Cosmological meaning of geometric curvatures

Abstract

In this paper, we analyzed the physical meaning of scalar curvatures for a generalized Riemannian space. It is developed the Madsen?s formulae for pressures and energy-densities with respect to the corresponding energy-momentum tensors. After that, the energy-momentum tensors, pressures, energy-densities and state-parameters are analyzed with respect to different concepts of generalized Riemannian spaces. At the end of this paper, linearities of the energy-momentum tensor, pressure, energy-density and the state-parameter are examined.

Generalized Riemannian spaces with respect to 4-velocity vectors and functions of state parameters

“…for the pressure p, the energy-density ρ, h i j = i j − u i u j , π iα u α = π αi u α = 0, u α q α = 0, where u i , q i , π i j are the components of 4-velocity û, the energy flux vector q and the anisotropic stress tensor π, respectively. In [5,14], it is obtained…”

“…The curvatures of generalized Riemannian space GR 4 are correlated with symmetric and anti-symmetric parts of the metric tensor [14].…”

“…In [14], it is obtained that the energy-momentum tensor T, the pressure p and the energy-density ρ satisfy the equalities…”