Definition of The Solution
The metric tensor of the unique exact solution modeling a linearly polarized electromagnetic plane wave with amplitude and frequency can be written, in terms of Rosen coordinates, in the form
where is the first positive root of where . In this chart, are null coordinate vectors while are spacelike coordinate vectors.
Here, the Mathieu cosine is an even function which solves the Mathieu equation and also takes the value . Despite the name, this function is not periodic, and it cannot be written in terms of sinusoidal or even hypergeometric functions. (See Mathieu function for more about the Mathieu cosine function.)
In our expression for the metric, note that are null vector fields. Therefore is a timelike vector field, while are spacelike vector fields.
To define the electromagnetic field, we may take the electromagnetic four-vector potential
We now have the complete specification of a mathematical model formulated in general relativity.
Read more about this topic: Monochromatic Electromagnetic Plane Wave
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