… does that mean our universe is basically on the surface of a four-dimensional sphere? I’ve was thinking about what it would mean for a two-dimensional space to be curved, and the most intuitive result would be the surface of a sphere. If you travel a certain distance in any direction on the surface of our planet, you’ll end up back where you started. So I think the most intuitive way for a three-dimensional space to be curved would be around the four-dimensional version of a sphere. Of course there’s not any dirt inside the sphere or anything. So I was wondering if this is how scientists who theorize curved space believe that it exists, or if our space would even show the same properties as being the skin of a 4D bubble.
It means that our universe is the interior of some 4-dimensional surface, but not necessarily a sphere. If the curvature of space is negative, as the current theory suggests, we’re living on the 4-dimensional analog of a hyperbolic paraboloid, IIRC.
I’d always heard that just because our space is (maybe) curved, that doesn’t necessarily mean that there’s anything “outside” our universe. The curvature can be -I’m not really sure of the correct way to put it- “internal” to our space.
For starters, it wouldn’t be a four-dimensional sphere, it’d be a three-dimensional one (at least, if we’re talking about just the spatial dimensions). Contrary to what you might think, the surface of the Earth isn’t three-dimensional, it’s two-dimensional (by which we mean that you can specify a point on the surface of the Earth using only two numbers).
And for the record, current observations seem to suggest that, though the Universe is curved on small scales, it’s flat on large scales. A two-dimensional analogy might be something like egg-crate foam, which has all sorts of bumps and valleys, but a sheet of it still lies flat on the floor.
The number of dimensions in our universe is disputed, but can’t be less than four. Some string theories say 10; some say 11 dimensions. But there’s no “if” about space being curved. It has been more than sufficiently proved that large masses (especially sun-sized and larger) distort the shape of space around them.
Space is flat. Einstein was curved.
[sub]Seen on a button at a Science Fiction convention. Also: ‘186,000 miles per second. Not just a good idea. It’s The Law.’ [/sub]
Interesting. I’d never heard about that. That would mean space isn’t wrapped, which is kind of disappointing.
Yeah, I know the surface of Earth is two-dimensional, but it’s wrapped around a three-dimensional space. If it was its own universe, there would still be a volume within it even if its inhabitants couldn’t access it. So I was asking if our universe is believed to wrap around a four-dimensional volume or if it curves in some other way like Lumpy was talking about.
Do you know what observations back this up? I’m curious. That seems to be what M-Theory theorizes, but as far as I know, there’s no proof whatsoever for it.
Matter does distort the space around it, no doubt, but does it curve it through another dimension? You can stretch a sheet of rubber without it curving.
Another thing about that. If space was curved like that, there would be a center to it, right?
Not wrapped in those directions, but if you believe the string theorists the other directions are wrapped up into a shape sorta like this. Actually, that’s a 3-d shadow of the real 6-d shape.
Indeed, because if space is wrapped (we say, “not simply connected”) then all sorts of neat topological effects start having influences on particle physics and make the whole subject a lot more mathematically interesting.
Basically if you take a universe filled with a pressureless “dust” (which is pretty accurate on an intergalactic scale) and assume it looks the same in any direction and at any point (again, no reason to assume there’s any “special” points or directions) then the amount of mass-energy inside it tells you whether it curves back in on itself (positive curvature), is flat, or spreads out like a hyperboloid (negative curvature). Looking at how fast various objects are moving away from us tells us how far away they are (Hubble) and combining that with estimates of how much “stuff” is out there, we get the idea that space is more or less flat, with bumps near big clumps of matter.
As for M-theory… let’s not even bring that up. What we’re talking about here is plain old general relativity and cosmology. Throwing a word like “M-theory” in to show you’ve heard of it won’t help at all.
No, there’s no “other dimension” outside of space for it to curve into. You’re (entirely understandably) confusing intrinsic and extrinsic curvature.
Consider a torus – the surface of a donut. This is a 2-dimensional surface. To you looking at it embedded the usual way in 3-d space it looks curved. This is extrinsic curvature, which depends on how the surface is situated in space. However, if you just made measurements on the surface of the donut itself you would find your space to be flat! This is intrinsic curvature, the curvature which comes from the shape of the surface itself and not from how it’s sitting in some larger ambient space. The torus shows us that a space can be intrinsically flat, but extrinsically curved. Most curvatures we talk about in relativity are intrinsic, since we can’t get outside of the 4-dimensional space to consider its extrinsic curvature. We don’t even have any evidence that there is such an exterior space.
In short: no.
Awesome answer. Yes, that’s what I was asking, if the curvature was believed to be extrensic or intrensic. I didn’t know the words. I assumed it was intrinsic for a long time, but then I thought it could make sense if it was extrensic too and that would make things pretty interesting. Intrensic curvature is pretty interesting itself, of course. It’s a good time for science though, being on the point of figuring a lot of basic properties of the universe.
If I understand it correctly such a center wouldn’t be in the universe though. The center of the surface of a globe is not on the surface either.
