S7....i/; inertial or normal coordinate system for this purpose. A normal coordinate
system can alv--, be found where the metric and its first derivatives match
the Minkowski metric on the particle's world-line F, and the coordinate time T
measures the proper time [19]. Normal coordinates for a geodesic, however, are
not unique and have an ambiguity at O(p3), where p is the proper distance from p
to F measured along the spatial geodesic which is orthogonal to F. For example,
differences of O(p3) distinguish Riemann normal from Fermi normal coordinates
[19]. For our purposes a normal coordinate system introduced by Thorne and
Hartle [20] and later extended by Zhang [21] (henceforth, referred to as THZ
normal coordinate system) is particularly advantageous. It will be shown later in
Subsection 4.3.2 that in this coordinate system the scalar wave equation takes a
simple form and that as a result we obtain
S= q/p + (p3/R4), (4-45)
where R represents a length scale of the background geometry. The approximation
in Eq. (4-45) is accurate enough for self-force regularization because
Va,) = Va(q/p) + O(p2/R4), (4-46)
and the O(p2/74) remainder vanishes in the coincidence limit p i p'.
The THZ coordinates XA = (T, X Z) associated with a given geodesic F
have the following features [21]:
(i) Locally inertial and Cartesian; more specifically, on F, gAB = rAB and
aC9AB = 0. And T measures the proper time along the geodesic F, and
X = = Z =0 on F. Also, the metric is expandable about F in powers of
p /2 + 2 + Z2 in a particular form like
gAB = TAB + Pf x (homogeneous polynomials in XI of degree q)
(4-47)