We take the metric of the background spacetime to be a solution of the Einstein field equations in vacuum. (We impose this condition globally.) We describe the gravitational perturbation produced by a point particle of mass in terms of trace-reversed potentials defined by
Equations of motion for the point mass can be obtained by formally demanding that the motion be geodesic in the perturbed spacetime with metric . After a mapping to the background spacetime, the equations of motion take the form of
We now remove from the retarded perturbation and postulate that it is the radiative field that should act on the particle. (Note that satisfies the same wave equation as the retarded potentials, but that is a free gravitational field that satisfies the homogeneous wave equation.) On the world line we have
The equations of motion of Equation (48) were first derived by Mino, Sasaki, and Tanaka , and then reproduced with a different analysis by Quinn and Wald . They are now known as the MiSaTaQuWa equations of motion. Detweiler and Whiting  have contributed the compelling interpretation that the motion is actually geodesic in a spacetime with metric . This metric satisfies the Einstein field equations in vacuum and is perfectly smooth on the world line. This spacetime can thus be viewed as the background spacetime perturbed by a free gravitational wave produced by the particle at an earlier stage of its history.
While Equation (48) does indeed give the correct equations of motion for a small mass moving in a background spacetime with metric , the derivation outlined here leaves much to be desired – to what extent should we trust an analysis based on the existence of a point mass? Fortunately, Mino, Sasaki, and Tanaka  gave two different derivations of their result, and the second derivation was concerned not with the motion of a point mass, but with the motion of a small nonrotating black hole. In this alternative derivation of the MiSaTaQuWa equations, the metric of the black hole perturbed by the tidal gravitational field of the external universe is matched to the metric of the background spacetime perturbed by the moving black hole. Demanding that this metric be a solution to the vacuum field equations determines the motion of the black hole: It must move according to Equation (48). This alternative derivation is entirely free of conceptual and technical pitfalls, and we conclude that the MiSaTaQuWa equations can be trusted to describe the motion of any gravitating body in a curved background spacetime (so long as the body’s internal structure can be ignored).
It is important to understand that unlike Equations (33) and (40), which are true tensorial equations, Equation (48) reflects a specific choice of coordinate system and its form would not be preserved under a coordinate transformation. In other words, the MiSaTaQuWa equations are not gauge invariant, and they depend upon the Lorenz gauge condition . Barack and Ori  have shown that under a coordinate transformation of the form , where are the coordinates of the background spacetime and is a smooth vector field of order , the particle’s acceleration changes according to , whereThe MiSaTaQuWa equations of motion are not gauge invariant and they cannot by themselves produce a meaningful answer to a well-posed physical question; to obtain such answers it shall always be necessary to combine the equations of motion with the metric perturbation so as to form gauge-invariant quantities that will correspond to direct observables. This point is very important and cannot be over-emphasized.
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