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  1. Event horizon is a region of space whereby nothing with information can escape, so I imagine that a photon falls into this region of space does it accelerates toward the singularity if it exists?

  2. Do photon inside event horizon exceeds speed of light?

  3. Case study: electron emits photon inside the event horizon, what is it frequency?

  4. I know this event can never be observed but what does the latest theory suggests?

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Locally, light always travels at the speed of light. If your laboratory were falling into a black hole, you would measure the speed of light to be $c$, after accounting for tidal effects. You wouldn't notice a difference regardless of how far you were from the singularity.

From another frame of reference, things are different. An easily examined case is two photons released at the event horizon, one directed away from the singularity and one directed towards: the first stays at the horizon, while the second falls into the black hole at at least $c$, as "measured" by an outside observer. Of course, the first photon can only be measured if the black hole evaporates so that the horizon shrinks, and if that happens, the outside observer will still locally measure the photon's speed as $c$ when it passes through the lab. It could be argues that this outcome is a result not of a changing speed of light, but of time dilation becoming infinite at the horizon - though in the context of special relativity, that's more a semantic argument than a physical one.

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    $\begingroup$ hi! special relativity sets the upper speed limit for light moving in a vacuum at c regardless of reference frame and you mentioned the second photon falls into the black hole at at least c, what does it means actually? $\endgroup$ – user6760 Dec 11 '15 at 2:40
  • $\begingroup$ This would have been a better answer if you had stopped after the first paragraph, where you restricted yourself, correctly, to local measurements. In the second paragraph, you put scare quotes on "measured" to show that you understand that this sort of measurement isn't really possible. Then why discuss it? It just confuses beginners, who assume based on experience with Newtonian physics that velocities can be measured over arbitrary distances. $\endgroup$ – user4552 Jun 10 '19 at 12:59

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