Is it plausible for spacetime to be shaped something like a torus?

Question

I have heard three theories for how space-time is shaped, flat, sphere-like, or saddle-like. Flat is the most likely, as all our measurements implies that space time has curvature close to 0.

Is it plausible for space-time to be shaped like some 3 or 4 dimensional analogue of a torus? A torus has an average curvature of 0. In fact, if you have something like a chess board, and glue the edges together, every square looks like the center, and it is flat around it. (again, I don't know how this generalizes to higher dimensions.)

Answer 1

I have heard three theories for how space-time is shaped, flat, sphere-like, or saddle-like. Flat is the most likely, as all our measurements implies that space time has curvature close to 0.

Inflation makes it so that a sphere like or saddle like spacetime evolves into a sphere like or saddle like spacetime that has a curvature very very close to zero. So actually the fact that the curvature is small now doesn't tell us how likely is that the universe has one of those three types of shapes.

And inflation wasn't made to do that, it was made to explain otherwise difficult to explain very large scale regularities.

Is it plausible for space-time to be shaped like some 3 or 4 dimensional analogue of a torus? A torus has an average curvature of 0. In fact, if you have something like a chess board, and glue the edges together, every square looks like the center, and it is flat around it.

It is possible to have a perfectly flat torus. But then in empty space you have a preferred frame of rest, the one that has a surface of simultaneity that appears just once on the chess set. Now haven't a globally distinguished frame isn't a problem with for general relativity, because it is a local theory. You still have Einstein's Field Equation hold in every neighborhood, and every local coordinate patch lacks a preferred frame.

Once you have sources, then it is common to have a preferred frame. For instance we have the frame in which the Cosmic Microwave Background Radiation looms the most isotropic (sometimes called a frame that is comoving with the Hubble flow). This is no different really than the fact that the surface of the earth gives us a natural up and down even though the laws of physics themselves prefer no direction, the arrangement of matter is simply simpler in one frame.

So back to the source free example, the flat torus for space is a perfectly fine solution to GR, it locally solves Einstein's Field Equation and is a perfectly fine global manifold. It just has a frame where is looks nicest even without any matter. That's because to be a flat torus like that you have to relate the end to be glued to another end at a particular time, and in one frame those are glued to the same time.

There isn't time travel (though if you did the flat torus trick in the time direction you could get that) but there are some things that go differently. For instance, like many spatially compact manifolds your past light cone can intersect yourself and can intersect the same world line over and over again. But again, this isn't a problem just because you are familiar with a different situation.

OK, but can we have matter and be isotropic and be homogeneous and everything like that and still be a flat torus type?

Yes. If you consider the metric

$$ds^2=dt^2-a(t)^2\left(dx^2+dy^2+dz^2\right),$$

then you must identifying say $x=0$ and $x=1$ and identify $y=0$ and $y=1$ and identify $z=0$ and $z=1$ rather than identifying places a certain distance apart in the comoving frame. But having them be a certain coordinate difference apart it is like your chess set that has the squares be coordinates it all locally looks the same as the infinite flat space so it still satisfies Einstein's Field Equation. It's just that the physical metrical size of each square gets bigger (and that happened already without a flat torus).

So just line up the coordinates and everything is fine because GR is a local theory.

Is it plausible? It's worth looking for evidence of repeats. We do look.

On the other hand, if we looked and saw evidence of an apparent repeat, how do we know it is a repeat rather than a regularity? So that's really where plausibility comes into play, if we saw evidence consistent with the universe looking similar (identical up to our experimental noise) in different directions would we favor theories that make it be similar or ones that make it be the same.

Probably some would consider each.

Answer 2

Is it plausible for spacetime to be shaped something like a torus?

I can't give a factual answer, so I will give an opinion. People have proposed that space or spacetime has a toroidal topology. This goes back to the old asteroids game, and there's plenty of papers on the arXiv. But there's nothing actually plausible about these proposals. There's no scientific evidence. The Planck mission didn't find anything. We have nothing to suggest that there's any kind of intrinsic curvature of hidden dimensions to space. Of course, some might claim it is plausible, but note that the torus has the same topology as a coffee cup. Is it plausible that the universe is shaped something like a coffee cup? Or how about a teapot? Or a whole tea set? IMHO once you allow one utterly speculative hypothesis to be "plausible" without any evidence whatsoever, you're on a slippery slope. So my answer is no.

I have heard three theories for how space-time is shaped, flat, sphere-like, or saddle-like.

And two out of three were always going to be wrong. Then on top of that, the toroidal universe is none of the above.

Flat is the most likely, as all our measurements implies that space time has curvature close to 0.

Yes, both WMAP and Planck indicate that the universe is flat.

A torus has an average curvature of 0. In fact, if you have something like a chess board, and glue the edges together, every square looks like the center, and it is flat around it.

Yes, you bend your sheet into a cylinder and glue the edges together, then you bend your cylinder into a torus and glue the ends together. But like I said, there's simply no evidence for it. There's no evidence that the universe has anything other than a 3D extent.

Again, I don't know how this generalizes to higher dimensions.

What higher dimensions? We have no evidence for any higher dimensions. When it comes to higher dimensions, my advice is to take careful note of higher dimensions have historically been the exclusive realm of charlatans, mystics, and science fiction writers.