The special and general relativistic effects on orbits around point masses

Abstract

In this paper I compare the orbits around a point mass in Newtonian mechanics with those for a relativistic test particle in a Coulomb potential and in the Schwarzschild metric. In Newtonian mechanics there is an infinite centrifugal barrier for any test particle with a finite angular momentum, so that the test particle cannot reach the point mass. For relativistic motion in the Coulomb potential the centrifugal barrier disappears at a small, but still finite, specific angular momentum. In the Schwarzschild metric finally the centrifugal barrier is always finite, meaning that sufficiently energetic particles can reach the origin independently of their angular momentum, and there are no stable circular orbits inside a finite radius, , or equivalently below a specific angular momentum, . These effects have been modelled in Newtonian mechanics by modifying the gravitational potential. I suggest that a more physical approach is to keep the Newtonian potential, and let it couple with the total energy of the test particle rather than its rest mass.