[10]
(E QJ)2 ( 3_) (- uubhb + Rubfab) (5-37)
5.4.2 Mode-sum Regularization
Both 2 and E QJ in Eqs. (5-35) and (5-37) contain the terms that depend
on hab or arhab, which imply the effects of self-force due to the perturbation of the
geometry by the presence of the point mass p. The contributions from these terms
can be evaluated via the same technique of mode-sum regularization, which was
pioneered by Barack and Ori [9], as used for the scalar field problem in Section 4.2.
BR. --. and Wheeler [33] and Zerilli [34] show how to obtain the metric per-
turbations of Schwarzschild via spherical harmonic decomposition. Both Tab and
hab are fourier analyzed in time, with frequency u, and decomposed in terms of
spherical harmonics, with multiple indices and m. Linear combinations of the
components of he~w satisfy ordinary differential equations which can be numeri-
, Ill/ integrated. The periodicity of a circular orbit makes a discrete set frequencies
Wm = -m [10].
Assuming that h 'W(r) can be determined for any and m, the sum of these
over all and m will constitute hac. If evaluated at the location of p, however, this
sum will diverge. Subtracting the singularity h emw) from h (mw) and summing
the difference over and m, we obtain a convergent sum [10]
hR -^ R(&m,w) _\ rYact(&nm,w) hS (m,c)~\ /.
hab ,hoab [hab (5-38)
which is the regular remainder field. Similarly, we can determine its derivative via
a R ( rR(,) a ^ )ct()m,w) h(& ,w) /
clab Y c'ab c'ab c'ab
In R.B 2.- and Wheeler's analysis [33], the individual components hi" are
classified according to their angular momentum properties under a rotation of the