The Geroch group in Einstein spaces

TL;DR

The paper extends Geroch's solution-generating method to Einstein spaces with a cosmological constant by reducing along a Killing vector to a three-dimensional coset and formulating a sigma-model with $\mathrm{SL}(2,\mathbb{R})$ symmetry acting on $\tau=\omega+i\lambda$. A dilaton-like conformal factor $\kappa$ is introduced to capture the full solution space, enabling reconstruction of 4D Einstein metrics from 3D data. The dynamics reduce to a particle on $\mathbb{R}\times H_2$ with a potential, and the Hamilton-Jacobi analysis shows that $\Lambda$ is a simple integration constant while a subgroup $N\subset SL(2,\mathbb{R})$ generates algebraic transformations of the physical parameters $m$, $n$, and $\Lambda$. The framework yields concrete solutions, including Schwarzschild-(A)dS and Taub-NUT-(A)dS families, and outlines paths toward Ernst-type two-Killing-vector extensions and Lax-pair techniques for Kerr-(A)dS geometries.

Abstract

Geroch's solution-generating method is extended to the case of Einstein spaces, which possess a Killing vector {{}and are thus asymptotically (locally) (anti-)de Sitter}. This includes the reduction to a three-dimensional coset space, the description of the dynamics in terms of a sigma-model and its transformation properties under the $SL(2,\mathbb{R})$ group, and the reconstruction of new four-dimensional Einstein spaces. The detailed analysis of the space of solutions is performed using the Hamilton--Jacobi method in the instance where the three-dimensional coset space is conformal to $\mathbb{R}\times \mathcal{S}_2$. The cosmological constant appears in this framework as a constant of motion and transforms under $SL(2,\mathbb{R})$.