Geometry, Topology and Destiny

By Mark Trodden | April 8, 2012 1:56 pm

I’ve reached the cosmology part of my General Relativity (GR) course, and one of the early points that comes up is my traditional rant against confusing three very distinct concepts when thinking about the universe. Roughly stated, these are; What is the shape of the universe? Is the universe finite or infinite? and Will the universe expand forever or recollapse.

When we apply GR to cosmology, we make use of the simplifying assumptions, backed up by observations, that there exists a definition of time such that at a fixed value of time, the universe is spatially homogeneous (looks the same wherever the observer is) and isotropic (looks the same in all directions around a point). We then specialize to the most general metric compatible with these assumptions, and write down the resulting Einstein equations with appropriate sources (regular matter, dark matter, radiation, a cosmological constant, etc.). The solutions to these equations are the famous Friedmann, Robertson-Walker spacetimes, describing the expansion (or contraction) of the universe.

It is important to take a moment to emphasize what we have done here. GR is indeed a beautiful geometric theory describing curved spacetime. But practically, we are solving differential equations, subject to (in this case) the condition that the universe look the way it does today. Differential equations describe the local behavior of a system and so, in GR, they describe the local geometry in the neighborhood of a spacetime point.

Because homogeneity and isotropy are quite restrictive assumptions, there are only three possible answers for the local geometry of space at any fixed point in time – it can be spatially positively curved (locally like a 3-dimensional sphere), flat (locally like a 3-dimensional version of a flat plane) or negatively spatially curved (locally like a 3-dimensional hyperboloid). A given cosmological solution to GR tells you one of these answers around a spacetime point, and homogeneity then tells you that this is the same answer around every spacetime point. This is what we mean when we say that GR tells us about geometry – the shape of the universe – as depicted in the NASA graphic below.

This raises a very different question that is often confused with the one above. If our solution tells us that the universe is locally a 3-sphere (or flat space, or a hyperboloid) around every point, then does that mean it is a 3-sphere, or an infinite flat 3-dimensional space, or an infinite hyperboloid. This is really a question of topology – how is it connected up – which also answers the question of whether the universe is finite or infinite. To illustrate the point, suppose we have solved the cosmological equations of GR, and discovered that at every spacetime point, the universe is locally a flat 3-dimensional space. This is, by the way, what observations actually indicate our universe is like. Then, just off the top of your head, you can think of many different spaces with precisely this same property. One example is, of course, that the universe is indeed a flat, infinite 3-dimensional space. Another is that the universe is a 3-torus, in which if you were to fix time and trace out a line away from any point along the x, y or z-axis, you traverse a circle and come right back to where you started. This is a finite volume space, that is connected up in a very specific way, but which is everywhere flat, just like the infinite example. In two dimensions, one might visualize it as

Of course, I could have only made one or two directions into circles (leaving it still infinite in some directions), or made the space into a finite one with more than one hole, or any number of other possibilities.

This is the beauty of topology, but it is not something that solving the equations of GR tells us. Rather it is an extra input into our solutions. It is, however, something we can test, most precisely through measurements of the Cosmic Microwave Background radiation, as I may discuss in a later post.

Completely independent of questions of topology, the geometry of a given cosmological solution raises another issue that is often mixed up with those of geometry and topology. Suppose that the universe contains only conventional matter sources (regular matter, dark matter and radiation, say), and suppose you know (you might question whether this is truly possible) that this is all it will ever contain. Then the equations easily predict that, in the case of positive spatial curvature, an expanding universe will ultimately reach a maximum size and recollapse in a big crunch, whereas flat or negatively curved universes will expand forever. These are predictions of the destiny of the universe, and often lead to the following connection

However, as I made clear, there are some assumptions that go into the connection between geometry and destiny, and although these may have seemed reasonable ones at one time, we know today that the accelerated expansion of the universe seems to point to the existence of some kind of dark energy (a cosmological constant, for example), that behaves in a way quite different from conventional mass-energy sources. In fact, we know that for sources like this, once acceleration begins, it is easily possible for a positively curved universe, for example, to expand forever. Indeed, in the case of a cosmological constant, this is precisely what happens.

