right hand side of Eq. (3-6) results from the particle's motion and the curvature
of spacetime, and will make the particle deviate from its original world-line to the
order of e2, even in the absence of an external incident field Fi2b. Hence, radiation
damping is expected to occur even for a particle in free fall.
3.1.3 Quinn: Radiation Reaction of Scalar Particles in Curved Space-
time
In a similar manner to Dewitt and Brehme, using the Hadamard expansion
techniques for the scalar field in curved spacetime, Quinn [6] was able to derive the
equation of motion for a scalar point particle moving in curved spacetime as
1 1 1
m a qaV'in + q2( a 2Za) + tq (R b + aRbcb ) q2R a
3 6 12
+ lim q2 VaGret (z(r), z(T')) dT, (3-9)
where Gret (z(r), z(r')) is a bi-scalar retarded Green's function for the scalar wave
equation in curved spacetime
v2 ret = -47, (3-10)
with the scalar charge density
=q j(-g)-1/2(4)(x- z())dr. (3-11)
Again, we have a 1 i!" term involving the Green's function in Eq. (3-9), which
gives the same implication as that of Dewitt and Brehme's. Similarly as in the case
of electromagnetic vector field, the last three terms including this tail term on the
right hand side of Eq. (3-9) result from the particle's motion and the curvature of
spacetime, and will be responsible for radiation reaction of the scalar particle in
free fall.