  We investigate the Teukolsky equation in horizon-penetrating coordinates to
study the behavior of perturbation waves crossing the outer horizon. For this
purpose, we use the null ingoing/outgoing Eddington-Finkelstein coordinates.
The first derivative of the radial equation is a Fuchsian differential equation
with an additional regular singularity to the ones the radial one has. The
radial functions satisfy the physical boundary conditions without imposing any
regularity conditions. We also observe that the Hertz-Weyl scalar equations
preserve their angular and radial signatures in these coordinates. Using the
angular equation, we construct the metric perturbation for a circularly
orbiting perturber around a black hole in Kerr spacetime in a
horizon-penetrating setting. Furthermore, we completed the missing metric
pieces due to the mass M and angular momentum J perturbations. We also provide
an explicit formula for the metric perturbation as a function of the radial
part, its derivative, and the angular part of the solution to the Teukolsky
equation. Finally, we discuss the importance of the extra singularity in the
radial derivative for the convergence of the metric expansion.
