Radial Motions: Exact Solution and Its Analysis

Abstract

We describe the picture of radial motions depending on the value of the total energy, and give the exact solution to the problem in the form of an elliptic integral. The detailed form of the Hubble law is derived. Two solutions in terms of elementary functions are found, one of them coinciding with the Friedmann relativistic solution for expanding universe filled with no-pressure matter and DE. Other cases are reduced to a combination of the standard Legendre elliptical integrals and elementary functions, with derivations partly in Appendix A.