  We study accelerating relativistic reference frames in Minkowski space-time
under the harmonic gauge. It is well-known that the harmonic gauge imposes
constraints on the components of the metric tensor and also on the functional
form of admissible coordinate transformations. These two sets of constraints
are equivalent and represent the dual nature of the harmonic gauge. We explore
this duality and show that the harmonic gauge allows presenting an accelerated
metric in an elegant form that depends only on two harmonic potentials. It also
allows reconstruction of the spatial structure of the post-Galilean coordinate
transformation functions relating inertial and accelerating frames. The
remaining temporal dependence of these functions together with corresponding
equations of motion are determined from dynamical conditions, obtained by
constructing the relativistic proper reference frame of an accelerated test
particle. In this frame, the effect of external forces acting on the observer
is balanced by the fictitious frame-reaction force that is needed to keep the
test particle at rest with respect to the frame, conserving its relativistic
linear momentum. We find that this approach is sufficient to determine all the
terms of the coordinate transformation. The same method is then used to develop
the inverse transformations. The resulting post-Galilean coordinate
transformations extend the Poincar\'e group on the case of accelerating
observers. We present and discuss the resulting coordinate transformations,
relativistic equations of motion, and the structure of the metric tensors
corresponding to the relativistic reference frames involved.
