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.