So the universe may be positively or negatively curved, or flat, and our solutions to GR tell us this. They may be finite or infinite, and connected up in interesting ways, but GR does not tell us why this is the case. And the universe may expand forever or recollapse, but this depends on detailed properties of the cosmic energy budget, and not just on geometry. Cosmological spacetimes are some of the simplest solutions to GR that we know, and even they admit all kinds of potential complexities, beyond the most obvious possibilities. Wonderful, isn’t it?

  • Sili

    This is probably a stupid question, but how can a universe be isotropic if it isn’t also homogenous? Doesn’t the former entail the latter? What would be an example of world that looks the same in all directions, but isn’t everywhere the same?

  • Mark Trodden

    One only needs a spacetime with a center but that looks the same in all directions from that one point. An example that is not a cosmological spacetime is the Schwarzschild spacetime describing a black hole or the spacetime around the Sun. This is isotropic around one point but not homogeneous.

    It is important to note that this is isotropy about a point. If we automatically demanded isotropy about every point, then we would, indeed, have homogeneity.

  • Fernando Curiel

    Truly wonderful. I actually forgot until now I had this confusion after my graduate course in GR. But the instructor did not seem to understand it better. I think this could make also for some interesting concept problems in a GR course. And it gave me a couple of ideas for my spanish blog. Glad you shared this!

  • Pedro J.

    Sili, take a look at
    One circle is homogeneus but not isotropic and the other is isotropic but not homogeneous.

  • Greg Egan

    Though it might not be applicable to our own universe, I think it’s worth mentioning Myers’ Theorem, which states that a manifold with positive Ricci curvature, bounded below by some non-zero value, must have a finite volume.

    I always thought it was intuitively reasonable to suppose that a manifold could be locally isometric to a 3-sphere and yet somehow find a way to avoid closing up on itself … but Myers’ Theorem says that really can’t happen.

  • Moshe

    One technical question I should know the answer to. Statements connecting local geometry to global topology must rely on some smoothness conditions, for example absence of singularities. But this seems to me problematic: not only can we not know for sure those conditions apply to any point in spacetime, we do know that some singularities (i.e. black hole singularities) do in fact exist in our very own spacetime. Are there weaker statements about the connection between topology and some suitably “averaged out ” geometry that one can make?

    • Sean Carroll

      Moshe– Black holes have spacelike singularities in the future, so in principle they wouldn’t prevent you from constructing nonsingular spacelike surfaces. But that’s probably not the answer you were looking for.

      I presume if you had a singularity on your spacelike surface, you could cut it out by removing a spherical region around it and replacing that with some nonsingular geometry that matched on the boundary. But I haven’t thought about it carefully.

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  • Moshe

    Thanks Sean. Black holes are a bad example, but I am I still puzzled by the issue of singularities, which are fairly common even if you start with non-singular initial data. I assume that smoothness can be replaced by a condition that some averaged curvature invariant is locally bounded. But, since we cannot possibly know that any statement like that is valid absolutely everywhere in spacetime, can we make any inference about the topology of space/spacetime?

    • Sean Carroll

      I doubt it. Not sure what right we would have to say anything about nontrivial handles etc. at the Planck scale, for example. And also I think we have little/no right to make assumptions about the geometry outside our horizon, so I would be loathe to say anything about the very largest scales either.

  • Trevor

    One thing I always wondered about — why does the curvature “k” in the FLRW metric have to be independent of time? Why can’t the Universe go from positively curved to negatively curved over time? Does that somehow violate homogeneity and isotropy because of the problem of trying to uniquely define a slice of constant time?

  • Igor Khavkine

    Moshe, this may be by now obvious, but the crucial assumption that allows us to make conclusions about global topology is, as already mentioned by Mark, the homogeneity hypothesis. Are you perhaps wondering to what extend this hypothesis could be relaxed yet still allow us to make global conclusions? One kind of answer was pointed out by Greg above: the positivity of Ricci curvature forces the topology to be compact. Homogeneity is relaxed, since the curvature is not everywhere the same, but not completely since it is still everywhere bounded from below. I belive that there are also results (I think due to M. Gromov and others) that restrict the topology by bounding the number of holes (Betti numbers) from above, provided the (sectional?) curvature is everywhere negative.

