Global properties of warped solutions in General Relativity with an electromagnetic field and a cosmological constant. II

TL;DR

This paper classifies all global solutions in general relativity with a cosmological constant and electromagnetic field, where spacetime is a warped product of constant curvature surfaces, revealing symmetry emergence and invariance under Lorentz groups.

Contribution

It provides a complete classification of solutions with warped product structure under the specified conditions, highlighting symmetry emergence in such spacetimes.

Findings

Solutions are invariant under Lorentz SO(1,2) or Poincare IO(1,1) groups.

At least one surface must have constant curvature, leading to symmetry.

All solutions with Lorentzian surface of constant curvature are classified.

Abstract

We consider general relativity with cosmological constant minimally coupled to the electromagnetic field and assume that the four-dimensional space-time manifold is a warped product of two surfaces with Lorentzian and Euclidean signature metrics. Field equations imply that at least one of the surfaces must be of constant curvature leading to the symmetry of the metric (``spontaneous symmetry emergence''). We classify all global solutions in the case when the Lorentzian surface is of constant curvature. These solutions are invariant with respect to the Lorentz SO(1,2) or Poincare IO(1,1) groups acting on the Lorentzian surface.