Time in Early Universe

[CliffEdge]

No there is not an observable centre. You like many others have misunderstood.

I'm not sure what you mean by "observable" centre. If you just mean that you cannot point your telescope at any unique spot and say that that is the centre from which everything else is moving apart then that is trivially true. It may appear that all the distant galaxies are rushing away from us and the milky way making us the centre but someone on a planet in one of those distant galaxies would also see all the distant galaxies rushing away from them like they were the centre.

However there is a centre in the past for the observable universe but it is a trivial solution as it is a single point, a singularity, which contains the whole observable universe at one point at one particular time - that of the big bang at the beginning of time. At no other time is there a centre to the observable universe *. In the balloon analogy the centre would be the centre of the spherical balloon rather than a point on its present surface with the radius of the balloon being the time dimension - and that centre would only be on the surface (trivially) at time zero.

* To complicate matters when we look vast distances in space we are looking back in time but even if we could look back far enough to see the big bang it would appear to be in every direction (as does the CMB) rather than appearing at one point because at the time of the big bang everything in the observable universe was in that singularity - the big bang happened everywhere in the observable universe.

A good effort. Pleasing to see you willing to learn. I'm not sure you understand that it is impossible to infer that there has ever been a centre to the universe. There seems now, thankfully , to be alternative theories to the singularity, eg Roger Penrose. The shape of the universe is unknown though it is believed to be flat. I suspect that your recent belief in three dimensional expansion leads you erroneously to believe the universe is a sphere. Of course the observable universe is a sphere, due to the invariance of C, in practical terms, which doesn't mean much really.

I will come back when I have time to help you gain a better understanding of time as a fourth dimension within space-time geometry. I think you have another common misconception there. But we'll discuss the failure of Euclidean geometry in the face of the invariance of C, and the elimination of the concept of simultaneity etc later.

[AsleepInYorkshire]

I'm not sure what you mean by "observable" centre. If you just mean that you cannot point your telescope at any unique spot and say that that is the centre from which everything else is moving apart then that is trivially true. It may appear that all the distant galaxies are rushing away from us and the milky way making us the centre but someone on a planet in one of those distant galaxies would also see all the distant galaxies rushing away from them like they were the centre.

However there is a centre in the past for the observable universe but it is a trivial solution as it is a single point, a singularity, which contains the whole observable universe at one point at one particular time - that of the big bang at the beginning of time. At no other time is there a centre to the observable universe *. In the balloon analogy the centre would be the centre of the spherical balloon rather than a point on its present surface with the radius of the balloon being the time dimension - and that centre would only be on the surface (trivially) at time zero.

* To complicate matters when we look vast distances in space we are looking back in time but even if we could look back far enough to see the big bang it would appear to be in every direction (as does the CMB) rather than appearing at one point because at the time of the big bang everything in the observable universe was in that singularity - the big bang happened everywhere in the observable universe.

A good effort. Pleasing to see you willing to learn. I'm not sure you understand that it is impossible to infer that there has ever been a centre to the universe. There seems now, thankfully , to be alternative theories to the singularity, eg Roger Penrose. The shape of the universe is unknown though it is believed to be flat. I suspect that your recent belief in three dimensional expansion leads you erroneously to believe the universe is a sphere. Of course the observable universe is a sphere, due to the invariance of C, in practical terms, which doesn't mean much really.

I will come back when I have time to help you gain a better understanding of time as a fourth dimension within space-time geometry. I think you have another common misconception there. But we'll discuss the failure of Euclidean geometry in the face of the invariance of C, and the elimination of the concept of simultaneity etc later.

Having asked you to expand on your recent post on the illegal Russian invasion of Ukraine, which met with no reply from you, I'm bound to have reservations about your promise to come back on the fourth dimension.

