CHAPTER 3
 CALCULATIONS OF SELF-FORCE: REVIEW OF GENERAL SCHEMES AND
 ANALYTICAL APPROACHES

 In C!i ipter 2 we studied general formal schemes of radiation reaction in a vari-

ety of contexts, from Dirac's radiating electrons in flat spacetime to Mino, Sasaki,

and Tanaka and also Quinn and Wald's gravitational radiation reaction in curved

spacetime [2, 3, 4, 5, 6]. These formal schemes are theoretically well developed and

provide a good foundation for radiation reaction in curved spacetime. However,

the practical, quantitative calculations of radiation reaction remain a challenge.

The difficulty lies in the "tail" integral terms appearing in the equations of motion:

it is extremely difficult to determine precisely the retarded Green's functions in

the integrals for general geometry and for general geodesic of particle's motion.

Some attempts were made to evaluate the self-force by computing those 1 I!"

integral terms directly, but their applications had to be limited to the problems

having certain symmetries and conditions that would simplify the Green functions

in the integrals [7, 8]. Hence, for more realistic physical problems, in which special

conditions and restrictions might not be alv--,v- expected, different schemes of

calculations would be demanded to compute the I il" integral terms, thence the

self-force.

 In Section 3.1 we revisit the general formal schemes and review briefly the

structure of the equations of motion for the self-force for each case from Dirac to

Mino, Sasaki, and Tanaka, and Quinn and Wald [2, 3, 4, 5, 6]. Then, Section 3.2

presents two examples of the purely analytic attempts to the self-force calculations,

in which the tail integral terms are directly calculated as the retarded Green's

functions are simplified by some special conditions. Dewitt and Dewitt [7] and