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

TL;DR

This paper explores 2+2 direct product spacetimes in Weyl conformal gravity, generalizing Birkhoff--Riegert theorem, and analyzes their solutions, symmetries, and relations to known models.

Contribution

It generalizes the Birkhoff--Riegert theorem to spacetimes with constant Gaussian curvature subspaces in Weyl gravity and classifies solutions via Weyl equivalence.

Findings

All such spacetimes admit at least two independent, commuting non-null Killing vectors.

Derived a generalization of the Birkhoff--Riegert theorem for these spacetimes.

Established connections between solutions in Weyl conformal gravity and Einstein gravity.

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.