- #36
timmdeeg
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Ah, that clarifies the issue, thanks.PAllen said:Just think light cones.
Ah, that clarifies the issue, thanks.PAllen said:Just think light cones.
timmdeeg said:Could you please elaborate a bit on the "other direction". By means of an Eddington-Finkelstein diagram I understand that the stern receives light from the bow but I fail to see the other way round. From the diagram it seems that the bow is in the past of the stern.
Wiki said:Kruskal–Szekeres coordinates have a number of useful features which make them helpful for building intuitions about the Schwarzschild spacetime. Chief among these is the fact that all radial light-like geodesics (the world lines of light rays moving in a radial direction) look like straight lines at a 45-degree angle when drawn in a Kruskal–Szekeres diagram (this can be derived from the metric equation given above. ... All timelike world lines of slower-than-light objects will at every point have a slope closer to the vertical time axis (the T coordinate) than 45 degrees. So, a light cone drawn in a Kruskal–Szekeres diagram will look just the same as a light cone in a Minkowski diagram in special relativity.
Your wish is my command:pervect said:Unfortunately, I didn't find any good diagrams of an infalling object in KS coordinates to illustrate this graphically.
pervect said:the ingoing Eddington-Finkelstein diagram is different than the outgoing one.
Ibix said:Your wish is my command:
I took the geodesic equations in Schwarzschild coordinates and substituted ##u## and ##v## for ##r## and ##t##. This works even at the horizon because the geodesic equation remains valid at and below the horizon - it's just its expression in Schwarzschild coordinates that goes wrong. I fed the result to one of scipy's numerical integrators and plotted results in pyplot. Confession: the light pulses were added by hand. My code can handle null geodesics, but I couldn't be bothered to do all the initialisation and intercept calculations in order to generate 45° straight lines.pervect said:Very nice! How did you calculalte and make the diagram?
Indeed, this is very convincing.pervect said:It shouldn't be too hard to see this that for a short enough space-ship, light from the stern can thus reach the bow, as well as the reverse, because locally the light is always traveling "faster" than any material object (the slope of the infalling object is more vertical).
timmdeeg said:In all the diagrams light goes up as time increases, so light emitted by B can't reach A. I'm not sure how to resolve this.
Heureka, I can see it now. https://www.researchgate.net/figure/Space-time-diagram-in-Eddington-Finkelstein-coordinates-showing-the-light-cones-close-to_fig4_260835665 is Fig. 93 in Geroch's book I've mentioned. It shows C freely falling and outgoing light, ingoing light according to the light cones. If I paint the worldline of B who jumped first together with the past light cone very close to that of C then it turns out that indeed B can receive light of C. It's a bit tricky though and as mentioned by others it's far better to be seen in a KS diagram.PeterDonis said:If B falls in long enough after A, then light signals he emits inward won't reach A before A hits the singularity. To have A see any signals from B, you have to make sure the time between them falling in isn't too large.
There is something wrong with that diagram. It purports to show a timelike world line that does not remain inside its own future light cone. That is impossible. Your confusion is caused by a quantitatively inaccurate diagram. Consider especially the future light cone emanating from W, and the proposed world line beyond W. This cannot be right. It would make the free fall spacelike.timmdeeg said:Heureka, I can see it now. https://www.researchgate.net/figure/Space-time-diagram-in-Eddington-Finkelstein-coordinates-showing-the-light-cones-close-to_fig4_260835665 is Fig. 93 in Geroch's book I've mentioned. It shows C freely falling and outgoing light, ingoing light according to the light cones. If I paint the worldline of B who jumped first together with the past light cone very close to that of C then it turns out that indeed B can receive light of C. It's a bit tricky though and as mentioned by others it's far better to be seen in a KS diagram.
Yes. The diagram which I mentioned in #43 doesn't show light cones. I was drawing the two parallel timelike geodesics close to each other and inside the respective light cones, with the ingoing null geodesics having an angle of 45° with the space axis.PAllen said:Consider especially the future light cone emanating from W, and the proposed world line beyond W. This cannot be right. It would make the free fall spacelike.
It only looks like this. But in this diagram the light cones tilt as you go along the green/red world line. The closer to the singularity you are the more horizontal (open to the right) they are.PAllen said:There is something wrong with that diagram. It purports to show a timelike world line that does not remain inside its own future light cone. That is impossible. Your confusion is caused by a quantitatively inaccurate diagram. Consider especially the future light cone emanating from W, and the proposed world line beyond W. This cannot be right. It would make the free fall spacelike.
I know, but don’t think that affects my critique. The black dashed lines are paths of radially outgoing light, thus they form the left boundary of light cones. Along one of these dashes lines, the width of a light cone is shown constant. Following these rules of the diagram, the light cone at event W does not enclose the purported free fall path.martinbn said:It only looks like this. But in this diagram the light cones tilt as you go along the green/red world line. The closer to the singularity you are the more horizontal (open to the right) they are.
I see, but I thought the diagram is only schematic.PAllen said:I know, but don’t think that affects my critique. The black dashed lines are paths of radially outgoing light, thus they form the left boundary of light cones. Along one of these dashes lines, the width of a light cone is shown constant. Following these rules of the diagram, the light cone at event W does not enclose the purported free fall path.
[edit: I should note that the issue is that this diagram is not remotely an accurate representation of Eddinton-Finkelstein coordinates. ]
Altogether the 17 diagrams in this Book are very instructive. The first one shows just light cones with decreasing r getting increasingly tilted while their width decreases. There are various scenarios with one and two observers, with a rope going in, etc. The only scenario missing is the one discussed here.martinbn said:I see, but I thought the diagram is only schematic.
At least in this case, had the diagram been a little more precise with angles, and perhaps shown light cones along the infaller, it would have been less misleading.martinbn said:I see, but I thought the diagram is only schematic.