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 nonlinear 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.

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 nonlinear 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.

typ: | article |
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journal: | Phys. Rev. D |

volume: | 95 |

pages: | 084025 |

year: | 2017 |

grant: | Centrum Alberta Einsteina pro gravitaci a astrofyziku, GAČR 14-37086G, 2014-2018 |

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physrevd.95.084025.pdf (182.63 kB) |