Is the Universe Infinite and Flat in Cosmology?

[Cosmo Novice]

Am i right in saying that consenses amongst the cosmological community is that the curvature of space is 0? So it is flat euclidean geometric space (I hope I am using correct terminology) which is neither +/- in curvature but exactly 0?

Am I also right in saying that flat cosmological models assume an infinite universe, an open universe that is spacially flat without any boundary?

That being said, the universes age has been approximated through various data to 13.7GY, and the receeding galaxies (where some are receeding >c due to expansion) can be approximated through redshift and other data.

If the universe is currently spacially flat and infinite then does this imply the universe is now infinite but was not infinite at the point of the BB? (I understand that U=the totality of everything) so I am not asking if there was anything external to U, just if U was finitely measurably (I think I have read inflationary models that approximate spatial distance of the U in early expansion).

If U was finitely measurably then does this imply that beyond the OU there are galaxies much much further out that are now receeding at an infinite speed? Some galaxies whose photons will never reach us due to recession >c?

So to summarise: Assuming U is euclidean space, and assuming U is infinite, how can U be infinite if it the sum of two finite variables. Namely Expansion rate and Time since expansion? Finite Recession+Finite Time=Inifinite U

Apologies this question is long and drawn out, I am not a student of cosmology, just have an amateur interest so please try to use laymans terms and any corrections are welcomd

 

[edgepflow]

It is diffucult to visualize, but I try to keep a few points in mind when thinking about this:

* After the big bang (t=0) and before the Planck Time (10^-43 sec), I do not take too seriously any suggestions about the size and shape of space time.

* After the Planck time, I fall back to the flat infinite piece of grid paper model with space coordinates expanding per the Friedmann Equation with dark energy (cosmological constant) and all that jazz. Hubble's relation v = Hr applies and so c = H r will establish the distance that objects are moving away from us at light speed. Beyond, the OU, I do not see why Hubble's law would not apply and recession velocities from us would increase unbounded, but we could never verify this with observational data.

 

[Cosmo Novice]

It is diffucult to visualize, but I try to keep a few points in mind when thinking about this:

* After the big bang (t=0) and before the Planck Time (10^-43 sec), I do not take too seriously any suggestions about the size and shape of space time.

* After the Planck time, I fall back to the flat infinite piece of grid paper model with space coordinates expanding per the Friedmann Equation with dark energy (cosmological constant) and all that jazz. Hubble's relation v = Hr applies and so c = H r will establish the distance that objects are moving away from us at light speed. Beyond, the OU, I do not see why Hubble's law would not apply and recession velocities from us would increase unbounded, but we could never verify this with observational data.

Thankyou that really helps. I was not implying any suggestions of space time physical dimensions prior to Planck time or pre inflation, as I understand it conventional physics breaks down at this point, although my understanded is limited to a few days reading. :)

I think it would be safe to make an assumption that recession velocities would increase unbounded, especially given U is homogeneous and isotropic, if I am right in thinking that is the accepted view? Then as there is no restriction on recession speeds, because GR defines them as in their own space time, and it is recession not kinetic movement, they are not breaking laws of physics. So if U is spacially flat and infinite then there would be no physical law restricting expansion rates, in fact as cosmology models dictate a flat U must be infinite then the rate of expansion has to be infinite.

lf the space between matter (if I am right in thinking its the empty space between galaxies) that is expanding, so all galaxies are moving away from each other then the BB would have had to happened at all points in space/time, someone else on this forum mentioned the same thing and i was wondering on opinion on this? That would actually fit into a flat topology, infinite matter, infinite expansion rates as geometrically it began everywhere anyway.

Please just clarify a few points for me:

1. Does standard cosmology model fit flat ueclidean space?
2. If so do cosmologists think the euclidean space is infinite in all directions?
3. If so does that not mean if there was a BB event, and a time (following Plancke time) where U can be physically approximated prior to or during the inflationary period (As I believe some people have commented)?
4. If all galaxies are receeding from each other does that mean geometrically the BB happened in all places at the same time and that expansion rates would be infinite as for U to be infinite expansion would have to be infinite?

Thanks a lot for your reply. I am finding it vey hard to try to visualise expansion, i have seen the balloon analogy and the dough and they helped a bit but its so much more difficult. I guess my mind cannot imagine more than 3D, even looking at the more D images i just see 3D or 2D :)

A curved and finite U seems to be more easy to accept for me, though I think cosmological models currently disagree with that and i would accept their knowledge.

