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 [39
], and then
reproduced with a different analysis by Quinn and Wald [49
]. They are now known
as the MiSaTaQuWa equations of motion. Detweiler and
Whiting [23
] 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 [39
] 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 [8
] 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
, where