Nonperturbative foundations of the self-force problem and application to self-gravitating extended objects

In general relativity, sufficiently small bodies are idealized as test particles moving along geodesics of the background spacetime. This picture becomes inadequate, however, once the object’s own gravitational field is taken into account. The self-force program resolves this by showing that compact objects still follow geodesic motion — not of the external spacetime, but of an effective metric that includes the influence of the object’s self-field. The central technical challenge is that the self-field is singular on the worldline and must be carefully regularized so that only its regular component contributes to the effective geometry. This construction is now well understood through first order in the mass ratio and has only recently been extended to second order.