5.3.2 Regular Field hb
From the decomposition of h t in Eq. (5-14), the i, l l.n remainder field is
defined by
hbt hb (5-31)
in a neighborhood of F where h satisfies the vacuum Einstein equations (5-16).
ha does not change over an O(p) length scale, and with the 0(p) corrections
included, the world-line of a point mass p through the background is a geodesic of
gab + hab. This statement is justified by the consistency of the matched wiI. *5.
expansions, and a detailed discussion of this is found in Detweiler [10].
5.4 An Example: Self-force Effects on Circular Orbits in the
Schwarzschild Geometry
Detweiler and Poisson [35] presents an elementary example of a self-force effect
using Newtonian analysis. A small mass p revolving around a more massive object
M in a circular orbit of radius R has an angular frequency 2 given by
M
Q2 = (5-32)
R3 (1 + /M)2
When p is infinitesimal, the larger mass M does not move, the radius of the orbit
R is equal to the separation between the masses and 22 = M/R3. However, when
p is still small but finite, the two masses orbit their common center of mass with
a separation of R (1 + p/M), and the angular frequency is as given in Eq. (5-32).
The finite p influences the motion of M, which then influences the gravitational
field within which p moves. This back action of p upon its own motion is the
typical characteristic of a self-force, and the p dependence of Eq. (5-32) is properly
described as a Newtonian self-force effect [10]. For p < M, Eq. (5-32) can be
expanded as
2 M [1 2p/M + O(p2/M2)] (5-33)
P,3