CHAPTER VIII - PLANE WAVE MANIFOLDS AND HUYGENS' PRINCIPLE

Abstract

This chapter presents the calculation of the fourth order moments for the scalar wave operator. The moments are trace-free, symmetric, and conformally invariant tensors with conformal weight -1 that depends polynomially on the metric tensors, the Riemann curvature tensor, and its covariant derivatives. There are only three linearly independent tensors with these properties and it is possible to determine them explicitly. In a four-dimensional Riemannian or pseudo-Riemannian space, the set of real conformally invariant polynomial covariant 4-tensors of weight -1, which are symmetric and trace-free, is given by the set of linear combinations. It is necessary to determine the coefficients α (υ). The moments and the Wünsch tensors are computed for suitable chosen metrics. The coefficients α(υ) can be determined by a simple comparison. In general relativity, the manifolds with vanishing Ricci tensor describe empty space-times and Maxwell's equations rule the propagation of electro-magnetic waves.