Exploring the Universe's Topology: Insights from a New Study | Space.com"

[Pavel]

Hi, I've read this article (http://www.space.com/scienceastronomy/mystery_monday_040524.html), which says:

"In the new study, researchers examined primordial radiation imprinted on the cosmos. Among their conclusions is that it is less likely that there is some crazy cosmic "hall of mirrors" that would cause one object to be visible in two locations. And they've ruled out the idea that we could peer deep into space and time and see our own planet in its youth."

My understanding is that the Universe must have some kind of topology in Hyperspace. It was widely believed to be some kind of toroid, which would make the universe finite, but unbounded. Correct? If so, this new study says the topology is not toroidal, but then what is it? Does this mean now we can talk about the "edge" of the universe in whatever sense the new topology might imply?

Thank you in advance for clearing up the confusion.

Pavel.

 

[Wallace]

The overall topology of the Universe is unknown. Current data is consistent with a Universe that is spatially flat everywhere on large scales, which implies that the Universe is infinite in extent. However, within the uncertainty regions permitted by current data, it is possible that the Universe has a small amount of positive or negative spatial curvature. If the curvature is positive, then the Universe would be finite and allow the 'Hall of mirrors' effect in principle. However given the constraints on any curvature are pretty tight, the Universe is very very big whatever the curvature actually is.

Of course of all this depends on the model that is being fitted to the data being correct, which is also a subject of debate.

 

[Pavel]

The overall topology of the Universe is unknown. Current data is consistent with a Universe that is spatially flat everywhere on large scales, which implies that the Universe is infinite in extent. However, within the uncertainty regions permitted by current data, it is possible that the Universe has a small amount of positive or negative spatial curvature. If the curvature is positive, then the Universe would be finite and allow the 'Hall of mirrors' effect in principle. However given the constraints on any curvature are pretty tight, the Universe is very very big whatever the curvature actually is.

Of course of all this depends on the model that is being fitted to the data being correct, which is also a subject of debate.

I thought the curvature was really a "local" thing and had little bearing on the overall topology. IN other words, all possible three curvatures are still consistent with a torus. So, even if the CMBR shows flatness as far as we can see, the overall universe can still be toroidal. Can it not? And if so, how can they say "they ruled out the idea of hall of mirrors"? Perhaps they mean that as far as we can see and we will ever be able to see, there won't be any Halls of Mirrors. Is that it?


Pavel.

 

[marcus]

that's an old article
it is space.com
full of confusion because popular journalism

you can easily find Neil Cornish journal article (peer-review professional grade)
free for download on arxiv.org
then you see what Cornish, Spergel et al REALLY said
not what the space.com popular journalist THOUGHT he heard them say

just go to arxiv.org and put in Cornish for the author and 2004 for the year or something like that

In the space.com article, if you read it, it quotes Cornish saying
they did NOT rule out
http://www.space.com/scienceastronomy/mystery_monday_040524.html

"Our results don't rule out a hall-of-mirrors effect, but they make the possibility far less likely," Cornish told SPACE.com, adding that the findings have shown "no sign that the universe is finite, but that doesn't prove that it is infinite."

You ask the question "how can they say they rule out?"

And if so, how can they say "they ruled out the idea of hall of mirrors"? ...

The answer is that they didn't say this.
=================

If I remember right we had a PF thread about this back in 2004, anyway I remember discussing it and reading the journal article.

He has written more recently (2006) on the same thing, extending the distance bound outwards with new data, IIRC,
or reducing the uncertainty.

IMHO it is not very interesting.

 

[Wallace]

I thought the curvature was really a "local" thing and had little bearing on the overall topology. IN other words, all possible three curvatures are still consistent with a torus. So, even if the CMBR shows flatness as far as we can see, the overall universe can still be toroidal. Can it not?

Nope, if the Universe is flat then it is flat. It can't be flat and a torus at the same time! Curvature is neither a strictly 'local' or 'global' thing, the Sun causes curvature that causes the Earth to orbit it, yet the overall average (spatial!) curvature of the Universe may be zero.

 

[Hurkyl]

Nope, if the Universe is flat then it is flat. It can't be flat and a torus at the same time! Curvature is neither a strictly 'local' or 'global' thing, the Sun causes curvature that causes the Earth to orbit it, yet the overall average (spatial!) curvature of the Universe may be zero.

