Adiabatic expansion for a metric perturbation and the condition to solve the gauge problem for the gravitational radiation reaction problem

We examine the adiabatic approximation in the study of a relativistic two-body problem with the gravitational radiation reaction. We recently pointed out that the usual metric perturbation scheme using a perturbation of the stress-energy tensor may not be appropriate for study of the dissipative dynamics of the bodies due to the radiation reaction. Over a time scale during which the usual perturbation scheme is valid, the orbits may not deviate substantially relative to the orbits of the background orbits. As a result, one can eliminate the orbital deviation through a gauge transformation. This is called the gauge problem of the gravitational radiation reaction exerted on the bodies,, and it has been reported that a careful gauge fixing may be necessary to produce a physically reasonable prediction for the evolution of the system. We recently proposed a possible approach to solve this problem with a linear black hole perturbation. This paper proposes a non-linear generalization of that method for a general application of this problem. We show that, under a specific gauge condition, the method actually allows us to avoid the gauge problem.