Gravitational radiation close to a black hole horizon: Waveform regularization and the out-going echo

Black hole perturbation theory for Kerr black holes is best studied in the Newman Penrose Formalism, in which gravitational waves are described as perturbations in the Weyl scalars $\psi_0$ and $\psi_4$, with the governing equation being the well-known Teukolsky equation. Near infinity and near horizon, $\psi_4$ is dominated by the component that corresponds to waves propagating towards the positive radial direction, while $\psi_0$ is dominated by the component that corresponds to waves that propagate towards the negative radial direction. Since gravitational-wave detectors measure out-going waves at infinity, research has been mainly focused on $\psi_4$, leaving $\psi_0$ less studied. But the scenario is reversed in the near horizon region where the in-going-wave boundary condition needs to be imposed. Thus, the near horizon phenomena, e.g., tidal heating and gravitational-wave echoes from Extremely Compact Objects (ECOs), require computing $\psi_0$. In this work, we explicitly calculate the source term for the $\psi_0$ Teukolsky equation due to a point particle plunging into a Kerr black hole. We highlight the need to regularize the solution of the $\psi_0$ Teukolsky equation obtained using Green's function techniques. We suggest a regularization scheme for this purpose and go on to compute the $\psi_0$ waveform close to a Schwarzschild horizon for two types of trajectories of the in-falling particle. We compare the $\psi_0$ waveform calculated directly from the Teukolsky equation with the $\psi_0$ waveform obtained by using the Starobinsky-Teukolsky identity on $\psi_4$. We also compute the out-going echo waveform near infinity, using the near-horizon $\psi_0$ computed directly from the Teukolsky equation and the Boltzmann boundary condition on the ECO surface. We show that this echo is quantitatively different (stronger) than the echo obtained using previous prescriptions. (abridged)