Invariance of the Massless Field Equations under Changes of the Metric

Abstract

It is shown that the Maxwell equations with sources, expressed in terms of the covariant tensor field Fijand the current density four-vector Ji, are invariant under the change of the metric gijby gij′= gij+ liljif liis a principal null direction of Fijand that an analogous result holds in the case of the massless Klein-Gordon equation if liis null and orthogonal to the gradient of the field and in the case of the null dust equations if liis parallel to the dust four-velocity. An elementary proof of the following generalization of the Xanthopoulos theorem is also given: Let (gij, Fij) be an exact solution of the Einstein-Maxwell equations and let libe a principal null direction of Fij, then (gij+ lilj, Fij) is also an exact solution of the Einstein-Maxwell equations if and only if (lilj, 0) satisfies the Einstein-Maxwell equations linearized about the background solution (gij, Fij). Furthermore, analogous theorems, where the source of the gravitational field is a massless Klein-Gordon field or null dust, are presented.