 

[phinds]

Like you, I'm an amateur at this. I want only to comment on one of your questions:

If U was finitely measurably then does this imply that beyond the OU there are galaxies much much further out that are now receeding at an infinite speed? Some galaxies whose photons will never reach us due to recession >c?

I believe that it is absolutely accepted that there are galaxies that are receding from us > c and that their photons are therefor forever out of our range, but they are not an infinite distance away. I'm NOT commenting on whether or not there ARE galaxies that are an infinite distance away, just that they don't have to be to be forever out of our OU.

Further, I have read, but cannot comment on the validity of, the statement that due to changes in the Hubble constant, there are some galaxies that are just out of our OU now, but which will move into our OU because our OU is expanding faster than they are receding.

I found your comments and questions quite interesting, by the way.

Paul

 

[edgepflow]

Please just clarify a few points for me:

1. Does standard cosmology model fit flat ueclidean space? Yes, from the observation that the expansion is accelerating from a cosmological constant (dark energy). Dark energy drives space flat.
2. If so do cosmologists think the euclidean space is infinite in all directions? Yes, this is considered most likely. The other possibility is a "non trivial topology." This is like the old video games where you go off one side of the screen and come back at the other. This avoids "edges" of space and thus fits the model.
3. If so does that not mean if there was a BB event, and a time (following Plancke time) where U can be physically approximated prior to or during the inflationary period (As I believe some people have commented)? Basically, yes.
4. If all galaxies are receeding from each other does that mean geometrically the BB happened in all places at the same time and that expansion rates would be infinite as for U to be infinite expansion would have to be infinite? I think so, but like many others, I resist the idea of infinite physical things.

Please see above blue responses.

 

[Cosmo Novice]

I believe that it is absolutely accepted that there are galaxies that are receding from us > c and that their photons are therefor forever out of our range, but they are not an infinite distance away. I'm NOT commenting on whether or not there ARE galaxies that are an infinite distance away, just that they don't have to be to be forever out of our OU.

If U is homogeneous and isotropic and has a flat euclidean geometry which comsological models indicate, then than means all OU's would be pretty much the same, that given there must be galaxies extending into infinity, if you could 'pause existence' and had no limitation on movement, then a flat cosmological model dictates you could travel infinitely in any direction and never reach an 'edge' or the same space, in fact however far you moved space ould look pretty much the same, recession speeds the same and all OU's the same this would be consistent with the idea of the BB geometrically occurring in all space/time all at once.

Further, I have read, but cannot comment on the validity of, the statement that due to changes in the Hubble constant, there are some galaxies that are just out of our OU now, but which will move into our OU because our OU is expanding faster than they are receding.

Yes I thought this too, but also given expansion is accelerating then all OUs will eventually become smaller as more and more distant galaxies begin to recede>c due to increasing spatial expansion and so their photons will never reach us (what we see now may in a billion years be receeding >c or even further into our subjective furute). I wonder if any cosmologists predict the possible size of the OU given current accelerating expansion rates? Would be interesting to know. Cosmologically all galaxies must eventually exist in their own OU. Given accelerated expansion (also assuming spatial expansion>the life of a galaxy, which i don't really know maybe some cosmologists do)

Thanks for your post

 

[Cosmo Novice]

Edgepflow,

I find the idea of "The other possibility is a "non trivial topology." This is like the old video games where you go off one side of the screen and come back at the other. This avoids "edges" of space and thus fits the model." just plain wrong. I do not know why just on an instinctive level I cannot believe space is so simple and to come back in the other "edge" would to me require a closed topology and therefore spatial curvature.

I also find the idea of intinite physical space quite intuitively incorrect, but if cosmological models dictate flat space then infinity is implied. If we assume U is isotropic as models dictate then literally all OUs to infinity would pretty much be the same (same sort of composition of matter etc) that being said as our OU begins to get smaller as it must due to expansion then that occurs across infinity.

So in a very long time as all the stars in all the galaxies in our OU begin to go supernovae/die(those where expansion <c so are still visible) that will effectively be happening across the entire U, given the assumption of isotropy. So heat death would be infinite.

I find it hard to accept.

Thanks for your post.