I won't hold my breath

AiY(D)

[ursaminortaur]

I'm not sure what you mean by "observable" centre. If you just mean that you cannot point your telescope at any unique spot and say that that is the centre from which everything else is moving apart then that is trivially true. It may appear that all the distant galaxies are rushing away from us and the milky way making us the centre but someone on a planet in one of those distant galaxies would also see all the distant galaxies rushing away from them like they were the centre.

However there is a centre in the past for the observable universe but it is a trivial solution as it is a single point, a singularity, which contains the whole observable universe at one point at one particular time - that of the big bang at the beginning of time. At no other time is there a centre to the observable universe *. In the balloon analogy the centre would be the centre of the spherical balloon rather than a point on its present surface with the radius of the balloon being the time dimension - and that centre would only be on the surface (trivially) at time zero.

* To complicate matters when we look vast distances in space we are looking back in time but even if we could look back far enough to see the big bang it would appear to be in every direction (as does the CMB) rather than appearing at one point because at the time of the big bang everything in the observable universe was in that singularity - the big bang happened everywhere in the observable universe.

A good effort. Pleasing to see you willing to learn. I'm not sure you understand that it is impossible to infer that there has ever been a centre to the universe. There seems now, thankfully , to be alternative theories to the singularity, eg Roger Penrose. The shape of the universe is unknown though it is believed to be flat. I suspect that your recent belief in three dimensional expansion leads you erroneously to believe the universe is a sphere. Of course the observable universe is a sphere, due to the invariance of C, in practical terms, which doesn't mean much really.

I will come back when I have time to help you gain a better understanding of time as a fourth dimension within space-time geometry. I think you have another common misconception there. But we'll discuss the failure of Euclidean geometry in the face of the invariance of C, and the elimination of the concept of simultaneity etc later.

Sorry,but although I mentioned the balloon analogy I wasn't in anyway implying that the Universe was a sphere and am well aware that the universe appears to be either flat or so close to being flat that we can't detect the difference. And as I tried to make clear when I was referring to a centre at the time of the big bang I was referring to a centre then of the observable universe in accordance with a big bang theory involving a singularity. I'd hoped that was clear especially when I went on to talk about the possibility of the universe extending beyond the observable universe potentially being infinite. I am fully aware of non-euclidean geometries and that relativity means that different observers disagree on the passage of time and simultaneity of events. What I was trying to correct was your apparent belief that the expansion of the universe in someway implied that the universe was expanding into some spatial fourth dimension as implied by your statement ( which is not what I had been saying at all).

Yes, the three dimensions of the universe are expanding equally in all directions - effectively it as though the universe is expanding in a fourth dimension as you say

And your original statement

The universe is not expanding in three dimensions.

[ursaminortaur]

A good effort. Pleasing to see you willing to learn. I'm not sure you understand that it is impossible to infer that there has ever been a centre to the universe. There seems now, thankfully , to be alternative theories to the singularity, eg Roger Penrose. The shape of the universe is unknown though it is believed to be flat. I suspect that your recent belief in three dimensional expansion leads you erroneously to believe the universe is a sphere. Of course the observable universe is a sphere, due to the invariance of C, in practical terms, which doesn't mean much really.

I will come back when I have time to help you gain a better understanding of time as a fourth dimension within space-time geometry. I think you have another common misconception there. But we'll discuss the failure of Euclidean geometry in the face of the invariance of C, and the elimination of the concept of simultaneity etc later.

Sorry,but although I mentioned the balloon analogy I wasn't in anyway implying that the Universe was a sphere and am well aware that the universe appears to be either flat or so close to being flat that we can't detect the difference. And as I tried to make clear when I was referring to a centre at the time of the big bang I was referring to a centre then of the observable universe in accordance with a big bang theory involving a singularity. I'd hoped that was clear especially when I went on to talk about the possibility of the universe extending beyond the observable universe potentially being infinite. I am fully aware of non-euclidean geometries and that relativity means that different observers disagree on the passage of time and simultaneity of events. What I was trying to correct was your apparent belief that the expansion of the universe in someway implied that the universe was expanding into some spatial fourth dimension as implied by your statement ( which is not what I had been saying at all).

