The Flat Critical Branch Between Nariai and Bertotti-Robinson Geometries as a Solution of Cosmological Einstein-Maxwell Theory

TL;DR

This paper investigates a special class of Einstein-Maxwell solutions with a flat two-dimensional Lorentzian factor, serving as a critical midpoint between Nariai and Bertotti-Robinson geometries, with universal properties across higher-curvature theories.

Contribution

It identifies and characterizes the critical flux-supported geometry as an algebraic midpoint between two well-known solutions, highlighting its universal solution properties.

Findings

The critical geometry occurs at the balance point where Lorentzian curvature vanishes.

The spacetime is Petrov type-D with constant scalar invariants.

It solves a broad class of higher-curvature gravity theories due to its algebraic curvature structure.

Abstract

We analyze a class of product geometries of the form $\mathbb{R}^{1,1}\times \Sigma_2$ supported by electric, magnetic, or dyonic flux in the Einstein-Maxwell-$\Lambda$ theory. These spacetimes belong to a unified family of direct products $(dS_2,\mathbb{R}^{1,1},AdS_2)\times \Sigma_2$ distinguished solely by the sign of the Lorentzian curvature of the two-dimensional factor. We focus on the critical configuration for which the Lorentzian curvature vanishes. At this balance point between the cosmological curvature and the Maxwell stress, the longitudinal geometry becomes exactly flat while the transverse sphere radius is fixed algebraically by the conserved flux. We refer to this geometry as the critical Maxwell flux string: a homogeneous flux-supported geometry curved only in the transverse directions and invariant along a two-dimensional null worldvolume. It represents the algebraic midpoint interpolating between the Nariai $(dS_2\times S^2)$ and Bertotti-Robinson $(AdS_2\times S^2)$ spacetimes. A qualitative structural change occurs precisely at this midpoint. The spacetime is Petrov type-D with constant scalar curvature invariants, placing it in the degenerate Kundt/CSI class. Because the curvature structure reduces any polynomial rank-two tensor to a linear combination of the metric and the Maxwell stress tensor, the same configuration solves a broad class of algebraic higher-curvature gravity theories. In this sense, the critical flux string and its aligned deformations constitute almost universal solutions.