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/sci/ - Science & Math

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4475131 No.4475131 [Reply] [Original]

Since a circle with an infinite radius is equivalent to a straight line, would an infinite universe with a spherical geometry be equivalent to a universe with a euclidean geometry?

>> No.4475138

That 'straight' line in your pic is messing with my OCD.

>> No.4475159
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4475159

I'd answer your question with yes OP, since the curvature really gets smaller in relation. However, in general relativity the interesting curvature comes from nearby objects, where by nearby for us I mean stars and planets in the galaxy. So the grand scheme of things, i.e. the question what the empty universe would look like, has no local consequences anyway.

>> No.4475166

>>4475159
But doesn't the influence of objects on the cuvature of space interact with the geometry of empty space? Thus, isn't the geometry of empty space in itself not important?

>> No.4475187

Oh come on.

>> No.4475196

All this talk about straight line is discriminating towards homosexual lines.

>> No.4475206

you mean an ELLIPSE. a circle's radius would grow infinitely in all directions - 360 degrees around. it would never become a straight line.

>> No.4475212

>>4475206

>implying a circle is not a type of ellipse

>> No.4475217

>a circle with an infinite radius is equivalent to a straight line

Nope.

>> No.4475242

>extending 2D results to the third dimension
Nope.

>> No.4475244

>>4475212
I want you to re-read your post.
And then delete it for being so dumb.

>> No.4475251
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4475251

>4475166
>isn't not
What?

Anyway.
The big sphere empty universe would not be distinguishable from the flat one in a local area (like our galaxy), so they are effectively the same. This is what you proposed and agree. In such a locally flat universe (we just established that it's locally flat in both cases) the interaction of energy with spacetime would look the same. Both curve it - but that hasn't to do with the question anymore.
In general relativity, the grad spatial scheme of things has little influence on things on earth. No strong version of Machs principle is incorporated into the geometric gravity theory we are dealing with here. Cry more, Einstein.

>> No.4475254

>>4475206
The curvature of a circle would approximate 0 if the radius approximates infinity, no?

>> No.4475259

>>4475251
OP here, I was away for a while. I didn't read the post that got deleted, it wasn't mine.

Anyway, this:
>The big sphere empty universe would not be distinguishable from the flat one in a local area (like our galaxy), so they are effectively the same.
is all I wanted to know. So thanks for your response.

>> No.4475261

"An infinite universe with spherical geometry" is self-contradictory, isn't it?

>> No.4475262

>>4475244

are you saying a circle is not an ellipse? Because you would be wrong.

>> No.4475277

>>4475261
Good point. I don't know if it is though, but maybe someone who knows topology can answer that.

>> No.4475296

Interesting thread? On /sci/?

>> No.4475310

>>4475277
A sphere is compact, and assuming an infinite universe does not have a compact topology means they can not be topologically equivalent. The whole discussion here though is based on incomplete and ill-defined premises. E.g. nobody every defined what they mean precisely by "infinite universe". Otherwise this should be answerable with simple topology.

>> No.4475316

>>4475310
Assuming though that by "infinite universe" you mean a metric space which is either not complete or not totally bounded (probably rather the latter), by the general Heine-Borel theorem it follows that it cannot be topologically (and surely thus not geometrically) equivalent to the sphere.

>> No.4475317

>>4475310
Ah, I see we have someone who knows what they're talking about.

Would you care to provide some guidance then? How would you define 'infinite' universe?

>> No.4475322

>>4475131
It's not about the size of the universe, but the strength of the curvature. If the curvature is infinitely weak, then yes, spherical geometry is equal to euclidean geometry. In that sense, euclidean geometry is a special case of spherical (and hyperbolic) geometry.

>> No.4475323

>>4475317
>How would you define 'infinite' universe?
I've given one very abstract possibility in >>4475316, but it should be noted that it's possible to endow appropriately unbounded spaces with almost arbitrary geometries. As an example you could consider hyperbolic space which is smoothly equivalent (precisely diffeomorphic) to Euclidean space. But you could construct many other weirder geometries.

The important point is this: if a space has spherical geometry it means it's isometrically isomorphic to the sphere. This in particular means that it's compact and hence any non-compact topological space cannot have spherical geometry. Thus the moment you want your universe to have spherical geometry you will have to take a compact space as a model.

>> No.4475336

>>4475323
>>4475322
Thanks guys, that cleared up a lot.