differentiability of the subsequent approximation for QR, and the self-force 0a R: if
the approximation for bs is in error by a C" function, then the approximation for
bR is no more differentiable than C" and the approximation for 0tQR is no more
differentiable than C"-1. This concern of differentiability is associated with our
last regularization parameters, Da-terms. According to Eq. (4-33), Da-terms are
determined by the el-order terms of 0,s8, and correspond to the accuracy ~ p2/.R3
in is. This allows errors in bs by O(p3/R4), which is C2 in the limit p -- 0. Then,
our self-force 0dR is no more differentiable than C1.
It would be instructive to give an intuitive interpretation of Eq. (4-80) using
the features of THZ coordinates. The scalar wave operator in THZ coordinates is
[18]
VA A a = A (ABaB ) A (HABaB)
f= TQABOAOB b HIJOIj 2HIO(10io)
-HOO, ,1', (4-81)
where the second equality follows from the de Donder gauge condition, Eq. (4-57).
When ) is replaced by q/p in Eq. (4-81) [18],
gV A A(q/p) -4q6(3)(X) + (p/R4), p/R 0, (4-82)
for which we used Eqs. (4-58)-(4-60) and the fact that p is independent of T. A
C2 correction to q/p, of O(p3/R4), would remove the order term on the right hand
side of Eq. (4-82) and we are led to the conclusion that s = q/p + O(p3/R4) is
an inhomogeneous solution of the scalar field wave equation [18]. The error in the
approximation of bs by q/p is C2.
4.3.3 The Determination of THZ Coordinates
The singular field is approximated in a simple form via THZ coordinates as
in Eq. (4-80). In our self-force problem, this bs depends on the geodesic of the