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