Pfenning and Poisson [8] are provided as the examples. An alternative scheme
for the self-force calculations, which has been devised to work for more general
problems, is a hybrid of both analytical and numerical methods. This will be
the main approach that this dissertation is going to take, and we leave its full
discussion for the next two C'! lpters.
3.1 General Formal Schemes Revisited
3.1.1 Dirac: Radiating Electrons in Flat Spacetime
For an electron of mass m moving in the flat spacetime region with the
incident electromagnetic field, Dirac [2] derived the following equation of motion
using the conservation of the stress-energy tensor inside a narrow world-tube
surrounding the particle's world-line,
m-" = eibFbi + e2 (- -a ) (3-1)_
where Fifb aAb ObAa represents the incident electromagnetic field and the
second term on the right hand side, known as the Abraham-Lorentz-Dirac (ALD)
force, results from the radiation field produced by the moving electron. In this
analysis, the retarded electromagnetic field is decomposed into two parts:
1 1
ab.&_ / ab ab ab ab Nt7& av i( 2 (FTb F (3-)
ret + adv () + Fret Fadv) (i) (3-2)
The first term (i) on the right hand side of Eq. (3-2) is the solution of the inhomo-
geneous equation
OA, = -47Ja, (3-3)
with the charge-current density
Ja I a(T)6(4) (x z(r)dr, (3-4)
and corresponds to the field resembling the Coulomb q/r piece of the scalar
potential near the particle, which does not contribute to the force on the particle