Yes, the three dimensions of the universe are expanding equally in all directions - effectively it as though the universe is expanding in a fourth dimension as you say

And your original statement

The universe is not expanding in three dimensions.

A pretty good explanation as to why the universe isn't expanding into anything is given here by Sabine Hossenfelder

http://backreaction.blogspot.com/2021/05/what-does-universe-expand-into-do-we.html

[Bubblesofearth]

A pretty good explanation as to why the universe isn't expanding into anything is given here by Sabine Hossenfelder

http://backreaction.blogspot.com/2021/05/what-does-universe-expand-into-do-we.html

My take from this is that the expansion can be measured from within the universe itself but doesn't preclude expansion into something else.

Or have I missed something?

BoE

[ursaminortaur]

A pretty good explanation as to why the universe isn't expanding into anything is given here by Sabine Hossenfelder

http://backreaction.blogspot.com/2021/05/what-does-universe-expand-into-do-we.html

My take from this is that the expansion can be measured from within the universe itself but doesn't preclude expansion into something else.

Or have I missed something?

BoE

There is no evidence that any embedding space exists or what properties it has if it were to exist. Hence Occam's razor (scientific parsimony) leads scientists to build their theories just based upon what we can measure from within the universe.

[Bubblesofearth]

There is no evidence that any embedding space exists or what properties it has if it were to exist. Hence Occam's razor (scientific parsimony) leads scientists to build their theories just based upon what we can measure from within the universe.

OK, but there is still a difference between saying there is no evidence for an embedding space and stating categorically that one doesn't exist. IMO it's especially important when dealing with concepts as complex and difficult to visualise as space-time. The analogies of spots on balloons and currents in cake break-down because the expansion of both does rely on the existence of an embedding space. Is there a better analogy, i.e. one that does not rely on an ES?

BoE

[mc2fool]

There is no evidence that any embedding space exists or what properties it has if it were to exist. Hence Occam's razor (scientific parsimony) leads scientists to build their theories just based upon what we can measure from within the universe.

OK, but there is still a difference between saying there is no evidence for an embedding space and stating categorically that one doesn't exist. IMO it's especially important when dealing with concepts as complex and difficult to visualise as space-time. The analogies of spots on balloons and currents in cake break-down because the expansion of both does rely on the existence of an embedding space. Is there a better analogy, i.e. one that does not rely on an ES?

Any analogy of such is necessarily problematic 'cos of our personal experience of the world we live in. AIUI the general consensus (and, yes, there are other theories) is that the universe is all that exists.

So even if (big if) the universe is finite in size that's still it, there is no "outside" -- and that's not to say that there's "nothing" outside 'cos nothing, to us from our experience of the world, implies an empty space but an empty space still has existence. If you can think of an example of physical non-existence (if that's not an oxymoron) that could be used in an analogy let us know...

[XFool]

OK, but there is still a difference between saying there is no evidence for an embedding space and stating categorically that one doesn't exist. IMO it's especially important when dealing with concepts as complex and difficult to visualise as space-time. The analogies of spots on balloons and currents in cake break-down because the expansion of both does rely on the existence of an embedding space. Is there a better analogy, i.e. one that does not rely on an ES?

Any analogy of such is necessarily problematic 'cos of our personal experience of the world we live in. AIUI the general consensus (and, yes, there are other theories) is that the universe is all that exists.

That feels to me like a very ambiguous statement. Do you mean by that the word "universe" means - by definition - "everything that exists"? The "observable universe"? The entire extent of 'our' universe - observable and non observable - however big it may be (we don't know)? What if there are 'other' universes somehow in parallel with ours? Higher realities than we would normally understand by the term "universe"?

[mc2fool]

Any analogy of such is necessarily problematic 'cos of our personal experience of the world we live in. AIUI the general consensus (and, yes, there are other theories) is that the universe is all that exists.

