CHAPTER 5
PRACTICAL SCHEMES FOR CALCULATIONS OF SELF-FORCE (B):
EFFECTS OF GRAVITATIONAL SELF-FORCE
In this C!I ipter we provide a method to determine the effects of the gravi-
tational self-force on a point mass orbiting a Schwarzschild black hole. First, we
address the gauge issues in relation to MiSaTaQuWa Gravitational Self-force [4, 5].
Then we follow a recent analysis by Detweiler [10] to describe the gravitational
field, which is the perturbation created by the point mass from the background
spacetime. To avoid the gauge problem, rather than calculating the self-force di-
rectly, we focus on gauge invariant quantities and determine their changes due to
the self-force effects. Techniques involved in calculating the regularization param-
eters for the gravitational field case are more complicated than for the scalar field
case. We follow analyses by Detweiler and Whiting [11] to find the methods for
calculating the regularization parameters.
5.1 MiSaTaQuWa Gravitational Self-force and Gauge Issues
In Section 3.1 we briefly reviewed the gravitational self-force due to the
perturbation hab created by a point mass m from the background spacetime gab,
which is characterized by Eq. (3-12) and often referred to as \ !SaTaQuWa" self-
force. After a mapping to the background spacetime, MiSaTaQuWa equations take
the form [30]
mib b a -Tn (gab + -abb) jc d (Vc'tbi b'd) ,t (5-1)
where
V,^tail c ret 4 Vbret ddb) z), da' zr')) a'b d'. (5-2)
\ rO