Table 4-1. The fitted parameters of 7ret (by courtesy of Detweiler et al, 2003 [18])
k E4 Ek part of ,lf
1 1. 80504(4) x 10-4 2.78201(7) x 10-9
2 -1. 000(3) x 10-4 6.90(2) x 10-12
3 4. 3(3) x 10-5 3.1(2) x 10-14
4 -5. 6(5) x 10-5 7.7(6) x 10-16
2r2 (r 2M)] 1/2 M(ro 2M)F-12
Sr o-3M 2r4(ro -3M)
(ro 2M)(ro- 4M)Fi/2 (ro 3M)(5r2 7roM 14M2)F3/2
8r(ro- 2M) + 16r(ro 2M)2
3(ro- 3M)2(ro + M)F5/2 (4-263)
16T 2M)2 (4-263)
16r4(ro 2M)2
The fourth term of Eq. (4-260) was introduced to extrapolate the higher order
regularization parameters than Dr-term. Er are independent of and are to be
determined by numerical fitting. The use of these additional parameters will help
to increase dramatically the effective convergence to our final result of elff (see
Table 4-1 and Figure 4-1). More technical details regarding this are found in
Ref. [18].
From Eqs. (4-259) and (4-260), we see that our self-force is computed by
summing the residuals after removing the terms of Ar, B,, Dr and additional
terms of Ek from the numerical solution .Jt. Removing the contribution of
E4 improves the falloff of the residuals by an additional two powers of Nu-
merically determined E. coefficients and their contributions to the self-force
,self are given in Table 4-1. Figure 4-1 summarizes the results of this numer-
ical analysis. The curve labeled Fout represents Ff as a function of The
curves A, B and D show f ( + 1/2) A,, Tr [( + 1/2) A, + B,] and
gret [(+ 1/2) A, + B 2v2Dr/(2 1)(2f + 3)], respectively. The El to E4
curves show the residuals after numerically fitting from 1 to 4 of the E, coefficients
and removing their contributions cumulatively [18].