A torus can certainly be flat...

 

[Wallace]

More information and application to cosmology?

 

[marcus]

A torus can certainly be flat...

that's for darn sure!
the typical example cosmologists always give of a flat universe which is compact (finite volume) is a toroidal one.

in 2D it is easy to see. you take a flat piece of paper and declare N and S edges to be identified-----so you have a cylinder topologically but it is still a piece of paper lying flat on the table
and then you declere the E and W edges identified and you have a torus topologically, but it is flat.

the only time you would need to consider a curved torus surface is if you EMBEDDED the 2D torus into a higher dimension surrounding space

however our universe is not considered to need to be embedded in some higher D surrounding space. it is sufficient unto itself.

so it could be a flat 3D cube like the flat piece of paper, and you identified the N and S faces, and the E and W faces, and the U and D faces----and it would be a 3D toroid thing, and it would be flat, metrically speaking.

but all that fancy topology stuff is silly IMHO, probably the U is just infinite R-three or finite S-three

 

[Wallace]

Sounds silly to me, if you can't measure it, it doesn't exist!

 

[pervect]

More information and application to cosmology?

Consider the topology of the arcade game "Asteroids". If you're not familiar with the game, you have a flat square screen, and the top of the screen wraps around to the bottom, the left side wraps around to the right hand side.

The topology of this screen is that of a torus. However, it has no curvature. It's flat both in the intuitive sense, and in the mathematical sense that the curvature tensor vanishes everywhere.

I don't know if anyone is seriously considering this as a possible topology of the universe, but you can't rule it out from Einstein's field equations.

 

[Wallace]

I don't know if anyone is seriously considering this as a possible topology of the universe, but you can't rule it out from Einstein's field equations.

Really? That surprises me, I would have thought that this would violate something from standard GR. Something doesn't feel right. I'll have a think...

 

[chronon]

Something doesn't feel right. I'll have a think...

It doesn't feel right to me either - I can't help feeling that such a topology would make the twin paradox into a real paradox, and this could only be resolved by it having some local effect. But that doesn't mean that people aren't thinking about it.

One easy to read book about this topic is How the universe got its spots by Janna Levin. I've written a review of this book at http://www.chronon.org/Science/How_the_universe_got_its_spots.php

 

[Hurkyl]

It doesn't feel right to me either - I can't help feeling that such a topology would make the twin paradox into a real paradox, and this could only be resolved by it having some local effect.

You're referring to the the cosmological twin 'paradox', I presume? It's only a pseudoparadox -- like the usual twin paradox, it arises from making a conceptual mistake.

A geodesic is a local maximum for the proper duration along a timelike path between two points -- i.e. when traveling between a given pair of points in space-time, an inertial traveller will always age more than a non-inertial traveller that follows a similar trajectory.

In special relativity, space-time is affine -- there is a unique geodesic joining any pair of points. This implies that it must actually be a global maximum, and you can conclude that you will age more along an inertial trajectory than any other trajectory with the same endpoints.

In general relativity, space-time is not affine, and you can have more than one geodesic joining a given pair of points. The above shortcut doesn't work.




There is another common pseudoparadox involving a closed universe. Consider a flat, 1+1-dimensional space-time in the shape of a cylinder. It still makes sense to think of a "line of simultaneity" in this space-time, but it's very easy to make the mistake of assuming that lines of simultaneity are circles. In actuality, most are helixes.

If you try to analyze this space-time in a way as similar to special relativity as possible, you find that in any particular frame, each time you go "around the universe", you have to introduce a correction factor to coordinate time. Furthermore, this correction factor is a frame-dependent quantity.

In fact, while this space-time is locally isotropic, it is not globally isotropic -- space-time has an axis, and if you could perform an experiment involving the entire universe, you could determine whether or not a given observer is "stationary" with respect to the axis of space-time.

 

[Wallace]

Can you give any more details of the flat but toroidal space-time? I've never heard of that and can't see how it would work, particularly in an expanding FRW metric? I'd be interested any more info?

