r direct + tail direct + (' c det 1 tai) (4-11)
I eret+t ~tai ,
where irct 2 direct + ict) and dnect di- iect d- iret) such that
direct direct dect. We separate Ydirect from the rest on the right hand
side of the above equation since this term is singular and exerts no force on the
particle. Then, we single out ', ;C1 and ret from Eq. (4-11), which are the only
contributions to the force on the particle, and may write down
= ect ,- (4-12)
With the help of Eqs. (4-6), (4-7) and (4-10) together with the definition of ', ;jct
above, one can express Eq. (4-12) as
"_ ( qu r[p, '(T )] 1dv T ret
2(P) [- a -_q v [p,p'(r) ]dr, (4-13)
L 2a J et -
which gives our self-force via Eq. (4-1).
Although this traditional approach provides adequate methods to compute
the self-force, it does not share the physical simplicity of Dirac's analysis where the
force is described entirely in terms of an identifiable, vacuum solution of the field
equations [16]: unlike Dirac's radiation field, the ', '" in Eq. (4-13) is not a solution
of the vacuum field equation V2 = 0.
In addition, the ', '" is not fully differentiable on the world-line F. The first
term in Eq. (4-13) is finite and differentiable in the coincidence limit, p -- F. This
term, in fact, provides the curved spacetime generalization of the ALD force, and
is eventually expressed in terms of the acceleration of F and components of the
Riemann tensor via local expansions of u(p,p') and &(p,p') as in Refs. [3, 4, 5, 6].
The integral term in Eq. (4-13) comes from the tail part of the Green's function.