Space--time symmetries and linearization stability of the Einstein equations. II

In a previous paper we began a study of the Fischer–Marsden conditions for the linearization stability of vacuum space–times with compact, Cauchy hypersurfaces. We showed that a space–time of this class is linearization stable if and only if it admits no global Killing vector fields. In this paper we derive the general nonlinear constraints upon the perturbations which are necessary, whenever Killing symmetries occur, to exclude spurious perturbation solutions. We establish the hypersurface independence of these constraints by relating them to the conserved integrals of the perturbation equations associated with the Killing symmetries of the background. As a corollary of this result, we also establish the gauge invariance of the nonlinear constraints. We briefly discuss the noncompact case and mention a possible application of our results to the study of the Hawking process of quantum mechanical particle production by black holes.