An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry

Finster, Felix; Kamran, Niky; Smoller, Joel; Yau, Shing–Tung

doi:10.1007/s00220-005-1390-x

Cited by 24 publications

(66 citation statements)

References 15 publications

(33 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The same is true for the last summand due to our ODE estimates of Theorem 2.1. We conclude that n 0 (t) converges in L 2 loc as n 0 → ∞, and the limit coincides with the weak limit, which in [1,2] is shown to be the solution (t) of the Cauchy problem.…”

Section: Decay In L ∞ Locmentioning

confidence: 51%

See 1 more Smart Citation

Decay of Solutions of the Wave Equation in the Kerr Geometry

Finster

Kamran

Smoller

et al. 2008

Commun. Math. Phys.

Self Cite

132

View full text Add to dashboard Cite

As was recently pointed out to us by Thierry Daudé, there is an error on the last page of our paper [2]. Namely, the inequality (8.3) cannot be applied to the function (t) because it does not satisfy the correct boundary conditions. This invalidates the last two inequalities of the paper, and thus the proof of decay is incomplete. We here fill the gap using a different method. At the same time, we will clarify in which sense the sum over the angular momentum modes converges in [1, Theorem 1.1] and [2, Theorem 7.1], an issue which in these papers was not treated in sufficient detail. The arguments in these papers certainly yield weak convergence in L 2 loc ; here we will prove strong convergence. Our method here is to split the wave function into the high and low energy components. For the high energy component, we show that the L 2 -norm of the wave function can be bounded by the energy integral, even though the energy density need not be everywhere positive (Sect. 1). For the low energy component we refine our ODE techniques (Sect. 2). Combining these arguments with a Sobolev estimate and the Riemann-Lebesgue lemma completes the proof (Sect. 3).We begin by considering the integral representation of [2, Theorem 7.1], for fixed k and a finite number n 0 of angular momentum modes, The online version of the original article can be found under

show abstract

Section: Decay In L ∞ Locmentioning

confidence: 51%

“…We recall that for large ω, the WKB-estimates of [1,Sect. 6] ensure that the fundamental solutions b kωn go over to plane waves, and thus, since the initial data 0 is smooth and compactly supported, the functionˆ n 0 (ω, r, ϑ) decays rapidly in ω (for details on this method see [3, proof of Theorem 6.5]).…”

mentioning

confidence: 99%

Decay of Solutions of the Wave Equation in the Kerr Geometry

Finster

Kamran

Smoller

et al. 2008

Commun. Math. Phys.

Self Cite

132

View full text Add to dashboard Cite

show abstract

“…In this paper we shall consider the Cauchy problem analytically, giving a mathematically rigorous treatment of superradiance for scalar waves. Our analysis is based on the integral representation for the wave propagator obtained in [8,9].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

A Rigorous Treatment of Energy Extraction from a Rotating Black Hole

Finster

Kamran

Smoller

et al. 2009

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

The Cauchy problem is considered for the scalar wave equation in the Kerr geometry. We prove that by choosing a suitable wave packet as initial data, one can extract energy from the black hole, thereby putting supperradiance, the wave analogue of the Penrose process, into a rigorous mathematical framework. We quantify the maximal energy gain. We also compute the infinitesimal change of mass and angular momentum of the black hole, in agreement with Christodoulou's result for the Penrose process. The main mathematical tool is our previously derived integral representation of the wave propagator.

show abstract

“…Recently an integral representation was derived for solutions of the scalar wave equation in the Kerr black hole geometry [4]. This result relies crucially on a spectral representation for the oblate spheroidal wave operator for complex values of the aspherical parameter Ω (also referred to as "ellipticity parameter" or "semifocal distance").…”

Section: Introductionmentioning

confidence: 99%

“…Note that the potential in the spheroidal wave operator is in general complex, 4) and therefore A is symmetric only if Ω is real. In previous works, asymptotic expansions for individual eigenvalues are derived [5,11], and it is shown numerically that eigenvalues can degenerate for non-real Ω [7], but rigorous estimates or completeness statements are not given.…”

Section: Introductionmentioning

confidence: 99%

Spectral estimates and non-selfadjoint perturbations of spheroidal wave operators

Finster¹,

Schmid²

2006

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

We derive a spectral representation for the oblate spheroidal wave operator which is holomorphic in the aspherical parameter Ω in a neighborhood of the real line. For real Ω, estimates are derived for all eigenvalue gaps uniformly in Ω.The proof of the gap estimates is based on detailed estimates for complex solutions of the Riccati equation. The spectral representation for complex Ω is obtained using the theory of slightly non-selfadjoint perturbations.

show abstract

An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry

Cited by 24 publications

References 15 publications

Decay of Solutions of the Wave Equation in the Kerr Geometry

Decay of Solutions of the Wave Equation in the Kerr Geometry

A Rigorous Treatment of Energy Extraction from a Rotating Black Hole

Spectral estimates and non-selfadjoint perturbations of spheroidal wave operators

Contact Info

Product

Resources

About