# Black holes and other exact spherical solutions in Quadratic Gravity

## TL;DR

This paper investigates static, spherically symmetric vacuum solutions in Quadratic Gravity, revealing new classes of black hole solutions, including Schwarzschild-Bach black holes, and analyzing their physical and thermodynamical properties.

## Contribution

The study introduces a simplified conformal-to-Kundt metric approach and systematically classifies solutions, identifying new Schwarzschild-Bach black holes with unique properties.

## Key findings

- Identified six classes of solutions, including Schwarzschild and non-Schwarzschild types.
- Discovered a new family of Schwarzschild-Bach black holes with a nonzero Bach tensor.
- Analyzed thermodynamics and tidal effects of the new solutions.

## Abstract

We study static, spherically symmetric vacuum solutions to Quadratic Gravity, extending considerably our previous Rapid Communication [Phys. Rev. D 98, 021502(R) (2018)] on this topic. Using a conformal-to-Kundt metric ansatz, we arrive at a much simpler form of the field equations in comparison with their expression in the standard spherically symmetric coordinates. We present details of the derivation of this compact form of two ordinary differential field equations for two metric functions. Next, we apply analytical methods and express their solutions as infinite power series expansions. We systematically derive all possible cases admitted by such an ansatz, arriving at six main classes of solutions, and provide recurrent formulas for all the series coefficients. These results allow us to identify the classes containing the Schwarzschild black hole as a special case. It turns out that one class contains only the Schwarzschild black hole, three classes admit the Schwarzschild solution as a special subcase, and two classes are not compatible with the Schwarzschild solution at all since they have strictly nonzero Bach tensor. In our analysis, we naturally focus on the classes containing the Schwarzschild spacetime, in particular on a new family of the Schwarzschild-Bach black holes which possesses one additional non-Schwarzschild parameter corresponding to the value of the Bach tensor invariant on the horizon. We study its geometrical and physical properties, such as basic thermodynamical quantities and tidal effects on free test particles induced by the presence of the Bach tensor. We also compare our results with previous findings in the literature obtained using the standard spherically symmetric coordinates.