I’m just asking from memory so my figures are probably way off, but in the movie Interstellar, the crew of 3 goes to the surface the water planet, which is heavily affected by the gravity of a black hole.
They leave a crew member behind in the ship and go about their business for several hours of surface time. In the meantime, 23 years passes for the crew member who waited behind.
I can wrap my head around the physics of that as I’m familiar enough with Einstein’s theories on relativity, but a thought crossed my mind. What if the “away team” had worn cameras and broadcast a live feed of their surface mission to the ship that was farther outside of the influence of the black hole? How would the signal arrive at that ship? Would trickle in in slow motion or would it come in at normal speed as the 23 years ended for the Doctor on that ship? Basically, I was curious as to how the relative difference between the planet’s surface and the main ship would affect radio or video transmissions?
A relativistic correction to the frequency of the signal is the physical manifestation of time dilation, though it must be stressed time dilation effects are not the only cause of shifts in frequency.
Firstly the receiver would need to take into account they are picking up a lower frequency than emitted. More importantly to your question though, however you choose to modulate the signal to transmit the information, the lower reception frequency would result in the live feed appearing in slow motion.
If the faraway observer was approximately static relative to the BH, but the surface mission was on a planet orbiting the black hole, there would be additional complications (and it is possible at some points the signal may even be received at a higher frequency than the emission frequency and thus the live feed appear in fast motion), but overall any signal would have a net slowed-down effect if taken over a long enough time period.
There was actually a transmission sent from the planet (not by the ones who landed, apparently). They make a point of saying that Miller had been sending the same ‘planet good’ signal for the last decade, but when they land, they realize it was only because from her point of view, she’d just landed. If the signal was shifted, they’d have known immediately about the time dilation.
I consider it one of the bigger sacrifices of science for plot convenience, but maybe Kip Thorne can wave his hands and say they thought it was the black hole’s fault.