Recovering a Riemannian Metric from Cherenkov Radiation in Inhomogeneous Anisotropic Medium

Abstract

Although travelling faster than the speed of light in vacuum is not physically allowed, the analogous bound in medium can be exceeded by a moving particle. For an electron in dielectric material this leads to emission of photons which is usually referred to as Cherenkov radiation. In this article a related mathematical system for waves in inhomogeneous anisotropic medium with a maximum of three polarisation directions is studied. The waves are assumed to satisfy $P^k_j u_k (x,t) = S_j(x,t)$, where $P$ is a vector-valued wave operator that depends on a Riemannian metric and $S $ is a point source that moves at speed $\beta < c$ in given direction $\theta \in \mathbb{S}^2$. The phase velocity $v_{\text{phase}}$ is described by the metric and depends on both location and direction of motion. In regions where $v_{\text{phase}}(x,\theta) < \beta <c $ holds the source generates a cone-shaped front of singularities that propagate according to the underlying geometry. We introduce a model for a measurement setup that applies the mechanism and show that the Riemannian metric inside a bounded region can be reconstructed from partial boundary measurements. The result suggests that Cherenkov type radiation can be applied to detect internal geometric properties of an inhomogeneous anisotropic target from a distance.