Charged black holes in quadratic gravity

We study electrically charged, static, spherically symmetric black holes in quadratic gravity using the conformal-to-Kundt technique, which leads to a considerable simplification of the field equations. We study the solutions using a Frobenius-like approach of power-series expansions. The indicial equations restrict the set of possible leading powers to a few cases, describing, e.g., black holes, wormholes, or naked singularities. We focus on the black hole case and derive recurrent formulas for all series coefficients of the infinite power-series expansion around the horizon. The solution is characterized by electric charge \(q\), the black-hole radius \(a_0\), and the Bach parameter \(b\) related to the strength of the Bach tensor at the horizon. However, the Bach parameter has to be fine-tuned to ensure asymptotic flatness. The fine-tuning of \(b\) for a given \(q\) and \(a_0\) returns up to two values, describing two branches of asymptotically flat, static, spherically symmetric, charged black holes in quadratic gravity. This is in agreement with previous numerical works. We discuss various physical properties of these black holes, such as their asymptotic mass, temperature, photon spheres, and black-hole shadows. A straightforward generalization to dyonic black holes in quadratic gravity is also briefly mentioned.