From the self-force problem to the radiation reaction formula

We review recent theoretical progress in the so-called self-force problem of a general relativistic two-body system. Although a two-body system in Newtonian gravity is a very simple problem, some fundamental issues are involved in relativistic gravity. Besides, because of recent projects for gravitational wave detection, it has become possible to see those phenomena directly via gravitational waves, and the self-force problem becomes one of urgent and highly-motivated problems in general relativity. Roughly speaking, there are two approaches to investigate this problem; the so-called postNewtonian approximation, and a black-hole perturbation. In this paper, we review theoretical progress in the self-force problem using a black-hole perturbation. Although the self-force problem seems to be just a problem to calculate a self-force, we discuss that the real problem is to define a gaugeinvariant concept of a motion in a gauge-dependent metric perturbation. PACS numbers: 04.20.Cv, 04.25.Nx