That feels to me like a very ambiguous statement. Do you mean by that the word "universe" means - by definition - "everything that exists"? The "observable universe"? The entire extent of 'our' universe - observable and non observable - however big it may be (we don't know)? What if there are 'other' universes somehow in parallel with ours? Higher realities than we would normally understand by the term "universe"?

Yes, the entire extent of our universe ... but as I said, there are other theories (including multiple, possibly even an infinite number, of other universes).

As I say, AIUI that's the general consensus, so even if the (our) universe is finite in size there is no "outside" of it as the universe is all that exists, and that's problematic to find a "real world" analogy that we can relate to.

A "un-analogy" for what we experience as "nothing" or "empty" could be a sealed container. What's in the container you may ask. Nothing comes the reply. It's empty? Yes, not even air, nothing. But there isn't "nothing" in it, it has dimensions, there is space in it, the inside exists. But outside the universe (if finite) there isn't emptiness, there isn't nothing, there just isn't an outside, it doesn't exist.

As I understand the general consensus, at least ... (ideas of possible higher planes of existence etc notwithstanding)

[ursaminortaur]

There is no evidence that any embedding space exists or what properties it has if it were to exist. Hence Occam's razor (scientific parsimony) leads scientists to build their theories just based upon what we can measure from within the universe.

OK, but there is still a difference between saying there is no evidence for an embedding space and stating categorically that one doesn't exist. IMO it's especially important when dealing with concepts as complex and difficult to visualise as space-time. The analogies of spots on balloons and currents in cake break-down because the expansion of both does rely on the existence of an embedding space. Is there a better analogy, i.e. one that does not rely on an ES?

BoE

The only evidence we can gather is from within the universe. We can imagine embedding the universe in a higher dimensional space but that gains us nothing and without greater knowledge of the shape of the universe could be potentially very misleading. The first question would be how many extra dimensions should you add ? If you add one extra dimension then why stop there might not that 4d space be embedded in a 5d space ? *
Secondly is the embedding space a euclidean space or something else ?

You may think that you could always embed the universe in a euclidean space which is one dimension bigger but that isn't necessarily the case. For instance a Klein bottle is a two dimensional surface which cannot be embedded in 3d euclidean space ( but can be embedded in 4d euclidean space). And even if you can embed a surface in a larger euclidean space that embedding can cause distortions. So for instance embedding a flat torus in 3d euclidean space distorts distances compared with those you would measure intrinsically using the most natural metric.**

Similar problems would occur with the higher dimensional analogues of the Klein bottle and Torus which might conceivably correspond to the overall shape of our universe if it is finite.

* For this discussion I'm ignoring the time dimension.

**. Strictly speaking there is a way of embedding a flat torus in 3d Euclidean space whilst preserving distances (isometry) but it is a somewhat pathological solution which relaxes ideas of the smoothness of the surface. See

http://hevea-project.fr/ENPageToreDossierDePresse.html

The torus of revolution thus represents the square flat torus in tridimensional space. But this representation is far from ideal since it distorts distances. For instance, horizontals and verticals in the square flat torus all have the same length while this is not true for the corresponding latitudes and longitudes in the torus of revolution.
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In 1954, John Nash while examining the isometric embedding problem in four dimensional space (or in spaces with even larger dimensions) finds an unexpected result: the obstruction to the existence of such embeddings — i.e., the curvature — can be bypassed... provided we pay the price! What price? We will shortly see. One year latter, Nicolaas Kuiper extends the work of John Nash to the case of the three dimensional space and he deduces a somewhat paradoxical consequence: there exist isometric embeddings of the square flat torus in ambient space. This is in total contradiction with what we have just seen above. How is it possible?.
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The key point for resolving the apparent contradiction raised by John Nash and Nicolaas Kuiper is the following: if a surface is not regular enough then it becomes impossible to compute its curvature; in fact the very idea of curvature loses all meaning. That is precisely what is happening with surfaces of class C° and C¹ (in the above two first examples the curvature has no meaning along the connecting line). By contrast, the curvature is well-defined at every point of a surface of class C². For the square flat torus, its vanishing curvature prevents the existence of isometric embeddings with C² regularity. However, it does not obstruct the existence of an isometric embedding generating a surface of class C¹ only, as the curvature no longer exists for such an embedding... Indeed, John Nash and Nicolaas Kuiper show —inter alia— that isometric embeddings of the square flat torus in the ambient space do exist, but the counterpart, that is the price to pay, is that these embeddings belong to the class C¹ and can not be enhanced to belong to the class C². Surprisingly this price comes with a bonus, Nash and Kuiper prove that not only isometric embeddings in the class C¹ do exist but there are infinitely many.
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The way John Nash and Nicolaas Kuiper demonstrate the existence of isometric embeddings is not amenable to visualization. We are faced with a frustrating situation: we know that there exist numerous surfaces with fascinating properties but we are unable to picture a single one!
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Though as shown in the next part of the article you can using computers generate an approximate picture of the isometric embedding

