Can a wave exist within a black hole? Can a wave exist within a black hole?  A wave implies motion (amplitude and frequency) but motion can not occur without the passage of time.  A wave also implies wavelength that cannot occur because gravity does not allow for any outward motion.  Therefore, can a wave even exist inside a black hole?
In this forum we always speak of relative time and motion.  But is there a relative timeframe if time is completely stopped?  Can we speak of the frame of reference of being within a black hole if time does not move at all?
 A: This is an underspecified question as such but still it can be answered by deconstructing the question by pointing out conceptual misunderstandings I guess. So I'll give it a go. 


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*Wave doesn't imply motion. Only mechanical waves imply motion. Electromagnetic waves are also waves and there is no motion associated with it. It simply consists of oscillations of electric and magnetic fields. However, you're right that you need a notion of time to have a notion of wave (although, this would be true of any physically meaningful notion, a physics without time is something not much imaginable)

*Time doesn't cease to exist in a black hole (except at singularity where we don't know what happens). Time certainly exists inside a blackhole. In fact, according to general relativity, you won't even notice that you crossed the horizon if you're falling into a large enough black hole. 

*Yes, if you use the coordinates of an observer sitting at the asymptomatic infinity, it'd look like the time stops as you approach horizon and the metric blows up at the horizon. However, this is just a pathology of a bad coordinate system as has been well understood for a long time. If you choose a good set of coordinates such as the Kruskel coordinates, you see that nothing catastrophic happens at the horizon and an infalling observer wouldn't even notice that they have crossed the horizon as I mentioned. 

*Finally, electromagnetic waves can obviously exist inside the blackhole. No electromagnetic wave can come outside of the blackhole but radiation can certainly fall into blackhole. For example, see, the ingoing Vaidya metric which is a blackhole solution with radiation falling into the blackhole. 
