Quasi-local contribution to the gravitational self-force

The gravitational self-force on a point particle moving in a vacuum background spacetime can be expressed as an integral over the past worldline of the particle, the so-called tail term. In this paper, we consider that piece of the self-force obtained by integrating over a portion of the past worldline that extends a proper time ${\Delta}{\tau}$ into the past, provided that ${\Delta}{\tau}$ does not extend beyond the normal neighborhood of the particle. We express this ``quasi-local'' piece as a power series in the proper time interval ${\Delta}{\tau}$. We argue from symmetries and dimensional considerations that the $O({\Delta}{\tau}^0)$ and $O({\Delta}{\tau})$ terms in this power series must vanish, and compute the first two non-vanishing terms which occur at $O({\Delta}{\tau}^2)$ and $O({\Delta}{\tau}^3)$. The coefficients in the expansion depend only on the particle's four velocity and on the Weyl tensor and its derivatives at the particle's location. The result may be useful as a foundation for a practical computational method for gravitational self-forces in the Kerr spacetime, in which the portion of the tail integral in the distant past is computed numerically from a mode sum decomposition.