Gravitational Radiation and the Motion of Two Point Masses

TLDR

General‑relativistic field equations expanded in the gravitational coupling constant yield conservation laws of energy, momentum, and angular momentum, and in the nonrelativistic limit the radiation of energy and angular momentum is expressed through time derivatives of the quadrupole tensor. The authors employ both radiation‑reaction and large‑sphere flux techniques, which agree on time‑averaged results, and apply them to two point masses in elliptical orbits. They find that gravitational‑wave emission causes a loss of energy and angular momentum, leading to secular decay of the semimajor axis and eccentricity that can be integrated to predict the orbital decay as a function of initial conditions.

Abstract

The expansion of the field equations of general relativity in powers of the gravitational coupling constant yields conservation laws of energy, momentum, and angular momentum. From these, the loss of energy 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 these calculations and are shown to be in agreement over a time average. In the nonrelativistic limit, the energy and angular momentum radiation and angular distributions are expressed in terms of time derivatives of the quadrupole tensor ${Q}_{\mathrm{ij}}$. These results are then applied to a bound system of two point masses moving in elliptical orbits. The secular decays of the semimajor 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.