Newman-Penrose formalism in higher dimensions: vacuum spacetimes with a non-twisting geodetic multiple Weyl aligned null direction

TL;DR

This paper extends the Newman-Penrose formalism to higher dimensions to analyze vacuum spacetimes with specific null directions, revealing metric dependencies and singularities in various algebraic types, including explicit solutions.

Contribution

It develops a higher-dimensional Newman-Penrose formalism and characterizes metric and Weyl tensor dependence on affine parameters for different algebraic types, including explicit solutions.

Findings

Type III and N metrics are quadratic in r.

All spacetimes with non-zero expansion have curvature singularities.

Explicit five-dimensional Type N solution is derived.

Abstract

Vacuum spacetimes admitting a non-twisting geodetic multiple Weyl aligned null direction (WAND) are analyzed in arbitrary dimension using recently developed higher-dimensional Newman-Penrose (NP) formalism. We determine dependence of the metric and of the Weyl tensor on the affine parameter r along null geodesics generated by the WAND for type III and N spacetimes and for a special class of type II and D spacetimes, containing e.g. Schwarzschild-Tangherlini black holes and black strings and branes. For types III and N, all metric components are at most quadratic polynomials in r while for types II and D the r-dependence of the metric as well as of the Weyl tensor is determined by an integer m corresponding to the rank of the expansion matrix S_{ij}. It is shown that for non-vanishing expansion, all these spacetimes contain a curvature singularity. As an illustrative example, a shearing expanding type N five-dimensional vacuum solution is also re-derived using higher-dimensional NP formalism. This solution can be, however, identified with a direct product of a known four-dimensional type N metric with an extra dimension.