Monochromatic Electromagnetic Plane Wave - The Electromagnetic Field

The Electromagnetic Field

With respect to our frame, the electromagnetic field obtained from the potential given above is

This electromagnetic field is a source-free solution of the Maxwell field equations on the particular curved spacetime which is defined by the metric tensor above. It is a null solution, and it represents a transverse sinusoidal electromagnetic plane wave with amplitude and frequency, traveling in the direction. When we

• compute the stress-energy tensor for the given electromagnetic field,
• compute the Einstein tensor for the given metric tensor,

we find that the Einstein field equation is satisfied. This is what we mean by saying that we have an exact electrovacuum solution.

In terms of our frame, the stress-energy tensor turns out to be

Notice that this is exactly the same expression that we would find in classical electromagnetism (where we neglect 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. (See frame fields.)