Ricci identities in higher dimensions

TL;DR

This paper extends the Newman-Penrose formalism to higher-dimensional spacetimes, analyzing null congruences and Weyl tensor algebraic structures, revealing new geometric properties and classifications in dimensions greater than four.

Contribution

It provides the full set of Ricci identities in higher dimensions and explores their implications for null congruences and spacetime classifications.

Findings

Kundt spacetimes are of type II or more special in higher dimensions

Twisting geodetic WANDs in odd dimensions must be shearing

Extension of Newman-Penrose formalism to n>4 dimensions

Abstract

We explore connections between geometrical properties of null congruences and the algebraic structure of the Weyl tensor in n>4 spacetime dimensions. First, we present the full set of Ricci identities on a suitable "null" frame, thus completing the extension of the Newman-Penrose formalism to higher dimensions. Then we specialize to geodetic null congruences and study specific consequences of the Sachs equations. These imply, for example, that Kundt spacetimes are of type II or more special (like for n=4) and that for odd n a twisting geodetic WAND must also be shearing (in contrast to the case n=4).