Complex structures and representations of the Einstein equations

Abstract

The space-time model of general relativity is that of a four-dimensional manifold M, with a metric of Minkowski signature. The space of two-forms on M is endowed with a natural complex structure, J, generated by the star duality operator. The existence of such a structure is an accidental characteristic of the dimension four and of the metric signature. The full differential geometric structure equations are expressed in this two-form language and it is pointed out that the weakened Einstein empty space equations, i.e., Rab −(1/4)gabR =0, reduce to the condition that the curvature form commute with J. This fact, together with the isomorphism of the two-form space with the Lorentz Lie algebra, l0, are then shown to provide the basis for the importance of the various complex representations of l0, such as S O (3,C) and the spinor S L (2,C), in understanding the real geometry of Einstein spaces. In fact, the complexified Einstein structure equations naturally divide into two sets, each the complex conjugate of the other, each involving only one-half of the basis.