Generalized Birkhoff theorems and 2+2 direct pruduct spacetimes in Weyl conformal gravity

TL;DR

The paper investigates 2+2 direct-product spacetimes in Weyl conformal gravity sourced by electromagnetic and Yang–Mills fields, proving the existence of two independent, commuting non-null Killing vectors and deriving the general solution in a canonical form.A key advance is the generalization of the Birkhoff–Riegert theorem within WCG, including careful treatment of degenerate Weyl transformations that can alter causal and global structures, and the organization of solutions into Weyl-equivalence classes and finer subclasses.The work extends previous vacuum results to non-vacuum cases, showing how EM and YM fields can be consistently incorporated while preserving the 2+2 symmetry, and clarifies connections to known GR solutions through conformal mappings, including charged C-metrics and MK-type spacetimes.Together, these results provide a unified framework for analyzing the geometry, horizons, and curvature invariants of WCG 2+2 spacetimes and pave the way for further explorations of matter couplings and higher-dimensional generalizations.

Abstract

In this paper, we study 2+2 direct product spacetimes sourced by separated electromagnetic and Yang--Mills fields within Weyl conformal gravity. We prove that all such configurations admit at least 2 independent, commuting non-null Killing vectors, which we use to find general solutions. As a special case, we obtain a generalization of the Birkhoff--Riegert theorem to all spacetimes containing a two-dimensional subspace of constant Gaussian curvature, and we also revisit the original formulation of the theorem. We further analyze the resulting solutions in terms of Weyl equivalence classes. Their connections to known solutions in both Weyl conformal gravity and Einstein gravity are established through conformal relations. We also examine the fundamental physical and geometric properties of the newly obtained configurations and their equivalence classes.