Huygens' principle for relativistic wave equations on Petrov type III space-times

Abstract

This thesis makes a contribution to the solution of Hadamard's problem for some relativistic wave equations on Petrov type III space-times. For the conformally invariant scalar wave equation we show that if any one of the spin coefficients a, /3, tr or Ricci spin coefficient ~ 11 vanishes in an appropriate null tetrad, then Huygens' principle is not satisfied on Petrov type III space-times. We also show that the corresponding problem for the non-self-adjoint scalar wave equation can be reduced to the conformally invariant scalar equation case. Finally, we prove that there are no Petrov type Ill space-times on which either the conformally invariant scalar equation, Weyl 's neutrino equation or Maxwell's equations satisfy Huygens' principle in the strict sense. In order to obtain the above results we have employed Newman-Penrose spin coefficient formalism and Penrose's two-component spinor formalism, together with their implementations available in the computer algebra system Maple, to determine the components of the tensorial relations given by the imposition of Huygens' principle. The resulting system of polynomial equations is then analysed by using a variant of Buchberger's algorithm, available in Maple.