Monochromatic Electromagnetic Plane Wave - Caustics

Caustics

In the Brinkmann chart, our frame field becomes rather complicated:

and so forth. Naturally, if we compute the expansion tensor, electrogravitic tensor, and so forth, we obtain exactly the same answers as before, but expressed in the new coordinates.

The simplicity of the metric tensor compared to the complexity of the frame is striking. The point is that we can more easily visualize the caustics formed by the relative motion of our observers in the new chart. The integral curves of the timelike unit geodesic vector field give the world lines of our observers. In the Rosen chart, these appear as vertical coordinate lines, since that chart is comoving.

To understand how this situation appears in the Brinkmann chart, notice that when is large, our timelike geodesic unit vector field becomes approximately

Suppressing the last term, we have

We immediately obtain an integral curve which exhibits sinusoidal expansion and reconvergence cycles. See the figure, in which time is running vertically and we use the radial symmetry to suppress one spatial dimension. This figure shows why there is a coordinate singularity in the Rosen chart; the observers must actually pass by one another at regular intervals, which is obviously incompatible with the comoving property, so the chart breaks down at these places. Note that this figure incorrectly suggests that one observer is the 'center of attraction', as it were, but in fact they are all completely equivalent, due to the large symmetry group of this spacetime. Note too that the broadly sinusoidal relative motion of our observers is fully consistent with the behavior of the expansion tensor (with respect to the frame field corresponding to our family of observers) which was noted above.

It is worth noting that these somewhat tricky points confused no less a figure than Albert Einstein in his 1937 paper on gravitational waves (written long before the modern mathematical machinery used here was widely appreciated in physics).

Thus, in the Brinkmann chart, the world lines of our observers, in the shortwave case, are periodic curves which have the form of sinusoidals with period, modulated by much smaller sinusoidal perturbations in the null direction and having a much shorter period, . The observers periodically expand and recollapse transversely to the direct of propagation; this motion is modulated by short period small amplitude perturbations.