Thanks for your post

 

[bapowell]

Keep in mind that current cosmological observations yield information about the geometry of the observable universe. Such observations give us insight into the local geometry of the universe. Current CMB measurements put the spatial curvature of the universe to be flat to within about 1%. The universe could, and might well be, closed or open globally.

 

[yenchin]

Edgepflow,

I find the idea of "The other possibility is a "non trivial topology." This is like the old video games where you go off one side of the screen and come back at the other. This avoids "edges" of space and thus fits the model." just plain wrong. I do not know why just on an instinctive level I cannot believe space is so simple and to come back in the other "edge" would to me require a closed topology and therefore spatial curvature.Thanks for your post

Why is flat non-trivial topology "just plain wrong"? It is quite easy to imagine such a world [whether we are living in one is another matter all together of course] *without* any curvature. Sure if you want to glue opposite sides of square together to form a torus you need to curve them, but this is not necessary since toral topology is possible on flat space. It is not "plain wrong" but a mathematical fact that this is possible. [Again I make no assertion that the real world is like this]

Whether something sounds "instinctive" or not should be the least of our concern when judging Nature - quantum mechanics is nothing but intuitive for example. If intuitiveness counts, nontrivial topology is more intuitive than quantum mechanics.

By the way a recenthttp://arxiv.org/abs/1104.0015" claims that the most probable universe has topology T^2 x R^1, which is even stranger than a compact flat topology. Well, just to say that the study of topology in cosmology is still an active area of research...

 

[Cosmo Novice]

Why is flat non-trivial topology "just plain wrong"? It is quite easy to imagine such a world [whether we are living in one is another matter all together of course] *without* any curvature. Sure if you want to glue opposite sides of square together to form a torus you need to curve them, but this is not necessary since toral topology is possible on flat space. It is not "plain wrong" but a mathematical fact that this is possible. [Again I make no assertion that the real world is like this]

Whether something sounds "instinctive" or not should be the least of our concern when judging Nature - quantum mechanics is nothing but intuitive for example. If intuitiveness counts, nontrivial topology is more intuitive than quantum mechanics.

By the way a recenthttp://arxiv.org/abs/1104.0015" claims that the most probable universe has topology T^2 x R^1, which is even stranger than a compact flat topology. Well, just to say that the study of topology in cosmology is still an active area of research...

You say "toral topology is possible on flat space" is that a definite fact and can you link anything to prove it as I would find that really interesting, or if it is heavy math is it a safe assumption? To be honest i do not really know much regarding QM, I have only really been reading into GR and as my posts say that's at a basic level mainly restricted to forums.

Thanks



Thanks

 

[Cosmo Novice]

Keep in mind that current cosmological observations yield information about the geometry of the observable universe. Such observations give us insight into the local geometry of the universe. Current CMB measurements put the spatial curvature of the universe to be flat to within about 1%. The universe could, and might well be, closed or open globally.

Ah so if the OU is much smaller than U then it would be even harder to measure any curvature given error ratios? Thats really interesting, must curvature be 1% +/- for space to be curved or can it be any variable away from 0. So 0.000001 spatial curvature is still curved topology?

Thankyou

 

[bapowell]

Ah so if the OU is much smaller than U then it would be even harder to measure any curvature given error ratios? Thats really interesting, must curvature be 1% +/- for space to be curved or can it be any variable away from 0. So 0.000001 spatial curvature is still curved topology?

Technically speaking, a flat universe must satisfy $\Omega = \rho/\rho_{critical} = 1$ exactly.

 

[yenchin]

You say "toral topology is possible on flat space" is that a definite fact and can you link anything to prove it as I would find that really interesting, or if it is heavy math is it a safe assumption? To be honest i do not really know much regarding QM, I have only really been reading into GR and as my posts say that's at a basic level mainly restricted to forums.

Thanks



Thanks

That is a definite mathematical fact. All you need is basic topology. I don't know what you mean by math heavy being "safe assumption" though, there is not much to assume here when you construct such a space. See for example:

1. http://www.carliner-remes.com/jacob/math/project/math.htm, especially the section on "And Beyond: 3-Manifolds".

2. http://www.astro.utoronto.ca/~yzaguirre/CosmicTopology(mini).pdf .

