Monochromatic Electromagnetic Plane Wave - Relative Motion of The Observers

Relative Motion of The Observers

The Rosen chart is said to be comoving with our family of inertial nonspinning observers, because the coordinates are all constant along each world line, given by an integral curve of the timelike unit vector field . Thus, in the Rosen chart, these observers might appear to be motionless. But in fact they are in relative motion with respect to one another. To see this, we should compute their expansion tensor with respect to the frame given above. This turns out to be

where . The nonvanishing components are identical, and are

1. concave down on
2. vanish at .

Physically, this means that a small spherical 'cloud' of our inertial observers hovers momentarily at and then begin to collapse, eventually passing through one another at . If we imagine them as forming a three dimensional cloud of uniformly distributed test particles, this collapse occurs orthogonal to the direction of propagation of the wave. The cloud exhibits no relative motion in the direction of propagation, so this is a purely transverse motion.

For (the shortwave approximation), we have approximately

, we have

where the exact expressions plotted in red and the shortwave approximations in green.

The vorticity tensor of our congruence vanishes identically, so the world lines of our observers are hypersurface orthogonal. The three-dimensional Riemann tensor of the hyperslices is given, with respect to our frame, by

So the curvature splits neatly into wave (the sectional curvatures parallel to the direction of propagation) and background (the transverse sectional curvature).