The existence of black holes is a central prediction of general relativity and thus serves as a basic consistency test for modified theories of gravity. In spherical symmetry, only two classes of dynamic solutions are compatible with the formation of an apparent horizon in finite time of a distant observer. Moreover, the formation of black holes follows a unique scenario involving both types of solutions. To be compatible with their existence, any self-consistent theory of modified gravity must satisfy several constraints. We derive properties of the modified gravity terms of $\mathfrak{f}(R)$ and generic fourth-order gravity theories and find that they naturally accommodate both classes of solutions. Consequently, the observation of an apparent horizon by itself may not suffice to distinguish between general relativity and modifications including up to fourth-order derivatives in the metric.

• Received 1 November 2021
• Accepted 27 December 2021

DOI:https://doi.org/10.1103/PhysRevD.105.044051