The Spacetime Geometry of an Electromagnetic Wave

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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.

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