 

[Garth]

You're referring to the the cosmological twin 'paradox', I presume? It's only a pseudoparadox -- like the usual twin paradox, it arises from making a conceptual mistake.

A geodesic is a local maximum for the proper duration along a timelike path between two points -- i.e. when traveling between a given pair of points in space-time, an inertial traveller will always age more than a non-inertial traveller that follows a similar trajectory.

In special relativity, space-time is affine -- there is a unique geodesic joining any pair of points. This implies that it must actually be a global maximum, and you can conclude that you will age more along an inertial trajectory than any other trajectory with the same endpoints.

In general relativity, space-time is not affine, and you can have more than one geodesic joining a given pair of points. The above shortcut doesn't work.

Does this not actually deepen the paradox?

If the twins take separate and distinct geodesic paths (inertial trajectories) between two events, how do you decide which one is going to age more?

My resolution is that it is necessary to involve the distribution of the matter causing the space-time curvature, ie. invoke Mach's Principle.

Garth

 

[Hurkyl]

If the twins take separate and distinct geodesic paths (inertial trajectories) between two events, how do you decide which one is going to age more?

You integrate $d\tau$ along the trajectories and compare the results.

 

[pervect]

The serious proposal that comes to mind isn't quite a torus:

http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Aastro-ph%2F0310253 .

This uses much the same trick of gluing together edges, but rather than gluing the edges of a square (as in the screen on the game Asteroids in my earlier example), or the faces of a cube a cube (the obvious 3-d generalization of the 2-d example above), they glue together opposite sides (faces) of a dodecahedron to form something called the "Poincare dodecahedral space".

There's some info on this in the wikipedia.

A simple construction of this space, which makes clear the term "dodecahedral space", begins with a dodecahedron. Each face of the dodecahedron can be identified with its opposite face by using the minimal clockwise twist to line up the faces. Glue each pair of opposite faces together using this identification. After this gluing, the result is a closed 3-manifold.

 

[Garth]

You integrate $d\tau$ along the trajectories and compare the results.

Yes, obviously, but the paradox only exists if we consider the twins not being able to locally distinguish between their separate inertial frames of reference.

Integrating along the trajectories requires a global knowledge of the curvature of space-time that can only be done with knowledge of the source of that curvature - the matter in the rest of the universe.

I liked your helical surfaces of simultaneity, can you explain more - for example, how are such surfaces determined?

Garth

 

[Hurkyl]

Yes, obviously, but the paradox only exists if we consider the twins not being able to locally distinguish between their separate inertial frames of reference.

Integrating along the trajectories requires a global knowledge of the curvature of space-time that can only be done with knowledge of the source of that curvature - the matter in the rest of the universe.

This only involves the part of space-time lying along the trajectory. The corresponding physical experiment would be "each observer uses their wristwatch to measure the time elapsed between the endpoints of their journeys".

I don't need to see the Earth in order to feel its gravitational pull!

I liked your helical surfaces of simultaneity, can you explain more - for example, how are such surfaces determined?

I think the Euclidean geometry of the cylinder has all of the same relevant properties as the Minkowski geometry on the cylindrical 1+1 space-time... so I will propose some experiments you can do with the Euclidean geometry of real-life cylinders to get an idea for what's going on.



The easiest way to write down coordinates for a cylinder is, of course, cylindrical coordinates. Each point of the cylinder can be given $(z, \theta)$ coordinates. Infinitely, in fact, since $(z, \theta)$ and $(z, \theta + 2\pi)$ describe the same point.

You could draw "coordinate axes" for this coordinate system -- the z axis is the line $\theta = 0$ parallel to the axis of the cylinder. The $\theta$ axis is the circle $z = 0$.

There are three kinds of straight lines on the cylinder:
(1) Ordinary lines parallel to the axis of the cylinder
(2) Circles whose plane is perpendicular to the axis of the cylinder
(3) Helixes

Cylindrical coordinates are very convenient... and very special... because the axes are a line and a circle. What happens if you try to set up a different orthogonal coordinate system for a cylinder?

This is something you can do yourself: pick any point on the cylinder and draw two perpendicular "lines" through it, neither one parallel to the axis. They will be helixes. You can use these as some sort of "coordinate axes". They are periodic like cylindrical coordinates, but the periodicity involves both coordinates: there is some nonzero a and b such that $(z', \theta')$ and $(z' + a, \theta' + b)$ define the same point.



