Global properties of warped solutions in general relativity with an electromagnetic field and a cosmological constant

Abstract

We consider general relativity with cosmological constant minimally coupled to electromagnetic field and assume that four-dimensional space-time manifold is the warped product of two surfaces with Lorentzian and Euclidean signature metrics. Einstein's equations imply that at least one of the surfaces must be of constant curvature. It means that the symmetry of the metric arises as the consequence of equations of motion (`spontaneous symmetry emergence'). We give classification of global solutions in two cases: (i) both surfaces are of constant curvature and (ii) the Riemannian surface is of constant curvature. The latter case includes spherically symmetric solutions (sphere S^2 with SO(3)-symmetry group), planar solutions (two-dimensional Euclidean space R^2 with IO(2)-symmetry group), and hyperbolic solutions (two-sheeted hyperboloid H^2 with SO(1,2)-symmetry). Totally, we get 37 topologically different solutions. There is a new one among them, which describes changing topology of space in time already at the classical level.