Bianchi identities in higher dimensions

Abstract

Abstract. A higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension n. It follows that the principal null congruence is geodesic and expands isotropically in two dimensions and does not expand in n − 4 spacelike dimensions or does not expand at all. It is shown that the existence of such principal geodesic null congruence in vacuum (together with an additional condition on twist) implies an algebraically special spacetime. We also use the Myers-Perry metric as an explicit example of a vacuum type D spacetime to show that principal geodesic null congruences in vacuum type D spacetimes do not share this property.

Alignment and Algebraically Special Tensors in Lorentzian Geometry

“…In higher dimensions, differential consequences of the Bianchi identities in type III and N spacetimes have been considered in [15].…”

Algebraic classification of higher dimensional spacetimes

“…Ultimately, we seek a higher-dimensional version of the Goldberg-Sachs theorem. A first step was taken in [29], in which the Bianchi identities in higher dimensions were studied. Here we simply make some comments on the properties of the L-tensor for the spacetimes that have been classified.…”

On Kerr-Schild spacetimes in higher dimensions

“…In fact, it has been pointed out that this can not be done in the most direct way [7,15,16,17,18]. Our results below will suggest a possible weak generalization of the shearfree condition, and a partial extension of the Goldberg-Sachs theorem to n > 4 (limited to KS solutions).…”

“…Along with the Bianchi identities [16], the optical contraint also imply that expanding vacuum KS solutions can not be of the type III or N, so that in arbitrary dimension n ≥ 4 …”