The Spacetime Geometry of an Electromagnetic Wave
Since the 1920's it has been known how to characterize almost all solutions to the Einstein-Maxwell equations in terms of geometric conditions built solely from the spacetime metric. These conditions are known as the "Rainich conditions"; they provide a generalization to electrovacuum spacetimes of the geometry of vacuum (Ricci-flat) spacetimes. With the aid of modern computer algebra systems, the Rainich conditions also provide a novel approach to solving the Einstein-Maxwell equations. The Rainich conditions fail to describe solutions of the Einstein-Maxwell equations which have a null electromagnetic field, e.g., electromagnetic plane waves. In this talk I will review Rainich geometry and then describe geometric conditions on a spacetime which are necessary and sufficient for the existence of a solution to the Einstein-Maxwell equations with a null electromagnetic field. These conditions can be viewed as the analog of the Rainich conditions for null electrovacua, and they are equally amenable to computer implementation.
Torre, Charles G., "The Spacetime Geometry of an Electromagnetic Wave" (2013). Presentations and Publications. Paper 3.