Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions (original) (raw)

Scalar spheroidal harmonics in five dimensional Kerr(A)dS

2011

We derive expressions for the general five-dimensional metric for Kerr-(A)dS black holes. The Klein-Gordon equation is explicitly separated and we show that the angular part of the wave equation leads to just one spheroidal wave equation, which is also that for charged five-dimensional Kerr-(A)dS black holes. We present results for the perturbative expansion of the angular eigenvalue in powers of the rotation parameters up to 6th order and compare numerically with the continued fraction method.

Asymptotic iteration method for spheroidal harmonics of higher-dimensional Kerr(A)dS black holes

Physical Review D, 2009

In this work we calculate the angular eigenvalues of the (n+4)-dimensional simply rotating Kerr-(A)dS spheroidal harmonics using the asymptotic iteration method. We make some comparisons between this method and that of the continued fraction method and use the latter to check our results. We also present analytic expressions for the small rotation limit up to O(c3) with the coefficient of each power up to O(α2), where c=aω and α=a2Λ (a is the angular velocity, ω the frequency, and Λ the cosmological constant).

Angular Eigenvalues of Higher-Dimensional Kerr(A)dS Black Holes with Two Rotations

2011

In this paper, following the work of Chen, L\"u and Pope, we present the general metric for Kerr-(A)dS black holes with two rotations. The corresponding Klein-Gordon equation is separated explicitly, from which we develop perturbative expansions for the angular eigenvalues in powers of the rotation parameters with Dgeq6D\geq 6Dgeq6.

Theory and computation of spheroidal wavefunctions

Journal of Physics A: Mathematical and General, 2003

In this paper we report on a package, written in the Mathematica computer algebra system, which has been developed to compute the spheroidal wave functions of Meixner [J. Meixner and R.W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen, 1954] and is availlable online (www.physics.uwa.edu.au/\~falloon/spheroidal/spheroidal.html). This package represents a substantial contribution to the existing software, since it computes the spheroidal wave functions to arbitrary precision for general complex parameters m, n

On the Eigenvalues of the Fermionic Angular Eigenfunctions in the Kerr Metric

Entropy

In view of a result recently published in the context of the deformation theory of linear Hamiltonian systems, we reconsider the eigenvalue problem associated with the angular equation arising after the separation of the Dirac equation in the Kerr metric, and we show how a quasilinear first order PDE for the angular eigenvalues can be derived efficiently. We also prove that it is not possible to obtain an ordinary differential equation for the eigenvalues when the role of the independent variable is played by the particle energy or the black hole mass. Finally, we construct new perturbative expansions for the eigenvalues in the Kerr case and obtain an asymptotic formula for the eigenvalues in the case of a Kerr naked singularity.

Linear waves in the Kerr geometry: A mathematical voyage to black hole physics

Bulletin of the American Mathematical Society, 2009

This paper gives a survey of wave dynamics in the Kerr spacetime geometry, the mathematical model of a rotating black hole in equilibrium. After a brief introduction to the Kerr metric, we review the separability properties of linear wave equations for fields of general spin s = 0, 1 2 , 1, 2, corresponding to scalar, Dirac, electromagnetic fields and linearized gravitational waves. We give results on the long-time dynamics of Dirac and scalar waves, including decay rates for massive Dirac fields. For scalar waves, we give a rigorous treatment of superradiance and describe rigorously a mechanism of energy extraction from a rotating black hole. Finally, we discuss the open problem of linear stability of the Kerr metric and present partial results.

Quasinormal modes of Kerr black holes in four and higher dimensions

Physical Review D, 2008

We analytically calculate to leading order the asymptotic form of quasinormal frequencies of Kerr black holes in four, five and seven dimensions. All the relevant quantities can be explicitly expressed in terms of elliptical integrals. In four dimensions, we confirm the results obtained by Keshest and Hod by comparing the analytic results to the numerical ones.

An analytic representation for the quasi-normal modes of Kerr black holes

Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 1985

The gravitational quasi-normal frequencies of both stationary and rotating black holes are calculated by constructing exact eigensolutions to the radiative boundary-value problem of Chandrasekhar and Detweiler. The method is that employed by Jaffé in his determination of the electronic spectra of the hydrogen molecule ion in 1934, and analytic representations of the quasi-normal mode wavefunctions are presented here for the first time. Numerical solution of Jaffé’s characteristic equation indicates that for each l -pole there is an infinite number of damped Schwarzschild quasi-normal modes. The real parts of the corresponding frequencies are bounded, but the imaginary parts are not. Figures are presented that illustrate the trajectories the five least-damped of these frequencies trace in the complex frequency plane as the angular momentum of the black hole increases from zero to near the Kerr limit of maximum angular momentum per unit mass, a = M , where there is a coalescence of th...

Energy and angular momentum in D-dimensional Kerr-Ads black holes — New formulation

International Journal of Geometric Methods in Modern Physics

Recently, it was shown that the conserved charges of asymptotically anti-de Sitter spacetimes can be written in an explicitly gauge-invariant way in terms of the linearized Riemann tensor and the Killing vectors. Here, we employ this construction to compute the mass and angular momenta of the [Formula: see text]-dimensional Kerr-AdS black holes, which is one of the most remarkable Einstein metrics generalizing the four-dimensional rotating black hole.