Seeing the edge of an unbounded, closed universe

[DaveC426913]

Kaku's 'Hyperspace' is currently inspiring my higher-dimensional imaginings. I'm trying to imagine what one would see in a closed, unbounded hyperspherical universe. I know of the conceot that, if you look far enough, your image can wrap around the universe so you're looking at the back of your head. It works in 3D but just as well as in 2D, but I'll start with the ant-on-the-balloon.

An ant lives on the surface of a balloon universe. His movement, rays of light and everything else are constrained to follow the curvature of its surface.

If the ant looks ahead, he'll see an ant way in the distance, facing away from him - it is in fact, him. If he looks to his left, he'll see an ant way in the distance looking away - again, in fact, him.

Right? Right.

But here's my difficulty. No matter what direction he looks, he's going to see himself. If he turns his head by ever decreasing amounts, 1 degree, half a degree, one-zillionth of a degree he's going to see himself.

You quickly realize that he's not going to see an entire copy of himself at every one of those angles, lined up like a chorus line. No, he's only going to see a piece of himself - the exact opposite piece from where he's looking. Imagine he looks directly ahead; in the exact center of his vision, he will see the exact rear tip of his tail. What will be 1/10th of a degree to the right of that point? Well, if we trace the light ray to is source we see that it is coming from a spot 1/10th of a degree to the left of the tip of his tail. What he sees 2/10th of a degree to the right of the tip of his tail, is the rear-end 2/10ths of a degree to the left of his tail. You can do this across his entire vision.

The net effect is that his view will be that of one *single* giant ant filling his entire vision, no matter which way he looks. His entire universe will be a shell of an (albeit concave and reversed) ant.

Are you still with me?

Note that it is irrelevant how *big* his balloon universe is. If it is a foot in diameter or 20 yards in diameter, he will see the same thing - while the "ant shell" will be much farther away, it will be correspondingly larger, cancelling out. Even if his balloon is a mile in diameter, the image he sees is no larger or smaller than "filling his entire vision with a single ant image".

You can verify all the above by drawing the diagram on a piece of paper.

An aside: The only mitigating factor is intervening objects. Obviously, large objects would block the view of the ant's universe-edge. But additionlly, if the ant's balloon universe were not perfectly transparent (say it had an atmosphere, or interstellar dust and qas), then the ant would see a correspondingly faded edge-of-universe. If his universe were large enough, the edge of his universe be lost in the distance and he would not see it at all.


Now let's look at what we would see in our 3D/4D universe if it really were closed on itself.

The extrapolation here is that, if we really lived in a universe that were closed on itself in a 4th dimension, we would (barring blocking objects, gas and particles) see an infinitely large backdrop of - ourselves. It would not be too small to see due to distance, no - it would be scaled up by just the right amount, so that the entire boundary of our universe would look like our own body (or planet that we're standing on).

This means that the ONLY thing preventing us from seeing the edge of our universe - and seeing our ourselves - is intervening gas and dust.

 

[RandallB]

An aside: The only mitigating factor is intervening objects.

This means that the ONLY thing preventing us from seeing the edge of our universe - and seeing our ourselves - is intervening gas and dust.

Oh I’d say there are a few other mitigating factors getting in the way.

How’s this for a BIG one.
Let’s draw a map of the world on your balloon putting the ant on equator in Ecuador.
And at the same time put cube of sugar on Cuba.
Problem is the ant needs to wait for the light showing the sugar to show up.
During which time you’re blowing up the balloon a little bigger.

No problem -Not that far not that long –
BUT, by looking carefully the ant can see a faint light coming from behind the sugar. Identified as the North Pole giving off a beacon from WHEN you first started blowing up the balloon – that’s way before the ant was put on said balloon!

Now if the ant can only see to the North Pole now, how long does it need to wait for the it’s image to make it all the way around the balloon to see his back side? Especially since you’re still blowing it up, and nothing seems to be behind the light from the North Pole.

