Quasilocal contribution to the gravitational self-force

The gravitational self-force on a point particle moving in a vacuum background space-time can be expressed as an integral over the past world line 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 world line 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 quasilocal 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 nonvanishing 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 space-time, in which the portion of the tail integral in the distant past is computed numerically from a mode-sum decomposition.