# Light beam in a flat universe

I know this is a commonly adressed topic, but it is one in which things are hazy to me.

If the universe is finite (which we don't know), and flat (which we are becoming ever more increasingly certain about), then how could a light beam fired from a point in the universe not ever reach its starting point? I know that most make analogies, like balloons, but this is for a universe with some curvature.

If a light beam is fired to the farthest reaches of space, then it is commonly said that the beam would end up where it started. However, this scenario seems to me as to be described only from a stand-point of the beam curving its way around the spherical shape of the universe. If one were to imagine a flat universe witha finite size, then a light beam would travel to the farthest reaches of space, and then what? It certainly can't curve back in and reverse its trajectory.. the universe is flat, remember. It wouldn't be able to reverse its trajectory, so what would happen to it?

Thanks.

Chronos
Gold Member
How long would it take for that beam to return? Very possibly longer than the age of the universe. Even in a finite universe, it may never have enough time to return if the universe is expanding. See http://www.sciencenews.org/sn_arc98/2_21_98/bob1.htm for an interesting discussion.

How long would it take for that beam to return? Very possibly longer than the age of the universe. Even in a finite universe, it may never have enough time to return if the universe is expanding. See http://www.sciencenews.org/sn_arc98/2_21_98/bob1.htm for an interesting discussion.

Let's suppose, just for the sake of theory, that the light would have enough time. How would it return?

Let's suppose, just for the sake of theory, that the light would have enough time. How would it return?

Like Magellan? http://en.wikipedia.org/wiki/Ferdinand_Magellan" [Broken]

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atyy
Wallace
There are a number of important concepts here, when you say 'flat' you are referring to a particular solution to the equations of General Relativity that have, as step one, the assumption that the Universe is the same everywhere (homogenous) and looks the same in every direction in every place (isotropic). Within this solution, derived under these assumptions, a flat universe there MUST be infinite.

That is not to say that the real universe is neccessarily infinite, but the mathematic model that you implicitally refer to when you say 'flat universe' cannot sensibly be used to describe a finite universe.

It may be possible that the observable universe is very close to being spatial flat, but beyond that the properties of the universe change (the average density is less for instance) in which case the broader universe could be finite, but there is no way for us to know this now (we could learn this if we wait long enough, depending on what the properties of the universe are). Therefore the ultimate answer to your question in unknown.

The important distinction to make though is between reality and mathematical models. A spatially flat universe following a Friedmann-Robertson-Walker solution to General Relatitivity must be infinite, but this model could describe our universe very very well, even if on some level the universe is in fact finite.

Edit: And on the question of light returning, there is again no sensible answer in the case of a flat universe, the light ray will never return. This is no problem since in a flat universe, as explained, the universe must be infinite.

sylas
There are a number of important concepts here, when you say 'flat' you are referring to a particular solution to the equations of General Relativity that have, as step one, the assumption that the Universe is the same everywhere (homogenous) and looks the same in every direction in every place (isotropic). Within this solution, derived under these assumptions, a flat universe there MUST be infinite.

All good points... but I need to raise a quibble. (Sorry!)

There's also a theoretical possibility of unusual topologies, in which the universe is connected back on itself in some way. Such a universe can be finite, even if flat or with positive curvature. An interesting example if the universe is a bit like a torus (donut) but in three dimensions. A torus is actually "flat" in the technical sense that applies for spacetime geometry. It's a bit like having a finite flat sheet of paper, for each each edge is identified with the opposite edge. What goes off one edge comes in at the other.

In three dimensions there are some interesting ways in which such topological effects could occur. A couple of years ago there was briefly a stir in the popular press when public imagination was captured by a serious proposal for a finite "docedahedral" universe. See http://news.nationalgeographic.com/news/2003/10/1008_031008_finiteuniverse.html, in National Geographic News, Oct 2003; or the scientific paper:

This is a serious possibility, although this particular hypothesis was disproved almost as soon as it was published. But it does illustrate a way in which you can have a finite flat universe, and what would happen is that a light beam fired off in one direction might come back to the co-moving emission source from a different direction, much much later.

