Finalizing the classification of type II or more special Einstein spacetimes in five dimensions

TL;DR

This paper completes the classification of five-dimensional Einstein spacetimes of type II or more special, demonstrating that all such spacetimes satisfy the optical constraint and characterizing their null directions.

Contribution

It finalizes the classification of 5d Einstein spacetimes of type II or more special by proving constraints on their null directions and optical matrices.

Findings

Unique geodesic mWAND cannot have rank 1 or 3 optical matrix.

All such spacetimes satisfy the optical constraint.

Classification of all type II or more special Einstein spacetimes in 5d is completed.

Abstract

Einstein spacetimes in 5d that are of genuine type II in the null alignment classification are considered. It is shown that the unique geodesic multiple Weyl aligned null direction (mWAND) cannot have an optical matrix of rank 1 or 3. This finalizes the classification of all type II or more special Einstein spacetimes in 5d (i.e., those allowing for an mWAND), and also proves that they all satisfy the so-called optical constraint.