Black holes and other spherical solutions in quadratic gravity with a cosmological constant

We study static spherically symmetric solutions to the vacuum field equations of quadratic gravity in the presence of a cosmological constant A. Motivated by the trace no-hair theorem, we assume the Ricci scalar to be constant throughout a spacetime. Furthermore, we employ the conformal-to-Kundt metric ansatz that is valid for all static spherically symmetric spacetimes and leads to a considerable simplification of the field equations. We arrive at a set of two ordinary differential equations and study its solutions using the Frobenius-like approach of (infinite) power series expansions. While the indicial equations considerably restrict the set of possible leading powers, careful analysis of higher-order terms is necessary to establish the existence of the corresponding classes of solutions. We thus obtain various non-Einstein generalizations of the Schwarzschild, (anti-)de Sitter [or (A)dS for short], Nariai, and Plebanski-Hacyan spacetimes. Interestingly, some classes of solutions allow for an arbitrary value of A, while other classes admit only discrete values of A. For most of these classes, we give recurrent formulas for all series coefficients. We determine which classes contain the Schwarzschild-(A)dS black hole as a special case and briefly discuss the physical interpretation of the spacetimes. In the discussion of physical properties, we naturally focus on the generalization of the Schwarzschild-(A)dS black hole, namely the Schwarzschild-Bach-(A)dS black hole, which possesses one additional Bach parameter. We also study its basic thermodynamical properties and observable effects on test particles caused by the presence of the Bach tensor. This work is a considerable extension of our Letter [Phys. Rev. Lett. 121, 231104 (2018)].

Automated summary

This study extends a series of non-Einstein solutions for the Schwarzschild black hole and other spacetime structures through the investigation of static spherically symmetric solutions to the vacuum field equations of quadratic gravity. Analysis of ordinary differential equations and power series expansions reveals restrictions on the values of the cosmological constant A, leading to discussions on the physical interpretation of these solutions. The study also introduces the Schwarzschild-Bach-(A)dS black hole, with additional parameters, and examines its thermodynamical properties and effects on test particles.