Symmetry in Einstein-Maxwell Space-Time

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Abstract
By using the complex null tetrad as basis for the tangent space, a Killing vector field (``symmetry'') is introduced into the system of Einstein's equations with Maxwell's equations. The two bivectors Fμν and Kμ;ν (the associated Killing bivector) are assumed to have a principal null direction in common. Killing's equations, Maxwell's equations, and Einstein's equations are then written down for the case where this special direction is also a principal null geodesic for the Weyl conformal tensor. A certain analog of the Goldberg‐Sachs theorem is proved. The static cases, plus a sizeable class of the static algebraically special cases are examined, to wit: where the special direction is also shear‐free. In particular, all such algebraically special spaces must be Petrov Type D as a result of a coupling of the principal null directions for Fμν. This algebraically special metric is derived as an example of the static classes and is a static generalization of the Reissner‐Nordström metric.

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