On the ergoregion instability

Rotating, ultra-compact stars in general relativity can have an ergoregion, in which all trajectories are dragged in the direction of the star's rotation. The existence of the ergoregion leads to a classical instability to emission of scalar, electromagnetic and gravitational radiation from the star. In this paper we calculate eigenfrequencies (including e-folding times) for stable and unstable modes of a scalar field on a background metric which has an ergoregion. Within a W.K.B.J. approximation for modes with angular dependence exp (\textimf), we find that unstable modes exist for all |m| > m0 (m0 depending upon the star), but that the e-folding time is asymptotically t = t0 exp (2bm), where b is of order 1. Typically, t0 is several orders of magnitude longer than the age of the universe. However, the techniques evolved here should be applicable to other 'rotational dragging' instabilities in general relativity. Particularly useful should be the result that links the eigenfrequencies to resonances in the effective potentials governing photon motion in the metric; these potentials are rotationally 'split'