The Radiation Reaction Effects in the Solutions of the Perturbed Einstein Equations

The gravitational radiation reaction effects in the systems described by the perturbations of given solutions of the Einstein equations are considered. There are two kinds of perturbations to be considered; one is the perturbation induced by no external source and the other is the perturbation due to the presence of a source particle. For the former case, we find that there exists a conserved current constructed from a quadratic combination of the solutions to the linearly perturbed equations, provided that the unperturbed geometry admits a Killing vector. Thus, some effects of radiation reaction are found to be included in the linear approximation. For the latter case, it is found that the usual perturbation expansion scheme fails but there is a possible approach analogous to the one in the Lorentz-Dirac theory of charged particles in order to include the reactive effects. By this approach we find that a naive argument on the energy conservation leads an additonal reactive term which contributes to the energy equation. However this term is found to be negligible if the particle is under a quasi-periodic motion.