In classical mechanics by choosing the right trajectory you can approach a planet arbitrarily closely, if there is no atmosphere or anything to slow you down, you can approach the surface then fly away without firing your rockets.
Does the same works in general relativity with black holes? Can you do close approach of the event horizon without falling in?
(My guess is that it's the distance of the photon sphere, but it's just a guess.)
To be more specific:
- observer much smaller than the black hole, so you can assume it's a point particle. And won't be torn apart by tidal forces above the horizon.
- Assume a flyby: The observer must be able to escape to the infinity unpowered.
- Assume the black hole doesn't have accretion disk.
- I don't mind if it's rotating, but if it's simpler to make calculations in terms of a non-rotating black hole I accept it too.