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

Berti, Emanuele; Cardoso, Vítor; Casals, Marc

doi:10.1103/physrevd.73.024013

Cited by 271 publications

(192 citation statements)

References 72 publications

Supporting

Mentioning

182

Contrasting

Order By: Relevance

“…For Kerr spacetime, ∆ θ = 1 and the above equation becomes a spheroidal equation which has been studied in detail in [25][26][27][28][29][30][31][32]. And λ can be expanded as a series of aω for small aω, whose explicit form can be found in [26,27,32]. For Kerr-de Sitter BH, eq.…”

Section: Perturbation Equationsmentioning

confidence: 99%

Superradiant instability of Kerr-de Sitter black holes in scalar-tensor theory

Zhang

Wang

2014

J. High Energ. Phys.

View full text Add to dashboard Cite

We investigate in detail the mechanism of superradiance to render the instability of Kerr-de Sitter black holes in scalar-tensor gravity. Our results provide more clues to examine the scalar-tensor gravity in the astrophysical black holes in the universe with cosmological constant. We also discuss the spontaneous scalarization in the de Sitter background and find that this instability can also happen in the spherical de Sitter configuration in a special style.

show abstract

Section: Perturbation Equationsmentioning

confidence: 99%

Superradiant instability of Kerr-de Sitter black holes in scalar-tensor theory

Zhang

Wang

2014

J. High Energ. Phys.

View full text Add to dashboard Cite

show abstract

“…At least for small aω, we may expect these contributions to be small. In fact, the corrections turn out to be small even for moderately large values of aω (see [30] for an explicit calculation of the inner products at the QNM frequencies). Nevertheless, using spherical harmonics instead of spheroidal harmonics can induce a small amount of mode-mixing in the initial data.…”

Section: Rotational Mode Mixingmentioning

confidence: 99%

“…The coefficients c ℓ ′ ℓm are related to the more familiar Clebsch-Gordan coefficients [30,31]. As a result of (23), and because of the orthogonality of the (spin-weighted) spherical harmonics, inner products of different spheroidal harmonics will be given by inner products of spherical harmonics with higher-order corrections in aω.…”

Section: Rotational Mode Mixingmentioning

confidence: 99%

“…II D we described our initial data family sets, which were expanded in spherical harmonics. Since the Kerr background is not spherically symmetric we should not expand the perturbation in terms of spherical harmonics, but (more rigorously) in terms of the spin-weighted spheroidal harmonics s S ℓm (aω), where s is the spin weight of the perturbing field, a = jM is the black hole spin parameter, and ω is the frequency in a Fourier expansion of the perturbation (a quantitative discussion of spin-weighted spheroidal harmonics and more references can be found in [30]). However, as first shown by Press and Teukolsky [31], the s S ℓm 's may be expanded as a power series in aω:…”

Section: Rotational Mode Mixingmentioning

confidence: 99%

See 1 more Smart Citation

Numerical study of the quasinormal mode excitation of Kerr black holes

Dorband

Berti²,

Diener

et al. 2006

Phys. Rev. D

Self Cite

View full text Add to dashboard Cite

We present numerical results from three-dimensional evolutions of scalar perturbations of Kerr black holes. Our simulations make use of a high-order accurate multi-block code which naturally allows for fixed adaptivity and smooth inner (excision) and outer boundaries. We focus on the quasinormal ringing phase, presenting a systematic method for extraction of the quasinormal mode frequencies and amplitudes and comparing our results against perturbation theory.The detection of a single mode in a ringdown waveform allows for a measurement of the mass and spin of a black hole; a multimode detection would allow a test of the Kerr nature of the source. Since the possibility of a multimode detection depends on the relative mode amplitude, we study this topic in some detail. The amplitude of each mode depends exponentially on the starting time of the quasinormal regime, which is not defined unambiguously. We show that this time-shift problem can be circumvented by looking at appropriately chosen relative mode amplitudes. From our simulations we extract the quasinormal frequencies and the relative and absolute amplitudes of corotating and counterrotating modes (including overtones in the corotating case). We study the dependence of these amplitudes on the shape of the initial perturbation, the angular dependence of the mode and the black hole spin, comparing against results from perturbation theory in the so-called asymptotic approximation. We also compare the quasinormal frequencies from our numerical simulations with predictions from perturbation theory, finding excellent agreement. For rapidly rotating black holes (of spin j = 0.98) we can extract the quasinormal frequencies of not only the fundamental mode, but also of the first two overtones. Finally we study under what conditions the relative amplitude between given pairs of modes gets maximally excited and present a quantitative analysis of rotational mode-mode coupling. The main conclusions and techniques of our analysis are quite general and, as such, should be of interest in the study of ringdown gravitational waves produced by astrophysical gravitational wave sources.

