Let's put an observer hovering near the event horizon of a charged black hole. As the black hole is charged we can change its velocity from zero to 10 m/s in one second. But we can not send a message to the observer in one second. So the observer does not know that the black hole that he is sitting on started moving. So I guess he is moving with the black hole. Why does he not see stars moving? I mean those stars that were still in the black hole's sky when the black hole was not moving, why do those stars stay still when the black hole and the observer start moving?