Simply connected vs. non-simply connected

[RModule]

Hello,

I'm a mathematics student specializing in (semi-)Riemannian geometry and relating this to relativity. My main reference is O'neill's "Semi-Riemanninan Geometry with Applications to Relativity".

This book defines the Robertson-Walker spacetime as follows:

Let $$S$$ be a connected three-dimensional Riemanninan manifold of constant curvature $$k = -1,0,1$$. Let $$f>0$$ be a smooth function on an open interval $$I$$ in $$R_1^1$$ Then the warped product $$M(k,f) = I \times_f S$$ is called a RW spacetime. Explicitly $$M(k,f)$$ is the manifold $$I\times S$$ with line element $$-dt^2+ f^2(t)d\sigma^2$$ where $$d\sigma^2$$ is the line element of $$S$$.

Is this a non-standard definition? From what I can see on wiki, the manifold should be simply connected. Further, does this definition imply that $$M$$ is orientable?

Because, from this definition $$\mathbb{R}P^3 \times I$$ would be a RW spacetime, even though it is not simply connected. Which leads me to the following; $$S^3$$ and $$\mathbb{R}P^3$$ are locally "the same" but globally quite different. What relativistic implications would not being simply connected have on the spacetime? I guess you essentially could see the same stars from two(or more) different directions. Are there any other important differences?

Thank you.

 

[George Jones]

Usually, "standard" cosmological models are required to be spatially homogeneous and spatially isotropic (see Wald for mathematical definitions) (the Copernican principle). You might want to look at the references in

https://www.physicsforums.com/showthread.php?p=2587993#post2587993.

 

[marcus]

Module,
Starting from what GJ said (taking for granted we agree on that) there is still the question of "simply connected". I do not think that S is required to be simply-connected: for example a 3-torus would be OK, and is sometimes taken as an illustrative example. That would not be simply connected. (GJ uses it as an example of flat, or zero curvature, in that other thread.)

So if T³ (cross a time interval) is allowed then the real projective space that you mentioned would also be allowed. I would certainly think!

That doesn't mean that working cosmologists seriously consider exotic topologies like that.
RW spacetimes are a broad category, about which Robertson Walker proved a theorem.

The standard cosmo model is not some very general thing like an RW. It is more like "FLRW" for Friedmann Robertson Walker Lemaître, and more specifically the case they mostly considered is called LCDM for Lambda (the cosmo const.) Cold Dark Matter.

An observational effort has been made (Cornish Spergel and others) to rule out spatial periodicities out to the greatest possible distance. But of course one cannot say anything past a certain distance, there could be periodicity on such a large scale that we can't see it and never will be able to see it.

As a mathematician you have the freedom to consider such possibilities.

I have nothing substantive to say beyond what GJ said already. Just want to communicate the idea that I'm mildly open to non-simply-connected spatial topologies and appreciate your question.

 

[RModule]

Thank you for your replys. I did read about the dodacahedral, and it is indeed really interesting.

Now, instead of asking if our universe in fact has a non-trivial topology, I ask - what would be different if we lived on T^3 or RP^3?

As an example: a difference between a closed universe and a flat universe is that we can see mirror images of stars in a closed one.

Can you think of "similar" examples between simply connected and multiply connected?

(I know there are problems around the existence of Dirac particles, but I would prefer an answer which is more in layman's terms)

 

[sardar]

Hello,

Thank you for your question. The concept of simply connected vs. non-simply connected is an important one in mathematics and has implications in various fields, including relativity. Simply connectedness refers to the topological property of a space in which every loop can be continuously deformed into a single point. Non-simply connected spaces have loops that cannot be continuously deformed into a single point.

In the context of Robertson-Walker spacetime, the definition provided in O'neill's book is a standard one. The reason for this is that this definition allows for a wider range of spacetimes to be considered, including non-simply connected ones. This is important in the study of relativity as it allows for the consideration of more complex and realistic scenarios.

The fact that a spacetime is not simply connected does not necessarily imply that it is not orientable. Orientability refers to the ability to assign a consistent direction to every point in a space, and it is possible for a non-simply connected space to be orientable. In the case of the Robertson-Walker spacetime, orientability is not specified in the definition, so it would depend on the specific choice of S and f.

The difference between S^3 and \mathbb{R}P^3 in terms of simply connectedness has important implications in relativity. For example, in a non-simply connected spacetime, there exist closed timelike curves (CTCs) which are paths that loop back in time. This can lead to paradoxes and inconsistencies in the theory of relativity. In the case of S^3, CTCs are not possible, but in \mathbb{R}P^3, they are. This highlights the importance of considering simply connectedness in the study of relativity.

In terms of other differences, non-simply connected spacetimes can exhibit more complex topological and geometric properties, which can have implications in the behavior of matter and energy in these spacetimes. For example, in a non-simply connected spacetime, the paths of particles can be affected by the curvature of the space in different ways, leading to observable differences in the behavior of matter and energy.

I hope this helps to clarify the concept of simply connected vs. non-simply connected and its implications in the study of relativity. Thank you for your question.