geometry. This requires us to change the stress-energy tensor in a way which is
dependent upon the first order metric perturbations before solving the second order
equations. This correction to Tab is said to result from the "self-force" on the object
from its own gravitational field and includes the dissipative effects of i ,, [i11
reaction" as well as other nonlinear aspects of general relativity [10].
To focus on those details of the self-force which are independent of the
structure of the object, we model the object by an abstract point particle with no
spin angular momentum or other internal structure. The stress-energy tensor of a
point particle is
Tab u UU x Xa(s)) ds, (5-12)
where X"(s) describes the world-line F of the particle in some coordinate system
as a function of the proper time s along the world-line. This modeling of a small
object by a delta-function distribution for the stress-energy tensor is satisfactory
in the first order perturbation analysis. The integrability condition at the first
order as described by Eq. (5-11) implies that the world-line F of the particle is
approximately a geodesic of the background geometry gab, with an acceleration
bVbU" = O(p). This can be proved as below [10, 32]
(g0 + U Ua) VbTa (gC + Ut ) (v ) 7U (Xu Xa(s))
+uaVb r 64 (xa Xa()) } ds
J~ (VbU) 6 (X X"(s)) ds, (5-13)
where the second equality follows from properties of the projection operator
ga + uta. If VbTab = 0, then we must have Ub bUa = 0 from this equation. The
integrability condition at the second order, however, presents a difficulty. At the
second order, the particle is to move along a geodesic of gab+hab, but hab is singular
at the location of the particle and not differentiable on F. This difficulty can be