Exact solutions to quadratic gravity

TL;DR

This paper explores non-Einstein vacuum solutions in quadratic gravity, proving conditions under which solutions are Kundt spacetimes and demonstrating how conformal transformations generate solutions from Einstein backgrounds.

Contribution

It establishes that certain traceless Ricci and Weyl tensor aligned solutions are necessarily Kundt spacetimes and develops a method to construct solutions via conformal transformations from Einstein backgrounds.

Findings

Vacuum solutions with traceless Ricci tensor of type N are Kundt spacetimes.

Solutions with traceless Ricci type III and aligned Weyl tensor are also Kundt.

Explicit Kundt and Robinson-Trautman solutions are constructed using conformal transformations.

Abstract

Since all Einstein spacetimes are vacuum solutions to quadratic gravity in four dimensions, in this paper we study various aspects of non-Einstein vacuum solutions to this theory. Most such known solutions are of traceless Ricci and Petrov type N with a constant Ricci scalar. Thus we assume the Ricci scalar to be constant which leads to a substantial simplification of the field equations. We prove that a vacuum solution to quadratic gravity with traceless Ricci tensor of type N and aligned Weyl tensor of any Petrov type is necessarily a Kundt spacetime. This will considerably simplify the search for new non-Einstein solutions. Similarly, a vacuum solution to quadratic gravity with traceless Ricci type III and aligned Weyl tensor of Petrov type II or more special is again necessarily a Kundt spacetime. Then we study the general role of conformal transformations in constructing vacuum solutions to quadratic gravity. We find that such solutions can be obtained by solving one non-linear partial differential equation for a conformal factor on any Einstein spacetime or, more generally, on any background with vanishing Bach tensor. In particular, we show that all geometries conformal to Kundt are either Kundt or Robinson-Trautman, and we provide some explicit Kundt and Robinson-Trautman solutions to quadratic gravity by solving the above mentioned equation on certain Kundt backgrounds.