Tadv Tret = 2p + O(p2/R7), the integral term in Eq. (4-20) is

 1
 qpv(p,p) + O(p2/R3) qpR(p) + O(p2/R3), p F. (4-21)
 12

Taking the derivative of the right hand side of this equation gives

 1 qR(p)
 qR(p)Vp + O(p/R) ( x') + O(p/R), p (4-22)
 12 12p

When this result is combined with Eq. (4-14) via Eq. (4-20), the troublesome part

of Va', '" in Eq. (4-14) is canceled by its negative counterpart in Eq. (4-22), and

we simply end up with


 V, = V,, '" + O(p/R3), p F, (4-23)

where the remainder term O(p/R3) vanishes in the limit that p approaches F and

gives no contribution to the self-force.

 For the rest of this C'! Ilpter, QR replaces ', '' for an explicit computation of

the self-force, and the alternative split of Qret is adopted, namely

 ,ret s + R (4-24)


where bs is termed the Singular Source field, and CR the Regular Remainder field.

We determine an analytical approximation of bs via a multiple expansion, then

subtract this from the numerical solution of ,, for the ultimate calculation of the

self-force.

 4.2 Mode-sum Decomposition and Regularization Parameters

 The self-force problems having more general conditions, such as a strong

gravitational field in the background and particle velocities beyond the non-

relativistic limit, are not approachable fully analytically via direct calculation of

the Green's function in the tail part. Barack and Ori [9] sI----. -1 .1 a method to

analyze such problems when the background spacetime is spherically symmetric