Then we find that the finite mass ratio p/M brings a change in the orbital fre-
quency by a fractional amount
A(5-34)
2 M
The extension of this Newtonian problem to general relativity would be
the simplest and interesting example for our relativistic gravitational self-force
problem. In this Section we focus on a small mass p in a circular geodesic about
a Schwarzschild black hole of mass M and study the effects of self-force on some
gauge invariant quantities such as the angular frequency Q and the combined
quantity E QJ where E and J are reminiscent of the particle's energy and
angular momentum per unit rest mass-as the perturbation breaks the symmetries
of the Schwarzschild geometry, there is no naturally defined energy or angular
momentum for the particle. Our attempt of evaluating the changes in these gauge
invariant quantities as the effects of self-force will avoid the ambiguity posed by the
gauge freedom in MiSaTaQuWa's approach as described in Section 5.1.
5.4.1 Gauge Invariant Quantities
Detweiler [10] presents the examples of gauge invariant quantities in a few
different categories. Among them, we first focus on the angular frequency Q, which
will correspond to a direct observable measured at infinity. Ref. [10] gives the
expression of f for a circular orbit as measured at infinity
Q2 A/4)2 ( /t)2 = M R 3M ab
l_ 2t t3 2R2 U U ar/b, (5-35)
where Ua represents the four-velocity of the particle moving in a circular orbit on
the equatorial plane, whose components are defined by
ut = -E, u = J, u = 0 and u = 0. (5-36)
Another gauge invariant quantity we are interested in is E fJ, which is the
contraction of Ua with the Killing vector a. Its expression for a circular orbit is