A question about curved space

Factual Questions
1

I’ve seen the image (or variations of it) a thousand times - showing how the gravity of a star or a planet or whatever ‘curves’ space: http://meta-religion…s/dead_star.jpg

It always bugged me. I can’t ‘put my finger on’ why but spaced is not 2 dimensional. Gravity would bend space in on something from all around it. It would look nothing like that stupid picture. In fact, since I don’t see how space can ‘stretch’ (you can’t create more space and a curve takes up more material than a straight line) the whole idea seems off to me.

I am sure tons of math can be thrown at me to say I’m wrong, but something sure seem to not fit about it.

What is space anyway? It’s nothingness, right? How do you bend nothingness? Or is this where dark matter and other such stuff comes into play?

Anyway, color me confused.

2

You are right, but it’s just a way to visualize things. Human brains don’t think in terms of a four dimensional bent spacetime; so we need to either use math to model it, or flawed analogies like a bent two dimensional sheet.

Actually more space is created all the time; the universe is expanding after all.

Spacetime isn’t nothingness; it’s spacetime. What exactly spacetime is though, as far as I know no one really knows.

3

Basically (as I understand it), space is not Euclidean as our simple intuition tells us. Mass warps the space around it. Even though much of that space might be empty just at the moment, its warped nature (its “curvature”) will still affect whatever moves through it later.

For example, you’d expect that if you travel in a straight line for 1 light year, turn right exactly 90 degrees, and repeat those steps three more times, you’d be tracing out a perfect square. You should end up exactly where you started, and facing the same direction. In a perfectly Euclidean space, you would. In the actual universe we inhabit, you don’t.

4

This appears to be your problem. Space is not nothingness. It’s space. We may not be able to visualize it, but it’s a real thing and can be mathematically represented at every point as the stress-energy tensor. And that’s the key. If you can mathematically represent something, then you can mathematically represent when it changes responding to a force. That’s what happens when gravity “bends” space. The better way of thinking about it is that the mathematical representation of space at that point changes. Since gravity drops off at the square of distance, the changes involved can be tracked. Put all the changes together and you have a mathematical picture of how gravity “warps” space. You don’t need to understand deeply what space “is” for a good approximation of reality.

Dark matter doesn’t enter into this, except as a contribution to local gravity that affects the magnitude of the change. Dark energy isn’t space either, but does have a more direct effect as Chronos explained in this recent post. Note that gravity is only one of many components. And that’s why you can’t visualize what space is. It’s not a material in the everyday sense and isn’t subject to “common sense” understanding.

5

I can’t believe that no-one referenced the recent perfect XKCD.

6

Just a nitpick, but the stress-energy tensor isn’t space, it’s something that keeps track of, roughly, the contents of space – matter, energy, momentum, those kinds of things. Space (or spacetime, rather) is represented by the metric tensor, which is the dynamical variable of Einstein’s field equation – gravity itself, so to speak --, which roughly keeps track of the geometry (stress-energy tensor represents the source, the metric represents the field). Euclidean geometry was a good clue by Bytegeist – originally, it was thought of as ‘the’ geometry, but people kept wondering about the parallel postulate, and eventually, came round to the fact that it can be replaced, yielding a framework just as consistent as Euclidean geometry – elliptic and hyperbolic geometry, respectively. The rescinding of the parallel postulate means, basically, that parallel lines that started out a fixed distance from each other, do not stay that way, but may converge or diverge. Particles moving along those lines thus may approach one another, or increase their distance – they’d experience a force, in contrast to the Euclidean case, where they’d stay a constant distance. These geometries are curved.

The yet more general framework appropriate to general relativity is called Riemannian geometry – basically, a Riemannian geometry is anything that looks like Euclidean geometry on a sufficiently small scale (just like the surface of the Earth looks flat if you zoom in enough), but may diverge wildly on larger scales – the way how it diverges is encoded in the metric, and the reason for this divergence is given by the stress-energy tensor, i.e. roughly matter; and like in the elliptical and hyperbolic cases, in general, parallel moving objects won’t stay at a fixed distance, but rather, will experience forces.

This is a very superficial account of course, and there’s various technical things I skipped over, but I hope it gets the general idea across – matter dictates the choice of geometry to describe space, and the geometry describes how things move through that space; and in anything but the Euclidean (flat) case, (i.e. the ‘curved’ cases) things will typically move as if influenced by a force, which we call gravity. The whole rubber sheet thing is really just a (necessarily incomplete) analogy.

7

I think the best way to ‘imagine’ curved 3-space and spacetime is terms of geodesics. Geodesics are paths that minimize the distance between two points*.

In flat space straightlines are geodesics and has already been noted, the parallel postulate applies to straight lines. In general curved space the parallel postulate does not apply.

Making the connection between geodesics and curvature is doubly important in general relativity as the idea that free-falling objects have geodesic worldlines is one of the most fundamental principles and it’s essential to understand this to have any understanding of the general theory of relativity.

*infact that’s not strictly true, they are locally minimizing which means they minimize the distance in some ‘chunk’ (neighbourhood, to be technical) of sapce/spacetime.

8

Hm. Interesting. I never thought about it that way. I always think of it as the stuff in space moving away from all the other stuff, all emanating from a ‘center’ [the big bang event]; that space itself is not expanding, but all the stuff in space is moving out closer to the edge to occupy heretofore empty space.

Side track, I know. Sorry.

