref [41]. …”

Proper time reparametrization in cosmology: Möbius symmetry and Kodama charges

“…On the other hand, it would be interesting to understand how these symmetries generalize to more general symmetry-reduced gravitational systems such as black holes. See [11][12][13][14] for recent results in these two directions. In the end, we expect this structure to play a guiding role when attempting to describe these simplified gravitational fields from a suitable hydrodynamical limit of a more fundamental theory of quantum gravity.…”

“…On the other hand, it was shown in [11,12] that the SL(2, R) symmetry stands as a subsector of a larger SO (3,2) group of symmetry which mixes the gravitational and matter degrees of freedom. Finally, this symmetry was found to hold for more general systems, such as the Schwarzschild-(A)dS black hole mechanics (described by the Kantowski-Sachs midi-superspace model) for which the symmetry is upgraded to the group SL(2, R) R 3 [13,14]. These different generalizations therefore suggest the existence of a deeper structure to be uncovered and beg for additional investigations.…”