Curvature properties of spacelike hypersurfaces in a RW spacetime

Abstract

Curvature invariants of both intrinsic and extrinsic nature play a significant role in elucidating the geometry of a spacetime. In particular, these invariants are useful in detecting event horizon of black holes. Notable examples of spacetimes are provided by the generalized Robertson-Walker (GRW) models. An (m+1)-dimensional GRW spacetime is a Lorentzian warped product I×fN endowed with the warped metric gˆ=−dt2+f2(t)g, having as base an open real interval I equipped with the metric −dt2 and as fiber any Riemannian space (N,g) of dimension m, where f is a smooth positive-valued function on I. In this article, we focus our study on the GRW spacetimes having the fiber a Riemannian space with metric g=gk of constant sectional curvature k and denoted by L1m+1(k,f), i.e. RW spacetimes. Using an approach originally developed in Decu et al. (2008) [46], we obtain in this work the lower bound of the (generalized) normalized Casorati curvatures for a spacelike hypersurface H in the GRW spacetime L1m+1(k,f) in terms of the (normalized) scalar curvature of H, the constant sectional curvature k of the fiber, the normal hyperbolic angle of the hypersurface and the warping function f. We also derive the conditions under which this lower bound is reached. We finally apply the results to some basic cosmological models, namely Lorentz-Minkowski, de Sitter, Einstein-de Sitter, anti de Sitter and steady state spacetimes.