Radiation Reaction, Gravitational Self-Force and Higher-Order Perturbation Theory in General Relativity*

In General Relativity a small object of mass $m$ moves along a geodesic. This elementary fact implies that even under the influence of ``gravitational radiation reaction'' while the object orbits, say, a large black hole whose metric is $g_{ab}$, the motion of $m$ is geodesic---but not a geodesic of $g_{ab}$! The retarded metric perturbation $h^{\textrm{ret}}_{ab}$ caused by $m$ is determined using perturbation analysis. In a neighborhood of the small object, $h^{\textrm{ret}}_{ab}$ may be decomposed into two parts $h^{\textrm{ret}}_{ab} = h^S_{ab} + h^R_{ab}$ where the ``singular source term'' $h^S_{ab}$ appears, in coordinates local to the object, as the part of the Schwarzschild metric of mass $m$ which is linear in $m$ along with some other terms linear in $m$ that reflect the tidal distortion of the object. The ``regular remainder'' $h^{\textrm{R}}_{ab}$ is also linear in $m$ and is known to be differentiable in a neighborhood of the small object. The effect of radiation reaction and, more generally, the gravitational self-force then requires that the object move along a geodesic of $g_{ab} + h^{\textrm{R}}_{ab}$, which is a source free solution to the Einstein equations in a neighborhood of $m$. This description is extendable to include higher order perturbation analysis.