Light-like parallel vector fields and Einstein equations of gravity

TL;DR

This paper demonstrates the existence of real gravitational fields with light-like parallel vector fields satisfying Einstein's equations, revealing a class of wave-like solutions distinct from spherical waves.

Contribution

It provides a complete solution to Einstein's equations under the geometric constraint of light-like parallel vector fields, highlighting a new class of wave-like gravitational solutions.

Findings

Existence of real gravitational fields with light-like parallel vector fields.

Complete solutions to Einstein's equations under this constraint.

Identification of a class of wave-like solutions different from spherical waves.

Abstract

We prove that, contrary to the situation with time-like and space-like parallel vector fields, there are real gravitational fields satisfying Einsteins equations of gravity and admitting nontrivial light-like parallel vector fields; we solve completely the field equations under this geometric constraint and obtain through Bianchi identities an interesting property for the Riemann curvature tensor corresponding with a real gravitational field admitting a nontrivial parallel vector field. The solutions we obtained turn out to be generically wave-like and then we prove by a geometric method that this class of solutions is quite different from the class of solutions corresponding with spherical waves.