Stability Study of a Model for the Klein–Gordon Equation in Kerr Space-Time

The current early stage in the investigation of the stability of the Kerr metric
is characterized by the study of appropriate model problems. Particularly interesting is
the problem of the stability of the solutions of the Klein–Gordon equation, describing
the propagation of a scalar field of mass μ in the background of a rotating black hole.
Rigorous results prove the stability of the reduced, by separation in the azimuth angle
in Boyer–Lindquist coordinates, field for sufficiently large masses. Some, but not all,
numerical investigations find instability of the reduced field for rotational parameters
a extremely close to 1. Among others, we derive a model problem for the
equation which supports the instability of the field down to a/M ≈ 0.97.