# Type D Einstein spacetimes in higher dimensions

## TL;DR

This paper classifies higher-dimensional static and stationary Einstein spacetimes by their Weyl types, analyzes type D vacuum solutions, and explores specific examples like Myers-Perry black holes, revealing geometric properties and principal null directions.

## Contribution

It provides a comprehensive classification of higher-dimensional Einstein spacetimes by Weyl type, including static, stationary, and warped cases, and analyzes properties of type D vacuum solutions.

## Key findings

- All static higher-dimensional spacetimes are of Weyl types G, I_i, D, or O.
- Type D vacuum spacetimes have geodetic principal null directions in generic cases.
- Examples include higher-dimensional Myers-Perry black holes and Kerr-NUT-AdS metrics.

## Abstract

We show that all static spacetimes in higher dimensions are of Weyl types G, I_i, D or O. This applies also to stationary spacetimes if additional conditions are fulfilled, as for most known black hole/ring solutions. (The conclusions change when the Killing generator becomes null, such as at Killing horizons.) Next we demonstrate that the same Weyl types characterize warped product spacetimes with a one-dimensional Lorentzian (timelike) factor, whereas warped spacetimes with a two-dimensional Lorentzian factor are restricted to the types D or O. By exploring the Bianchi identities, we then analyze the simplest non-trivial case from the above classes - type D vacuum spacetimes, possibly with a cosmological constant, dropping, however, the assumptions that the spacetime is static, stationary or warped. It is shown that for ``generic'' type D vacuum spacetimes the corresponding principal null directions are geodetic in any dimension (this applies also to type II spacetimes). For n>=5, however, there may exist particular cases of type D spacetimes which admit non-geodetic multiple principal null directions and we present such examples in any n>=7. Further studies are restricted to five dimensions, where the type D Weyl tensor is described by a 3x3 matrix \Phi_{ij}. In the case with ``twistfree'' (A_{ij}=0) principal null geodesics we show that in a ``generic'' case \Phi_{ij} is symmetric and eigenvectors of \Phi_{ij} coincide with those of the expansion matrix S_{ij}, providing us with three preferred spacelike directions of the spacetime. Similar results are also obtained when relaxing the twistfree condition and assuming instead that \Phi_{ij} is symmetric. The n=5 Myers-Perry black hole and Kerr-NUT-AdS metrics in arbitrary dimension are briefly studied as specific examples of type D vacuum spacetime.