Rotating spacetimes with a cosmological constant

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.