Rotating spacetimes with a cosmological constant

TL;DR

The paper addresses the problem of generating rotating, axisymmetric solutions in $D$-dimensional gravity with a cosmological constant and relates them to lower-dimensional Einstein–Maxwell–dilaton systems. It extends the Ernst–Papapetrou framework to arbitrary dimensions and nonzero $ abla$, using a Kaluza–Klein reduction to a $d=D-1$ EMD system with a Liouville potential, and identifies conditions for decoupling in the $ abla=0$ case along with a generalized Ernst equation for $ abla\neq0$. Key contributions include (i) a decoupling-based uplift method that produces infinite higher-dimensional solutions from 4D seeds when $ abla=0$, (ii) a generalized Ernst formulation for $ abla\neq0$ with sigma-model target-space symmetries that generate new solutions, and (iii) explicit deformations of AdS solitons and planar AdS black holes plus seed-based examples for 5D black rings and Myers–Perry black holes. The work provides a versatile toolkit for constructing and analyzing higher-dimensional rotating spacetimes relevant to AdS/CFT and braneworld contexts, while clarifying the role of dualities and coordinate choices in AdS geometries and outlining directions for future coordinate-system developments.

Abstract

We develop solution-generating techniques for stationary metrics with one angular momentum and axial symmetry, in the presence of a cosmological constant and in arbitrary spacetime dimension. In parallel we study the related lower dimensional Einstein-Maxwell-dilaton static spacetimes with a Liouville potential. For vanishing cosmological constant, we show that the field equations in more than four dimensions decouple into a four dimensional Papapetrou system and a Weyl system. We also show that given any four dimensional 'seed' solution, one can construct an infinity of higher dimensional solutions parametrised by the Weyl potentials, associated to the extra dimensions. When the cosmological constant is non-zero, we discuss the symmetries of the field equations, and then extend the well known works of Papapetrou and Ernst (concerning the complex Ernst equation) in four-dimensional general relativity, to arbitrary dimensions. In particular, we demonstrate that the Papapetrou hypothesis generically reduces a stationary system to a static one even in the presence of a cosmological constant. We also give a particular class of solutions which are deformations of the (planar) adS soliton and the (planar) adS black hole. We give example solutions of these techniques and determine the four-dimensional seed solutions of the 5 dimensional black ring and the Myers-Perry black hole.