Can a sphere have flat geometry?

[Hoku]

I know that the vocabulary for topology/geometry is complex. Please bear with me if I'm not using the appropriate terminology. I'm trying to get a grasp on my question as much as I am trying to find an answer to it.

Is it possible for something to disperse throughout a spherical space if the basic stucture of that space is flat? I mean flat in 3-dimensions, like a lattice structure that is flat instead of curved.

I'll try rephrasing the question:

If the shape of space is spherical, does it matter whether or not it's geometry is flat or curved? If so, what are the consequences of each scenario?

 

[Ben Niehoff]

A topological sphere can have a geometry that is flat except at a finite number of points: for example, the regular polyhedra are spheres that are flat except at their vertices. At the vertices themselves, the curvature becomes infinite, in such a way that its integral gives the deficit angle at that vertex.

There are "index theorems" which relate the total curvature integrated over a manifold to that manifold's topological invariants. In particular, for a 2-manifold, the integral of the Ricci scalar gives (2 pi times) the Euler characteristic. The Euler characteristic of a sphere is 2, so there must be some curvature somewhere if a manifold is to be topologically a sphere.

 

[Hoku]

Ben, thank you for your insights! As I said, I'm also trying to understand my question here, so I'd like to begin refining it...

What if you're dealing with a 3-dimensional sphere that lacks a 2-dimentional "skin"? Such a thing exists, doesn't it? Wouldn't it somehow be similar to a 3-torus if it lacked the skin?

 

[Studiot]

If you mean what I think you mean by 'skin' you need to be careful of terminology here.

Technically a sphere is a surface.

Check your definitions of

Sphere
Open Ball
Closed Ball

http://en.wikipedia.org/wiki/Ball_(mathematics )

 

[Hoku]

Thanks for clearing that up, Studiot. That's why I gave warning that I might not have the right terminology. I did look at the wiki link but the math is over my head. Is the math describing an internal geometry? If an open ball even has geometry, can it have FLAT geometry?

I'm trying to apply this to an absolute spacetime - spacetime that the big bang happened in instead of spacetime that the big bang created. People say that spacetime is flat except where mass curves it. I'm trying to understand, on a simple level, if that means spacetime must be infinite in all directions or if flat space with points of curvature can also be an open ball.

As Ben pointed out, A sphere must have at least some curvature somewhere but I think a ball can be described with flat geometry. Am I right?

 

[Studiot]

Spacetime.

I'm sorry but you are talking to the wrong guy here: I don't fully buy into the Minkowski scenario. It's a bit like a horse with three legs and one wing. Pegasus on the other hand was a fully formed (albeit mythical) horse.

It would be really good for some others to contribute here.

 

[mma]

Since we don't know the topology of the spacetime, we always talk only about a part of it. If this part is homeomorphic with a 4-dimensional ball, then of course you can endow it by flat metric if you want. But in physics metric is not arbitrary because it is used to describe gravitation. Perhaps gravitation can also be described by a metric having torsion instead of curvature (see http://en.wikipedia.org/wiki/Teleparallelism" ). But "flat" means curvature-free and torsion-free. So I'm afraid that a flat metric of spacetime can't have any physical significance. However, gravitational-like phenomena can occur in flat geometry too, for example in an accelerating elevator, being far from any masses, the spacetime is flat. Inertial forces are due to the nonvanishing components of the Christoffer-symbols, rather than curvature.

 

[Studiot]

Since we don't know the topology of the spacetime, we always talk only about a part of it. If this part is homeomorphic with a 4-dimensional ball, then of course you can endow it by flat metric if you want.

I think one fundamental issue is that in Mathematics you can divorce the frame from the objects it contains; in Physics you cannot.

So in Mathematics the 4d ball has existence and properties, regardless of the points within it.

Modern theories in Physics are equivalent to saying that you can start with a pure unadulterated, but empty, spactime universe and start placing objects in it.
As soon as we place some object (matter, energy, dark matter, whatever you like) in our universe they interact with it and you no longer have the original universe that was homeomorphic to the ball.
In Mathematics the ball is still there, but not in Physics.

 

[mma]

I don't think that the local topological properties of the spacetime are affected by matter (except perhaps singularities, like a black hole). I think that generally, this is only metric that is affected.
(I mean spacetime the set of the possible events endowed with a topology)

 

[Studiot]

Viewing time (or space) as a set of 'events' is inadequate. That was what my horse analogy was about.

 

[mma]

Viewing time (or space) as a set of 'events' is inadequate.

Yes, you are right. Neither space, nor time is that. But spacetime is. (at least in the theory of relativity). What else could it be?

 

[Studiot]

I call it the 'salami theory'

A salami has a readily identifiable cohesion or identity of its own.
That means it has length (between say A and B) and many other properties, including location, measured in any way you desire.

If I were to change the location I would not dream of cutting a slice from somewhere in the middle (ie between A and B ) and transporting just that slice and claiming I still have a salami. I would move the entire object. This is equivalent to changing the Minkowski spacelike coordinates from A_B to C_D.

I can of course accomplish the move in spacelike coordinates, because there are three of them so I can manouver the salami round other objects also occupying the same spacelike universe.

This is all very well for spacelike coordinates, but what about timelike ones? Minkowski has only one.

What of the cohesion and identity of my salami?
What if I again try to slice the middle and move it to another timelike location?
Does the continuity and cohesion require the the salami to have an extent on the timelike axis, in the same way has dimensional extent on the spacelike?

In Mathematics we can arbitrarily change the map, assigning points (events) in one frame to points in another, even in non sequential jumbled fashion.

We cannot do this in the 'real universe', we are constrained to take sets of points and treat them as 'objects'.

 

[mma]

I don't think that the existence of salamies disproves the theory of relativity.

 

[Studiot]

I don't think that the existence of salamies disproves the theory of relativity.

Did I say it did?

 

[mma]

Then I'm afraid, I don't understand what did you want to say. Perhaps somebody else will do.

 

[Hoku]

I have to say, comparing the universe to a salami is most entertaining! I think the analogy could use some clarification but I think the basis of your objection is, as you said, that three dimensions of space and one of time is "ugly". I agree. It is the common description of spacetime but I believe it is errored. I think it's a perception we have trouble shaking off because, we are so dependant and bound to the Earth's movement through space, that it's hard for us to maintain an objective view on it. I'm about to begin a thread in the "special and general relativity" portion of the forums, if you're interested. I think I'll call it "Spacetime...3+1?". There's another thread there entitled, "Minkowski 4 space? yes or no." that you might find interesting. It's a bit too mathematical to be of use to me, though.

The main question to this "ball geometry" thread has been satisfactorily answered, though new insights are always welcome. Thank you.

 

[Studiot]

All true Physics and Physicists should be able to handle salami!

It's a cop out to suggest otherwise.

But I didn't compare the whole universe to a salami, just any old material object within the universe.