Unstable fields in Kerr spacetimes

We show that both the interior region r < M - root M-2 - a(2) of a Kerr black hole and the a(2) > M-2 Kerr naked singularity admit unstable solutions of the Teukolsky equation for any value of the spin weight. For every harmonic number, there is at least one axially symmetric mode that grows exponentially in time and decays properly in the radial directions. These can be used as Debye potentials to generate solutions for the scalar, Weyl spinor, Maxwell and linearized gravity field equations on these backgrounds, satisfying appropriate spatial boundary conditions and growing exponentially in time, as shown in detail for the Maxwell case. It is suggested that the existence of the unstable modes is related to the so-called time machine region, where the axial Killing vector field is timelike, and the Teukolsky equation, restricted to axially symmetric fields, changes its character from hyperbolic to elliptic.