Ernst equation and spheroidal coordinates with a cosmological constant term

TL;DR

This paper extends solution generating techniques for stationary, axially symmetric spacetimes to include a cosmological constant, introducing adapted spheroidal coordinates and demonstrating solution construction in higher dimensions.

Contribution

It develops an extended Ernst equation incorporating the cosmological constant and proposes spheroidal coordinates suitable for asymptotically de-Sitter and anti-de Sitter spacetimes.

Findings

Extended Ernst equation with cosmological constant formulated

Spheroidal coordinates adapted to de-Sitter and anti-de Sitter spacetimes introduced

Higher-dimensional solutions constructed from 4D stationary and static solutions

Abstract

We discuss solution generating techniques treating stationary and axially symmetric metrics in the presence of a cosmological constant. Using the recently found extended form of Ernst's complex equation, which takes into account the cosmological constant term, we propose an extension of spheroidal coordinates adapted to asymptotically de-Sitter and anti de-Sitter static spacetimes. In the absence of a cosmological constant we show in addition that any higher dimensional metric parametrised by a single angular momentum can be given by a 4 dimensional solution and Weyl potentials parametrising the extra Killing directions. We explicitly show how a stationary, and a static axially symmetric spacetime solution in 4 dimensions, can be {\it added} together to give a 5 dimensional stationary and axisymmetric solution.