3.1.4 Mino, Sasaki, and Tanaka; Quinn and Wald: Gravitational
Radiation Reaction of Particles in Curved Spacetime
Extending Dewitt and Brehme's approach to the gravitational tensor field
hab, the perturbation created by a point mass m from the background spacetime
gab, Mino, Sasaki, and Tanaka [4] and Quinn and Wald [5] obtained the following
equation of motion
a1 ( h 21ad
m a = m V 7a bcin Vbh acin ad
2 2
+lim m2 b c [Va Grret (z(T) b ()) V bGret"a a'b' ((), ))
_- a dVdGe (b ), b)) a'()b' (')d', (3-12)
where Gab, (z( ), z(r')) is a bi-tensor retarded Green's function for the tensor
wave equation in curved spacetime
V2ab + 2Rabdhcd = -1Tab, (3-13)
with the trace-reversed field defined by
1
hab hab (hcC) gab (3-14)
2
and the stress-energy tensor given by
Tab = aa ga (x, z(T)) gbl, z(X, )) a')b')(4) (x z(T)) d, (3-15)
Jr
and with the harmonic gauge condition
hab;b = 0. (3-16)
It should be noted that the Abraham-Lorentz-Dirac (ALD) term ~ ('" x2a)
is absent from the equation of motion (3-12) unlike the other cases: it has been
effectively dropped off by the reduction of order procedure. The "tail" term