Indeed, for the fifth corrugations wave, the amplitudes are so small that they are not visible to the naked eye. With this in mind, the pictures obtained at the fourth step really show an isometric embedding of the square flat torus in three dimensional space.

[CliffEdge]

There appears to be another fundamental misapprehension here. The universe is not a container within which stuff exists. The universe is the stuff. Think of what we see as stuff as a manifestation of behaviours within an object, the object being the universe.
Imagine a TV screen with a tree on it, you cannot take the tree out into the room, it is a manifestation of behaviour of the pixels within the object ie the screen.
I said the expansion of the universe is as though it's expanding into a fourth dimension. I believe
it's meaningless to ask what the universe is expanding into. The expansion is internal.
Worth reading Penrose on conservation of angles v conservation of scale.

[ursaminortaur]

There appears to be another fundamental misapprehension here. The universe is not a container within which stuff exists. The universe is the stuff. Think of what we see as stuff as a manifestation of behaviours within an object, the object being the universe.
Imagine a TV screen with a tree on it, you cannot take the tree out into the room, it is a manifestation of behaviour of the pixels within the object ie the screen.
I said the expansion of the universe is as though it's expanding into a fourth dimension. I believe
it's meaningless to ask what the universe is expanding into. The expansion is internal.
Worth reading Penrose on conservation of angles v conservation of scale.

But your first post said

The universe is not expanding in three dimensions.

which seemed to rule out the 3d internal expansion you now seem to accept.

[mc2fool]

There appears to be another fundamental misapprehension here. The universe is not a container within which stuff exists.

If you're referring to my "un-analogy" above, it wasn't referring to or describing the universe (that's why it's an un-analogy ), but on our difficulty of coming up with analogies for a universe that can't have an "outside" because it is all that exists.

[ursaminortaur]

There appears to be another fundamental misapprehension here. The universe is not a container within which stuff exists. The universe is the stuff. Think of what we see as stuff as a manifestation of behaviours within an object, the object being the universe.
Imagine a TV screen with a tree on it, you cannot take the tree out into the room, it is a manifestation of behaviour of the pixels within the object ie the screen.

Sorry but I do really find it difficult to follow your arguments and understand what point you are trying to make.
The tree on the TV screen is an image of a tree and with the right equipment you can certainly project that image onto a screen elsewhere in the room or on a modern TV you can use screen casting or mirroring to project the image onto another device. With a bit of fiddling and the right software, if you cast it to a phone or computer, you might even be able to isolate the image of the tree from the background and other items on the screen.

[Bubblesofearth]

The only evidence we can gather is from within the universe. We can imagine embedding the universe in a higher dimensional space but that gains us nothing and without greater knowledge of the shape of the universe could be potentially very misleading. The first question would be how many extra dimensions should you add ? If you add one extra dimension then why stop there might not that 4d space be embedded in a 5d space ? *
Secondly is the embedding space a euclidean space or something else ?

