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In this paper we study the problem of uniqueness for spacelike hypersurfaceswith constant higher order mean curvature in generalized Robertson-Walker (GRW)spacetimes. In particular, we consider the following question: Under whatconditions must a compact spacelike hypersurface with constant higher ordermean curvature in a spatially closed GRW spacetime be a spacelike slice? Weprove that this happens, esentially, under the so called convergence condition.Our approach is based on the use of the Newton transformations (and theirassociated differential operators) and the Minkowski formulae for spacelikehypersurfaces.

Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes