While the equations of motion for the other fields, (3-6) and (3-9) are generally
covariant, Eq. (5-1) is not and reflects a specific choice of coordinate system and
would not preserve its form under an infinitesimal coordinate transformation.
That is to -i, the gravitational self-force calculated via MiSaTaQuWa equations
is not gauge invariant and depends upon the gauge condition given by Eq. (3-16).
According to Ref. [31], under a coordinate transformation
xa + a, (5-3)
where Xa are the coordinates of the background spacetime and a is a smooth
vector field of O(m), the particle's acceleration changes accordingly
+ z[]", (5-4)
where
[] (ab + ai b) (b + RbCdeC cde (5-5)
is the ,, '-1' acceleration" and b (~b;a a) ,c is the second covariant derivative
of $b in the direction of the world-line. This implies that a gauge transformation
can alter the particle's acceleration: a special choice of a could even make 2" = 0,
which is just back to the original geodesic of motion [30].
From this observation we come to the conclusion that the MiSaTaQuWa
equations of motion are not gauge invariant and cannot provide by themselves a
meaningful interpretation to the gravitational self-force. To obtain a physically
meaningful answer to the gravitational self-force problem, we should be aiming at
the quantities that must be describable in a manner which is gauge invariant [30].
5.2 First Order Perturbation Analysis
Perturbation analysis provides the framework for an understanding the
self-force and radiation reaction on an object of small mass and size in general
relativity [10]. Suppose a particle of small mass p moves along a geodesic F of