Exact wave-optical imaging of a Kerr-de Sitter black hole using Heun's equation

authored by: Felix Willenborg, Dennis Philipp, Claus Lämmerzahl
Abstract: Spacetime perturbations due to scalar, vector, and tensor fields on a fixed background geometry can be described in the framework of Teukolsky's equation. In this work, wave scattering is treated analytically, using the Green's function method and solutions to the separated radial and angular differential equations in combination with a partial wave technique for a scalar and monochromatic perturbation. The results are applied to analytically describe wave-optical imaging via Kirchhoff-Fresnel diffraction, leading to, e.g., the formation of observable black hole shadows. A comparison to the ray-optical description is given, providing new insights into wave-optical effects and properties. On a Kerr-de Sitter spacetime, the cosmological constant changes the singularity structure of the Teukolsky equation and allows for an analytical, exact solution via a transformation into Heun's differential equation, which is the most general, second-order differential equation with four regular singularities. The scattering of waves originating from a point source involves a solution in terms of the so-called Heun's function Hf. It is used to find angular solutions that form a complete set of orthonormal functions similar to the spherical harmonics. Our approach allows to solve the scattering problem while taking into account the complex interplay of Heun's functions around local singularities.
External Organisation(s): University of Bremen
Center of Applied Space Technology and Microgravity (ZARM)
Type: Article
Journal: Physical Review D
Volume: 109
ISSN: 2470-0010
Publication date: 15.02.2024
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Nuclear and High Energy Physics
Electronic version(s): https://doi.org/10.48550/arXiv.2310.12917 (Access: Open)
https://doi.org/10.1103/PhysRevD.109.044056 (Access: Closed)

BibTeX

@article{29e3a2bc222c42c3b4fb06810deb19ec,
title = "Exact wave-optical imaging of a Kerr-de Sitter black hole using Heun's equation",
abstract = "Spacetime perturbations due to scalar, vector, and tensor fields on a fixed background geometry can be described in the framework of Teukolsky's equation. In this work, wave scattering is treated analytically, using the Green's function method and solutions to the separated radial and angular differential equations in combination with a partial wave technique for a scalar and monochromatic perturbation. The results are applied to analytically describe wave-optical imaging via Kirchhoff-Fresnel diffraction, leading to, e.g., the formation of observable black hole shadows. A comparison to the ray-optical description is given, providing new insights into wave-optical effects and properties. On a Kerr-de Sitter spacetime, the cosmological constant changes the singularity structure of the Teukolsky equation and allows for an analytical, exact solution via a transformation into Heun's differential equation, which is the most general, second-order differential equation with four regular singularities. The scattering of waves originating from a point source involves a solution in terms of the so-called Heun's function Hf. It is used to find angular solutions that form a complete set of orthonormal functions similar to the spherical harmonics. Our approach allows to solve the scattering problem while taking into account the complex interplay of Heun's functions around local singularities.",
author = "Felix Willenborg and Dennis Philipp and Claus L{\"a}mmerzahl",
note = "Publisher Copyright: {\textcopyright} 2024 American Physical Society.",
year = "2024",
month = feb,
day = "15",
doi = "10.48550/arXiv.2310.12917",
language = "English",
volume = "109",
journal = "Physical Review D",
issn = "2470-0010",
publisher = "American Institute of Physics",
number = "4",
}