On Huygens' principle for the Hodge-de Rham equations with Lorentzian gauge
type: Item
instance of: scholarly article
title: On Huygens' principle for the Hodge-de Rham equations with Lorentzian gauge
zbMATH Open document ID: 0876.58049
DOI: 10.4171/ZAA/749
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1997-11-25T00:00:00Z
review text: 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 \(R(R_{ab}-1/4 Rg_{ab}) =0\).
Mathematics Subject Classification ID: 58J45
Mathematics Subject Classification ID: 83C50
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 53B30
zbMATH DE Number: 1011100
zbMATH Keywords: Lorentzian manifold
zbMATH Keywords: Hodge-de Rham equations
zbMATH Keywords: Lorentzian gauge
zbMATH Keywords: Huygens' principle
author: Volkmar Wünsch
reviewed by: Volker Perlick
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/749
OpenAlex ID: W2057481153
description: scientific article; zbMATH DE number 1011100