Determination of black holes by boundary measurements

Abstract

For a wave equation with time-independent Lorentzian metric consider an initial-boundary value problem in [Formula: see text], where [Formula: see text] is the time variable and [Formula: see text] is a bounded domain in [Formula: see text]. Let [Formula: see text] be a subdomain of [Formula: see text]. We say that the boundary measurements are given on [Formula: see text] if we know the Dirichlet and Neumann data on [Formula: see text]. The inverse boundary value problem consists of recovery of the metric from the boundary measurements. In the author’s previous works a localized variant of the boundary control method was developed that allows the recovery of the metric locally in a neighborhood of any point of [Formula: see text] where the spatial part of the wave operator is elliptic. This allows the recovery of the metric in the exterior of the ergoregion. The goal of this survey paper is to recover the black holes. In some cases the ergoregion coincides with the black hole. In the case of two space dimensions we recover the black hole inside the ergoregion assuming that the ergosphere, i.e. the boundary of the ergoregion, is not characteristic at any point of the ergosphere.