Type D Einstein spacetimes in higher dimensions
Abstract
We show that all static spacetimes in higher dimensions are of Weyl types G, Ii, 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 ij. In the case with ``twistfree'' (Aij=0) principal null geodesics we show that in a ``generic'' case ij is symmetric and eigenvectors of ij coincide with those of the expansion matrix Sij, providing us with three preferred spacelike directions of the spacetime. Similar results are also obtained when relaxing the twistfree condition and assuming instead that 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.