particle as well as on the geometry of the background spacetime. Thus, in order
to derive the regularization parameters from the multiple components of Va8s,
where a denotes the coordinates of the background geometry, one requires that p
in Eq. (4-80) be expressed in terms of the coordinates of the background geometry.
For this purpose, given the definition p2 X2 + y2 + Z2 in the THZ coordinates,
we need clarify the relationship between the background coordinates (t, r, 0, 4) and
the THZ coordinates (T, X, Y, Z) associated with an event p' on the world-line
F. In this Subsection we provide the expressions of the THZ coordinates in the
vicinity of the source p' moving on a general orbit F, in terms of expansions of the
Schwarzschild background coordinates up to the quartic order. The procedure to
complete this task can be summarized in the following steps:
(i) Find initial inertial coordinates XA (A 0, 1,2, 3) in the neighborhood of
the event p' on F in terms of Taylor expansions of the background coordi-
nates Xa = (t, r, 0, 4) about x', where x' represents p' in the background
coordinates (henceforth, the subscript o denotes the evaluation at p'). This
coordinate frame is static in the sense that the event p' is not in motion along
F yet.
(ii) Construct Fermi normal coordinates, which have vanishing ('!i 1-i.!! !
symbols along F. With the metric components being expandable about F in
powers of proper distance to F for all time, Fermi normal coordinates provide
a standardized way in which freely falling observer can report observations
and local experiments [23].
(iii) Determine the transformation from Fermi normal coordinates to THZ
coordinates and finally combine this with the results of Steps (i) and (ii).
Step (i).
We build initial inertial coordinates XA via the expansions of the Schwarzschild
coordinates x" = (t, r, 0, 4) about xt. Weinberg's [24] Eq. (3.2.12) shows that