In the conventional simplest topology for FRW solutions, light cannot get right around a finite (hyperspherical) critical density universe before it collapses again in a big crunch. I'm not sure about a universe with positive curvature and dark energy combined

Cheers -- sylas

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Wallace
Hmm, I don't know a lot about these topological theories you mention (I've heard vague mentions, but haven't looked at the details). I'm not sure that these would be compatible with GR, in the most basic sense, so that while this is certainly possible, it requires a different theory of gravity (which of course could be the true theory, if GR is not complete).

If you parrallel transport any arbitirary vector around a loop in a flat torus you don't (I think?) always get back the same vector. By definition then (unless I'm mistaken) this space-time is not equivalent to flat FRW solution.

It would be interesting to test this though. Can you (or anyone else?) write down the metric of a FRW-like torus? It would have to have some additional sine or cosine term in the spatial part, to give you the torus and allow a path to get back to the same place. If such a metric gives you the same stress energy tensor when you put it through the field equations then indeed this type of Universe would be permitted by GR. The trick is coming up with an appropriate metric? (maybe it is in the literature?)

I'm not sure about a universe with positive curvature and dark energy combined

In this case the curvature vanishes as t goes to infinitey, i.e. the universe becomes closer and closer to flat. I'm pretty sure you couldn't get a photon to do a loop in such a universe, or if you could it would have to be early on when the universe is curvature dominated.

sylas
Hmm, I don't know a lot about these topological theories you mention (I've heard vague mentions, but haven't looked at the details). I'm not sure that these would be compatible with GR, in the most basic sense, so that while this is certainly possible, it requires a different theory of gravity (which of course could be the true theory, if GR is not complete).

My understanding is that GR works unchanged in a spacetime with unusual topologies; and that it does not require a different theory of gravity. That is, the theory we have at present does not require a simple topology, and works unchanged with a more complex topology.

We need a different theory of gravity anyway, to properly combine QM and GR, and in that case there might be some useful constraints on possible topologies... I don't know. As it stands, we have no good theoretical reason in GR to prefer a simple cosmology over a complex one.

I think the matter of what topology you have for space is more like a boundary condition for the universe, rather than something actually constrained by relativity or by theories of gravity. You consider some topological defect, and then see what the implications are from that using GR, without change.

If you parrallel transport any arbitirary vector around a loop in a flat torus you don't (I think?) always get back the same vector. By definition then (unless I'm mistaken) this space-time is not equivalent to flat FRW solution.

The simplest form of the flat FRW solution has the simplest possible topology, but that's not intrinsic to the FRW equations, I think. Just a convenient assumption. I think you can apply the FRW solutions also in more complex topologies, without much change. You still have the boundary conditions of curvature, and matter density, and radiation density, and dark energy; and on top of that you try some multiply-connected topology, which "expands" and "contracts" in the much the same way as the simply singly connected topology.

There's what amounts to a book on this, which is quite readable early on, and explains how GR is a local theory which has no implications one way or another for topology, and looks at some of the history of multiply connected topologies, which goes back a long way.

See Cosmic Topology by M. Lachieze-Rey and J.P.Luminet, as arXiv:gr-qc/9605010v2.

I'm not an expert on this, but I'm sure some of our in house cosmologists know a bit about such ideas.

Cheers -- sylas

All good points... but I need to raise a quibble. (Sorry!)

There's also a theoretical possibility of unusual topologies, in which the universe is connected back on itself in some way. Such a universe can be finite, even if flat or with positive curvature. An interesting example if the universe is a bit like a torus (donut) but in three dimensions. A torus is actually "flat" in the technical sense that applies for spacetime geometry. It's a bit like having a finite flat sheet of paper, for each each edge is identified with the opposite edge. What goes off one edge comes in at the other.
Cheers -- sylas

I'm not sure I understand this. The torus is shaped like a donut, so there must be some type of curvature, otherwise it would be Euclidean, right? Or are you saying that the angles of a triangle put on a torus add up to 180 degrees?

