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An explicit determination of the Petrov type N space-times on which the conformally invariant scalar wave equation satisfies Huygens’ principle. (English) Zbl 0595.35067

The linear hyperbolic partial equation \(g^{ab} u_{;ab}+A^ a u_{,a}+C u=0\) for a function u on the n-dimensional Lorentz space with \(n\geq 4\) and metric tensor components \(g^{ab}\) is considered. This equation is said to satisfy Huygens’ principle (in the strict sense), or to be a Huygens’ differential equation, if and only if the solution for every Cauchy problem depends merely on the data in an arbitrarily small neighborhood of the intersection of the past null conoid with a space- like (n-1)-dimensional submanifold.
Formerly, the authors [”Gravitation, geometry and relativistic physics”, Lect. Notes Phys. 212, 138-142 (1984; Zbl 0557.53046)] conjectures that every space-time, on which the conformally invariant wave equation \(g^{ab} u_{;ab}+R u/6=0\) \((R:=g^{ab},R_{ab})\) based on a generalized plane wave space-time (with R denoting the curvature scalar) satisfies Huygens’ principle, is conformally flat, i.e. conformally related to the exact plane wave space-time whose metric was given by J. Ehlers and K. Kundt [Article in ”Gravitation: An introduction to current research” ed. by L. Witten (1964; Zbl 0115.431)]. The authors [Phys. Lett., A 105, 351-354 (1984)] already stated this conjecture to be true for space-times of the Petrov type N, one of the five possible types of the Weyl conformal curvature tensor \(C_{abcd}\) of space-time, equivalent to the existence of a necessarily null vector field \(\ell\) such that \(C_{abcd} \ell^ d=0\) at every point.
The present paper provides a detailed proof of this result cast in three theorems. Incidentally, some related intermediate results are also given concerning the validity of Huygens’ principle for Maxwell’s equations and Weyl’s neutrino equation for a two-spinor.
Reviewer: M.Biermann

MSC:

35L10 Second-order hyperbolic equations
83C15 Exact solutions to problems in general relativity and gravitational theory
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References:

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