A Generalization of the Goldberg–Sachs theorem and its consequences

Abstract

The Goldberg-Sachs theorem is generalized for all four-dimensional manifolds endowed with torsion-free connection compatible with the metric, the treatment includes all signatures as well as complex manifolds. It is shown that when the Weyl tensor is algebraically special severe geometric restrictions are imposed. In particular it is demonstrated that the simple self-dual eigenbivectors of the Weyl tensor generate integrable isotropic planes. Another result obtained here is that if the self-dual part of the Weyl tensor vanishes in a Ricci-flat manifold of (2,2) signature the manifold must be Calabi-Yau or symplectic and admits a solution for the source-free Einstein-Maxwell equations. In this paper the term "complexified manifold" means a manifold in which the metric can be complex, so that the Weyl tensor is also generally complex. The here called "real manifolds" are the ones with real metric and, consequently, real Weyl tensor. In general the tangent bundle of the real manifolds will be assumed to be complexified. Finally, the term "complex manifold" will mean a manifold that can be covered by complex charts with analytic transition functions, these manifolds are sometimes called Hermitian.

Class of integrable metrics and gauge fields

“…So, for any gauge group, the most general solution for the class of spacetimes considered here are the solutions obtained in the previous section supplemented by the condition (34). Particularly, in the abelian case the condition (34) represents no constraint at all, since in such a case all elements of the Lie algebra commute with each other, so that Q 1 and Q 2 can be completely arbitrary elements. Thus, if k is the dimension of the gauge group, the number of charge parameters in the abelian case is 2k, inasmuch as a general element of the Lie algebra has k independent components.…”

Pure subspaces, generalizing the concept of pure spinors

“…When the metric on the even-dimensional space has split signature the results are also valid without complexification, since in this case the isotropic subspaces can have dimension equal to half of the dimension of the full vector space. The results on real even-dimensional vector spaces can be extracted from the complex case by choosing suitable reality conditions, in the spirit of [9,12].…”

“…Thus, if v = v b e b and ϕ = ϕ βψ β are general vector and spinor fields respectively, then Eqs. (9) and (8) imply that:…”

“…Thus, if I is integrable then (I ∩ I) generates a null geodesic congruence. In four dimensions, 2n = 4, if the Ricci tensor vanishes then this geodesic congruence is shear-free and the Weyl tensor is algebraically special [9].…”

“…Such approaches, in which isotropic distributions play a prominent role, brought much progress to general relativity. Notably, Kinnersley was able to find analytically all solutions of Einstein's vacuum equation in 4 dimensions for the case of the space-time admitting two independent integrable distributions of isotropic planes [7,8,9]. Particularly, all known 4-dimensional black-holes are contained in this class of solutions.…”