Adiabatic Expansion for Metric Perturbation and the condition to solve the Gauge Problem fo.pdf

a r X i v : g r - q c / 0 6 0 1 0 1 9 v 1 5 J a n 2 0 0 6 Adiabatic Expansion for Metric Perturbation and the condition to solve the Gauge Problem for Gravitational Radiation Reaction Problem Yasushi Mino ? mail code 130-33 California Institute of Technology Pasadena CA 91125 USA February 7, 2008 Abstract 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. I. INTRODUCTION Many-body problems are fundamental problems in general relativity and have a long history of theoretical investi- gation. Starting from the famous paper of Einstein, Infeld and Hoffman, [1] the equation of motion of point particles was studied by various authors. The pioneering work of Einstein, Infeld and Hofman [1] used Dirac’s delta functions as particles (the point particle approximation), and assumed a coordinate system in which the spacetime is weakly curved and the motion of the particles is sufficiently