Cauchy's problem and Huygens' principle for the linearized Einstein field equations

Abstract

The linearized Einstein field equations with Einstein space-times as background are studied by use of the harmonic gauge. By means of Riesz' integration method a representation theorem for the solution of Cauchy's problem, using the constraints of the Cauchy data and the calculus of symmetric differential forms, is proved. We introduce some linear differential operators, which map the set of symmetric differential forms into the subset with vanishing divergence and trace and use these operators to derive necessary conditions for the validity of Huygens' principle from which it follows that the linearized field equations satisfy Huygens' principle only in flat space-times.