Special relativistic reference frames are orthogonal coordinate for Minkowski geometry -- for the 1+1-dimensional cylindrical space-time, they will have the same kind of qualitative properties (but with a sign flip) as the nonstandard Euclidean coordinates for a cylinder that I described above. This space-time is locally flat, so for any observer, you can draw a local line of simultaneity which is orthogonal to his trajectory -- such lines usually extend to helixes. Such observers cannot view the universe as being "circular space with linear time", because the periodicity of the universe involves both his spatial and his temporal coordinate.

 

[Pavel]

Thank you for your help. Would I be correct then to summarize what you say, in down to Earth terms, as following:

The recent observations, such as done by WMAP suggest that, as far as we can see, the Universe is flat, save a few very small irregularities here and there. However, such observations do not warrant any particular overall topology of the Universe, which might very well be a torus exhibiting the "Hall of Mirrors" effect, albeit on a scale beyond any empirical verification. There are reasons to believe though, according to WMAP, that the topology might be Poincare's dodecahedral sphere.

Thanks,

Pavel.

 

[pervect]

While some authors have certainly proposed the dodecahedral sphere topology, I'd personally say the evidence is pretty weak at this point, and that for the most part we just don't know the overall topology.

 

[Garth]

This only involves the part of space-time lying along the trajectory. The corresponding physical experiment would be "each observer uses their wristwatch to measure the time elapsed between the endpoints of their journeys".

I don't need to see the Earth in order to feel its gravitational pull!

You do not feel its gravitational pull if you are in free fall - as our twins are.

The point of the 'paradox' is to explore the principle of "no preferred frames" on which not only is SR based, where in an empty and flat space-time it is quite appropriate, but also GR, where in the presence of curvature and matter it might be problematic. Mach's Principle might lead us to speculate whether the presence of matter introduces a 'preferred' or 'special' frame that can be defined from the centre of mass-momentum (centroid) of the system.

The twins pass close by each other at an event E₁, and they happen to be in the presence of a black hole that neither are aware of, or they are circumnavigating a static and flat space cylindrical universe. Some (long) time later they pass close-by each other again and are able to compare clocks.

They each believe that they are the stationary one and the other traveling at high velocity, consequently as "each observer uses their wristwatch to measure the time elapsed between the endpoints of their journeys" believes their watch is the one that would measure the greatest elapse of time.

One turns out to be correct in this expectation and the other not, but without referring to the presence of the BH or distribution of matter in the 'rest of the universe' which is which?

Special relativistic reference frames are orthogonal coordinate for Minkowski geometry -- for the 1+1-dimensional cylindrical space-time, they will have the same kind of qualitative properties (but with a sign flip) as the nonstandard Euclidean coordinates for a cylinder that I described above. This space-time is locally flat, so for any observer, you can draw a local line of simultaneity which is orthogonal to his trajectory -- such lines usually extend to helixes. Such observers cannot view the universe as being "circular space with linear time", because the periodicity of the universe involves both his spatial and his temporal coordinate.

Of course, but I thought you were saying the surfaces of simultaneity had to be cylindrical. What about the observer co-moving along the universe's z-axis?

Garth

 

[Hurkyl]

They each believe that they are the stationary one and the other traveling at high velocity, consequently as "each observer uses their wristwatch to measure the time elapsed between the endpoints of their journeys" believes their watch is the one that would measure the greatest elapse of time.

This statement is wrong; your alledged consequence does not follow from the premise. "I traveled inertially" is not sufficient cause to conclude "my wristwatch measured more time than anyone else whose worldline had the same endpoints". Therefore

One turns out to be correct in this expectation and the other not

neither of your observers has a valid argument. They both made an invalid argument -- one of them simply got lucky.

but without referring to the presence of the BH or distribution of matter in the 'rest of the universe' which is which?

We don't have to refer to the presence of the BH or the distribution of matter in the 'rest of the universe' -- we simply have to refer to the time elapsed on the wristwatches.

Of course, but I thought you were saying the surfaces of simultaneity had to be cylindrical. What about the observer co-moving along the universe's z-axis?

