# Mass bending space makes me think there's another side to space

1. Apr 3, 2013

### Seminole Boy

Mass bending space makes me think there's another "side" to space

Perhaps I'm not understanding space curvature very well, but how could it curve if the universe, or space, is an open system?

If mass is bending it, this makes me believe the universe is a limited and closed system. In other words, space is some kind of elastic boundary for matter. And this doesn't make sense because I thought space was a type of property without boundaries.

Edit: Or is it simply local space curvature? Silly me...if that's what it is.

Last edited: Apr 3, 2013
2. Apr 3, 2013

### Staff: Mentor

I'm not sure how to interpret your descriptions, but spacetime is not bending itself "in something else" - I think you mean the right thing with "simply local space curvature" (but spacetime, not just space). Mathematically, it is a (Pseudo-)Riemannian manifold.

3. Apr 3, 2013

### WannabeNewton

It seems you are confusing extrinsic curvature with intrinsic curvature. Note that while mathematically we can always embed riemannian manifolds in $\mathbb{R}^{n}$ for an appropriate $n$, as far as GR is concerned physically this is nonsense. There is no ambient space in which to embed space-time, space-time is all there is. In such a case, we deal with the intrinsic curvature of space-time. So, for example, you could think of the 2-sphere as embedded in $\mathbb{R}^{3}$ (which is what you naturally picture when you think of the 2-sphere) and look at its extrinsic curvature e.g. mean curvature or you can think of the 2-sphere abstractly, as just a manifold on its own, and look at its intrinsic curvature via the Riemannian curvature. In the case of space-times it doesn't make physical sense, as far as GR goes, to think of space-time as embedded in some ambient space and to look at its extrinsic curvature.

The riemann curvature is given by a tensor field that at each point assigns a tensor which measures the extent to which the metric tensor "fails" to be locally isometric to euclidean space.

4. Apr 3, 2013

### zn5252

watch this :It may clarify the mathematics...

5. Apr 3, 2013

### Staff: Mentor

Do you really mean "open" and "closed", or do you mean "infinite" and "finite"?

All manifolds are open, by definition. However, "open" and "closed" are not opposites, so you can have manifolds which are both closed and open (aka clopen: http://en.wikipedia.org/wiki/Clopen_set).

I suspect that isn't what you are talking about, but rather finite or infinite. In any case, you can have finite flat manifolds, finite curved manifolds, infinite flat manifolds, and infinite curved manifolds.

6. Apr 3, 2013

### Passionflower

Care to explain?

7. Apr 3, 2013

### WannabeNewton

Actually all manifolds (in fact all topological spaces) are both closed and open in their own topologies since both the empty set and the entire space must be in the topology. Whether a manifold is compact or non compact on the other hand (which is what I think you mean by finite vs infinite) will depend entirely on the topology endowed on it.

8. Apr 3, 2013

### WannabeNewton

Any topological space is open in itself by definition of a topology.

9. Apr 3, 2013

### Passionflower

So you are saying that for instance a torus is an open manifold?

10. Apr 3, 2013

### Seminole Boy

DaleSpam:

Yes, finite and infinite is what I meant.

11. Apr 3, 2013

### micromass

Staff Emeritus
The torus is a topological space that is open in itself. This is not the same as being an open manifold.

By definition, an "open manifold" is a manifold that has no compact connected component and that does not have a manifold boundary. So the torus is not a open manifold.

You might be confused because the term "open" can be used in many different ways. Open in "open manifold" is not the same as open in "open set in a topological space".

12. Apr 3, 2013

### WannabeNewton

Open set has a very strict meaning: a subset of a topological space is open iff it is an element of the topology and by definition of a topology, the overarching set must be in the topology. Don't confuse a compact set with an open set. A topological space is compact if every open cover of the space has a finite subcover. An open manifold is also a totally different thing from an open set.

13. Apr 3, 2013

### Passionflower

Who is confused here?

14. Apr 3, 2013

### micromass

Staff Emeritus
Manifolds are topological spaces. So saying that a manifold is open in itself makes sense and is a useful concept.

The confusion is that there are two concepts (that are totally unrelated) that both have the name "open". Personally, I will never use the term "open manifold" for exactly this reason. I will rather use something like noncompact or something similar.

15. Apr 3, 2013

### Seminole Boy

Guys:

I have another question, and I'll just post in on this thread. Do you sometimes believe that physics is getting overcomplicated with words and all these new findings? I mean, there seems to be a billion different words out there explaining a trillion different things, but it seems like it's all going in different directions. Our current society seems to be a product of high entropy. Everything is getting more and more disordered, which is leading to more and more confusion (Wild Goose Chases). Last time I checked, the basic goal of physics is to describe the physical nature of the universe. And when I hear highly paid lecturers at certain big-time schools talk about this stuff, they can run the math and use big words, but they seem not to have a clue what they're talking about. Einstein was the opposite. He used paper and pen and told the establishment to take a hike. In fact, when he started relying on mathematical formalism, he lost his touch. His greatest findings came when he was a young whippersnapper with a strong identity. I hope this doesn't come off as a rant. I just see all these grand equations and so forth being posted, but I just wonder if people are running around in circles, or just impressed that they can solve complicated equations. Math is great, but insight is greater, in my opinion. We have four fundamental forces, and while we are spending billions and billions and billions on atom crushers and so forth, it seems that we're no closer to unifying these forces than we were when Albert passed.

16. Apr 3, 2013

### Passionflower

By definition, an "open manifold" is a manifold that has no compact connected component and that does not have a manifold boundary. So the torus is not a open manifold.

Consistent?

Bye the way this is not the first time, it seems the 'thou shall not call a mentor wrong even if he is really wrong' is the social norm on this forum as others seem glad to rush in to 'correct' the 'confusion'.

How silly!

17. Apr 3, 2013

### micromass

Staff Emeritus
Who said this?? Do you say this? Why do you think they have no clue what they're talking about?

You can't have insight without the mathematics. The mathematics is what quantifies the natural world. Without mathematical formalism and complicated equations, physics would not exist.

18. Apr 3, 2013

### Seminole Boy

Micromass:

How much math did Einstein's SR papers contain?

19. Apr 3, 2013

### WannabeNewton

First of all, this is totally veering away from Seminole's question.
Secondly, stop nitpicking. Anyone with a basic understanding of topology would have known what DaleSpam meant. You arent proving anything by criticizing semantics.

20. Apr 3, 2013

### micromass

Staff Emeritus
You don't seem to be understanding that there are two concepts that are both called open.

Do you know what an open set is in a topological space? Do you know that every manifold is a topological space? Do you know that a manifold is an open set in that topological space?

It is indeed true that a torus is not an open manifold, but a torus is also an open set in itself.

Dalespam wasn't wrong. He just used one definition of an open set. You seem to be using another (different) definition of open. This is what the confusion is about.