Type III and N solutions to quadratic gravity

TL;DR

This paper identifies exact vacuum solutions of quadratic gravity for Weyl types N and III, revealing conditions under which Einstein spacetimes solve QG and exploring the properties of more general solutions.

Contribution

It demonstrates that all Weyl type N Einstein spacetimes with a suitable cosmological constant are solutions to quadratic gravity, and analyzes constraints for type III solutions, expanding understanding of exact solutions in QG.

Findings

Type N Einstein spacetimes are solutions to QG with specific cosmological constants.

Type III solutions satisfy additional constraints, but pp-waves do not solve QG.

QG field equations influence optical properties, restricting solutions to the Kundt class.

Abstract

We study exact vacuum solutions to quadratic gravity (QG) of the Weyl types N and III. We show that in an arbitrary dimension all Einstein spacetimes of the Weyl type N with an appropriately chosen effective cosmological constant $\Lambda$ are exact solutions to QG and we refer to explicitly known metrics within this class. For type III Einstein spacetimes, an additional constraint follows from the field equations of QG and examples of spacetimes obeying such constraint are given. However, type III pp-waves do not satisfy this constraint and thus do not solve QG. For type N, we also study a wider class of spacetimes admitting a pure radiation term in the Ricci tensor. In contrast to the Einstein case, the field equations of generic QG determine optical properties of the geometry and restrict such exact solutions to the Kundt class. We provide examples of these metrics.