show abstract

“…For an arbitrary form α we denote 18) where e a 1 ... a D is a totally antisymmetric tensor. The co-derivative δ is defined as follows:…”

Section: Cky Tensors As Differential Formsmentioning

confidence: 99%

Hidden Symmetries of Higher-Dimensional Rotating Black Holes

2007

View full text Add to dashboard Cite

In this thesis we study higher-dimensional rotating black holes. Such black holes are widely discussed in string theory and brane-world models at present.We demonstrate that even the most general known Kerr-NUT-(A)dS spacetime, describing the general rotating higher-dimensional asymptotically (anti) de Sitter black hole with NUT parameters, is in many aspects similar to its fourdimensional counterpart. Namely, we show that it admits a fundamental hidden symmetry associated with the principal conformal Killing-Yano tensor. Such a tensor generates towers of hidden and explicit symmetries. The tower of Killing tensors is responsible for the existence of irreducible, quadratic in momenta, conserved integrals of geodesic motion. These integrals, together with the integrals corresponding to the tower of explicit symmetries, make geodesic equations in the Kerr-NUT-(A)dS spacetime completely integrable. We further demonstrate that in this spacetime the Hamilton-Jacobi, Klein-Gordon, and stationary string equations allow complete separation of variables and the problem of finding parallel-propagated frames reduces to the set of the first order ordinary differential equations. Moreover, we show that the Kerr-NUT-(A)dS spacetime is the most general Einstein space which possesses all these properties. We also explicitly derive the most general (off-shell) canonical metric admitting the principal conformal Killing-Yano tensor and demonstrate that such a metric is necessarily of the special algebraic type D of the higher-dimensional algebraic classification. The results presented in this thesis describe the new and complete picture of the relationship of hidden symmetries and rotating black holes in higher dimensions. PrefaceWhen in 1963 Kerr discovered an astrophysically relevant but relatively complicated metric describing the gravitational field of a rotating black hole, it seemed that no analytical predictions were possible even for the simplest particle geodesic motion. However, a 'miracle' happened, and it turned out that not only the geodesic motion can be analytically solved, but also the equations describing various perturbations of this background can be 'drastically' simplified. This opened a way for studying astrophysical processes, such as the plasma accretion around black holes, the radiation produced by infalling matter, the origin of jets, the production and propagation of waves produced in the vicinity of black holes, and it even led to estimates of the gravitational wave production in star collisions and galaxy merges. It also facilitated the study of more theoretical problems, such as the problem of stability of the Kerr solution, the calculation of the quasinormal modes, or the study of the Hawking radiation. A hidden symmetry responsible for this miracle can be mathematically described by a simple antisymmetric object, called the Killing-Yano tensor.Recently, higher-dimensional rotating black hole spacetimes have become of high interest due to various developments in gravity and high energy physic...

show abstract

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

Cited by 271 publications

References 72 publications

Superradiant instability of Kerr-de Sitter black holes in scalar-tensor theory

Superradiant instability of Kerr-de Sitter black holes in scalar-tensor theory

Numerical study of the quasinormal mode excitation of Kerr black holes

Hidden Symmetries of Higher-Dimensional Rotating Black Holes

Contact Info

Product

Resources

About