I Shape of the universe -- Does it change with time?

[MathematicalPhysicist]

Are there any cosmological models that say that the shape of the universe changes with time?

As in it may oscillate between being a sphere, torus , saddle or a plane?

I remember the course in Astrophysics that I took in my BSc studies, and we had it that the shape of the universe depends on the cosmological constant, $k$ either it's positive, negative or zero.

And the options were if we had k>0 then it's a sphere if it's k<0 it's a saddle and for k=0 a plane.

Why isn't there an option of a torus?

 

[Chalnoth]

There are two separate things that are meant by "shape". First, there's the local spatial curvature. This is what is represented by $k$. In a homogeneous, isotropic universe, General Relativity says that the value $k$ is a constant.

The second meaning of shape is the overall connectedness of the universe. Such as whether it's a sphere that wraps back on itself, a torus, or something else. Locally a torus and a flat plane are identical (as an idealized torus has no spatial curvature). They only differ in their global properties, and unfortunately it's very difficult to determine what the global topology of our universe is when we can only ever observe a small corner of it.

 

[MathematicalPhysicist]

@Chalnoth are there any models that attest that the shape of the universe oscillates between different shapes?

As in the overall connectedness of the universe.
Why is homogeneity and isotropy will imply that the value of $k$ should be constant?
Lay on me the maths, I hope next year to return to Schutz's book on GR, so I'll understand your answer better. (in my undergrad studies I learned of Differential Geometry, so perhaps I just need a bit of a refresh).

 

[Chalnoth]

@Chalnoth are there any models that attest that the shape of the universe oscillates between different shapes?

I'm pretty sure the answer is no.

As in the overall connectedness of the universe.
Why is homogeneity and isotropy will imply that the value of $k$ should be constant?

The short answer is that the homogeneity and isotropy guarantee that the curvature will evolve in a simple manner over time. I believe the spatial curvature at any given time is actually $k/a(t)^2$ (though the precise formula depends a bit upon conventions).

Another way to put it is that since we have the assumption of homogeneity and isotropy, the spatial curvature must be identical everywhere (or else it would violate the assumption of homogeneity). But it could still change with time, which we can use to reformulate as a function of the scale factor instead. Solving Einstein's Equations shows that that function is proportional to $1/a^2$.

 

[phinds]

@Chalnoth are there any models that attest that the shape of the universe oscillates between different shapes?

It seems that that would require a state change of some radical sort, wouldn't it? Seems exceedingly unlikely.

 

[newjerseyrunner]

No, how could it? Time is a fundamental part of the universe, so how can the universe change in respect to it?

The observable universe is usually represented as this shape: Starting at the bottom and extending infinitely upwards, as well as starting at a point and expanding outwards in 3 spatial dimensions. In this representation time is the y axis.

http://www.thephysicsmill.com/blog/wp-content/uploads/Universe_history.jpg

The shape of observable SPACE changes over time (it expands.). Any given moment is described by taking the image I posted and it along a 3D slice of it.

 

[timmdeeg]

The shape of observable SPACE changes over time (it expands.).

It seems you mean size, not shape. A change in shape would be e.g. sphere to dodecahedron.

 

[martinbn]

You can try to search for topology change in GR. I have a vague memory that in classical/non-quantum GR there are results that forbid topology change, but I am not very sure what the assumptions are.

 

[timmdeeg]

You can try to search for topology change in GR.

Right, this is what I've found:
https://arxiv.org/pdf/gr-qc/9406006.pdf

"Nevertheless, our results may be interpreted as providing evidence that topology change
is suppressed."

Otherwise one should have an idea regarding the cause for such a change.

 

[Chalnoth]

It seems you mean size, not shape. A change in shape would be e.g. sphere to dodecahedron.

I don't know if this is what you were referring to, but I'm pretty sure a change from a sphere to a dodecahedron would not be a change in topology. It would be a change in local curvature at various locations. I could easily imagine something like that might happen with the right matter distribution in the universe.

A change in topology would be a change in the large-scale connectedness, such as a change from a sphere to a torus. I don't think that's likely to be possible, and definitely could not be achieved by any classical means.

 

[timmdeeg]

I don't know if this is what you were referring to, but I'm pretty sure a change from a sphere to a dodecahedron would not be a change in topology. It would be a change in local curvature at various locations. I could easily imagine something like that might happen with the right matter distribution in the universe.

Indeed, my search wasn't constructive. So one has to be careful to distinguish 'shape' from 'topology'.
Would you agree that a change of the shape can only be excluded in the case of the ideal fluid model?

 

[Chalnoth]

Indeed, my search wasn't constructive. So one has to be careful to distinguish 'shape' from 'topology'.
Would you agree that a change of the shape can only be excluded in the case of the ideal fluid model?

Are you asking whether an inhomogeneous universe model might have the average spatial curvature change by some function other than $k/a^2$? I suppose it's conceivable that large-scale inhomogeneities might do that. But we don't have large-scale inhomogeneities that are large enough to do this.

 

[timmdeeg]

Are you asking whether an inhomogeneous universe model might have the average spatial curvature change by some function other than $k/a^2$?