I think we need to stick a needle in the balloon to let the air out first.
Then we will have a Big Bang.

Till then the only edge we can expect to see is the one we are standing on, everything else we see is in the past and has moved on to where we cannot see.

 

[DaveC426913]

Yes. Of course, the boundary defined by light's travel would be paramount.

I was setting that aside, but it's not really a detail that can simply be 'supposed' away.

 

[|Orion's Thought|]

Never heard of Kaku's 'Hyperspace', but what I can say is, good ole ants! Once again, saving the day in online demonstrations! But seriously, is there anyplace I can look to find out more about Kaku's Hyperspace?

 

[SpaceTiger]

In such a hypothetical universe, I don't see anything immediately wrong with your reasoning. I think symmetry pretty much forces the conclusion that the image of yourself is seen in all directions. That can't be the case for our universe, of course, because we measure it to be flat. That doesn't mean it can't be closed, but it does mean that the topology scale would have to be much larger than assumed here, meaning that the particle horizon would be much smaller than the size of the "entire" universe.

There are, however, flat universes that can wrap around. Take, for example, a cylinder. It has zero curvature everywhere, but still wraps around. One can construct many different finite universes that are flat (within the errors) and produce multiple images on the sky. I don't think you can produce an image in every direction with any of these universes because that implies isotropy which, in turn, implies the FRW metric. The only flat universe consistent with the FRW metric is infinite, so there wouldn't be multiple images.

We've been looking for multiple images in the microwave background and have so far come up empty. This puts some constraints on the topology scale that are given in detail here:

http://lanl.arxiv.org/abs/astro-ph/0310233"

 

[Soul Surfer]

Probably the most interesting shape is the torus or toroidal surface. you wouldn't get the problem that you would with a sphere.

See also the "view from a ring singularity" that I posted in astrohysics a few days ago

re microwave background images. I thought some of the latest results had turned up some suspicious correlations

 

[DaveC426913]

Probably the most interesting shape is the torus or toroidal surface. you wouldn't get the problem that you would with a sphere.

I'm not convinced that's true. Can someone confirm?

 

[SpaceTiger]

I'm not convinced that's true. Can someone confirm?

Here's an old paper on searching for signs of a toroidal universe:

http://arxiv.org/abs/gr-qc/9602039"

The basic gist is that testing for the toroidal universe isn't particularly useful because such tests don't constrain other topologies very well. We can't rule it out as the topology of our universe, but there hasn't yet been any indication of multiple images (or "matching circles") in the microwave background. Also, more recent tests:

http://arxiv.org/abs/astro-ph/0307282"

 

[DaveC426913]

1]

"Probably the most interesting shape is the torus or toroidal surface. you wouldn't get the problem that you would with a sphere."

"I'm not convinced that's true. Can someone confirm?"

What I mean is, I'm not convinced that a torus would not suffer the same problem. You'd still get the wraparound.

2]
"There are, however, flat universes that can wrap around. Take, for example, a cylinder. It has zero curvature everywhere, but still wraps around."

How does a cylinder have zero curvature? Or do you mean "zero in one direction but wraps around in the other"?

 

[SpaceTiger]

What I mean is, I'm not convinced that a torus would not suffer the same problem. You'd still get the wraparound.

Yes, but in a universe with a finite particle horizon, you only get wraparound in specific directions -- that is, you'd get images. In a spherical universe, it would be a uniform haze. I don't know if that's what he meant by "problem".

How does a cylinder have zero curvature? Or do you mean "zero in one direction but wraps around in the other"?

It has non-zero extrinsic curvature, but zero intrinsic curvature (in all directions). See here, for example:

http://www.mth.uct.ac.za/omei/gr/chap6/node8.html"

 

[Chronos]

If inflationary and expansion models are correct, any wrap around effects would not be observable. The particle horizon would prevent you from viewing any past images of yourself.