    These results appear to be consistent with the intuition that conclusions about global topology survive a slight weakening of the homogeneity hypothesis.

  • Rhys

    Other commenters have already touched on this, but…

    Mark, your distinction between geometry and topology is an important one, but they are not completely independent. In two dimensions, every undergraduate knows the Gauss-Bonnet theorem, relating the Euler number of a surface to the average of its curvature; in particular, the only positively-curved two-dimensional manifold is the two-sphere. In three dimensions, things are more complicated (and I’m no expert); for example, the three-sphere admits an infinite number of free quotients (the lens spaces), the local geometry of which is therefore identical. Nevertheless, I think there are still relatively fewer possibilities when the curvature is positive or zero, than when it is negative.

    One other comment, which just occurred to me, is the two meanings of “homogeneous”. In mathematics, a homogeneous space is one with a transitive isometry group, whereas in physics it seems that we mean something weaker: any two points have isometric neighbourhoods. Obviously the first implies the second, but it’s unclear to me, what conclusions can be drawn from the second alone.

  • cormac

    Superb post. This is exactly the sort of point that gets left out (by necessity) in popular accounts, and that leads to all sorts of questions in the mind of the reader. Are the course notes online?

  • Phil

    How is a saddle shaped geometry positive curvature? Isn’t it positive on one axis and negative on the other? So wouldn’t this look like a bowl, vs a saddle?

  • Jose

    Trevor (#11),

    Changing the sign of the curvature implies changing the spatial topology, and there are theorems forbidding that, see e.g.

    Incidentally, if you allow k to vary in space you get the inhomogeneous Lemaitre-Tolman-Bondi cosmologies:

  • Moshe

    Igor, I was just wondering how much these conclusions (or even definition of topology) depends on their assumptions holding to arbitrarily short distances, which is what you have to do when you use mathematical theorems relating geometry to topology. For example, can we define and reach any conclusion on something we may call the averaged “large scale” topology, when we assume homogeneity only on average, and allow for exceptions on measure zero set (with some conditions on such exceptions)? Those would be more realistic assumptions, I think, because I don’t think we can assume that spacetime is smooth and is described by the usual GR structures to arbitrarily short distances, though it certainly does “on average”. As Sean points out, we also cannot know what is going on outside the horizon, but I think this is a separate issue.

  • Sili

    Oh, so it’s just one special point that need be isotropy around? I’d missed that completely. Which one is it, if that makes sense?

    What book are you using by the way?

  • Tintin

    Jose (#16) Personally, I do not see anything wrong with the Lemaitre-Tolman metrics (but I haven’t looked very hard).

  • Mark Trodden

    Getting in late from work and it’s nice to see people have provided better answers to some of these than I could have. Moshe, I don’t know the answer in general – I don’t think we can say much, and of course, once we get to very small scales, we’ll need to depart from GR anyway, as Sean said.

    Rhys – indeed there can be connections, but the point I was making was that in general they are different concepts. I also think the possibilities are fewer for positive curvature. For compact hyperbolic manifolds, by comparison, there are connections between the volume (in units of the curvature radius) and the topology.

    Sili – For a general space that is just isotropic there is indeed a special point – the one about which the space is isotropic. But our favorite cosmological spaces are isotropic and homogeneous, and so are isotropic about every point.

  • Sili

    Doh. Now I get it. I think. Thanks.

  • don’t kill the messenger

    I’d love to see a post on how the CMB probes the topology..

  • Jose

    Tintin (#19),

    The comment on the LTB metrics was just an aside. There is nothing wrong with these metrics, they are just inhomogeneous generalizations of the standard FLRW metrics. As in the FLRW case, the sign of the spatial curvature (which is the sign of the function E) is fixed for the LTB metrics, so you cannot go from positive to negative spatial curvature (nor can you have topology change).

  • Mark Trodden

    dktm #22: I’ll try to write one soon.

