where u" =. ( 0, 0, J) ;is the four-elocitty of the particle parameterizednr by proper time
(7), ro0 is the radius of the orbit and 02 = ~. For circular equatorial geodesics
r = ro, (6-8)
0 = (6-9)
dt (r, + a2)
r + a(L aE), (6-10)
Odr a
ad4 aT
r = aE + (6-11)
Sd-r a)
with
T = (r, + a2 E aL, (6-12)
where E = E/p and L = L/p are the energy and angular momentum per unit mass,
respectively. We can recover the corresponding result for the Schwarzschild spacetime
by simply taking a 0 Because the integration in Equation 6-6 is purely radial, it
is clear that the only terms that actually participate in the integral on the left side are
those involving two radial derivatives. This is where our form of the perturbed Einstein
equations comes in. While it is generally quite tedious and impractical to compute the
perturbed Einstein tensor for a background more general than Schwarzschild and pick out
the terms involving two derivatives, it is a quite trivial task for the Einstein equations in
GHP form. All we need to do is pick out the pieces involving two of P and P' (a mindless
task with the aid of GHPtools), plug in our favorite tetrad and voila! Note that these
conditions on the second derivatives are generally invariant with respect to choice of
tetrad. Because of this, we will write the jump conditions out in the symmetric tetrad,
which is obtainable from the K~innersley tetrad by a simple spin-hoost (Equation 2-16)