Rotating spacetimes with a cosmological constant

TL;DR

This paper develops solution-generating techniques for higher-dimensional rotating spacetimes with a cosmological constant, extending classical methods and providing new explicit solutions like deformations of AdS black holes.

Contribution

It generalizes solution-generating methods to arbitrary dimensions with a cosmological constant and constructs new solutions from four-dimensional seeds.

Findings

Decoupling of field equations in higher dimensions for zero cosmological constant

Extension of Ernst and Papapetrou methods to higher dimensions with cosmological constant

Explicit solutions including deformations of AdS soliton and black hole

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.