On the Goldberg–Sachs theorem in higher dimensions in the non-twisting case

Abstract

We study a generalization of the ‘shearfree part’ of the Goldberg–Sachs theorem for Einstein spacetimes admitting a non-twisting multiple Weyl aligned null direction (WAND) ℓ in n ⩾ 6 spacetime dimensions. The form of the corresponding optical matrix ρ is restricted by the algebraically special property in terms of the degeneracy of its eigenvalues. In particular, there necessarily exists at least one multiple eigenvalue, and further constraints arise in various special cases. For example, when ρ is non-degenerate and certain (boost weight zero) Weyl components do not vanish, all eigenvalues of ρ coincide and such spacetimes thus correspond to the Robinson–Trautman class. On the other hand, in certain degenerate cases all non-zero eigenvalues can be distinct. We also present explicit examples of Einstein spacetimes admitting some of the permitted forms of ρ, including examples violating the ‘optical constraint’. The obtained restrictions on ρ are, however, in general not sufficient for ℓ to be a multiple WAND, as demonstrated by a few ‘counterexamples’. We also discuss the geometrical meaning of these restrictions in terms of integrability properties of certain totally null distributions. Finally, we specialize our analysis to the six-dimensional case, where all the permitted forms of ρ are given in terms of just two parameters. In the appendices, some examples are given and certain results pertaining to (possibly) twisting multiple WANDs of Einstein spacetimes are presented.