I Spacetime topology of the Universe

[cianfa72]

TL;DR Summary

Speculations about the topology of the Universe as spacetime

The question might be a bit weird: which are the current "speculations" about the topology of the Universe as spacetime ?

I'm aware of, from the point of view of spacelike hypersurfaces of constant cosmological time, the topology of such "spaces" might be nearly flat on large scale.

What about the topology of spacetime itself? Thanks.

 

[martinbn]

I wouldn't say that flatness is a topological property. It requires more structure. If the space slices are homeomerphic to $X$, the the space time is homeomorphic to $X\times \mathbb R$.

 

[cianfa72]

I wouldn't say that flatness is a topological property. It requires more structure.

Ah yes, I should say spacelike hypersurfaces of constant cosmological time homeomorphic to a "flat" 3D space.

If the space slices are homeomerphic to $X$, the the space time is homeomorphic to $X\times \mathbb R$.

Why ? According to you the spacetime as whole should have the topology of a sort of "sheet" or "cylinder".

 

[PeterDonis]

the topology of such "spaces" might be nearly flat on large scale.

That's not topology, that's geometry. In standard FRW models of the universe, the topology of spacetime is $\mathbb{R}^4$ if the universe is spatially infinite, or $\mathbb{S}^3 \times \mathbb{R}$ if it is spatially finite (closed).

 

[PeterDonis]

spacelike hypersurfaces of constant cosmological time homeomorphic to a "flat" 3D space.

That's no better than your previous statement; "homeomorphic" says nothing whatever about geometry, "flat" or otherwise. If it's homeomorphic to $\mathbb{R}^3$, it's homeomorphic to any space with topology $\mathbb{R}^3$, whether it's flat or not.

 

[cianfa72]

In standard FRW models of the universe, the topology of spacetime is $\mathbb{R}^4$ if the universe is spatially infinite, or $\mathbb{S}^3 \times \mathbb{R}$ if it is spatially finite (closed).

In standard FRW models, the timelike congruence of comoving observers is hypersurface orthogonal (let's say it is "irrotational", i.e. it has zero vorticity), yet in general it isn't stationary (spacelike hypersurfaces of constant cosmological time don't have the same geometry).

Does "expanding universe" mean that the "proper distance" between galaxies, evaluated as the length along geodesic curves connecting them in any of such spacelike hupersurfaces (geodesic curves when restricted to any of such hypersurfaces), increases with cosmological time ?

 

[PeterDonis]

In standard FRW models, the timelike congruence of comoving observers is hypersurface orthogonal (let's say it is "irrotational", i.e. it has zero vorticity), yet in general it isn't stationary (spacelike hypersurfaces of constant cosmological time don't have the same geometry).

None of these things have anything to do with topology. This thread is about topology, isn't it? What do you want to talk about?

Does "expanding universe" mean that the "proper distance" between galaxies, evaluated as the length along geodesic curves connecting them in any of such spacelike hupersurfaces (geodesic curves when restricted to any of such hypersurfaces), increases with cosmological time ?

None of this has anything to do with topology either. What is the topic of this thread supposed to be?

 

[cianfa72]

None of this has anything to do with topology either. What is the topic of this thread supposed to be?

You are right, it was a bit OT. It was just to better understand the meaning of "expanding universe".

Coming back to spacetime topology, FRW models (or better FLRW ?) with topologies $\mathbb R^4$ or $\mathbb{S}^3 \times \mathbb{R}$ allow in principle Closed Timelike Curves (CTC) as topological manifolds. However the relevant structure for their actual existence is the metric (metric tensor $g_{ab}$) i.e. let me say the actual "distribution" of light cones through spacetime.

Do exist examples of FWR models that have CTC ?

 

[PeterDonis]

It was just to better understand the meaning of "expanding universe".

Then start a separate thread on that question.

 

[PeterDonis]

FRW models (or better FLRW ?) with topologies $\mathbb R^4$ or $\mathbb{S}^3 \times \mathbb{R}$ allow in principle Closed Timelike Curves (CTC) as topological manifolds.

This is nonsense. There is no such thing as a CTC as a topological manifold; "timelike" requires a metric, which is not a topological concept.

I am closing this thread because you can't even pose a question that is on the topic you yourself chose for the thread.