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