What would physics be like in a wrap-around universe?

Question

This question has some close votes but has been edited/clarified, and has attracted more comments and discussion over the possible answers, and the geometry has overshadowed the actual question being raised. There are prerequisites!

There have been other questions to work out details and reality-check of different universe geometries, but specific things like how to keep the dirt and air where it belongs given other inputs. I don't mean to pick on that specific Question, as there have been others before that need to laboriously describe the geometry in question before getting to the real point, and then has Comments to clarify options, and then has Answers the need to cover the basics and don't (technically) get to his real question!

This is a more meta question, concerning wrap-around universes in general. Don't worry about where to put the air, how the sun moves, etc.: those kinds of questions use this as a starting point. Given the geometry itself, what would be the ramifications of having a universe with one, two, or all three “compact” spacial dimensions?

This is meant to stay as a reference to be useful for all such scenarios. More specific questions can refer to the general info here and then refer to the ideas an nomenclature and illustrations, when explaining how to keep the air on one side or whatever.

Consider a classic video arcade game 2D world, where opposite edges are identified (that means each pair of edges are the same one really).

If you fly off the right, you appear on the left. Likewise up and down. There is nothing special about this edge position actually, and you could scroll the entire view (keep the ship in the center and pan the screen instead, if the game had such a mode) without changing how the points are connected and the geometry of this space.

The Asteroids game (and most screen games like this) is topologically a torus, but unlike the surface of a torus in normal 3D space, the space does not have internal curvature! There is a distinction, and you could do things either way.

You'll notice that the path of a ship is non-repeating if you fly off at an angle that's not horizontal or vertical, as are the hazards: they cross the screen along one path, then a different path, each time. This is because there are two preferred axes in the torus.

An alternative is a sphere, like the surface of the earth. Again, no intrinsic curvature! But you go some distance in any direction and find you're back where you started. But this time all directions are the same, and the original point is the same distance away in every direction.

More complex shapes are possible. It can be odd shaped and lumpy; it can have some intrinsic curvature, or be intrinsically curved in one direction but not the other, and many other choices are possible.

Now consider another wrinkle. What if the opposite edges were identified but flipped upside down? What if the edge gluing was a Möbius strip, as in this delightful video?

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The Answers to this question should elucidate the ramifications of this kind of universe. So you might consider as the formal question to answer, “What is an interesting feature?” Help people understand what strange things will occur that are mind-bending, both for the other world-builders mental preparation and to pass along in their own descriptions to their readers/players. Illustrations are encouraged.

You can show something that is different from our deep instincts and might not realize with our experience in this universe, or go into issues concerning conservation of angular momentum and quantum mechanics, explain what things would look like from the inside, etc. This should be a reference for all things wrap-around, and multiple answers from the same person are OK, to keep different subjects separate if you like.

Next time someone asks a question set in this kind of universe, he should be able to (referring back here) know what to call the specific variety and not have to describe it from scratch. So “what is the proper nomenclature, in a catalog of different universes what are the different options called?” is also a specific questions to answer here. (and anyone using it subsequently is expected to upvote that Answer!)

Answer 1

Some other place on Physics Stack Exchange we discussed the physics of gravitational attraction in the world of Asteroids. For the most part, it's not a problem.

In order to predict gravity from an object (like a planet), you just sum up the field from an infinite lattice of such objects repeated infinitely. It's easy to see that with simple Newtonian approximations, the left/right and up/down fields do not diverge to infinity. The field is mathematically defined, and indeed, you can calculate it yourself. But while the field is well-behaved, the gravitational potential isn't for an infinitely repeating space. There's no reason that you can't still have the expansion of space in a world like this. Another resolution might be that gravity does not perfectly follow the 1/r^2 pattern, and falls significantly below it at far-off distances. But a divergent value for absolute gravitational potential isn't something that you necessarily have to concern yourself anyway because it won't affect any local properties.

