# Remind me why FLRW curvature can't switch between cases

• Matterwave
This is because the degree of freedom for this parameter (k) is a fixed constant, and as the universe expands, the curvature gets larger and larger.f

#### Matterwave

Gold Member
Hi, as we all know, the FLRW metric has 3 types of spatial curvatures, spherical, flat, and hyperbolic. I understand that of course due to homogeneity, this curvature must be spatially everywhere the same and so can not depend on the spatial coordinates. However, I can't recall what is the reason the curvature can't go from flat to spherical or spherical to hyperbolic during the dynamic evolution of the universe, as is assumed. Obviously changing from a compact manifold to a non-compact one seems like the manifold must "tear" or something, so it seems to be a good assumption that the universe can't switch between the 3 cases, but what is the actual mathematical or physical argument for why? It is escaping me at the moment.

I think you can examine this by looking at the number of degrees of freedom available.

For a general symmetric metric in four dimensions, there are ten degrees of freedom. This metric can always be diagonalized, which reduces the number of degrees of freedom to four once we allow for coordinate changes. Now, we have decided that our metric will represent a homogeneous and isotropic space. This means that the three spatial dimensions must be identical (if we're using a Euclidean coordinate system at every point), so we can fully describe the system with just two functions. Those functions also cannot depend upon space, so they must be functions that depend only upon time.

And the two functions that we usually choose to make use of are $a(t)$ and $\rho(t)$. We also have two differential equations which we can use to solve for these functions. One option available to us is to make use of the conservation of stress-energy, which given the matter content of the universe at a given time gives us $\rho(a)$, and the first Friedmann equation, which relates the rate of expansion to $\rho$.

So, now that we know that we can describe the system with just these two functions, we know that any additional function we would slip into the equations could only ever be dependent upon these. And this is the case with the spatial curvature: this parameter happens to have an impact on the expansion rate proportional to $k/a^2$, where $k$ is a constant.

Obviously changing from a compact manifold to a non-compact one seems like the manifold must "tear" or something, so it seems to be a good assumption that the universe can't switch between the 3 cases, but what is the actual mathematical or physical argument for why?

Yes, this can be looked at from a topological point of view.

Topologically, k = -1 and k = 0 FLRW universes both have spatial sections that are (homeomorphic to) ##\mathbb{R}^3##, so, topologically, these spacetimes are both ##\mathbb{R}^4##, while k = 1 FLRW universes have spatial sections that are ##\mathbb{S}^3##, so, topologically, these k = 1 spacetimes are ##\mathbb{R} \times \mathbb{S}^3##.

There are theorems that show spacetimes that have topology change also have singularities, possible naked; see Visser's advanced book on wormholes.

These theorems don't apply to k = -1 and k = 0 universe, however, since they have the same topology. For a k = -1 universe, pick a particular spatial section, and discard everything to the future of this section. For a k = 0 universe, pick a particular spatial section, and discard everything to the past of this section. Join the universes along these spatial hypersurfaces using the hypersurface junction formalism; see Poisson's book.

I haven't done this, but I am sure that you find at the joining hypersurface that, mathematically, the spacetime curavture is singular, and that, physically, the stress-energy tensor is singular. To join these spacetimes, a (hyper)surface layer of stress-energy is needed analogous to a surface layer of charge in electromagnetism.

There are theorems that show spacetimes that have topology change also have singularities, possible naked; see Visser's advanced book on wormholes.

Hmm...this confuses me. It seems to contradict the following:

Borde, 1994, "Topology Change in Classical General Relativity," http://arxiv.org/abs/gr-qc/9406053

First, it is stressed that topology change is kinematically possible; i.e., if a field equation is not imposed, it is possible to construct topology-changing spacetimes with non-singular Lorentz metrics. Simple 2-dimensional examples of this are shown.

He also says that without violating an energy condition, GR

does not permit topology change even at the price of singularities [...].

My understanding is that you need CTCs or a violation of the WEC, but singularities are neither necessary nor sufficient. (There is a 1967 paper by Geroch on this, http://adsabs.harvard.edu/abs/1967JMP...8..782G .)

Maybe I'm misunderstanding something...? Is it possible that you're misremembering Visser's results? Or if not, then do you know of any place, e.g., a paper on arxiv, where I could see what he says without buying his book?

In answer to the OP's question, even without getting into the technical stuff (which I haven't waded into in detail), a topology change from a closed FLRW spacetime to an open FLRW spacetime seems like a very implausible type of topology change. It would involve making a transition from a finite universe to an infinite universe, which just at the gut level strikes me as not reasonable: how do you start with some number of degrees of freedom, and then from them evolve into a system that has a number of degrees of freedom that is greater by an infinite factor? A type of topology change that to me is much more intuitively plausible would be something like a "pants diagram," where the universe splits into two parts.

Or, applying Geroch's 1967 result, we can say that FLRW spacetimes don't have CTCs, and we usually assume their matter content to obey the WEC, and therefore this topology change can't happen.

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Torbjorn_L
If the problem seems to be the "spherical -> hyperbolic/flat universe" implies some sort of tearing, or some sort of infinite increase in the degrees of freedom, then what about "flat <-> hyperbolic" changes? Are these possible within the standard FLRW framework? It does seem to imply an infinite change in the "radius of curvature" (whatever the equivalent is for Hyperbolic spaces is) though.

My understanding is that you need CTCs or a violation of the WEC, but singularities are neither necessary nor sufficient. (There is a 1967 paper by Geroch on this, http://adsabs.harvard.edu/abs/1967JMP...8..782G .)

Maybe I'm misunderstanding something...? Is it possible that you're misremembering Visser's results?

Yes, this is exactly what happened. I misremembered results that I read in Visser years ago. I don't have my copy of Visser home with me, and I don't' to go into the office until Monday,