Electromagnetic waves generated by a dielectric moving at a constant speed

Abstract

We consider a regular and bounded dielectric body moving at a speed $|V|$, following a constant vector field $V$, with respect to a reference frame. In this frame, the special relativity implies that the Maxwell system is derived through the constitutive equations linking the moving speed $|V|$ and the speed of light $c$ in the background medium (as the vacuum for instance). Based on this model, we derive the well-poseness of the related forward scattering problem in the natural regime where $\frac{|V|}{c}\leq C_{te}$ with an appropriate constant $C_{te} <1$ that we estimate. In particular, we show the invertibility of the related Lippmann-Schwinger system in this regime and state the corresponding Born series in terms of the ratio $\frac{|V|}{c}$. As an application, we state and show the unique identifiability of the inverse problem of detecting the dielectric body, without knowing the moving speed $|V|$ or $V$, by illuminating it with incident electromagnetic waves propagating at the speed $c$. Such identifiability result makes sense in the regime under consideration.