Monochromatic electromagnetic plane wave
Of these, the most important examples are the electromagnetic plane waves, in which the radiation has planar wavefronts moving in a specific direction at the speed of light.
Of these, the most basic is the monochromatic plane waves, in which only one frequency component is present.
This is precisely the phenomenon that this solution model, but in terms of general relativity.
The metric tensor of the unique exact solution modeling a linearly polarized electromagnetic plane wave with amplitude q and frequency ω can be written, in terms of Rosen coordinates, in the form where
This group is generated by a six-dimensional Lie algebra of Killing vector fields.
generate the Euclidean group, acting within each planar wavefront, which justifies the name plane wave for this solution.
For future reference, note that this six-dimensional group of self-isometries acts transitively so that our spacetime is homogeneous.
It is a null solution, and it represents a transverse sinusoidal electromagnetic plane wave with amplitude q and frequency ω, traveling in the e1 direction.
In terms of our frame, the stress-energy tensor turns out to be This is the same expression that one would find in classical electromagnetism (where one neglects the gravitational effects of the electromagnetic field energy) for the null field given above; the only difference is that now our frame is a anholonomic (orthonormal) basis on a curved spacetime, rather than a coordinate basis in flat spacetime.
The Rosen chart is said to be comoving with our family of inertial nonspinning observers, because the coordinates ve − u, x, y are all constant along each world line, given by an integral curve of the timelike unit vector field
This turns out to be where The nonvanishing components are identical and are Physically, this means that a small spherical 'cloud' of our inertial observers hovers momentarily at u = 0 and then begin to collapse, eventually passing through one another at u = u0.
If one imagines 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 vorticity tensor of our congruence vanishes identically, so the world lines of our observers are hypersurface orthogonal.
In contrast, the Bel decomposition of the Riemann curvature tensor, taken with respect to
Looking back at our graph of the metric tensor, one can see that the tidal tensor produces small sinusoidal relative accelerations with period ω, 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 nonspinning observers.
That is, if one looks through oncoming wavefronts at distant objects, one will see no optical distortion, but if one turns and look through departing wavefronts at distant objects, one will see optical distortions.
has vanishing optical scalars, but the null geodesic congruence generated by
has vanishing twist and shear scalars but nonvanishing expansion scalar This shows that when looking through departing wavefronts at distant objects, our inertial nonspinning observers will see their apparent size change in the same way as the expansion of the timelike geodesic congruence itself.
One way to quickly see the plausibility of the assertion that u = u0 is a mere coordinate singularity is to recall that our spacetime is homogeneous, so that all events are equivalent.
In this chart, one can see that an infinite sequence of identical expansion-recollapse cycles occurs!
The simplicity of the metric tensor compared to the complexity of the frame is striking.
The point is that one can more easily visualize the caustics formed by the relative motion of our observers in the new chart.
To understand how this situation appears in the Brinkmann chart, notice that when ω is extensive, our timelike geodesic unit vector field becomes approximately Suppressing the last term, the result is One immediately obtains an integral curve that exhibits sinusoidal expansion and reconvergence cycles.
See the figure, in which time is running vertically and one uses the radial symmetry to suppress one spatial dimension.
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 (concerning 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).
, modulated by much smaller sinusoidal perturbations in the null direction ∂v and having a much shorter period,
Comparing our exact solution with the usual monochromatic electromagnetic plane wave as treated in special relativity (i.e., as a wave in flat spacetime, neglecting the gravitational effects of the energy of the electromagnetic field), one sees that the striking new feature in general relativity is the expansion and collapse cycles experienced by our observers, which one can put down to background curvature, not any measurements made over short times and distances (on the order of the wavelength of the electromagnetic microwave radiation).
