[Paper Review] Inverse scattering at fixed energy for massive charged Dirac fields in de Sitter-Reissner-Nordström black holes

Damien Gobin|arXiv (Cornell University)|Dec 2, 2014

Black Holes and Theoretical Physics20 references3 citations

TL;DR

This paper establishes the unique recovery of the mass, charge, and cosmological constant of a de Sitter-Reissner-Nordström black hole from the partial wave scattering operators $ S(\lambda,n) $ at a fixed energy $ \lambda $, using analytic continuation in complex angular momentum $ z $ and the Müntz condition $ \sum_{n\in\mathcal{L}} \frac{1}{n} = \infty $. The result extends inverse scattering theory to massive, charged Dirac fields in curved spacetime with full uniqueness up to diffeomorphism.

ABSTRACT

In this paper, we consider massive charged Dirac fields propagating in the exterior region of de Sitter-Reissner-Nordström black holes.We show that the parameters of such black holes are uniquely determined by the partial knowledge of the corresponding scattering operator $S(λ)$ at a fixed energy $λ$.More precisely, we consider the partial wave scattering operators $S(λ,n)$ (here $λ\in \mathbb{R}$ is the energy and $n \in \mathbb{N}^{\star}$ denotes the angular momentum) defined as the restrictions of the full scattering operator on a well chosen basis of spin-weighted spherical harmonics.We prove that the knowledge of the scattering operators $S(λ,n)$, for all $n \in \mathcal{L}$, where $\mathcal{L}$ is a subset of $\mathbb{N}^{\star}$ that satisfies the Müntz condition $\sum\_{n \in \mathcal{L}} \frac{1}{n} = + \infty$, allows to recover the mass, the charge and the cosmological constant of a dS-RN black hole.The main tool consists in the complexification of the angular momentum $n$ and in studying the analytic properties of the "unphysical" corresponding data in the complex variable $z$.

Motivation & Objective

• To determine whether the physical parameters of a de Sitter-Reissner-Nordström black hole can be uniquely reconstructed from inverse scattering data at a single energy.
• To extend previous inverse scattering results for massless, uncharged Dirac fields to the case of massive, charged Dirac fields in curved spacetime.
• To establish uniqueness of the black hole parameters—mass $ M $, charge $ Q $, and cosmological constant $ \Lambda $—from partial knowledge of the scattering operator at fixed energy.
• To show that the scattering data for a subset $ \mathcal{L} \subset \mathbb{N}^* $ satisfying the Müntz condition $ \sum_{n\in\mathcal{L}} \frac{1}{n} = \infty $ suffice for reconstruction.

Proposed method

• Complexify the angular momentum $ n \in \mathbb{N}^* $ to $ z \in \mathbb{C} $, treating the scattering data as analytic functions in $ z $.
• Define the reflection matrix $ R(\lambda,n) $ and its analytic continuation $ \hat{R}(\lambda,n) $, relating them to the scattering matrix via phase factors involving $ \beta $.
• Use the Phragmén-Lindelöf and Liouville theorems to prove that the matrix $ P(X,\lambda,z) $, constructed from Jost functions and scattering data, is constant on $ \mathbb{C} $.
• Establish that $ P(X,\lambda,z) = \pm I_4 $ using asymptotic expansions of Jost functions and scattering data at infinity.
• Apply the Müntz condition to ensure density of the set $ \mathcal{L} $, enabling uniqueness of analytic continuation and parameter recovery.
• Compare scattering data from two black holes and derive equality of integrals $ A = \tilde{A} $, leading to full parameter identification.

Experimental results

Research questions

• RQ1 Can the parameters of a de Sitter-Reissner-Nordström black hole be uniquely determined from scattering data at a single fixed energy?
• RQ2 Does the knowledge of partial wave scattering operators $ S(\lambda,n) $ for $ n \in \mathcal{L} $, where $ \sum_{n\in\mathcal{L}} \frac{1}{n} = \infty $, suffice to reconstruct the black hole's mass, charge, and cosmological constant?
• RQ3 How does the inclusion of a mass term and electric charge affect the inverse scattering problem compared to the massless, uncharged case?
• RQ4 What role does complex analytic continuation of angular momentum play in recovering the black hole parameters?

Key findings

• The mass $ M $, charge $ Q $, and cosmological constant $ \Lambda $ of a de Sitter-Reissner-Nordström black hole are uniquely determined by the scattering operators $ S(\lambda,n) $ at a fixed energy $ \lambda $, for $ n \in \mathcal{L} $ with $ \sum_{n\in\mathcal{L}} \frac{1}{n} = \infty $.
• The scattering data at fixed energy uniquely determine the scalar functions in the Dirac equation up to a certain diffeomorphism, extending beyond just the black hole parameters.
• The proof relies on analytic continuation in complex angular momentum $ z $, with the matrix $ P(X,\lambda,z) $ shown to be constant $ \pm I_4 $ via Phragmén-Lindelöf and Liouville theorems.
• The equality $ A = \tilde{A} $, where $ A = \int_{\mathbb{R}} a(t) dt $, is derived from matching scattering data, enabling full parameter recovery.
• The result holds without requiring knowledge of the scattering operator over an energy interval, unlike previous works, making it more physically relevant.