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Type D Einstein spacetimes in higher dimensions

We show that all static spacetimes in higher dimensions n > 4 are necessarily of Weyl types G, Ii, D or O. This also applies to stationary spacetimes provided 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, on which we briefly comment.) 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 algebraic consequences of 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 (as defined in the text) the corresponding principal null directions are geodetic in arbitrary dimension (this in fact also applies to type II spacetimes). For n ⩾ 5, however, there may exist particular cases of type D vacuum spacetimes which admit non-geodetic multiple principal null directions and we explicitly present such examples in any n ⩾ 7. Further studies are restricted to five dimensions, where the type D Weyl tensor is fully described by a 3 × 3 real matrix Φij. In the case with ‘twistfree’ (Aij = 0) principal null geodesics we show that in a ‘generic’ case Φij is symmetric and eigenvectors of Φij coincide with eigenvectors of the expansion matrix Sij providing us thus in general with three preferred spacelike directions of the spacetime. Similar results are also obtained when relaxing the twistfree condition and assuming instead that Φij is symmetric. The five-dimensional Myers–Perry black hole and Kerr–NUT–AdS metrics in arbitrary dimension are also briefly studied as specific illustrative examples of type D vacuum spacetimes.