The Causal, Relativistic Version of Newton's Law of Gravity in the Weak-Field Approximation and Its Implications for Cosmology

The standard weak field perturbative gravity calculation is extended to highly relativistic observers in standard General Relativity by including the finite speed of gravity implied by the retarded time in the Green function for the solution of the perturbative metric and the presence of the observer’s four-velocity in the Geodesic Equations. This gives rise to an anisotropic gravitational field seen by the observer and a gravitational horizon for the source when placed in an expanding cosmology. The gravity is weaker if the observer is moving directly away or toward the source and stronger if the observer is moving tangentially to the source. This also happens to the Coulomb field of an electric charge seen by a relativistic observer. The gravitational field produced by our galaxy on a distant galaxy moving away relativistically is shown to be weaker than what the classical Newtonian calculation predicts. The Geodesic Equation is also used to show that the trajectories for energetic time-like particles deviate from Kepler’s Laws and cannot approach the null geodesic paths for photons. These relativistic gravitational effects may be able to reduce the requirements for dark energy and dark matter.