You may think that you could always embed the universe in a euclidean space which is one dimension bigger but that isn't necessarily the case. For instance a Klein bottle is a two dimensional surface which cannot be embedded in 3d euclidean space ( but can be embedded in 4d euclidean space). And even if you can embed a surface in a larger euclidean space that embedding can cause distortions. So for instance embedding a flat torus in 3d euclidean space distorts distances compared with those you would measure intrinsically using the most natural metric.**

If a 1D line is curved it can only be so if a 2nd dimension exists. If a 2D surface is curved it can only be so because of the existence of a 3rd dimension. If the universe is curved then surely there must exist a 4th dimension in which that curvature takes place? Can time be considered the 4th dimension? Not sure how that works though!

I don't really see how using a Klein bottle helps as that is only a theoretical construct?

BoE

[ursaminortaur]

The only evidence we can gather is from within the universe. We can imagine embedding the universe in a higher dimensional space but that gains us nothing and without greater knowledge of the shape of the universe could be potentially very misleading. The first question would be how many extra dimensions should you add ? If you add one extra dimension then why stop there might not that 4d space be embedded in a 5d space ? *
Secondly is the embedding space a euclidean space or something else ?

You may think that you could always embed the universe in a euclidean space which is one dimension bigger but that isn't necessarily the case. For instance a Klein bottle is a two dimensional surface which cannot be embedded in 3d euclidean space ( but can be embedded in 4d euclidean space). And even if you can embed a surface in a larger euclidean space that embedding can cause distortions. So for instance embedding a flat torus in 3d euclidean space distorts distances compared with those you would measure intrinsically using the most natural metric.**

If a 1D line is curved it can only be so if a 2nd dimension exists. If a 2D surface is curved it can only be so because of the existence of a 3rd dimension. If the universe is curved then surely there must exist a 4th dimension in which that curvature takes place? Can time be considered the 4th dimension? Not sure how that works though!

I don't really see how using a Klein bottle helps as that is only a theoretical construct?

BoE

There are two types of curvature

1). Intrinsic curvature which can be measured from properties inherent to the surface itself. Eg. Whether the angles of a triangle on the surface add up to exactly 180 degrees, less than180 degrees or greater than180 degrees indicate whether the surface is flat, positively curved or negatively curved in that region.

2). Extrinsic curvature. This curvature depends upon the embedding space.

A 1d curve such as a circle embedded in 2d euclidean space has an extrinsic curvature but has no intrinsic curvature. A theoretical inhabitant of that 1d curve has no angles to measure and would only be able to deduce that his line is somehow curved by traversing the entire line and finding that they had returned to their starting point.

A cylinder embedded in 3d euclidean space has zero intrinsic curvature (is flat) but has non zero extrinsic curvature.

Extrinsic curvature depends upon what you are embedding it in - so you would have a different extrinsic curvature if you embed a sphere in euclidean space to if you embedded it in a curved space eg into a hypersphere.

As to a Klein bottle being a theoretical construct. These are all mathematical concepts. In the real world a point with no dimensions, a line with one single dimension and a surface with two dimensions do not exist - they are all abstractions made by taking a real world object and shrinking one or more dimensions of that object to a limit which cannot actually exist in our universe. I assume you accept the existence of a mobius strip. A Klein bottle is just two mobius strips connected together via their edge.( Remember that a mobius strip has only one edge). And of course if you dismiss the idea of a Klein bottle because it is a "theoretical construct" then why are you even considering the idea that the Universe could be embedded in a higher dimensional space as that is even more of a "theoretical construct".

https://sites.nova.edu/mjl/graphics/spaced-out/klein-bottle/two-mbius-bands/

[Bubblesofearth]

There are two types of curvature

1). Intrinsic curvature which can be measured from properties inherent to the surface itself. Eg. Whether the angles of a triangle on the surface add up to exactly 180 degrees, less than180 degrees or greater than180 degrees indicate whether the surface is flat, positively curved or negatively curved in that region.

