What would you see if you looked at the sky in a universe with periodic boundary conditions?

Question

The be more concrete:

Suppose the universe was a cube with a side of length 2 light years, and that every point on a surface would be connected to the closest point on the parallel surface, such that if x>1 ly, x=x-2 (to put it as a programer of a simulation).

Suppose our solar system was all the matter in the universe. Looking at the sky, you'd be able to see the sun directly, and also, you'd find other directions in which you would see a mirror image of the sun, over a lightyear away.

Would the sky seem overcrowded with stars? would there be a patern, i.e. areas with more stars? would the result depend on the position of the sun and the viewer in the cube? how?

Answer 1

You would see lots of clouds that would remind you of different shapes and numbers. Hope it helps.

Answer 2

I think this is a problem more suited to simulation than analysis because of the images of images of images of images, etc.  After looking at the picture, some pattern will probably suggest itself that you can work backwards to find analytically.

If the universe had a finite age (on the order or its actual age), then you wouldn't see too much overcrowding of images.  What age are you going to use?

There probably is a pattern of sorts (just as there's a pattern when you hold a mirror up in front of a mirror and look at it).  Of course your position is going to matter in a way, although the pattern will be similar I think.  (I meant similar in the normal, speaking sense of the word, but it's probably also similar in the mathematical sense as well).

If you can make a picture and post it when you've done it, it should be cool to see.  When you plot it, make sure you plot the images with a relative brightness that goes down inversely to the square of the total distance to the image.

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Just thinking for the easy case or the box (which you could also think of as a universe with a lattice of suns).  I think the images will make lines.  You'll see a bright spot and then next to it, a less bright spot, and then a few more (how many depends on your universe's age) spots that get dimmer and dimmer and closer together.  And they'll be lined up.  I think I can imagine in my head pretty much what I'll see.  It's like a cubic lattice of suns.  The size will be

n = 2*age * c / boxsize.  So it's nxnxn and you're sitting somewhere near the very center.  The patterns you see are like the ones you see in an orchard of trees planted in perfect rows.  But it's just a 3-D orchard.  Pretty easy to visualize even without simulation.  As you move around the orchard, the rows of trees periodically become colinear with you, so the lines of images come together in a dot and then spread out again.  If you don't know what I mean, go run or drive by an orchard and watch the trees.

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The spherical case is going to be interesting.

You have a point producing luminous energy.

Then you have an equal amount of light coming out of a spherical shell of radius 2r (obviously much dimmer than the point because the light is spread out more).

Then an equal amount of light coming out of a spherical shell of radius 4r (dimmer still).

And so on and so on up to the age of your universe.

So you see light wherever you look.  But if you can see gradations of brightness, there are interesting patterns.  If you look right at the sun, that's an extremely bright spot.  (In your simulation, give the sun some non-zero size not TOO much smaller than r, so it has finite brightness).  If you look sort of tangent to one of the spheres, it looks a little brighter--Not so bright if you look at a surface along a normal line.  So you might be able to notice the brightness patterns of the second sphere through the first one, particularly if you are out to the edge of the universe.  Near the middle, things look pretty constant every way.  The best way to make the picture for your simulation will be to use color if possible. Make the brightest spots red and the dimmest spots violet or vice-versa.  That should look cool.

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What I wrote above about the orchard being nxnxn was wrong.  The cubic lattice is bounded by a sphere of radius age*c--it's the sphere inscribed into the nxnxn cube I described above.

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Also, unlike our universe, your universes have a true center.  When we look at the sky, it's the same in every direction.  Not so in your case.  If you go venture out towards the edges, things are generally less bright looking outward than looking inward.  As the universe gets older and older, this effect gets less and less obvious--at least for the "orchard" universe.  The center of the "shell" universe will always be special because of the geometry of it.