Inverse scattering at fixed energy in de Sitter-Reissner-Nordstr\"om black holes

TL;DR

This paper demonstrates that the metric of de Sitter-Reissner-Nordström black holes can be uniquely reconstructed from partial scattering data at a fixed energy, using complex analysis of the scattering matrix.

Contribution

It introduces a novel method of complexifying angular momentum and analyzing the analytic properties of the scattering matrix to achieve unique inverse reconstruction.

Findings

Unique determination of black hole parameters from scattering data.

Development of a complex analysis approach for inverse scattering.

Derivation of formulas for surface gravities of horizons.

Abstract

In this paper, we consider massless Dirac fields propagating in the outer region of de Sitter-Reissner-Nordstr\"om black holes. We show that the metric of such black holes is uniquely determined by the partial knowledge of the corresponding scattering matrix $S(\lambda)$ at a fixed energy $\lambda \ne 0$. More precisely, we consider the partial wave scattering matrices $S(\lambda,n)$ (here $\lambda \ne 0$ is the fixed energy and $n \in \N^*$ 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 $\Lambda$ of a dS-RN black hole (and thus its metric) can be uniquely determined from the knowledge of either the transmission coefficients $T(\lambda, n)$, or the reflexion coefficients $R(\lambda, n)$ (resp. $L(\lambda, n)$), for all $n \in {\mathcal{L}}$ where $\mathcal{L}$ is a subset of $\N^*$ that satisfies the M\"untz 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 $S(\lambda,z)$ 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 $\{z \in \C, \ Re(z) >0 \}$ for which we have analytic uniqueness theorems at our disposal. Eventually, as a by-product of our method, we obtain reconstrution 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.