So I've heard that light escaping a massive body such as the Earth becomes red-shifted the further its gets away from Earth's gravity but always travels at exactly the speed of light (is this correct?) My question is, though, let's say that we're sitting around the event horizon of a black hole. We have light waves right at the border there trying to escape the grips of the black hole's gravity. Some make it and some don't. Say we have two photons very close together right at the horizon. Both are struggling to get out of the black hole's grip. One makes it and one doesn't. What happens there? Were they both traveling at the speed of c and then one just stops moving and becomes part of the black hole? Do neither of these photons separating their trajectories just around the event horizon ever change their speeds relative to one another?