No, I was considering the homogeneous model with everywhere locally the same value for $k/a(t)^2$. Given topology and $k$ don't change, would this scenario still allow certain changes of shape? I seems unthinkable, but I have no better example than torus vs. Klein bottle in the moment.

The Einstein Equations don't say anything about topology and shape. Usually a change of state 1 to state 2 is driven by something. But perhaps it's already wrong to ascribe the term 'state' to the torus and the Klein bottle respectively.

 

[Chalnoth]

No, I was considering the homogeneous model with everywhere locally the same value for $k/a(t)^2$. Given topology and $k$ don't change, would this scenario still allow certain changes of shape? I seems unthinkable, but I have no better example than torus vs. Klein bottle in the moment.

I don't see how. Certainly General Relativity has no dynamics which could conceivably be involved in a change in shape.

If you are instead thinking of a lower-level theory, such as string theory, then on the surface it seems at least plausible for a change in shape to occur if the universe is a brane, and that brane wraps back and collides with itself, tearing into a new shape. If that were viable, it would seem to me to be a very violent event. But I think at a minimum it would require large extra dimensions, which could exist, but are definitely speculative.

 

[Khashishi]

If we assume global homogeneity and isotropy, then any change in topology would have to be a completely catastrophic event. I imagine it would look something like the Big Bang. I suppose another possibility would be something like a Big Rip, with infinite expansion in finite time.

 

[Khashishi]

On the other hand, the topology of the fine grained universe is constantly changing as black holes collide, right?

 

[PeterDonis]

the topology of the fine grained universe is constantly changing as black holes collide, right?

No. A pair of colliding black holes is just a single black hole with an event horizon which, if we imagine time as vertical and space as horizontal, looks like a pair of trousers instead of a cylinder, roughly speaking. The overall topology of spacetime is the same.

 

[Khashishi]

But if you look at time slices, the topology of space changes at the inseam of the trousers, no?

 

[PeterDonis]

if you look at time slices, the topology of space changes at the inseam of the trousers, no?

No. The topology of a spacelike slice is $S^2 \times R$ everywhere. The horizon is just a null surface, it doesn't do anything to the topology.

 

[timmdeeg]

If you are instead thinking of a lower-level theory, such as string theory, then on the surface it seems at least plausible for a change in shape to occur if the universe is a brane, and that brane wraps back and collides with itself, tearing into a new shape. If that were viable, it would seem to me to be a very violent event. But I think at a minimum it would require large extra dimensions, which could exist, but are definitely speculative.

Ok, thanks.

 

[phinds]

But if you look at time slices, the topology of space changes at the inseam of the trousers, no?

Think of flatland. Two circles merge into an oval. It's still flatland.

 

[Khashishi]

Think of flatland. Two circles merge into an oval. It's still flatland.

It's the same genus, but you've gone from two holes to one hole. That's a change in topology.

 

[PeterDonis]

It's the same genus, but you've gone from two holes to one hole.

I think you've misunderstood my description. I was describing the event horizon, not the spacetime as a whole.

Even with that said, your statement is not correct as applied to the topology of the event horizon. We are talking about a surface in spacetime, not its intersection with any spacelike slice. But in any case, the topology of the EH is not the same as the topology of the spacetime as a whole.

 

[Khashishi]

No. The topology of a spacelike slice is $S^2 \times R$ everywhere. The horizon is just a null surface, it doesn't do anything to the topology.

I'm staring at the diagrams in https://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates
It looks to me like the topology of a spacelike slice is $S^2 \times R$ only for times $-1 < T < 1$.
For $T > 1$, it looks like $S^2 \times (\text{R with the center cut out})$
So it looks like even just a single black hole represents a change in topology between spacelike slices. Of course, the singularity is just where GR breaks down, and the singularity is only a single point, despite occupying a region of X values in K-S coordinates.

 

[PeterDonis]

It looks to me like the topology of a spacelike slice is $S^2 \times R$ only for times $-1 < T < 1$.

If you treat "spacelike slice" as being coordinate-dependent, yes. But there is a coordinate-independent way of specifying what a "spacelike slice" is: it's a Cauchy surface, which is a spacelike hypersurface that intersects every timelike curve in the spacetime exactly once. The slices for $T > 1$ or $T < -1$ in Kruskal coordinates are not Cauchy surfaces. The topology of every Cauchy surface in this spacetime is $S^2 \times R$, and since there are $R$ of them, the topology of the spacetime as a whole is $S^2 \times R^2$. And since the topology of the spacetime as a whole can't change depending on your choice of coordinates, the answer using Cauchy surfaces must be the right one.

 

[Khashishi]

Thanks. I think I understand that.

 

[rootone]

Are there any cosmological models that say that the shape of the universe changes with time?

I don;t think so, but I think phase transitions are possible
Ice is another form of water stuff.

 

[timmdeeg]

I don;t think so, but I think phase transitions are possible
Ice is another form of water stuff.

This idea seems to assume that there exist shapes of the universe with different energy levels so that such a transition would produce heat. Hmm, hard to imagine.