Spacetime symmetries and linearization stability of the Einstein equations. I

Abstract

We consider the Marsden−Fischer conditions for linearization stability applied to vacuum spacetimes with compact Cauchy hypersurfaces. We show that if a vacuum spacetime S admits a Killing vector field, then the Marsden−Fischer criterion fails to be satisfied at any Cauchy surface for S. We also show that if the Marsden−Fischer criterion fails to hold on an initial surface, then there is a Cauchy development of this intial data which admits one or more Killing vectors. The number of independent Killing fields present is shown to equal the dimension of the kernel of the linear map defined by Marsden and Fischer.