EVOLVING TEST FIELDS IN A BLACK-HOLE GEOMETRY

We consider the initial value problem for a massless scalar field in t he Schwarzschild geometry. When constructed using a complex-frequency approach the necessary Green's function splits into three components. We discuss all of these in some detail. (1) The contribution from the singularities (the quasinormal modes of the black hole) is approximate d and the mode sum is demonstrated to converge after a certain well-de fined time in the evolution. A dynamic description of the mode excitat ion is introduced and tested. (2) It is shown how a straightforward lo w-frequency approximation to the integral along the branch cut in the black-hole Green's function leads to the anticipated power-law falloff at very late times. We also calculate higher order corrections to thi s tail and show that they provide an important complement to the leadi ng order. (3) The high-frequency problem is also considered. We demons trate that the combination of the obtained approximations for the quas inormal modes and the power-law tail provide a complete description of the evolution at late times. Problems that arise (in the complex-freq uency picture) for early times are also discussed, as is the fact that many of the presented results generalize to, for example, Kerr black holes.