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.