3. http://plus.maths.org/content/os/issue10/features/topology/index
"Notice that we started with a flat sheet of paper and then built a torus. However, the torus we drew in figure 4 is not flat. It is curved in a way that is not constant across its surface. This is an artifact of living in 3-dimensions and bending the 2-dimensional sheet of paper into 3 to help our 3-dimensional selves visualize the compactness and topology of the surface. Strictly speaking, a two-dimensional flat space can exist which has the fundamental shape of a rectangle with its edges glued together in pairs, but which doesn't bend up into three dimensions."

Here's another to think about it, if you want to imagine the construction by literally gluing the sides and thus creating so-called bending -- for our 3-dimensional universe you can imagine that flat but nontrivial topology means that it is "curved in the 4th dimension of space" but flat to us inhabitants in the 3-dimension which cannot see the 4th. But again this is only for *imagining*, since we can't see outside of 3 dimension of space, we don't know if this 4th dimension exist. But regardless of whether it exists, such a nontrivial topology constructed with identifying sides (or faces in 3-dimension) is equivalent to literally bending them in 4th dimension.

Think of the cylinder. It is 2-dimensional and it has zero Gaussian curvature, meaning any creature living on the cylinder will measure sum of angles in a triangle to be 180 degrees and thus as far as they are concerned their space is flat. We see the cylinder as curved because it bends in 3-dimension, this curvature is extrinsic (property of the embedding space around it) as opposed to Gaussian curvature being intrinsic (property of the space itself). If we cut the cylinder open and flatten it, but *still topologically identifying" the opposite sides, we still have a good old cylinder according to the inhabitants. There is no magic to "gluing the sides", it is the same as declaring the sides to be the same but without bending the manifold to literally glue them.

 

[Cosmo Novice]

That is a definite mathematical fact. All you need is basic topology. I don't know what you mean by math heavy being "safe assumption" though, there is not much to assume here when you construct such a space. See for example:

1. http://www.carliner-remes.com/jacob/math/project/math.htm, especially the section on "And Beyond: 3-Manifolds".

2. http://www.astro.utoronto.ca/~yzaguirre/CosmicTopology(mini).pdf .

3. http://plus.maths.org/content/os/issue10/features/topology/index
"Notice that we started with a flat sheet of paper and then built a torus. However, the torus we drew in figure 4 is not flat. It is curved in a way that is not constant across its surface. This is an artifact of living in 3-dimensions and bending the 2-dimensional sheet of paper into 3 to help our 3-dimensional selves visualize the compactness and topology of the surface. Strictly speaking, a two-dimensional flat space can exist which has the fundamental shape of a rectangle with its edges glued together in pairs, but which doesn't bend up into three dimensions."

Here's another to think about it, if you want to imagine the construction by literally gluing the sides and thus creating so-called bending -- for our 3-dimensional universe you can imagine that flat but nontrivial topology means that it is "curved in the 4th dimension of space" but flat to us inhabitants in the 3-dimension which cannot see the 4th. But again this is only for *imagining*, since we can't see outside of 3 dimension of space, we don't know if this 4th dimension exist. But regardless of whether it exists, such a nontrivial topology constructed with identifying sides (or faces in 3-dimension) is equivalent to literally bending them in 4th dimension.

Think of the cylinder. It is 2-dimensional and it has zero Gaussian curvature, meaning any creature living on the cylinder will measure sum of angles in a triangle to be 180 degrees and thus as far as they are concerned their space is flat. We see the cylinder as curved because it bends in 3-dimension, this curvature is extrinsic (property of the embedding space around it) as opposed to Gaussian curvature being intrinsic (property of the space itself). If we cut the cylinder open and flatten it, but *still topologically identifying" the opposite sides, we still have a good old cylinder according to the inhabitants. There is no magic to "gluing the sides", it is the same as declaring the sides to be the same but without bending the manifold to literally glue them.

Thankyou for you post, I will look into these articles when I have a little more time and respond properly - although I think I understand what you are saying. Its quite possible I cannot mathemtaically visualise higher dimensional space but I will gve those articles a go.

Thanks

 

[Cosmo Novice]

Technically speaking, a flat universe must satisfy $\Omega = \rho/\rho_{critical} = 1$ exactly.

Can you translate this into words as I do not fully understand the symbology, or provide me with a key so I can have another look.

Thankyou

 

[phinds]

I wonder if any cosmologists predict the possible size of the OU given current accelerating expansion rates?

The current model with accelerating expansion says that OUR OU will become our local galactic cluster and I believe that is true for EVERY galaxy. They will either stay together themselves if not in a cluster, or the cluster will stay together. All else will recede so far that its light will no longer reach the galaxy(ies).

