Journal
CLASSICAL AND QUANTUM GRAVITY
Volume 40, Issue 23, Pages -Publisher
IOP Publishing Ltd
DOI: 10.1088/1361-6382/ad0495
Keywords
linear perturbation in Kerr spacetime; horizon penetrating coordinates; Teukolsky equation; confluent Heun ODE; metric reconstruction; metric completion
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In this study, we investigate the behavior of perturbation waves in the Teukolsky equation using horizon-penetrating coordinates. We find that the radial functions satisfy physical boundary conditions and the Hertz-Weyl scalar equations preserve their characteristics in these coordinates. Using the angular equation, we construct the metric perturbation for a perturber orbiting a black hole in Kerr spacetime in a horizon-penetrating setting and provide an explicit formula for the metric perturbation.
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.
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