General Kerr–NUT–AdS metrics in all dimensions

Abstract

The Kerr-AdS metric in dimension D has cohomogeneity [D/2]; the metric\ncomponents depend on the radial coordinate r and [D/2] latitude variables \\mu_i\nthat are subject to the constraint \\sum_i \\mu_i^2=1. We find a coordinate\nreparameterisation in which the \\mu_i variables are replaced by [D/2]-1\nunconstrained coordinates y_\\alpha, and having the remarkable property that the\nKerr-AdS metric becomes diagonal in the coordinate differentials dy_\\alpha. The\ncoordinates r and y_\\alpha now appear in a very symmetrical way in the metric,\nleading to an immediate generalisation in which we can introduce [D/2]-1 NUT\nparameters. We find that (D-5)/2 are non-trivial in odd dimensions, whilst\n(D-2)/2 are non-trivial in even dimensions. This gives the most general\nKerr-NUT-AdS metric in $D$ dimensions. We find that in all dimensions D\\ge4\nthere exist discrete symmetries that involve inverting a rotation parameter\nthrough the AdS radius. These symmetries imply that Kerr-NUT-AdS metrics with\nover-rotating parameters are equivalent to under-rotating metrics. We also\nconsider the BPS limit of the Kerr-NUT-AdS metrics, and thereby obtain, in odd\ndimensions and after Euclideanisation, new families of Einstein-Sasaki metrics.\n