Erratum to: The upperbound of the volume expansion rate in a Lorentzian manifold

Abstract

where s(Ht ) and s M denote, respectively, the scalar curvature of level hypersurface Ht and M at the point γ (t) and θ(t) is the mean curvature of Ht along γ (t). To calculate the upperbound of the volume expansion rate along a timelike geodesic γ in a Lorentzian manifold, we used the above differential Eq. (1). But we neglected the sign when we take the trace of the Gauss equation (p. 408). Taking the sign of an unit speed timelike geodesic into account, Eq. (1) must be corrected to