On Huygens' principle for the Hodge-de Rham equations with Lorentzian gauge
On a 4-dimensional Lorentzian manifold \((M,g)\) the Hodge-de Rham equations in Lorentzian gauge (*) \(\Delta u=w\) are considered. Here \(w\) is a given \(p\)-form with \(\delta w=0\), and \(u\) is a \(p\)-form to be determined with \(\delta u=0\). As usual, \(\Delta= -d\delta -\delta d\) denotes the Hodge-de Rham operator, \(d\) is the exterior derivative and \(\delta\) is the codifferential determined by the Lorentzian metric \(g\). The goal is to find out for which Lorentzian metrics the Hodge-de Rham equation (*) satisfies Huygens' principle. By definition, this is the case if and only if the value of \(u\) at a point is locally determined by initial data in an arbitrarily small neighborhood of the past light cone of this point. The following results are proven. (i) If \(u\) and \(w\) are 1-forms, (*) satisfies Huygens' principle if and only if \(g\) is either a plane wave metric or a metric with \(C_{abcd}=0\) and \(R_{ab} =1/4 Rg_{ab}\). (ii) If \(u\) and \(w\) are 2-forms, (*) satisfies Huygens' principle if and only if \(g\) is either a plane wave metric or a metric with \(C_{abcd} =0\) and \(R(R_{ab}-1/4 Rg_{ab}) =0\).
scientific article; zbMATH DE number 1011100
scientific article