scalar field and as in Eq. (5-27) for the gravitational field in the Lorenz gauge.
Then, the structure analyses of the singular field, as shown in Eq. (4-159) along
with Eqs. (4-160)-(4-168), (4-187)-(4-209), (4-228)-(4-237) and (4-238)-(4-254) for
the case of the scalar field, have facilitated the calculations of the regularization
parameters. With these analyses the strategies of calculations have been developed
for each regularization parameter, Aa, Ba, Ca and Da. Similar analyses apply to
the case of the gravitational field. However, techniques of calculations involved
in the gravitational field case are more complicated than in the scalar field case.
This is due to the distinct geometrical properties of hab under a rotation on the
two-sphere in a R. *-_---Wheeler style decomposition as shown in Subsection 5.4.2.
The difficulties with regard to the geometrical properties have been resolved by
following the analyses by Detweiler and Whiting [11], and the different techniques
of calculating the regularization parameters for each different group, scalars
{Jt, hS,, h}, vectors {h7, h,, h h} and tensors {hSo, hS, h }, all defined on
the two-sphere have been developed as in Subsection 5.4.3.
The applications of these regularization parameters in the mode-sum self-force
calculation have shown nice results as in Section 4.5 for the case of the scalar field.
In particular, the use of Da-terms in the mode-sum computation has provided more
rapid convergence and more accurate final results. Calculations of the gravitational
self-force effects are currently in progress, and determinations of the regularization
parameters for chsb and the higher order regularization parameters Dab for hb
will be demanded, such as for the mode-sum calculations of the changes in gauge
invariant quantities like Q and E QJ as in Eqs. (5-35) and (5-37).
Also, the regularization parameters for the Kerr spacetime will be calculated
via similar strategies to the Schwarzschild case with slight modifications to the
local analysis of spacetime for the different background geometry.