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.