Metric Perturbation Construction in Kerr Spacetime in Horizon Penetrating Coordinates

In this project, the Teukolsky equation have been studied in a horizon penetrating coordinates and tetard,

followingly the radial equation and its first derivative have been studied, the former showed

to be a Confluent Heun with three singularity while the latter is a obeying a Fuchsian differential

equation with one extra singularity. The radial function obey the physical boundary condition

without applying any regularity condition. We verified that Hertz-Weyl scalars equations preserve

their angular and radial signature even in the Horizon penetration coordinates. Using the angular

equation, we have constructed the he metric perturbation in Kerr spacetime for a pertuber moving

in a circular orbit around the blackhole. Finally, we discussed how the extra singularity will be

crucial for the metric expansion convergence.