I have some black hole questions I’ve been struggling with for a week (well, years actually, I just thought about it more than usual during the last week or so) that I couldn’t find a satisfactory explanation for. I don’t think I’m asking about really unknown things, rather all explanations I see are either pop-sci explanations that don’t go deep enough, or detailed descriptions in terms of tensor equations that are too deep for what math I remember from university. I’m hoping that you could hit closer to the sweet spot :-)
I’ll split this into two comments to simplify threading. This first one is sort of a meta question:
I think I understand the what of the image. What I don’t quite get is the when and where of the thing.
That is, given that time and space bend in weird and wonderful ways around the black holes, and more importantly, they bends differently at different spots around them, what exactly are the X, Y and Z coordinates that are projected to the image plane (and, in the case of the video, the T coordinate that is “projected” on the duration of the video), given that the object in the image(s) is supposed to display the shape of time and space?
The closest I got trying to find answers:
(1) I saw Penrose diagrams of matter falling into a black hole, though I couldn’t find one of merging black holes. I couldn’t manage to imagine what one would look like, and I’m not quite sure it makes sense to ask for one: Since the X coordinate in a Penrose diagram is supposed to be distance from the singularity, I don’t see how you can put two of those, closing to each other, in one picture. Also, my brain knotted itself when trying to imagine more than one “spot” where space turns into time, interacting. On the other hand, that does look a bit like the coalescence simulations I’ve seen, so I might not be that far from the truth.
(2) I suppose the images might be space-like slices through the event, perhaps separated by equal time-like intervals at infinity in the case of the video. I don’t want to speculate more, in case I’m really far from the mark, so I’ll wait for an answer first.
(In case it helps with the answer: I do know what an integral is (including path, surface, and volume integrals), though I probably can’t do much with a complicated one mathematically. Similarly for derivatives, gradient, curl and divergence, though I have to think quite carefully to interpret the last two. If you say “manifold” and don’t have a good picture my eyes tend to glaze over, though. I sort of understand space curvature and frame-dragging, when they’re not too “sharp”, qualitatively if not quantitatively. I can visualize either of them—again, as long as they’re not “sharp” enough to completely reverse space and time dimensions; i.e., I have an approximate idea of what happens when you’re close to an event horizon, but not what goes on as you “cross” one. (Actually, I’m not sure I understand what “crossing an EH” means, again it’s the “when” and “where” the seem to be the trouble rather than the “what”; most simple explanations tend to indicate that there’s not much of a “what”, as in “nothing much happens as you cross one that doesn’t happen just before or just after”.) I can’t quite visualize a general tensor field, but when you split the Riemann tensor into tidal and frame-dragging components I can interpret the tendex and vortex lines on a well-drawn diagram if I think carefully.)
I saw Penrose diagrams of matter falling into a black hole, though I couldn’t find one of merging black holes.
I’ll try to draw one and post it, might take some time, given that you need more dimensions than just 1 space + 1 time on the original Penrose diagram, because you lose spherical symmetry. The head-on collision process still retains cylindrical symmetry, so a 2+1 picture should do it, represented by a 3D Penrose diagram, which is going to take some work.
I can’t believe nobody needed to do that already. Even if people who can draw one don’t need it because they do just fine with the equations, I’d have expected someone to make one just for fun...
Hi shminux, thanks for your offer!
I have some black hole questions I’ve been struggling with for a week (well, years actually, I just thought about it more than usual during the last week or so) that I couldn’t find a satisfactory explanation for. I don’t think I’m asking about really unknown things, rather all explanations I see are either pop-sci explanations that don’t go deep enough, or detailed descriptions in terms of tensor equations that are too deep for what math I remember from university. I’m hoping that you could hit closer to the sweet spot :-)
I’ll split this into two comments to simplify threading. This first one is sort of a meta question:
Take for instance FIG. 1 from http://arxiv.org/pdf/1012.4869v2.pdf or the video at http://www.sciencemag.org/content/suppl/2012/08/02/337.6094.536.DC1/1225474-s1.avi
I think I understand the what of the image. What I don’t quite get is the when and where of the thing.
That is, given that time and space bend in weird and wonderful ways around the black holes, and more importantly, they bends differently at different spots around them, what exactly are the X, Y and Z coordinates that are projected to the image plane (and, in the case of the video, the T coordinate that is “projected” on the duration of the video), given that the object in the image(s) is supposed to display the shape of time and space?
The closest I got trying to find answers:
(1) I saw Penrose diagrams of matter falling into a black hole, though I couldn’t find one of merging black holes. I couldn’t manage to imagine what one would look like, and I’m not quite sure it makes sense to ask for one: Since the X coordinate in a Penrose diagram is supposed to be distance from the singularity, I don’t see how you can put two of those, closing to each other, in one picture. Also, my brain knotted itself when trying to imagine more than one “spot” where space turns into time, interacting. On the other hand, that does look a bit like the coalescence simulations I’ve seen, so I might not be that far from the truth.
(2) I suppose the images might be space-like slices through the event, perhaps separated by equal time-like intervals at infinity in the case of the video. I don’t want to speculate more, in case I’m really far from the mark, so I’ll wait for an answer first.
(In case it helps with the answer: I do know what an integral is (including path, surface, and volume integrals), though I probably can’t do much with a complicated one mathematically. Similarly for derivatives, gradient, curl and divergence, though I have to think quite carefully to interpret the last two. If you say “manifold” and don’t have a good picture my eyes tend to glaze over, though. I sort of understand space curvature and frame-dragging, when they’re not too “sharp”, qualitatively if not quantitatively. I can visualize either of them—again, as long as they’re not “sharp” enough to completely reverse space and time dimensions; i.e., I have an approximate idea of what happens when you’re close to an event horizon, but not what goes on as you “cross” one. (Actually, I’m not sure I understand what “crossing an EH” means, again it’s the “when” and “where” the seem to be the trouble rather than the “what”; most simple explanations tend to indicate that there’s not much of a “what”, as in “nothing much happens as you cross one that doesn’t happen just before or just after”.) I can’t quite visualize a general tensor field, but when you split the Riemann tensor into tidal and frame-dragging components I can interpret the tendex and vortex lines on a well-drawn diagram if I think carefully.)
I’ll try to draw one and post it, might take some time, given that you need more dimensions than just 1 space + 1 time on the original Penrose diagram, because you lose spherical symmetry. The head-on collision process still retains cylindrical symmetry, so a 2+1 picture should do it, represented by a 3D Penrose diagram, which is going to take some work.
Oh, thank you very much for the effort!
I can’t believe nobody needed to do that already. Even if people who can draw one don’t need it because they do just fine with the equations, I’d have expected someone to make one just for fun...