When we look at visualizations of gravity “bending” the “fabric of spacetime”, it always gets depicted as a 2D sheet being bent by an object. But what happens when one tries to visualize it in 3D?
I picture a ball (heavy body in hydrostatic equilibrium) in a liquid. The liquid represents “inertial clamping” (Newton’s first law of motion), the body then rarifies the spacetime around it, resulting in, one might say, more inertia toward the outside (away from the body). This seems like a reasonable way to view gravity: an object experiences less inertia closer to the body than away from it and hence is attracted to the body (when you imagine a liquid, it seems to me that this is how things would behave when faced with gradients in density - not sure about waves).
So, one could view gravity as a sort of repulsive force, with large bodies pushing spacetime away from them. What, then, happens if spacetime is treated as a conserved medium (aether)? One could imagine that the repulsive effect of gravity, pushing spacetime away, might partially explain the very mechanism of universal expansion: the “aether” seeks equilibrium, hence it naturally expands in order to flatten out from the distortions created by large bodies.
Are there any maths that address this issue? How would “dark matter/energy” (galactic balance models) be affected by the idea conservation of spacetime?
There isn’t anything but math that addresses this issue. All of the stuff about rubber sheets is just a vague analogy-- The real work is done exclusively via the math. Any attempt at coming up with a better analogy must start from an understanding of the math, because we’ve verified by experiment that the math is correct (or at least, very, very close to it).
As for the specific maths that deal with the “2D rubber sheet” idea in the full 4 dimensions (3 spatial dimensions and 1 temporal dimension): differential geometry. The bulk of that wiki link is fairly technical, but the opening paragraph is more accessible and has links to the various fields of mathematics that underly differential geometry.
Actually Isaac Newton came up with something quite similar to this as a (speculative and qualitative) account of the cause of gravitational attraction. I believe it is in his Optiks (which is not, by any means, all about optics). The long range gravitational force was suggested to be the product of short range repulsive forces between the particles of a space filling aether and both other aether particles and the particles of regular matter. His ideas about short-range forces (repulsive and attractive) between particles probably derived from his researches in chemistry/alchemy, where he seems to have been groping towards a conception of something like the chemical bond. He was careful to say, however, that these ideas about both gravity (and chemistry) were merely hypotheses, and were not proven facts in the way he considered his laws of motion and inverse square law of gravity to be.
Of course it is Newton’s law of gravity that is speculatively explained by this, not Einstein’s, which has superseded Newton’s and makes slightly different predictions.
The rubber sheet analogy of gravity bugs the hell out of me, because in order to work, it requires gravity (not only to deform the sheet - which I could tolerate - but also to make the other ball roll down the depression). It explains something without actually explaining anything.
Nitpick it does not require gravity, just gravity is handy to use for it, but it would work the same in zero g in a system/frame that was accelerating at a constant rate. It may even work in a rotating frame.
I think you misunderstand me. Gravity is the thing being explained - the depression in the rubber sheet is caused by the mass of the object warping the sheet (OK, because of gravity). The other ball rolls into the depression because the rubber sheet is warped so as to form a downhill slope.
Why do things roll downhill? Gravity.
I would say that it explains a complicated thing we don’t have a lot of personal experience with, (gravity as a differential force based on the proximity of mass) in terms of a simple abstraction people understand on an intuitive level, (gravity as a constant force that always goes ‘down’)
Your mileage may vary.
And I think the other reason for the 2d sheet analogies is that we don’t have brains that are constructed to intuitively understand anything higher than 3-dimensional spaces naturally. Therefore, you swap out one of the standard dimensions that isn’t really necessary for the problem at hand for one that’s doing something new and interesting. A little like those ways of projectingor unfolding tesseracts into 3-d space. (Which can then be flattened into 2d images for the web )
Keep in mind that those visualizations are of a 4D object, and that they’re printed on a 2D page or projected on a 2D monitor. This is inevitably gonna be almost as helpful as making a visualization of a 3D object in a 1D place (i.e., a line). But that’s not the real answer, only an introduction. Because if I gave you an actual 3D solid rubber sheet with those hills and valleys, you’d have the exact same questions.
So here’s the real answer: You’re never going to get a fully satisfying visualization of a 4D object. Not as long as you remain a 3D human, anyways. The most you can hope for is to gain an appreciation for what each of these dimensions means. For this purpose, I strongly recommend the book Flatland, which introduces the reader to a 2D world and the beings inhabiting it. These are not cartoon characters who are able to go behind and in front of each other. They are totally restricted to their flat land, and no sort of overlapping is possible - or even imaginable. Then, once the reader has been made comfortable with these characters, they are visited by a 3D character. Mind-blowing hilarity ensues, as even the vocabulary of the 3D character is incomprehensible to the local 2D population, who simply cannot conceive of a direction other than north, south, east and west.
Oooh! I had not realized that the copyright on Flatland has run out, and that free on-line copies are now easily available. Check out the bottom section of that Wikipedia link!
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