I said most were helical. None are cylindrical; in a 1+1-dimensional universe, hypersurfaces of simultaneity are 1-dimensional.

 

[Garth]

They each believe that they are the stationary one and the other traveling at high velocity, consequently as "each observer uses their wristwatch to measure the time elapsed between the endpoints of their journeys" each believes their watch is the one that would measure the greatest elapse of time.

This statement is wrong; your alledged consequence does not follow from the premise. "I traveled inertially" is not sufficient cause to conclude "my wristwatch measured more time than anyone else whose worldline had the same endpoints".

As you said above, in SR space-time is an affine space whereas in GR it is generally not. Nevertheless in an empty universe GR space-time reduces to an affine space in which the paradox apparently exists.

Therefore in order to know that they are not in an affine space they have to be aware of the presence of gravitating sources in the rest of the universe.

We don't have to refer to the presence of the BH or the distribution of matter in the 'rest of the universe' -- we simply have to refer to the time elapsed on the wristwatches.

Let me make myself clear. There is no real paradox here, it is only if the observers are unaware of the presence of gravitating sources in the rest of the universe and then meet up again that a paradox seems to exist.

Once they meet up of course they compare clocks and that observation would resolve the paradox.

But until they make that observation, and without being aware of or referring to the presence of the gravitating sources, they would each predict it is they who are on the maximal time-like trajectory.

In order for them to meet up again each would have to assume it was the other who had decelerated and turned round.

I said most were helical. None are cylindrical; in a 1+1-dimensional universe, hypersurfaces of simultaneity are 1-dimensional.

Sorry, my mistake! I didn't read the "most". (Making a mental note to operate brain before responding.)

Garth

 

[Hurkyl]

As you said above, in SR space-time is an affine space whereas in GR it is generally not. Nevertheless in an empty universe GR space-time reduces to an affine space in which the paradox apparently exists.

"Empty" isn't enough -- you also need flat and simply connected. (These are certainly necessary and are not consequences of "empty". I don't know if these are sufficient)

Therefore in order to know that they are not in an affine space they have to be aware of the presence of gravitating sources in the rest of the universe.

If space-time is not flat, then you don't have to know anything at all about the rest of the universe to determine that fact; you just perform an experiment to measure the Riemann curvature tensor.

Let me make myself clear. There is no real paradox here, it is only if the observers are unaware of the presence of gravitating sources in the rest of the universe and then meet up again that a paradox seems to exist.

In other words -- if the observers make unjustified, incorrect assumptions, they are likely to get incorrect answers.

But until they make that observation, and without being aware of or referring to the presence of the gravitating sources, they would each predict it is they who are on the maximal time-like trajectory.

If they understood GR, they would do no such thing. They would know that they are on a "locally maximal time-like trajectory", and they wouldn't conjecture about trajectories that are not small perturbations about their trajectory.


Compare to a similar Euclidean example: on a Euclidean sphere, there is a geodesic from 0°N 0°E to 0°N 30°E that travels due West from the starting point to the ending point. This is a locally minimal path between these points... but would anyone think that it is actually the shortest path between those points?

It's easy to construct similar examples on Euclidean cylinders and cones.

 

[Garth]

Thank you Hurkyl, I am using this apparent paradox to try to tease out the consequences of the principle of 'no preferred frames of reference' and the Equivalence principle on which GR is based with relation to Mach's Principle.

The wiki definition of the EP, (which appears in this case to be accurate):

The outcome of any local non-gravitational experiment in a laboratory moving in an inertial frame of reference is independent of the velocity of the laboratory, or its location in spacetime.

Here local has a very special meaning: not only must the experiment not look outside the laboratory, but it must also be small compared to variations in the gravitational field, tidal forces, so that the entire laboratory is moving inertially.

So we have a laboratory sufficiently small that tidal forces cannot be detected. Bob and Alice both pass through this small space 'simultaneously' on inertial trajectories and compare clocks. Later they pass through again and compare clocks, each thinks they are stationary and the other moving at high speed.

In order to know which one is going to record the greater lapse of time they have to know about the presence and distribution of gravitating sources in the rest of the universe.

I find this conclusion to be Machian in nature, IMHO.

Garth