THE PHASE-INTEGRAL METHOD AND BLACK-HOLE NORMAL-MODES

The phase-integral method has proved to be a powerful tool for studyin g the quasinormal modes of black holes. A generalization of the WKB me thods of quantum mechanics, its treatment of the complex coordinate pl ane brings a number of important simplifications and potentially power ful computational aids to bear on the problem of computing eigenfreque ncies with large imaginary parts. It holds great promise of further ap plications to related problems, such as the quasinormal modes of relat ivistic stars. However, in some respects the method is incomplete, par ticularly in its assessment of error bounds. This paper makes availabl e to researchers in the field of relativity a simple and self-containe d introduction to the fundamental concepts of the phase-integral metho d, in which we also point out areas that seem to need further developm ent. As an example of the use of the method, we derive the two-transit ion-point phase-integral formula for quasinormal modes of the Schwarzs child black hole, which is an accurate asymptotic approximation for th e first modes. The present paper provides the foundation for related p apers in which we use the method to develop accurate asymptotic expres sions for highly damped modes.