Motion of extended bodies in general relativity and gravitational radiation

Abstract

In the framework of general relativity, we consider the motion of extended spherically symmetric bodies coupled together only by gravitational interactions. The self-gravitational radiation of the bodies is taken into account. The problem is solved by the Fock method of successive approximations in harmonic coordinates, assuming slow motion and weak fields (v²/c² ∿ U/c² ∿ ɛ² ≪ 1) with terms up to and including the fifth order in ɛ taken into account. The equations of motion of the two bodies are derived, including the gravitational radiation reaction terms. It is shown that the system is conservative up to fifth order but to second order in e the six classical integrals of the motion are replaced by only two (the total energy and total angular momentum). An induced rotation effect in a system of initially nonrotating bodies is obtained. It is shown that in the fifth order approximation the system is nonconservative because of gravitational radition. An expression is obtained for the rate of loss of energy from the system directly from the equations of motion.