Complexification of the algebraically special gravitational fields

Abstract

The fact that the Maxwell equations can be analytically extended into complex Minkowski space is used to show that a class of solutions to the real Maxwell equations exists which can be viewed as arising from a monopole source moving along a complex world line in the complex Minkowski space. This class of solutions is the natural analog of the class of regular, algebraically special type II, twisting metrics in Einstein's general theory of relativity, in that the two cases are characterized geometrically by the fact that the Maxwell and Weyl tensors, respectively, both possess a shear-free, diverging, geodesic principal null vector field l, which is twisting. By analytically extending the algebraically special metrics into a complex manifold, we show that the analogy runs even deeper than this. Aside from the constants, charge and mass, the solutions in both cases are completely determined by a single complex function φ. In both analytically extended manifolds the surfaces, φ=const, are complex null surfaces and the complexified versions of both the Maxwell and Weyl tensors now have a nontwisting principal null vector field l*, equal to the gradient of φ. We introduce the natural coordinate and tetrad systems associated with l and l* and show the relationship between them in both the flat and curved complex manifolds. The class of solutions to the Maxwell equations is solved in both systems. The algebraically special metrics are treated in detail, and the Kerr metric is given as an explicit example.