On the integrability of Einstein-Maxwell-(A)dS gravity in presence of Killing vectors

TL;DR

This paper investigates the symmetry and integrability properties of Einstein-Maxwell-(A)dS gravity with Killing vectors, showing conditions under which the equations of motion are integrable and extending previous results to include electromagnetic fields.

Contribution

It demonstrates the integrability of Einstein-Maxwell-(A)dS equations under specific symmetry assumptions, generalizing earlier work to include electromagnetic fields and cosmological constant effects.

Findings

Residual symmetry group is a semidirect product of Heisenberg and translation groups.

Equations of motion are integrable when fields depend on a single coordinate.

Solutions include those previously constructed by Krasinski.

Abstract

We study some symmetry and integrability properties of four-dimensional Einstein-Maxwell gravity with nonvanishing cosmological constant in the presence of Killing vectors. First of all, we consider stationary spacetimes, which lead, after a timelike Kaluza-Klein reduction followed by a dualization of the two vector fields, to a three-dimensional nonlinear sigma model coupled to gravity, whose target space is a noncompact version of $\mathbb{C}\text{P}^2$ with SU(2,1) isometry group. It is shown that the potential for the scalars, that arises from the cosmological constant in four dimensions, breaks three of the eight SU(2,1) symmetries, corresponding to the generalized Ehlers and the two Harrison transformations. This leaves a semidirect product of a one-dimensional Heisenberg group and a translation group $\mathbb{R}^2$ as residual symmetry. We show that, under the additional assumptions that the three-dimensional manifold is conformal to a product space $\mathbb{R}\times\Sigma$, and all fields depend only on the coordinate along $\mathbb{R}$, the equations of motion are integrable. This generalizes the results of Leigh et al. in arXiv:1403.6511 to the case where also electromagnetic fields are present. In the second part of the paper we consider the purely gravitational spacetime admitting a second Killing vector that commutes with the timelike one. We write down the resulting two-dimensional action and discuss its symmetries. If the fields depend only on one of the two coordinates, the equations of motion are again integrable, and the solution turns out to be one constructed by Krasinski many years ago.