 

[narrator]

Please just clarify a few points for me:

1. Does standard cosmology model fit flat ueclidean space? Yes, from the observation that the expansion is accelerating from a cosmological constant (dark energy). Dark energy drives space flat.
2. If so do cosmologists think the euclidean space is infinite in all directions? Yes, this is considered most likely. The other possibility is a "non trivial topology." This is like the old video games where you go off one side of the screen and come back at the other. This avoids "edges" of space and thus fits the model.
3. If so does that not mean if there was a BB event, and a time (following Plancke time) where U can be physically approximated prior to or during the inflationary period (As I believe some people have commented)? Basically, yes.
4. If all galaxies are receeding from each other does that mean geometrically the BB happened in all places at the same time and that expansion rates would be infinite as for U to be infinite expansion would have to be infinite? I think so, but like many others, I resist the idea of infinite physical things.

Does this amateur (me) have the following right?

If space is infinite yet flat (or even curved), in simple terms that seems to mean infinite along x and y-axis yet finite along z axis. If the U is expanding, that leaves 2 possibilities.

a. That the expansion is only happening on the z axis. But if dark energy is driving space flat, that means it's either holding back the expansion or slowing it.

b. That, using the video game analogy above, the screen is expanding in x and/or y directions.

... or some sort of combination of the two possibilities..

If space is flat, then I suppose it's like a huge rug that has no edges, which means that coordinates for one object can only ever be given relative to another object (or observer).

If space is curved, (saucer shaped) then it either has to be a relatively local phenomena or the curves must eventually meet at some finite point. But if the curve, top and bottom, are relatively parallel (frisbee shaped), and infinite, then the curves will eventually intersect, either by joining like a hollow ball or crossing creating layers like an onion or perhaps layers which curve away from the center, unlike an onion.

The latter is almost analogous of the video game analogy. If the curved space meets like a hollow ball, it could cross, continuing to curve, then it's the ball that's expanding. If they cross, creating layers (whether onion-like or not), then perhaps these layers have other dimension qualities.

If a hollow ball, then what's in the middle? lol.

I know these analogies are crude and probably don't reflect the way GR works, but it's hard to divorce our reality from such meanderings.

 

[bapowell]

Spatial flatness refers to the 3 spatial dimensions, so in an infinite space this would mean infinite x, y, and z dimensions. Such a surface is flat if the geometry is Euclidean -- parallel lines never intersect and the interior angles of a triangle sum to 180 degrees. This is the shortcoming of the 'rubber sheet' or 'balloon' analogies, because we are not really dealing with a 2D surface, but a 3D surface.

 

[narrator]

Spatial flatness refers to the 3 spatial dimensions, so in an infinite space this would mean infinite x, y, and z dimensions. Such a surface is flat if the geometry is Euclidean -- parallel lines never intersect and the interior angles of a triangle sum to 180 degrees. This is the shortcoming of the 'rubber sheet' or 'balloon' analogies, because we are not really dealing with a 2D surface, but a 3D surface.

Ohhh.. now that makes sense.. thank you.. So what's meant by the curved space idea then? Does it mean that this 3D space is uneven in time or spatial distortion respects? Or is it some function of c?

 

[bapowell]

The 2D surface of a sphere in regular 3D space is an example of a curved surface. The closest we can get to a notion of parallel lines on a sphere are lines of longitude -- but clearly these intersect at the poles. Also, the sum of the interior angles of a triangle is greater than 180 degrees on the surface of the sphere. Now, to generalize to the 3D universe, you need to imagine a the 3D surface of a 4D sphere! This is the way the universe would look if it was closed. (Don't go worrying about what the 4th dimension is here...it need not exist! The 3D spherical surface is actually well-defined in 3D -- you don't actually need the 4th dimension to define the geometry and topology of the 3-sphere. We are just used to visualizing spheres as existing in a higher dimensional space, because well, that's how they look to us in our 3D space.)

The spacetime manifold can also be 'curved' in the time dimension. This is essentially what expansion is. The current model of the universe is a flat, homogeneous, and isotropic universe -- an ordinary Euclidean space -- that happens to be expanding. The space is flat, but the spacetime is 'curved'. But that's a funny way to talk about it; typically one simply says that the spacetime is dynamical.

 

[StateOfTheEqn]

A spatially flat universe with no boundary must be infinite. A spatially flat finite universe must have a boundary. That is just mathematics.

