Abstract
We determine the most general solution of the five-dimensional vacuum Einstein equation, allowing for a cosmological constant, with (i) a Weyl tensor that is type II or more special in the classification of Coley et al., and (ii) a non-degenerate “optical matrix” encoding the expansion, rotation and shear of the aligned null direction. The solution is specified by three parameters. It is locally isometric to the 5d Kerr–de Sitter solution, or related to this solution by analytic continuation or taking a limit. This is in contrast with four dimensions, where there exist infinitely many solutions with properties (i) and (ii).
Highlights
There is a long tradition of classifying solutions of the four-dimensional Einstein equation according to the algebraic type of the Weyl tensor
In an algebraically special spacetime, the Einstein equation simplifies considerably and one can determine the explicit dependence of the metric on one of the coordinates
The Einstein equation reduces to PDEs in 3 dimensions
Summary
There is a long tradition of classifying solutions of the four-dimensional Einstein equation according to the algebraic type of the Weyl tensor. A solution is type II or more special in this classification if it admits a multiple Weyl aligned null direction (multiple WAND). The Myers–Perry black hole solution [5] (“higher dimensional Kerr”) is known to have a Weyl tensor of type D [6,7,8], which means that it admits two distinct multiple WANDs. The Myers–Perry solution has been generalized to include a cosmological constant in five [9] and higher [10] dimensions. We will refer to these as “Kerr–de Sitter" solutions (for any value of ) These have a Weyl tensor of type D [7].
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