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We examine the adiabatic approximation in the study of a relativistictwo-body problem with the gravitational radiation reaction. We recently pointedout that the usual metric perturbation scheme using a perturbation of thestress-energy tensor may not be appropriate for study of the dissipativedynamics of the bodies due to the radiation reaction.  We recently proposed a possible approach to solve this problem with a linearblack hole perturbation. This paper proposes a non-linear generalization ofthat method for a general application of this problem. We show that, under aspecific gauge condition, the method actually allows us to avoid the gaugeproblem.Comment: accepted by Progress of Theoretical Physic

Adiabatic Expansion for a Metric Perturbation and the Condition to Solve the Gauge Problem for the Gravitational Radiation Reaction Problem