A common example to describe how mass warps space is to imagine a taut rubber sheet onto which a ball is placed. The weight of the ball deforms the sheet to produce an indentation. That indentation illustrates the gravity well of mass in space.
However, that’s just a 2-D example. The 3-D ball is pushing the 2-D sheet into the 3rd dimension. But how does it work in reality? A planet deforms 3-D space to produce a gravity well. Is the deformed space just a compression of space? Or does it actually push space into the 4th dimension? Or something else completely?
One can model it as bulging into other dimensions, but one doesn’t need to. Curvature of spacetime, as discussed in the Theories of Relativity, is based entirely on measurements within the spacetime. You can determine the presence of curvature by doing things like measuring the angles of triangles (if space isn’t flat, then the angles of a triangle won’t add up to 180 degrees). You could interpret this in terms of space bulging out into other dimensions, but you could just as well interpret it in terms of space being stretched or squished, or all our measuring devices within space being distorted in particular ways. It doesn’t matter; you can’t tell the difference between the interpretations.
My understanding is that this is how a two-dimensional being might realize that his space is curved. But doesn’t this non-euclidean triangle bulge into the third dimension? There’s no way a flat triangle would add up to something other than 180, and if it is not flat then by definition it is bulging into the third dimension, no?
And by analogy, if we 3D beings would make such measurements, wouldn’t that be proof that we’ve bulged into the fourth dimension?
Jeez, reading this really gave me start. I thought in GR it was indisputable that there was most definitely not a fourth spatial dimension, and that the curvature was strictly intrinsic. This, I take it, is now gershtunken? Or are you talking about loop or strings or something?
You can always just postulate some embedding, if you like, to turn what would otherwise be regarded as intrinsic curvature into extrinsic curvature. But this is purely formal.
Consider the game Asteroids. (Now, the universe of Asteroids doesn’t have curvature in the particular sense being discussed here, but it does have some wrap-around properties…) Because of those wrap-around properties, you might want to take the perspective that the universe displayed in the game is just the surface of a particular torus in a much larger, unshown 3d universe. You can do that, if you like, or you can just take what you see as what you get, and say there’s not some hidden embedding. Either perspective works; it’s all up to you. [Though, certainly, an ontology-minimizing, Occam’s Razor-style approach would go with the latter perspective]
I.e., you can always postulate a hidden embedding, but if all you can ever interact with are things on the same embedded surface, then what’s the difference between this and there not being anything else anyway? So, speak in terms of an embedding if it makes the math more manifest to you, and don’t if it obfuscates things [and I think you’ll find, in most cases, that presentations requiring dragging around a hidden embedding do just obfuscate things], but recognize that the actual situation is the same either way.
Can someone explain in layman’s terms how the curvature could be intrinsic in 3-D. Is there a corresponding way to visualize the 2-D sheet warping without bending into the 3rd dimension?
One visualization is from Einstein himself. Suppose we’re measuring a tabletop, and our measurement apparatus is a bunch of standardized metal rods. I could do something like arrange my rods into an equilateral triangular lattice, and cover the entire tabletop (if it’s flat) with that lattice, with everything matching up just right. If my table had a big bulge in the middle, I wouldn’t be able to cover it with such a regular lattice, so this method can indeed detect the bulgy kind of curvature. But now, suppose that instead of the table bulging, suppose that it’s heated. Different parts of the table are at different temperatures, and cause the standard rods placed on the table to expand or contract by different amounts. In this case, also, I will be unable to cover the table completely with my lattice. Ought I to say that this table is curved? Yes, if I’m using “unable to cover the table with a Euclidean lattice” as my definition of curvature. Yet it’s not bulging out in the third dimension.
Though intuitively this is a bit strange, in principle there’s little difference from uncurved space in that regard. You can visualize both a curved and uncurved 2D surface as embedded in 3D, or both can be expressed purely in terms of two dimensions. The embeddedness is strictly a visual aid and not mathematically necessary in either case.
In this image you can see a 2D grid deformed. The angles are not all straight. This without needing a 3rd dimension.
Now imagine it not being a grid on paper but a 3D scaffolding. You can put all kinds of deformations on it without needing a 4th dimension.
A large mass, or so the story goes, would compress some of that scaffolding so that the corners are all closer to each other than in the rest of it. This, of course, would also mean that there is a region around this compressed part where the scaffolding would be somewhat stretched to make up for the lost distance on the compressed part.
All this just as in the image of the ball stretching the rubber membrane.
This may be a silly question, but what compels matter to move along those gridlines?
That is, matter curves space around it, but as another mass is passing by, what forces it to follow that curvature and not go straight? And by “straight”, I mean, as you are looking at the linked image above, draw a straight line using a straight egde, such that it connects two points and crosses over the curved lines instead of following them.
Since, as Chronos says “you can, if you choose, visualize space as being compressed/stretched by mass”, which means that you can choose not to and still get similar results, another way to pose the above question is: As one mass passes by another mass, what does it “care” that the other mass is there, and in fact how does it even “know” that the other mass is there, so that it ends up changing its direction of travel due to that other mass?
I am really out of my depth here, but I think that those curved lines are “straight”. That is, if you took a rope from point A to point D and stretch it, it would go through B and C along the curve. The curve is such for our vantage point outside space but someone in there sees it straight.
Once you start talking about actual motion of objects through the space, you need to include time (which is also curved) into the mix, as well. But yes, to the extent that one can talk about “straight lines” through spacetime at all, the paths followed by objects not acted on by nongravitational forces are in fact straight lines.