Monochromatic electromagnetic plane wave

In general relativity, the monochromatic electromagnetic plane wave spacetime is the analog of the monochromatic plane waves known from Maxwell's theory. The precise definition of the solution is quite complicated, but very instructive. In general relativity, the monochromatic electromagnetic plane wave spacetime is the analog of the monochromatic plane waves known from Maxwell's theory. The precise definition of the solution is quite complicated, but very instructive. Any exact solution of the Einstein field equation which models an electromagnetic field, must take into account all gravitational effects of the energy and mass of the electromagnetic field. Besides the electromagnetic field, if there lacks matter and non-gravitational fields present, we must simultaneously solve the Einstein field equation and the Maxwell field equations. In Maxwell's theory of electromagnetism, one of the most important types of an electromagnetic field are those representing electromagnetic radiation. 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 are the monochromatic plane waves, in which only one frequency component is present. This is precisely the phenomenon which our solution will model 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 ξ = u 0 ω {displaystyle xi ={frac {u_{0}}{omega }}} is the first positive root of C(a, 2a, ξ) = 0 where a = q 2 ω 2 {displaystyle a={frac {q^{2}}{omega ^{2}}}} . In this chart, ∂u, ∂v are null coordinate vectors while ∂x, ∂y are spacelike coordinate vectors. Here, the Mathieu cosine C(a, b, ξ) is an even function which solves the Mathieu equation and also takes the value C(a, b, 0) = 1. 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 ∂u, ∂v are null vector fields. Therefore, ∂u + ∂v is a timelike vector field, while ∂u − ∂v, ∂x, ∂y are spacelike vector fields. To define the electromagnetic field, we may take the electromagnetic four-vector potential