The Definition of Density in General Relativity

According to general relativity the geometry of space depends on the distribution of matter or energy fields. The relation between the local geometrical parameters and the volume enclosed in given limits varies with the distribution of matter. Thus properties like particle number, mass or energy density, defined in the Euclidean tangent space, cannot be integrated to give conserved integral data like total number, mass or energy. To obtain integral conservation, a correction term must be added to account for the curvature of space. For energy this correction term is the equivalent of potential energy in Newtonian gravitation. With this correction the formation of black holes in the sense of singularities by gravitational collapse does no longer occur and the so called dark energy finds its natural explanation as potential energy of matter itself.