resolved via a decomposition of hab, in which the -,:uo.lai, part hSb is identified and
removed from hab to leave the remaining hbR only. The point particle would then
move along a geodesic of gab + hb [10].
5.3 Decomposition of the Perturbation Field hab
In the previous Section, we briefly introduced a decomposition of the actual
metric perturbation h hab,
hc = hs+ hR, (5-14)
where hSb is termed the Singular field and h b is termed the Regular field. This
follows from the same spirit as we had for the scalar field in Section 4.1 of C'!h p-
ter 4. Analogous to bs and bR, hb and h R are natural solutions of the perturbed
Einstein equations (5-6) in a neighborhood of p: hS has only the mass p as its
source, while h b is a vacuum solution [10], i.e.
Eab(hS) = -8Tab, (5-15)
Eab(hR) = 0. (5-16)
An alternative way of splitting hb is
hact hdir tail (5-17)
ab -ab + 'ab
However, if h bi were inserted into Eq. (5-6), it would yield a physically infeasible
quantity Ttab Further, as pointed out in Ref. [16], unless i,: = 0 at the
location of the particle, htil is non-differentiable there. Thus, the approach based
on this decomposition does not clearly explain the self-force in terms of geodesic
motion in an actual gravitational field.
Our calculations of gravitational self-force effects will be based on the decom-
position (5-14), and as in the case of scalar field self-force the singular part hSb
is regarded as non-contributing to the self-force, while the remaining hRb is seen