timmdeeg said:
TL;DR Summary: How would an observer who is fixed at the upper end of a light chain see the lights above the horizon, at crossing the horizon and below the horizon?
My simple reasoning is that he can see all lights as long as he is above the horizon , during crossing the horizon an overlap of all lights and below the horizon all lights again until the lowest light vanishes because it hits the singularity before it reaches him.
How would an Eddington-Finkelstein diagram show this situation? Would the worldlines of the lights be upwards parallel in the diagram?
It is disregarded that the chain could break and that only the central part of the chain is in free fall.
You'll always see / photograph all the lights at the same time. See is ambiguious, I'm interpreting that as "photograph", rather than some sort of mental image you have by analyzing the photograph. Pretty much by definition, the photograph takes a picture at some instant, though I suppose that's idealized and you'd have to trace the optical path lenghts of the camera for path length variations under some specific defintion of "simultaneity" if you really wanted to get all the fine details exactly correct. But enough of that, I'm getting off-track.
You'd probably want your camera offset of the straight radially line to make the lights have some angular separation to get a photograph that did NOT have all the lights overlapping. A radial camera would always photograph all the lights as overlapping, without any angular separation between them, making it rather annoying to look at.
A detailed analysis would be involved esp. with non-radial lights, but for a large black hole and a short (in comparison) string of lights, you won't notice any difference in how they behave due to the black hole. . I am assuing a free-falling observer falling into a black hole. The only difference will be due to the tidal forces of the black hole, and you can ignore those for the case I am calling "short strings". If you are interested in the tidal forces, and also want to put your camera off the radial line , I suppose you could do something equivalent to ray tracing, but it'd be a lot of work.
Note that you won't be able to tell when you reach the horizon by looking at the local lights, there will be nothing that special going on, so that limiting case should guide your thinking. I think you can tell by photographing the distant stars though, though I don't recall how they looked at the hyorizon crossing.
If you are interested in an accelerating observer, the approximation without tidal forces goes from flat Minkowskii space-time to the Rindler metric. Somewhere way back when I reparameterized the Schwarzschild metric radial coordinate to illustrate the connection between the Rindler and Schwarzschild metrics. Alternatively, we can say that when the acceleration goes to zero, the approximation goes to flat space-time.
See for example
https://www.physicsforums.com/threa...ies-advanced-discussion.1007892/#post-6551199. There was a motivational analysis that led me to the specific form of the transformation equations I wrote, I might be able to dig up more if there was interest but I suspect I'm getting off-tract again.
