The eikonal equation, envelopes and contact transformations

Abstract

We begin with an arbitrary but given conformal Lorentzian metric on an open neighbourhood, U, of a four-dimensional manifold (spacetime) and study families of solutions of the eikonal equation. In particular, the families that are of interest to us are the complete solutions. Their level surfaces form a two-parameter (points of S2) family of foliations of U. We show that, from such a complete solution, it is possible to derive a pair of second-order PDEs defined solely on the parameter space S2, i.e., they have no reference to the spacetime points. We then show that if one uses the classical envelope method for the construction of new complete solutions from any given complete solution, then the new pair of PDEs (found from the new complete solution) is related to the old pair by contact transformations in the second jet bundle over S2. Further, we demonstrate that the pair of second-order PDEs obtained in this manner from any complete solution lies in a subclass of all pairs of second-order PDEs defined by the vanishing of a certain function obtained from the pair and is referred to as the generalized-Wünschmann invariant. For completeness we briefly discuss the analogous issues associated with the eikonal equation in three dimensions. Finally we point out that conformally invariant geometric structures from the Lorentzian manifold have natural counterparts in the second jet bundle over S2 on which the pair of PDEs lives.