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Complete solution of Hadamard’s problem for the scalar wave equation on Petrov type III space-times. (English) Zbl 0951.35131

This very interesting paper is devoted to the solution of Hadamard’s problem on Petrov type III space-times, for the conformally invariant scalar wave equation \[ \square u+ 1/6 Ru= 0,\tag{1} \] and the non-selfadjoint scalar wave equation \[ \square u+ A^a\partial_a u+ Cu= 0, \] where \(\square\) is the Laplace-Beltrami operator corresponding to the metric \(g_{ab}\) of the background space-time \(V_4\), \(u\) is the unknown function, \(R\) is the Ricci scalar, \(A^a\) – the components of a given contravariant vector field and \(C\) – a given scalar function. The background manifold, metric tensor, vector field and scalar function are assumed to be \(C^\infty\). The main result is that there exists no Petrov type III space-times on which the conformally invariant scalar wave equation (1) satisfies Huygens’ principle. Also there exist no Petrov type III space-times on which the non-selfadjoint scalar wave equation satisfies Huygens’ principle.

MSC:

35Q75 PDEs in connection with relativity and gravitational theory
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
53Z05 Applications of differential geometry to physics

Software:

NP; NPspinor; Maple
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Full Text: arXiv Numdam EuDML

References:

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