Evolution of scalar-field perturbations inside a Kerr black hole

We analyze the evolution of an axially symmetric, linear, massless scalar field inside a Kerr black hole. The initial data we assume correspond to incident inverse-power tails at the event horizon. Using the late-time expansion, we obtain the asymptotic behavior near the Cauchy horizon (CH). We find that at the early portion of the CH the scalar field is dominated by inverse powers of the Eddington-like coordinates u and $v,$ with the same power index as the ingoing perturbations at the event horizon. We also calculate the two leading-order coefficients in the expansion in $1/u$ and $1/v$ near the CH, which may be interpreted as the transmission and reflection coefficients for the internal scattering problem.