Hyperbolic Differential Equations in Two Dimensions

Abstract

We analyze the general two-dimensional hyperbolic differential equation of second order by means of a substitution method. Our main interest lies in the support of the solutions, i.e., in an answer to the question: under what circumstances can a signal be transmitted along null rays? It turns out that, in general, a signal spreads, i.e., fills the entire future of an event. However, reasonably large classes of differential equations do permit nonspreading (characteristic propagation) solutions. As examples it is shown that multipole solutions of the flat space-time scalar wave equation and Maxwell equations fall into the non-spreading class, whereas multipole solutions of the corresponding equations in a curved Schwarzschild background always show spreading (or continuous reflection).