Radiation Reaction and the Self-Force for a Point Mass in General Relativity

A point particle of mass $\mu$ moving on a geodesic creates a perturbation $h^{\mu }$, of the spacetime metric $g^0$, that diverges at the particle. Simple expressions are given for the singular $\mu /r$ part of $h^{\mu }$ and its quadrupole distortion caused by the spacetime. Subtracting these from $h^{\mu }$ leaves a remainder $h^R$ that is $C^1$. The self-force on the particle from its own gravitational field corrects the world line at $O\left(\mu \right)$ to be a geodesic of $g^0{+h}^R$. For the case that the particle is a small nonrotating black hole, an approximate solution to the Einstein equations is given with error of $O\left({\mu }^2\right)$ as $\mu \to 0$.