I would have said the surface of the Earth is the two dimensional surface of a three dimensional sphere.
And the analogue would be to say that the universe might be the three dimensional surface of a four dimensional sphere.
This is how its talked about in the books I’ve read, as well.
-FrL-
Chronos’s and Mathochist’s points were basically that there is no reason to introduce the presence of the 3D sphere. The 2D surface is a valid space in its own right. A sphere only enters because you’ve chosen to represent the 2-space as a subset of some flat 3-space.
The WMAP experiment sets the best bound on the total energy density of the universe at [symbol]W[/symbol][sub]tot[/sub] = 1.003 [sup]+0.013[/sup][sub]-0.017[/sub]. If this number is 1.0, the universe is globally flat.
What measurements? As the highest math I know is basic calculus, please speak slowly and use small words. 
Okay, I just reread Chronos’ post carefully. Is that basically it, the number of coordinates needed to specify a point?
No, that just indicates that the space is two dimensional, not that it’s flat.
But I’m not sure what is meant by the claim that measurements made on the surface of a torus, “internally” to it, so to speak, would make the torus seem “flat.”
The curvature of a space, whether that curvature is intrinsic or extrinsic, must make a difference to measurements, or else, what the heck does it even mean to call the space curved?
If it only makes a difference to “external” measurements (whatever that means) then I can’t see how we could say anything sensible about our own space being “flat” or “curved” since we can’t make “external” measurements of it.
-FrL-
Here’s a question: which lines count as the straight ones on the surface of a torus?
-FrL-
The (conceptually) simplest way to internally measure the curvature of a space is to draw big triangles, and measure the angles. For instance, on the surface of the Earth, I might draw a triangle with vertices at the North Pole, in Ecuador, and in Gabon. If I do this, all three angles will be right angles. So the sum of the angles of this triangle will then be 270 degrees. But in a plane (which is flat), the sum of the angles of a triangle will always be exactly 180 degrees. The fact that a triangle on the surface of the Earth has more than 180 degrees tells us that the surface is curved, and more specifically that it has what’s called positive curvature. I might also have a space where the angles of a triangle would be less than 180 degrees; such a space is said to be negatively curved.
A torus can actually be curved, and in fact the surface of a bagel is curved. Some parts (on the outside) are positively curved, and some parts (around the hole) are negatively curved, but the overall average curvature (for suitable definition of “average”) is zero. But you can also deal with a torus that is truly flat, such as the “world” of many video games. In many video games, if you go off the top of the screen, you end up on the bottom, and if you go off the left side of the screen, you end up on the right. This is not like the surface of the Earth: If you get to the North Pole, you don’t end up at the South Pole. It is in fact topologically equivalent to the surface of a bagel.
Quoth Mathochist:
Not exactly. That’d work in principle, but we know so little about dark matter (and even less about dark energy) that our estimates of the amount of “stuff” out there are completely worthless, unless we start by taking the observed curvature and working the calculation backwards from there. As Pasta says, the current best data come from the MAP sattelite, but there’s also some data from the Hubble Keystone Project (measuring distant supernovas) and other sources that backs it up. And theoretically, inflationary cosmology models predict that the Universe should be very close to flat, if not perfectly so: I think that this is what snailboy was thinking of when he mentioned M-theory (which is completely unrelated, and I don’t think actually says anything about the global curvature in our familiar dimensions).
[QUOTE=Chronos]
A torus can actually be curved, and in fact the surface of a bagel is curved. Some parts (on the outside) are positively curved, and some parts (around the hole) are negatively curved, but the overall average curvature (for suitable definition of “average”) is zero. But you can also deal with a torus that is truly flat, such as the “world” of many video games. In many video games, if you go off the top of the screen, you end up on the bottom, and if you go off the left side of the screen, you end up on the right. This is not like the surface of the Earth: If you get to the North Pole, you don’t end up at the South Pole. It is in fact topologically equivalent to the surface of a bagel.
[QUOTE]
Can’t a sphere be flat, too, then?
For example, suppose in our hypothetical video game, whenever the character crosses an edge, it emerges at the edge radially symmetrical from the edge it crossed?
So, for example, if it leaves some spot on the right side of the top edge, it emerges from (a corresponding point on) the left side of the bottom edge?
It looks to me that this surface would be topologically equivalent to a sphere, and would also be flat. Is that correct?
-FrL-
It would have large flat regions, but it’ll also have two infinite-curvature singularities (one at each corner). To illustrate this, consider drawing a circle with radius 1, centered on a corner: It’ll have a circumference of only pi, not 2pi.
I’m not quite certain whether it’d be topologically equivalent to a sphere, but it would at least be topologically equivalent to something that looks a lot like a sphere. If you deformed your rectangle to a hemisphere, you’d get rid of the singularities, and end up with a homogeneous space with uniform positive curvature (which would therefore have local geometry identical to the sphere). Mathematicians, a little help on the topology of this beastie?