Energy-momentum/Cotton tensor duality for AdS4 black holes

We consider the theory of gravitational quasi-normal modes for general linear perturbations of AdS4 black holes. Special emphasis is placed on the effective Schrödinger problems for axial and polar perturbations that realize supersymmetric partner potential barriers on the half-line. Using the holographic renormalization method, we compute the energy-momentum tensor for perturbations satisfying arbitrary boundary conditions at spatial infinity and discuss some aspects of the problem in the hydrodynamic representation. It is also observed in this general framework that the energy-momentum tensor of black hole perturbations and the energy momentum tensor of the gravitational Chern-Simons action (known as Cotton tensor) exhibit an axial-polar duality with respect to appropriately chosen supersymmetric partner boundary conditions on the effective Schrödinger wave-functions. This correspondence applies to perturbations of very large AdS4 black holes with shear viscosity to entropy density ratio equal to 1/4π, thus providing a dual graviton description of their hydrodynamic modes. We also entertain the idea that the purely dissipative modes of black hole hydrodynamics may admit Ricci flow description in the non-linear regime.