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

We provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzchild background. Our estimate implies that such solutions have asymptotic behavioras long as the source term is bounded in the normIn particular this gives scattering at small amplitudes for non-linear scalar fields of the form ✷gφ = λ|φ| p φ for all 2 < p.

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We study solutions of the decoupled Maxwell equations in the exterior region of a Schwarzschild black hole. In stationary regions, where the Schwarzschild coordinate r ranges over 2M < r1 < r < r2, we obtain a decay rate of t −1 for all components of the Maxwell field. We use vector field methods and do not require a spherical harmonic decomposition.In outgoing regions, where the Regge-Wheeler tortoise coordinate is large, r * > ǫt, we obtain decay for the null components with rates of |φ+| ∼ |α| < Cr −5/2 , |φ0| ∼ |ρ| + |σ| < Cr −2 |t − r * | −1/2 , and |φ−1| ∼ |α| < Cr −1 |t − r * | −1 . Along the event horizon and in ingoing regions, where r * < 0, and when t + r * > 1, all components (normalized with respect to an ingoing null basis) decay at a rate of Cu+ −1 with u+ = t + r * in the exterior region.In Einstein's equations, the energy-momentum tensor of the matter fields should influence the curvature. By decoupled, we mean that the electromagnetic field does not influence the Schwarzschild solution, which is taken as a fixed background manifold. We call the Schwarzschild solution the Schwarzschild manifold and use the word solution to refer to solutions of the Maxwell equations (1)-(2).Since F is a tensor, there is no coordinate independent norm with which to measure it (or, at least, not all components of it). To discuss the decay of F , we make a choice of basis and show that the corresponding components decay. A simple choice of basis consists of the coordinate vector fields rescaled so that they have unit length (|g(X, X)| = 1). The rescaled vectors arêGiven a time-like vector, there is a natural decomposition of the Maxwell field into electric and magnetic components. Since the Schwarzschild manifold has a time-translation symmetry, this provides a natural choice of time-like direction,T . The corresponding electric and magnetic components are