An explicit determination of the empty space-times on which the wave equation satisfies Huygens' principle

The validity of Huygens' principle in the sense of Hadamard's ‘minor premise’ is investigated for scalar wave equations on curved space-time. A new necessary condition for its validity in empty space-time is derived from Hadamard's necessary and sufficient condition using a covariant Taylor expansion in normal coordinates. A two component spinor calculus is then employed to show that this necessary condition implies that the plane wave space-times and Minkowski space are the only empty space-times on which the scalar wave equation satisfies Huygens' principle.