Annales Henri Poincaré

, Volume 12, Issue 1, pp 1–47 | Cite as

Inverse Scattering at Fixed Energy in de Sitter–Reissner–Nordström Black Holes

  • Thierry Daudé
  • François NicoleauEmail author


In this paper, we consider massless Dirac fields propagating in the outer region of de Sitter–Reissner–Nordström black holes. We show that the metric of such black holes is uniquely determined by the partial knowledge of the corresponding scattering matrix S(λ) at a fixed energy λ ≠ 0. More precisely, we consider the partial wave scattering matrices S(λ, n) (here λ ≠ 0 is the fixed energy and \({n \in \mathbb{N}^{*}}\) denotes the angular momentum) defined as the restrictions of the full scattering matrix on a well chosen basis of spin-weighted spherical harmonics. We prove that the mass M, the square of the charge Q 2 and the cosmological constant Λ of a dS-RN black hole (and thus its metric) can be uniquely determined from the knowledge of either the transmission coefficients T(λ, n), or the reflexion coefficients R(λ, n) (resp. L(λ, n)), for all \({n \in {\mathcal{L}}}\) where \({\mathcal{L}}\) is a subset of \({\mathbb{N}^{*}}\) that satisfies the Müntz condition \({\sum_{n \in{\mathcal{L}}}\frac{1}{n} = +\infty}\) . Our main tool consists in complexifying the angular momentum n and in studying the analytic properties of the “unphysical” scattering matrix S(λ, z) in the complex variable z. We show, in particular, that the quantities \({\frac{1}{T(\lambda,z)}}\), \({\frac{R(\lambda,z)}{T(\lambda,z)}}\) and \({\frac{L(\lambda,z)}{T(\lambda,z)}}\) belong to the Nevanlinna class in the region \({\{z \in \mathbb{C}, Re(z) > 0 \}}\) for which we have analytic uniqueness theorems at our disposal. Eventually, as a by-product of our method, we obtain reconstruction formulae for the surface gravities of the event and cosmological horizons of the black hole which have an important physical meaning in the Hawking effect.


Black Hole Event Horizon Dirac Equation Dirac Operator Partial Wave 
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© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontréalCanada
  2. 2.Département de Mathématiques, UMR CNRS 8088Université de Cergy-PontoiseCergy-PontoiseFrance
  3. 3.Laboratoire Jean Leray, UMR 6629Université de NantesNantes Cedex 03France

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