Wave and stability properties of black holes

In this paper we transform the wave equation governing gravitational perturbations of a Schwarzschild black hole from its standard Schrödinger or Regge-Wheeler form to a Klein-Gordon type wave equation. This latter form reveals immediately that incoming waves with frequencies () _cml , a critical frequency, are completely reflected (transmitted). This process is entirely due to the radial variation of the cut-off frequency inherent in the dispersive nature of the wave propagation properties of gravitational perturbations of the Schwarzschild metric. Moreover, those high-frequency waves ( _cml) which penetrate through the region near the Schwarzschild radiusr _sare, on crossing this event horizon, attenuated by a factor exp (–r _s/c), thereby dumping most of their energy and momentum into the black hole. It is shown that in the vicinity ofr _sthe metric is locally unstable. This feature and the wave absorption process indicate that the neighbourhood aroundr _sis dynamically active, and, as well as acting like a Hawking-type particle creator, will behave as a wave emitter in order to relax the stresses on the metric.