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We investigate the Teukolsky equation in horizon-penetrating coordinates tostudy the behavior of perturbation waves crossing the outer horizon. For thispurpose, we use the null ingoing/outgoing Eddington-Finkelstein coordinates.The first derivative of the radial equation is a Fuchsian differential equationwith an additional regular singularity to the ones the radial one has. Theradial functions satisfy the physical boundary conditions without imposing anyregularity conditions. We also observe that the Hertz-Weyl scalar equationspreserve their angular and radial signatures in these coordinates. Using theangular equation, we construct the metric perturbation for a circularlyorbiting perturber around a black hole in Kerr spacetime in ahorizon-penetrating setting. Furthermore, we completed the missing metricpieces due to the mass M and angular momentum J perturbations. We also providean explicit formula for the metric perturbation as a function of the radialpart, its derivative, and the angular part of the solution to the Teukolskyequation. Finally, we discuss the importance of the extra singularity in theradial derivative for the convergence of the metric expansion.

In Horizon Penetrating Coordinates: Kerr Black Hole Metric Perturbation Construction and Completion