In this chapter we present a status report of black hole-like solutions in
non-local theories of gravity in which the Lagrangians are at least quadratic
in curvature and contain specific non-polynomial (i.e., non-local) operators.
In the absence of exact black hole solutions valid in the whole spacetime, most
of the literature on this topic focus on approximate and simplified equations
of motion, which could provide insights on the full non-linear solutions.
Therefore, the largest part of this chapter is devoted to the linear
approximation. We present results on stationary metric solutions (including
both static and rotating cases) and dynamical spacetimes describing the
formation of non-rotating mini black holes by the collapse of null shells.
Non-local effects can regularize the curvature singularities in both scenarios
and, in the dynamical case, there exists a mass gap below which the formation
of an apparent horizon can be avoided. In the final part we discuss interesting
attempts towards finding non-linear black hole solutions in non-local gravity.
Throughout this chapter, instead of focusing on a particular non-local model,
we present results valid for large classes of theories (to a feasible extent).
This more general approach allows the comparison of similarities and
differences of the various types of non-local gravity models.Comment: Invited chapter for the Section "Nonlocal Quantum Gravity" of the
"Handbook of Quantum Gravity" (Eds. C. Bambi, L. Modesto and I.L. Shapiro,
Springer Singapore, expected in 2023). 29 page