3

If you look at a 2D representation of curved space-time around a black hole, it stretches out into an infinite tube, asymptotically approaching the Schwarzschild radius.

image from sciencenews.org image from sciencenews.org
The usual way to explain why light bends in the vicinity of a large object is to say that it actually goes straight along a geodesic if you consider space to be bent into a higher dimension (2D space bent into 3D in the picture, though its evidently mathematically equivalent to say the bending is into the dimension of time).

image from astronomynotes.com image from sciencenews.org
When you get closer to a black hole than the point at which light has a circular orbit (traveling a straight line according to this model), its orbit will decay. Any object with mass will actually fall in faster if you increase its orbital speed, as if centrifugal force is reversed.

The problem is that there doesn't actually seem to be a position in the diagram which has this property unless you make the end of the tube balloon outwards. I'd like to know what concepts have to be added to the model in order to make it accurately describe this case.

Dustin Soodak
  • 317
  • 4
    These rubber-sheet images are just crude visualizations to give beginners a little intuition. They’re not really models. If you want a visual aid that is actually correct and useful, learn about the “Schwarzschild effective potential”. – G. Smith Dec 20 '18 at 06:44
  • 2
    Your physical intuition is good! Yes, something is wrong with the rubber sheets: they are simply not valid in GR. Compare the classical and GR Schw. terms in the effective potential of a massive test particle (negligible but non-zero mass) on a momentarily circular orbit. The latter has a GR-specific attracting term related to orbital momentum as $-L^2/r^3$, or $-v^2/r$; the minus means that the "fast" particle will "steer" more on the sheet downwards, ever more parallel to radial gridlines (better think polar than cartesian grid). But graphing the potential is more revealing. – kkm -still wary of SE promises Dec 20 '18 at 12:50
  • @kkm Yes, something is wrong with the rubber sheets Something? Those are absolute mystification, a real shame. An "elastic" sheet instead of spacetime. A ball whose weight (coming from where?) should warp this pseudo-spacetime. I always wonder how could a minimally competent person have conceived such an idea, and how may it go on spreading. – Elio Fabri Dec 20 '18 at 15:02
  • OP asks I'd like to know what concepts have to be added to the model in order to make it accurately describe this case. My answer is: the only thing you can do is to throw anything away. – Elio Fabri Dec 20 '18 at 15:02
  • @ElioFabri, “you are right, too!” One way to understand what is wrong with a model is to attempt at fixing it (and fail at it, feeling how it resists fixing). Another is throw it away and go back to clear whiteboard--and often a blank confused mind. I've seen people arriving at a good answer and understanding in all twisted ways. It just seemed to me that the OP is already well underway in the first approach, and with a good intuitions, so why not? :) – kkm -still wary of SE promises Dec 20 '18 at 22:05
  • @ElioFabri And this is not to say that I do not despise the rubber sheets; I do. But I understand why I do, but just saying "this model is not even wrong" without any qualification is not an instructive advice, IMO, when it comes without explaining why. I dunno, maybe it's my thing--I like to learn by hitting my fingers with a hammer. – kkm -still wary of SE promises Dec 20 '18 at 22:09
  • @kkm just saying "this model is not even wrong" without any qualification is not an instructive advice. Sure. And actually I didn't limit to that. Within what is possible in a comment, I said what is wrong with that pseudo-model. An elastic sheet has nothing to do with a curved spacetime. – Elio Fabri Dec 21 '18 at 10:20
  • @kkm True, a ball rolling on it will describe a sort of "orbit", but for entirely different reasons. Warping of the sheet is due to a weighty ball at its centre, but what's that ball? Where does it live? In a third dimension. And it's weighty because of something attracting it downwards (what is this "downwards"?) – Elio Fabri Dec 21 '18 at 10:21
  • @kkm I'm unable to see how you could try to fix that "model", which isn't a model at all. It completely betrays the very ideas of GR: curved spacetime, geodesics, and so on. There is no way to amend, ameliorate that "model" so to allow approaching the real theory. It's only a stupid fable of no use, in any possibile sense. – Elio Fabri Dec 21 '18 at 10:22
  • @ElioFabri, I did not "try to fix" the rubber sheets. My point was only that the OP is on the right path: the Q states that he does not see how that model could work. IMO, it's natural for a physicist to try to fix a model before abandoning it. So I just noted he's on the right path (to failure at some point, indeed, but a very pedagogical one). It's easy to say "just think in geodesics on manifolds" to someone who thinks rubber sheets, but it's kinda hard to make this statement convincing. I've seen students doing math w/o a trace of physical imagination. To me, it's worse than rubber sheets. – kkm -still wary of SE promises Dec 23 '18 at 01:54
  • Related (but not a duplicate, as it's general, not about the extreme gravity): https://physics.stackexchange.com/q/90592/115253 – kkm -still wary of SE promises Dec 23 '18 at 02:01
  • Despite a common misunderstanding, "rubber sheet" diagrams are perfectly valid. They represent a radial distribution of the gravitational potential (or gravitational time dilation, if you prefer). You can take the proper formulas for these values based on the Schwarzschild metric and plot these diagrams to be mathematically correct without any simplifications. The misunderstanding comes from a wrong assumption that the diagrams represent curved spacetime. Obviously they don't. They can only represent a scalar value, not a tensor. – safesphere Jan 15 '19 at 08:56

0 Answers0