Bianchi identities in higher dimensions

TL;DR

This paper develops a higher-dimensional formalism to analyze Bianchi identities for the Weyl tensor in vacuum spacetimes, revealing properties of principal null congruences and their relation to spacetime algebraic types.

Contribution

It introduces a new higher-dimensional frame formalism to study Bianchi identities and characterizes the behavior of principal null congruences in various vacuum spacetime types.

Findings

Principal null congruence is geodesic and expands isotropically in two dimensions.

In certain conditions, the principal null congruence implies an algebraically special spacetime.

Vacuum type D spacetimes can have principal geodesic null congruences that do not share the same properties.

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.