Parallel Vector Fields and Einstein Equations of Gravity

Abstract

In this paper, we prove that no nontrivial timelike or spacelike parallel vector field exists in a region where the gravitational field created by macroscopic bodies and governed by Einstein’s equations does not vanish. In other words, we prove that the existence of such vector fields in a region implies the vanishing of the Riemann curvature tensor in that region. To prove this statement, we reduce the 4-dimensional problem to a 3-dimensional one. This enables us to use a link existing between the Riemann curvature tensor and the Ricci tensor in a 3-dimensional Riemannian manifold.