Like Magellan? http://en.wikipedia.org/wiki/Ferdinand_Magellan" [Broken]

Well this is definitely wrong. Magellan sailed around the spherical earth. Light in a flat universe can't do this.

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sylas
I'm not sure I understand this. The torus is shaped like a donut, so there must be some type of curvature, otherwise it would be Euclidean, right? Or are you saying that the angles of a triangle put on a torus add up to 180 degrees?

The angles on triangle for a toroidal space add up to 180 degrees, ideally.

Suppose you take a sheet of paper. Draw a triangle on it.

Now roll it up to make a cylinder. Still 180 degrees on the triangle... this is not "curvature" in the sense used in relativity.

Now comes a problem. We want to roll the cylinder as well, and join the ends... but we've run out of dimensions for rolling. In a 4D space, we could do it. Consider, for example, the surface inside the Euclidean space R4 defined by
$$1 = x_1^2 + x_2^2 = x_3^2 + x_4^2$$​

Topologists are not limited by models that can be embedded in a 3D Euclidean space. A torus can be nicely embedded in a 4D Euclidean space, or you can just define the 2D toroidal space as a mathematical object without bothering to embed in a higher dimensional space. Think of a computer game on a finite screen, in which anything moving off one edge reappears in the opposite edge. That game is using a toroidal space.

Or you can just bend and stretch the cylinder a bit to make a torus embedded in 3D space. You can also define your metric and curvature tensors on the sheet of paper, and they still work on the torus. In this sense, the torus is considered to be a "flat" space.

A 3D torus would be a like a cube, in which each face is identified with the one opposite.... a bit like the game with toroidal topology on a computer screen. It would be a flat space, and GR still works just fine since GR is a local theory, and you would have light able to hit you on the back of the head. There are serious attempts to see if our own universe has a complex topology by looking at the sky -- and the background radiation in particular -- to see if it looks suspiciously similar in different directions. That would be the tell-tale sign of a multiply connected topology for our own universe.

Cheers -- sylas

Cyosis
Homework Helper
A cylinder is flat and finite. Any metric that can be written in the form $ds^2=dx^2+dy^2$ is flat. A sphere isn't flat, because no matter how hard you try you can never reduce its metric to the previous mentioned form.

The angles on triangle for a toroidal space add up to 180 degrees, ideally.

Suppose you take a sheet of paper. Draw a triangle on it.

Now roll it up to make a cylinder. Still 180 degrees on the triangle... this is not "curvature" in the sense used in relativity.

Now comes a problem. We want to roll the cylinder as well, and join the ends... but we've run out of dimensions for rolling. In a 4D space, we could do it. Consider, for example, the surface inside the Euclidean space R4 defined by
$$1 = x_1^2 + x_2^2 = x_3^2 + x_4^2$$​

Topologists are not limited by models that can be embedded in a 3D Euclidean space. A torus can be nicely embedded in a 4D Euclidean space, or you can just define the 2D toroidal space as a mathematical object without bothering to embed in a higher dimensional space. Think of a computer game on a finite screen, in which anything moving off one edge reappears in the opposite edge. That game is using a toroidal space.

Or you can just bend and stretch the cylinder a bit to make a torus embedded in 3D space. You can also define your metric and curvature tensors on the sheet of paper, and they still work on the torus. In this sense, the torus is considered to be a "flat" space.

A 3D torus would be a like a cube, in which each face is identified with the one opposite.... a bit like the game with toroidal topology on a computer screen. It would be a flat space, and GR still works just fine since GR is a local theory, and you would have light able to hit you on the back of the head. There are serious attempts to see if our own universe has a complex topology by looking at the sky -- and the background radiation in particular -- to see if it looks suspiciously similar in different directions. That would be the tell-tale sign of a multiply connected topology for our own universe.

Cheers -- sylas

Woah... sweet!