9

This is a common misconception about the Big Bang and expanding space which is discussed on this board in many threads (and lots of other places too).The “stuff” isn’t simply spreading out; space itself is expanding. Everything is moving away from everything else; there is no center.

10

I always thought of space in terms of distance (the distance between here and there [which is apparently not a straight line] is ___).

11

Bitch, I am the center.

fluffs hair

12

Half Man Half Wit already caught the nitpick about the stress tensor, but now I have to catch one in his post: Riemannian geometry isn’t like Euclidean geometry on a small scale, but rather Minkowski geometry. In Euclidean geometry, time, if it’s addressed at all, would be treated in the same way as space, but in Minkowski geometry, it’s subtly different, in a way which accounts for all of the effects of Special Relativity.

Oh, and the post of mine which Exapno linked was addressing dark energy, not dark matter.

13

Actully Half Man Half Wit is correct. ‘Riemannian’ describes smooth manifolds where the tangent spaces are equipped with Euclidean inner products and ‘Lorentzian’ (a subcatergory of ‘pseudo-Riemannian’) describes smooth manifolds where the tangent spaces are equipped with Minkowski inner products. That’s just a nitpick though as it’s easy to forget :slight_smile:

14

Sorry. Stuck that in at the last minute because I wanted to set up Chronos, but I should have checked.

Describing space with a geometry seems to leave the basic question of what space is unresolved. The differences between geometries aren’t familiar and no matter which geometry actually describes the behavior of space it fails to explain what space is so as to be curved that way in the first place.

My representing space as fundamentally a series of mathematical points may not be going over with others, but it helps me. Representing a object through a grid of coordinates is as familiar an analogy as the sagging mattress. Topographical maps, isobar charts, latitude and longitude: we do it all the time. We don’t need to have a physical or a metaphysical understanding of what’s being represented. And changes become obvious. Without jargon, whcih is also a big plus.

Yes, and that’s what I said.

15

Okay, so based on some of the things I’ve read here, and some net research, what I gather is that I have stumbled in my question upon non-Euclidean geometry. These various forms of geometry all seem to hinge around the definition of the world parallel. There also seems to be considerable debate as to which, in any, are the ‘correct’ geometry for representing space-time. At any rate, from what I can see it looks like one big-ass mess that we are still trying to figure out :stuck_out_tongue:

This thread has given me a place to start exploring my thoughts anyway - and plenty of things to read. In retrospect I probably should have started questioning the nature of the universe before I turned 45 - but we don’t always get to choose our battles.

16

Not just that. There’s also the matter of how distances are defined — what sort of metric space you have.

(I probably should step aside and let the real physicists respond to this. But hey, I paid my $15.)

I believe General Relativity is pretty well settled on Riemannian geometry, and that this is by far our best model of space-time — at least at macroscopic scales, outside the realm of Quantum Mechanics, as well as outside the centers of black holes, where we don’t really know what’s going on yet.

The other geometries mentioned in this thread aren’t competitors to that; they’re special cases. Minkowski space for example comes up when you’re discussing Special Relativity — a restricted subset of GR that ignores gravity and accelerated motion. And Euclidean geometry comes up because (1) that’s the simple geometry our brains understand intuitively, and (2) whatever geometry we use to model the whole universe needs to appear Euclidean at the local level (because that’s what’s observed).

Hope I didn’t mangle that too badly.

17

What you have stumbled on is pseudo-Riemannian geometry (technically Riemannian geometry is a subcatergory of pseudo-Riemannian geometry on the basis that there’s no real reason to exclude Riemannian manifolds from the catergory of pseudo-Riemannian manifolds). Pseudo-Riemannian geometry has the power to describe Euclidan geometry and much, much more besides (the Euclidean plane is only a single example of the infinite number of pseudo-Riemannian manifolds).

On some pseudo-Riemannian manifolds it makes sense to define a notion of parallelism for curves in genera and on others it doesn’t. Thoguh that said on all pseudo-Riemannian manifolds there is a notion of parallel transport which is key to understanding curvature as it’s one of the ‘simplest’ ways to see the effect of curvature.

Spacetime is a matheamtical concept, I woudln’t worry too much about what it is and whether it’s real. It’s just a way of describing the relationships between ‘things’. Pretty much all concepts of spacetime (except those that are too simple to have the requiste structure) that pop up in phsyics are some sort of pseudo-Riemannian manifold. Thoguh more specifically 99% of the time when people are talking about spacetime they are talking about the general theory of relativity or it’s special case, the special theory of relativity in which case we’re in the specific area of Lorentzian manifolds (or for special relatvity, even more specifically, the Minkowski manifold).

18

Yep that’s pretty much correct (not that I’m claiming to be a physicist), thoguh as I mentioned earlier (thoguh re-reading my post it probably wasn’t clear at all), genera relativity uses pseudo-Riemannian (but not Riemannian) manifolds to describe spacetime and more specifically Lorentzian manifolds.

19

Euclid’s geometry was based on a handful of postulates (five or ten, in Euclid’s presentation of it, but a modern mathematician would give you a different count), and anything that violates any of those postulates is non-Euclidean. The space we live in may, actually, be flat (that is, satisfying the parallel postulate) on sufficiently large scales, and it’s certainly at the least very close to flat. It is not, however, uniform, since the distribution of mass in it is not uniform. By way of analogy, consider a big sheet of egg-crate foam: On a large scale, it’s flat, but on a small scale, it’s got a bunch of bumps and wiggles.

20

Oh, you’re not a physicist? For some reason, I just assumed from your posting history that you were. What, if you don’t mind me asking, do you actually do?