Hidden symmetries and decay for the wave equation on the Kerr spacetime

Energy and decay estimates for the wave equation on the exterior region of slowly rotating Kerr spacetimes are proved. The method used is a generalization of the vector-field method, which allows the use of higher-order symmetry operators. In particular, our method makes use of the second-order Carter operator, which is a hidden symmetry in the sense that it does not correspond to a Killing symmetry of the spacetime. The main result gives, in stationary regions, an almost inverse linear decay rate and the corresponding decay rate at the event horizon and null infinity. Except for the small loss in the decay rate, this generalizes the known decay results on the exterior region of the Schwarzschild spacetime.