Volume expansion rate of the Lorentzian manifold based on integral Ricci curvature over a timelike geodesic

Abstract

Recent astronomical observations show that the universe is not only expanding but also undergoing accelerated expansion [A.G. Riess, et al., The farthest known supernova, Astrophys. J. 560 (2001) 49–71; P.K. Townsend, M.N.R. Wohlfarth, Accelerating cosmologies from compactification, Phys. Rev. Lett. 91 (2003) 061302]. Then the timelike convergence condition does not hold every time, i.e. the Ricci curvature Ric ( v , v ) cannot be nonnegative for every timelike vector v . We obtain the volume expansion rate of the universe based on the integral norm of negative part of the Ricci curvature along a timelike geodesic.