Type III and N Einstein spacetimes in higher dimensions: general properties

TL;DR

This paper explicitly solves Sachs equations in higher dimensions to analyze Einstein spacetimes of type III and N, revealing their metric dependence, peeling behaviour, and potential curvature singularities, with explicit solutions provided.

Contribution

It provides the first explicit solutions for Sachs equations in higher dimensions and characterizes the properties of type III and N Einstein spacetimes, including their singularity structure.

Findings

Peeling behaviour of Weyl components depends on the affine parameter r.

Curvature singularities are generically present in expanding spacetimes.

Explicit examples of higher-dimensional type N/III Einstein solutions are constructed.

Abstract

The Sachs equations governing the evolution of the optical matrix of geodetic WANDs (Weyl aligned null directions) are explicitly solved in n-dimensions in several cases which are of interest in potential applications. This is then used to study Einstein spacetimes of type III and N in the higher dimensional Newman-Penrose formalism, considering both Kundt and expanding (possibly twisting) solutions. In particular, the general dependence of the metric and of the Weyl tensor on an affine parameter r is obtained in a closed form. This allows us to characterize the peeling behaviour of the Weyl "physical" components for large values of r, and thus to discuss, e.g., how the presence of twist affects polarization modes, and qualitative differences between four and higher dimensions. Further, the r-dependence of certain non-zero scalar curvature invariants of expanding spacetimes is used to demonstrate that curvature singularities may generically be present. As an illustration, several explicit type N/III spacetimes that solve Einstein's vacuum equations (with a possible cosmological constant) in higher dimensions are finally presented.