  Fermi normal coordinates provide a standardized way to describe the effects
of gravitation from the point of view of an inertial observer. These
coordinates have always been introduced via perturbation expansions and were
usually limited to distances much less than the characteristic length scale set
by the curvature of spacetime. For a plane gravitational wave this scale is
given by its wavelength which defines the domain of validity for these
coordinates known as the long-wavelength regime. The symmetry of this
spacetime, however, allows us to extend Fermi normal coordinates far beyond the
long-wavelength regime. Here we present an explicit construction for this
long-range Fermi normal coordinate system based on the unique solution of the
boundary-value problem for spacelike geodesics. The resulting formulae amount
to summation of the infinite series for Fermi normal coordinates previously
obtained with perturbation expansions. We also consider two closely related
normal coordinate systems: optical coordinates which are built from null
geodesics and wave-synchronous coordinates which are built from spacelike
geodesics locked in phase with the propagating gravitational wave. The
wave-synchronous coordinates yield the exact solution of Peres and Ehlers-Kundt
which is globally defined. In this case, the limitation of the long-wavelength
regime is completely overcome, and the system of wave-synchronous coordinates
becomes valid for arbitrarily large distances. Comparison of the different
coordinate systems is done by considering the motion of an inertial test mass
in the field of a plane gravitational wave.
