Maxwell’s Equations and Electromagnetic Waves over a Curved Space-Time

Abstract

Let (V ₄ , g) be a space-time, which is here an oriented differentiable manifold of dimension 4 , of class (C² , piecewise C⁴) provided with a metric tensor g of strictly hyperbolic type (signature + ...) of class (C¹ , piecewise C³) ; η is the volume element of (V ₄ , g). We suppose thus that in the intersection of the domains of two admissible charts, the coordinates of a point x ∈ V ₄ in the first chart are functions of class C² , with non zero jacobian, of the coordinates of the same point x in the second chart. The third and fourth derivatives of these functions exist, but are only piecewise continuous. We shall see that these hypothesis of differentiability are in strict relation with the field equations of general relativity and, as a consequence, with physics.