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  • Daina Taimina

    For those who want to learn more about the differences between geometry and topology let me suggest my book – 2012 Euler Book Prize winner:

    which is aimed to the general audience and mostly explains things visually.
    btw NASA picture depicts a surface with constant negative curvature incorrectly :-)

  • julianpenrod

    So often, crucial points go unrecognized and ignored among everything else. Mark Trodden speaks of a “fixed value of time”. In a universe governed by “Einstein’s relativity”, however, a fixed value of time is meaningless. Every particle moving in their own universe of measuremenst would see things differently from others. Indeed, what “shape” does the universe have for particles oving at different velocities? Shills for the “relativity” might say the universe has the same shape for all, overall. But, then, that establishes that uniform shape as a uniform frame of reference!

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  • Ray Gedaly

    Perhaps this is a ridiculous question (most of mine are), but does GR have amything to say about the geometry of time? At the very least, it could be either infinite or finite (finite at a singularity), and if finite than finite in either one or both directions. But why not curved time?

    Of course the case of a positively curved time dimension might suggest a closed circle, and that time repeats. Such a thing might produce a cyclic universe, although the more common idea of a cyclic universe assumes a flat time geometry.

  • Ray Gedaly

    Does GR differentiate in any way between normal and dark matter? It seems not to matter (no pun intended). Thus, can dark matter collapse to form a black hole?

    Of course the term black hole may not be a proper term for a type of matter that doesn’t interact with EM. But it would seem that a dense enough accumulation could collapse to form a singularity.

    A normal black hole would radiate Hawking radiation. Would not a dark matter black hole radiate as well, since Hawking radiation is a result of virtual particles falling/escaping the black hole?

    But then would not the dark matter black hole behave as a clasical blackbody and radiate EM energy?

  • the clayton peacock

    Anything can collapse to a black hole, and black holes should radiate democratically to all states that are thermally or kinematically accessible. The mechanism of production is virtual particle creation near the horizon – it does not come from the black hole itself.

    The point is that after the black hole forms, nature forgets what it was made of. It’s just a bunch of energy sitting there. Radiation is nature redistributing that energy thermally.

  • Mark Trodden

    Ray #30: As in the discussion of spatial geometry, GR has everything to say about curvature in the time direction (indeed, that’s what we have in cosmology) – and that is what geometry is about. Questions of whether time is a circle etc. are about topology, and can, at least in part, depend on initial or boundary conditions.

    Ray#31: peacock #32 is right in principle – anything can be a black hole. However, in the case of dark matter, it is quite hard to form a black hole, and the reason isn’t to do with GR, but with particle physics. We know dark matter is at best weakly interacting, and so unlike regular matter, which can lose energy, and hence collapse, by emitting radiation, dark matter has a hard time losing energy, and so doesn’t collapse easily. This is the basic reason why we have large, puffy dark matter halos around galaxies in which the regular matter is clumped up in to a disk and bulge.

  • Raskolnikov

    Are there theories predicting a definite topology for the universe? I’m not talking about phenomenological theories that just plug in some global topology and then make predictions based on that. No, I’m really asking for a theory that says “the universe should be a donut because…” or “If this principle holds, we should expect a Poincaré dodecahedral space…”.

  • Mark Trodden

    #34: Not that I know of.

  • Mano Singham

    Excellent post, Mark.

    The only thing that I would suggest is to expand slightly on how you can connect locally flat spaces in ways other than to get a flat infinite three-dimensional space. It is not obvious how, in the 2-D case where you have the donut, it is locally flat.

  • Chris Duston

    Great post; some of the issues of complete relativism in GR are discussed in a very nice article by Lee Smolin, which you can find in The Structural Foundations of Quantum Gravity (2006), pg 196. He talks about GR being “partially relative” in the sense that although the geometry is dynamically generated, the dimension, topology, smooth structure, and signature are not. A good read if anyone is interested.

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  • Agnes

    Very well written. Excellent observations. I might add the idea that it is something to consider the fact that any given object can potentially be of any given size – infinitely large or small – and that any event can potentially take any amount of time – infinitely long or infinitely short.


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About Mark Trodden

Mark Trodden holds the Fay R. and Eugene L. Langberg Endowed Chair in Physics and is co-director of the Center for Particle Cosmology at the University of Pennsylvania. He is a theoretical physicist working on particle physics and gravity— in particular on the roles they play in the evolution and structure of the universe. When asked for a short phrase to describe his research area, he says he is a particle cosmologist.


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