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 the 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, which 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.

29 pages, 16 figures

References in corpus (14)

• First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole
• First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way
• Calculating black hole shadows: Review of analytical studies
• What does a binary black hole merger look like?
• Kerr-de Sitter Universe
• Cosmological constant corrections to the photon sphere and black hole shadow radii
• Stellar-mass microlensing of gravitational waves
• Heun equation, Teukolsky equation, and type-D metrics
• Image formation process with the solar gravitational lens
• Exact solution for wave scattering from black holes: Formulation
• Spectroscopy of Kerr-AdS${}_5$ spacetime with the Heun function: Quasinormal modes, greybody factor, and evaporation
• Rainbow scattering of gravitational plane waves by a compact body
• Wave-optical study of the Einstein cross formed by a quadrupole gravitational lens
• Absence of Normalizable Time-periodic Solutions for The Dirac Equation in Kerr-Newman-dS Black Hole Background

Cited by in corpus (4)

• Spin wave optics for gravitational waves lensed by a Kerr black hole
• Charged massless scalar fields in a charged $C$-metric black hole: Exact solutions, Hawking radiation, and scattering of scalar waves
• The scalar angular Teukolsky equation and its solution for the Taub-NUT spacetime
• Analysis of scalar fields with series convolution