I think a black hole would move through space like stars and galaxies. But then the problem is, what will happen to the event horizon and photon sphere orbiting black hole? If a black hole moves through space does not it allow photons orbiting around it to escape or fall into it, so the black hole should always lose photon sphere?
Same for event horizon, does it move along with blackhole? How can it move? Doesn't it allow some photon at the event horizon to escape like a comet trail?
Best Answer
Black holes moves just like all other matter. The photon sphere, the event horizon, just like the inner-most stable circular orbit, etc, etc. all move along with the BH---as they are defined relative to the BH rest-frame, i.e. their relative velocity is zero.
Having the black hole be moving doesn't really effect the of (e.g.) particle trajectories much: it's very rare that a photon becomes trapped (even for a short time) at the photon-sphere, and it would be similar rare for a moving black hole, but the trajectories of the photons would simply be slightly different (in particular they would need have a velocity component in the direction of the black hole's motion. But there's definitely nothing strange or exotic happening here, and based on the fundamental principle of Lorentz invariance (things being the same from different velocity reference frames) nothing exotic is allowed to happen.