Monochromatic Electromagnetic Plane Wave - The Riemann Curvature Tensor

The Riemann Curvature Tensor

In contrast, the Bel decomposition of the Riemann curvature tensor, taken with respect to, is simplicity itself. The electrogravitic tensor, which directly represents the tidal accelerations, is

The magnetogravitic tensor, which directly represents the spin-spin force on a gyroscope carried by one of our observers, is

(The topogravitic tensor, which represents the spatial sectional curvatures, agrees with the electrogravitic tensor.)

Looking back at our graph of the metric tensor, we can see that the tidal tensor produces small sinusoidal relative accelerations with period, which are purely transverse to the direction of propagation of the wave. The net gravitational effect over many periods is to produce an expansion and recollapse cycle of our family of inertial nonspining observers. This can be considered the effect of the background curvature produced by the wave.

This expansion and recollapse cycle is reminiscent of the expanding and recollapsing FRW cosmological models, and it occurs for a similar reason: the presence of nongravitational mass-energy. In the FRW models, this mass-energy is due to the mass of the dust particles; here, it is due to the field energy of the electromagnetic field. There, the expansion-recollapse cycle begins and ends with a strong scalar curvature singularity; here, we have a mere coordinate singularity (a circumstance which much confused Einstein and Rosen in 1937). In addition, here we have a small sinusoidal modulation of the expansion and recollapse.