Gravitational Radiation and the Motion of Two Point Masses

The expansion of the field equations of general relativity in powers of the gravitational coupling constant yield conservation laws of energy, momentum, and angular momentum. From these laws, the loss of energy, momentum and angular momentum of a system due to the radiation of gravitational waves is found. Two techniques, radiation reaction and flux across a large sphere, are used in this calculation and are shown to be in agreement over a time average. These results are then applied to the system of two point masses moving in elliptical orbits around each other. The secular decays of the semi-major axis and eccentricity are found as functions of time and are integrated to specify the decay by gravitational radiation of such systems as functions of their initial conditions. For completeness, the secular change in the perihelion of the orbit for two arbitrary masses is found by a method which is in the spirit of the above calculations. The case of gravitational radiation when the bodies are relativistic is then considered, and an equation for the radiation similar to that of electromagnetic radiation is found. Also a proof is given that, regardless of coordinate systems or conditions, the energy of a system must decrease as a result of the radiation of gravitational waves, providing the potentials are inversely proportional to the distance from the source for large distances.