Evolution of linear gravitational and electromagnetic perturbations inside a Kerr black hole

We analyze the evolution of linear gravitational (s=±2) and electromagnetic (s=±1) perturbations inside a Kerr black hole, within the framework of the Newman-Penrose formalism. In particular, we derive explicit expressions for the asymptotic behavior of the perturbations at the early portion of the Cauchy horizon (CH). The calculation is carried out in the time domain, using late-time expansion. The initial data are the presumed inverse-power tails at the event horizon. We find that the ``outgoing'' fields $s<0$ are regular (though nonvanishing) at the CH. However, the ``ingoing'' fields $s>0$ diverge at the CH-like (r-r_-)^-s, where r is the radial Boyer-Lindquist coordinate and r_- is its value at the CH. This divergent term is multiplied by an inverse power of the ingoing Eddington coordinate $v.$ For nonaxially symmetric modes (m≠0), the divergence of the $s>0$ fields is also modulated by an oscillatory term e^imΩ_-v, where Ω_- is a fixed parameter and m is the magnetic number of the mode under consideration. This term exhibits an infinite number of oscillations on the approach to the CH. We also find that the nonaxially symmetric modes diverge faster than the axially symmetric ones. Based on the result of a previous nonlinear perturbation expansion, which showed that the nonlinear perturbation terms are negligible at the CH compared to the linear ones, we argue that the linear gravitational perturbations calculated here correctly describe the strength and features of the curvature singularity at the CH (to the leading order in $1/v$ and $1/u).$