One serious doubt I have about the flat space picture is its origin. Quantum cosmologists would like to have the universe start as a quantum tunneling event. That would involve a 4-sphere De Sitter instanton through which the universe quantum tunnels to a U x 3-sphere where U is cosmic time beginning at t=0. Inflation theories are then sometimes invoked to give us flatness but inflation theories keep failing.

I recall a paper by Atkatz and Pagels (1982) where their concluding remarks state:

A principal conclusion of our investigation is that only a spatially compact universe can originate as a quantum tunneling event.

If that conclusion still holds and if we assume space has no boundary then we must chose between a flat infinite universe which has not originated as a quantum tunneling event or a spatially compact (and not flat) one which has (or might have).

Another consideration about an infinite universe is that it would require infinite energy to happen. Where could infinite energy come from?

 

[narrator]

The space is flat, but the spacetime is 'curved'. But that's a funny way to talk about it; typically one simply says that the spacetime is dynamical.

Thanks bapowel.. that clears up a lot of my questions very elegantly. :)

Another consideration about an infinite universe is that it would require infinite energy to happen. Where could infinite energy come from?

Just a thought, but if edgepflow's video game analogy is reasonable, then could it be that both matter and energy is both infinite and in some respects finite? Kind of like the concept of space being folded back on itself?

 

[bapowell]

A spatially flat universe with no boundary must be infinite. A spatially flat finite universe must have a boundary. That is just mathematics.

But it's not correct mathematics. A torus is a spatially flat finite manifold without boundary.

... but inflation theories keep failing.

How so?

 

[StateOfTheEqn]

But it's not correct mathematics. A torus is a spatially flat finite manifold without boundary.

oops! you are right. The torus is homeomorphic to R[mod 2(pi)]xR[mod 2(pi)]. Another example would be the circle R[mod 2(pi)].

If we make both into differentiable manifolds with Riemannian metrics the torus is no longer flat but the circle is.

 

[StateOfTheEqn]

A spatially flat universe with no boundary must be infinite. A spatially flat finite universe must have a boundary.

I have been called on this so I will try to correct it.

Call a manifold M (equipped with a metric) bounded if there is a finite upper bound to the distance between any two points. Call it unbounded otherwise.

A manifold M (equipped with a metric) is either bounded or unbounded. If bounded it is called finite. If unbounded it is called infinite.

 

[narrator]

I have been called on this so I will try to correct it.

Call a manifold M (equipped with a metric) bounded if there is a finite upper bound to the distance between any two points. Call it unbounded otherwise.

A manifold M (equipped with a metric) is either bounded or unbounded. If bounded it is called finite. If unbounded it is called infinite.

Excuse my ignorance, but what is a manifold, in this context? And what is meant by "equipped with a metric"? Is the whole U considered a single manifold in the above quote? I tried to look the word up but could only guess at its meaning from the context of the few sites I read from. It sounds like the U is comprised of many "manifolds" each with it's own spacetime context.

I'm beginning to see just how inadequate words can be with this topic - a big bang that wasn't a bang (more like a sudden appearance of everything everywhere), flat space that is infinite in every direction, galaxies that can be seen despite moving away from us faster than light. I'm slowly catching on, but dumping lay paradigms isn't easy.. lol

 

[yenchin]

Excuse my ignorance, but what is a manifold, in this context? And what is meant by "equipped with a metric"? Is the whole U considered a single manifold in the above quote? I tried to look the word up but could only guess at its meaning from the context of the few sites I read from. It sounds like the U is comprised of many "manifolds" each with it's own spacetime context.

I'm beginning to see just how inadequate words can be with this topic - a big bang that wasn't a bang (more like a sudden appearance of everything everywhere), flat space that is infinite in every direction, galaxies that can be seen despite moving away from us faster than light. I'm slowly catching on, but dumping lay paradigms isn't easy.. lol

A manifold is a generalization of a surface but without thinking of them as being in Euclidean space. A manifold can be of arbitrary dimension. If my U you mean the universe, then yes, we model the universe as a 4-dimensional manifold. You can certainly talk about submanifold which is a manifold within a larger manifold. Equipped with a metric means that a manifold comes a priori without any mean to measure distance between two points (it's an abstract space) and we have to define how to define distances.

 

[narrator]

Thanks yenchin. That helps a lot :)