One of the most surprising things about such a world is that it doesn't necessarily have to be radically different from our own. We have a cosmological horizon that ends long before the universal wrap-around, so we experience no consequences of the wrap-around (if there is such a thing at all). But you could paint a picture of a world with completely observable wrap-around that people don't even notice until they have advanced telescope designs. That could have implications for a space-faring society, but none before leaving their planet.

Answer 2

Looking into the past

Depending on the geometry of the wrap, with a powerful enough telescope you'll be able to see yourself in the past. Possibly even several times and at different locations in time.

If the universe has a spherical wrap 1 light hour in radius, you'll be able to look up and see yourself 1 hour ago. If you then move over a bit you'll able to see all the way down the line with snapshots every hour. It's like standing in a hall of mirrors, but the mirrors don't reflect immediately.

Answer 3

To be honest, this question is way too broad, but I'm not going to be a jerk and close something that is over two years old. Instead, I will try to answer one part of this, which is what will the effect of gravity be in an infinitely wrapped universe.

2d sphere-surface universe

Lets start with a 2-d universe that wraps around; as in the surface of a sphere. The radius of the sphere is r, which means that if you $2\pi r$ in any direction you end up back where you started.

Since the circumference of a circle is $2\pi r$, the distance in the opposite direction will be $2\pi r-d$. The object will be $(2\pi r)*n + d$ distance away in the primary direction, for $n\in\{0,1,2...\}$, and $(2\pi r)*n + d$ distance away in the opposite direction for $n\in\{1,2,3...\}$.

Let M be the mass of an object in the sphere-surface universe, and G be the universal gravitation constant. Force of gravity ($F$) on an object will be

$$\begin{align}F &= \sum_{n\in0,1...}\frac{GM}{\left(2n\pi r + d\right)^2} - \sum_{n\in1,2...}\frac{GM}{\left(2n\pi r - d\right)^2}\\ &=\frac{GM}{d^2} - \sum_{n\in1,2...}\frac{8GM\pi nrd}{\left(\left(2\pi nr\right)^2-d^2\right)}\end{align}$$

This is somewhat interesting on cosmic scales. The reduction in gravity is proportional to the ratio between d (distance from the object) and r (radius of the universe). I could not solve that equation, but I numerically solved for this relationship:

$$F_{actual} = F_{expected}-F_{expected}\left(\frac{a}{\pi^3}\left(\frac{d}{r}\right)^3\right).$$

Here $F_{expected}$ is the gravity that we would expect from normal gravity, where $F_{actual}$ is what is actually felt in this universe. In this case $a$ has proven difficult to calculate; it is some function that depends on $d/r$. It is asymptotically equal to 0.600644 as $d/r\rightarrow\infty$ . My closest guess so far is

$$a = 1-0.39936\cdot e^{\frac{-d}{2r}}.$$

Anyways the details are not that important. What is relevant is that the force of gravity between two objects 1 universe-radian (i.e. a distance of $r$) apart is about 98% of what is expected; for the maximum possible distance of $\pi$ universe-radians apart, that is on opposite sides of the universe-sphere, gravity drops to zero.

Lets imagine for a minute that we are talking about a 3-d universe inscribed onto the surface of a 4-d sphere. All matter in the universe can be divided into two different universes with two non-overlapping event horizons; one in each hemisphere of the 4-d sphere. Once clustered there, the Schwarzchild radius of each universe will exclude the other. However, the effects of gravity of one universe will be felt in the other. In my limited cognition, this could be a reasonable explanation for dark energy. The unknown force driving galactic expansion is actually the force of gravity from the mirror universe, pulling out universe apart. Of course, it will not succeed, since each universe has its own Schwarzchild radius, excluding the other.

There is probably some flaw with this hypothesis, or some observable evidence that this is not the case, but its a proposal for an alien geometry, at least.

I have to go, but I'll come back at some point for more consequences, and to do gravity math for another type of infinite universe

Answer 4

Our universe is like this (Probably)! The universe your describing is compact without boundary, which means it has a finite volume but does not have an edge. The currently accepted model of the universe is called the standard model of cosmology which describes a universe that has these properties, so your universe would look like our universe.