Uniqueness and Nonuniqueness in Inverse Hyperbolic Problems and The Black Hole Phenomenon

TL;DR

This paper explores the conditions for black and white hole phenomena in inverse wave problems and demonstrates nonuniqueness of solutions through energy estimates in Lorentzian geometries.

Contribution

It provides new insights into the existence of black and white holes in inverse hyperbolic problems and establishes nonuniqueness results using energy estimates.

Findings

Conditions for black/white hole existence in wave equations

Nonuniqueness of inverse problems in Lorentzian settings

Energy estimates demonstrating nonuniqueness

Abstract

This paper consists of two parts. In the first part we describe the recent works on the inverse problems for the wave equation in $(n+1)$-dimensional space equipped with pseudo-Riemannian metric with Lorentz signature. We study the conditions of the existence of black (or white) holes for these wave equations. In the second part we prove energy type estimates on a finite time interval in the presence of black or white holes. We use these estimates to prove the nonuniqueness of the inverse problems.