Fermi-normal, optical, and wave-synchronous coordinates for spacetime with a plane gravitational wave

Abstract

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.