Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions

By Emanuele Berti, V. Cardoso, M. Casals. 2005

265 citations recorded by the source corpus.

Spin-weighted spheroidal harmonics are useful in a variety of physical situations, including light scattering, nuclear modeling, signal processing, electromagnetic wave propagation, black hole perturbation theory in four and higher dimensions, quantum field theory in curved space-time and studies of D-branes. We first review analytic and numerical calculations of their eigenvalues, and eigenfunctions in four dimensions, filling gaps in the existing literature when necessary. Then we compute the angular dependence of the spin-weighted spheroidal harmonics corresponding to slowly damped quasinormal mode frequencies of the Kerr black hole, providing numerical tables and approximate formulas for their scalar products. Finally we present an exhaustive analytic and numerical study of scalar spheroidal harmonics in (n+4) dimensions.

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Citations in the corpus

Superradiance: Energy Extraction, Black-Hole Bombs and Implications for Astrophysics and Particle Physics (2015, methodology): “76) The computation of the eigenvalues for generic spin can only be done numerically [184].”
Modeling ringdown: Beyond the fundamental quasinormal modes (2014, background): “where ω̃lmn is the complex QNM frequency, −2Slm are the spin weighted spheroidal harmonics, and Almn are the complex QNM amplitudes or excitation coefficients whose magnitude is contingent on where t is chosen to be zero [27, 40, 41].”
Holographic thermalization, quasinormal modes and superradiance in Kerr-AdS (2013, methodology): “The properties of spheroidal wavefunctions and eigenvalues are known analytically for some time in AF spacetimes [80].”
Kerr-CFT and gravitational perturbations (2009, background): “, [22]), with positive integer l specifying the number of zeros, ` − max{|m|, |s|}, of the eigenfunction.”
Black hole superradiance from Kerr/CFT (2009, background): “y Kℓ= ℓ(ℓ+1)+O(a2ω2). For a scalar at the superradiant bound ω= mΩH on an extreme black hole, we denote the separation constant by K¯ ℓ= Kℓ|a2ω2=m2/4 . (4.7) Numerical values of K¯ ℓ are tabulated in [24, 28]. 9 In terms of the rescaled frequency ˜nR = nR 2πTR and dimensionless Hawking temperature τ”
Instability of the massive Klein-Gordon field on the Kerr spacetime (2007, background): “The angular solutions are spheroidal harmonics [1, 6], Slm = S m l (cos θ; c).”
Testing the nature of dark compact objects: a status report (2019, background): “When χ = 0, the angular eigenvalues are λ = (l− s)(l+ s+ 1), whereas for χ 6= 0 they can be computed numerically or with approximated analytical expansions [354].”
Superradiance (2015, methodology): “26) can be separated by use of spin-0 spheroidal wavefunctions [243] as discussed in Sec.”