Conformal scattering on the Schwarzschild metric

Nicolas, Jean-Philippe

doi:10.5802/aif.3034

Cited by 38 publications

(67 citation statements)

References 32 publications

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“…Further, Dafermos and Shlapentokh-Rothmann, [24] obtained scattering maps for axisymmetric scalar waves on fixed Kerr backgrounds in the exterior as well as the interior with conclusion concordant with the above statements. For precise statements and further scattering results see [24,25,50] and references therein. For results concerning extremal black holes see [3][4][5] by Aretakis and [32,33] by Gajic.…”

Section: Scalar Wave Equation Without Symmetry In Black Hole Interiorsmentioning

confidence: 99%

Boundedness of Massless Scalar Waves on Kerr Interior Backgrounds

Franzen

2020

Ann. Henri Poincaré

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We consider solutions of the massless scalar wave equation g ψ = 0, without symmetry, on fixed subextremal Kerr backgrounds (M, g). It follows from previous analyses in the Kerr exterior that for solutions ψ arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast along the event horizon H + . Using the derived decay rate, we show that ψ is in fact uniformly bounded, |ψ| ≤ C, in the black hole interior up to and including the bifurcate Cauchy horizon CH + , to which ψ in fact extends continuously. In analogy to our previous paper, [30], on boundedness of solutions to the massless scalar wave equation on fixed subextremal Reissner-Nordström backgrounds, the analysis depends on weighted energy estimates, commutation by angular momentum operators and application of Sobolev embedding. In contrast to the Reissner-Nordström case the commutation leads to additional error terms that have to be controlled. FIG. 1: Penrose diagram of the maximal future development of a Cauchy hypersurface Σ in Kerr spacetime (M, g). What is depicted is in fact the range of a double null coordinate system which is global in the interior and will be discussed further in Section 2.2.3.The analysis of (1) for the exterior Kerr region J − (I + ), in the full subextremal range |a| < M , has already been accomplished in [27]. The purpose of the present work is to extend the investigation to the interior of the black hole, up to and including the Cauchy horizon CH + . The mathematical structure and notation of this paper closely follows our work on Reissner-Nordström backgrounds, see [30].As already mentioned, the analysis of (1) is motivated by the Strong Cosmic Censorship Conjecture. In order to investigate its validity we need to understand stability and instability properties of black hole interiors. A brief overview on this topic as well as a mathematical formulation of the conjecture were already given in Section 1.2 of [30]. References discussing the instability behavior in similar settings are [44,47]. Also refer to [62] for a numerical analysis and [6,52,58] for early discussions of heuristic models.

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Section: Scalar Wave Equation Without Symmetry In Black Hole Interiorsmentioning

confidence: 99%

Boundedness of Massless Scalar Waves on Kerr Interior Backgrounds

Franzen

2020

Ann. Henri Poincaré

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“…Then, more recently, J. Joudioux obtained the first result for a non linear wave equation [23] in non stationary situations. I came back to the topic a couple of years ago to propose an extension of these methods to black hole spacetimes [31], describing the construction in the Schwarzschild spacetime, which is static, and discussing the case of the Kerr metric and the associated difficulties.…”

Section: Conformal Scatteringmentioning

confidence: 99%

“…The method needs to be modified in order to deal with these singularities. The treatment of the difficulty at i 0 can be found in [26] and a conformal scattering on the Schwarzschild metric was developed in [31] with a way of dealing with timelike infinities. We give in the next two subsections the essential ingredients of the extension on the conformal scattering construction to spacetimes with singular i 0 and to black hole spacetimes.…”

Section: Conformal Scattering On Minkowski Spacetimementioning

confidence: 99%

“…Once this is done, the rest of the construction is mostly unchanged because the behaviour of the conformal metric at i + has no influence on the propagation of solutions from their future scattering data into the spacetime. We give in this subsection the essential features of the conformal scattering theory developed on the Schwarzschild metric in [31]. For more technical details as well as a discussion of the additional difficulties in the case of the Kerr metric, see [31] and references therein, in particular the recent work by M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman [11].…”

Section: Conformal Scattering On the Schwarzschild Metricmentioning

confidence: 99%

“…We give in this subsection the essential features of the conformal scattering theory developed on the Schwarzschild metric in [31]. For more technical details as well as a discussion of the additional difficulties in the case of the Kerr metric, see [31] and references therein, in particular the recent work by M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman [11].…”

Section: Conformal Scattering On the Schwarzschild Metricmentioning

confidence: 99%

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Abstract.This essay was written as an extended version of a talk given at a conference in Strasbourg on "Riemann, Einstein and geometry", organized by Athanase Papadopoulos in September 2014. Its aim is to present Roger Penrose's approach to asymptotic analysis in general relativity, which is based on conformal geometric techniques, focusing on historical and recent aspects of two specialized topics : conformal scattering and peeling.

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Abstract. We present a proof of the existence of the Hawking radiation for massive bosons in the Schwarzchild-de Sitter metric. It provides estimates for the rates of decay of the initial quantum state to the Hawking thermal state. The arguments in the proof include a construction of radiation fields by conformal scattering theory; a semiclassical interpretation of the blueshift effect; and the use of a WKB parametrix near the surface of a collapsing star. The proof does not rely on the spherical symmetry of the spacetime.

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Conformal scattering on the Schwarzschild metric

Cited by 38 publications

References 32 publications

Boundedness of Massless Scalar Waves on Kerr Interior Backgrounds

Boundedness of Massless Scalar Waves on Kerr Interior Backgrounds

The Conformal Approach to Asymptotic Analysis

A Quantitative Version of Hawking Radiation

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