2). Extrinsic curvature. This curvature depends upon the embedding space.

A 1d curve such as a circle embedded in 2d euclidean space has an extrinsic curvature but has no intrinsic curvature. A theoretical inhabitant of that 1d curve has no angles to measure and would only be able to deduce that his line is somehow curved by traversing the entire line and finding that they had returned to their starting point.

A cylinder embedded in 3d euclidean space has zero intrinsic curvature (is flat) but has non zero extrinsic curvature.

Extrinsic curvature depends upon what you are embedding it in - so you would have a different extrinsic curvature if you embed a sphere in euclidean space to if you embedded it in a curved space eg into a hypersphere.

As to a Klein bottle being a theoretical construct. These are all mathematical concepts. In the real world a point with no dimensions, a line with one single dimension and a surface with two dimensions do not exist - they are all abstractions made by taking a real world object and shrinking one or more dimensions of that object to a limit which cannot actually exist in our universe. I assume you accept the existence of a mobius strip. A Klein bottle is just two mobius strips connected together via their edge.( Remember that a mobius strip has only one edge). And of course if you dismiss the idea of a Klein bottle because it is a "theoretical construct" then why are you even considering the idea that the Universe could be embedded in a higher dimensional space as that is even more of a "theoretical construct".

https://sites.nova.edu/mjl/graphics/spaced-out/klein-bottle/two-mbius-bands/

All I'm saying about the existence of an embedding space is that it cannot be ruled out based only on observations within the universe. I agree there is no evidence for it.

When it comes to curvature, if this exists then whether intrinsic or extrinsic, it can only do so if a higher dimension exists. You can't have curvature of 3D space without that regardless of whether you can measure it or not. Or at least I can't see how you can.

BoE

[ursaminortaur]

There are two types of curvature

1). Intrinsic curvature which can be measured from properties inherent to the surface itself. Eg. Whether the angles of a triangle on the surface add up to exactly 180 degrees, less than180 degrees or greater than180 degrees indicate whether the surface is flat, positively curved or negatively curved in that region.

2). Extrinsic curvature. This curvature depends upon the embedding space.

A 1d curve such as a circle embedded in 2d euclidean space has an extrinsic curvature but has no intrinsic curvature. A theoretical inhabitant of that 1d curve has no angles to measure and would only be able to deduce that his line is somehow curved by traversing the entire line and finding that they had returned to their starting point.

A cylinder embedded in 3d euclidean space has zero intrinsic curvature (is flat) but has non zero extrinsic curvature.

Extrinsic curvature depends upon what you are embedding it in - so you would have a different extrinsic curvature if you embed a sphere in euclidean space to if you embedded it in a curved space eg into a hypersphere.

As to a Klein bottle being a theoretical construct. These are all mathematical concepts. In the real world a point with no dimensions, a line with one single dimension and a surface with two dimensions do not exist - they are all abstractions made by taking a real world object and shrinking one or more dimensions of that object to a limit which cannot actually exist in our universe. I assume you accept the existence of a mobius strip. A Klein bottle is just two mobius strips connected together via their edge.( Remember that a mobius strip has only one edge). And of course if you dismiss the idea of a Klein bottle because it is a "theoretical construct" then why are you even considering the idea that the Universe could be embedded in a higher dimensional space as that is even more of a "theoretical construct".

https://sites.nova.edu/mjl/graphics/spaced-out/klein-bottle/two-mbius-bands/

All I'm saying about the existence of an embedding space is that it cannot be ruled out based only on observations within the universe. I agree there is no evidence for it.

When it comes to curvature, if this exists then whether intrinsic or extrinsic, it can only do so if a higher dimension exists. You can't have curvature of 3D space without that regardless of whether you can measure it or not. Or at least I can't see how you can.

BoE

Mathematically intrinsic curvature is an internal property and doesn't depend upon the existence or properties of an embedding space. For the universe an embedded space can neither be ruled in or ruled out but is irrelevant scientifically because we have no way of knowing the properties of any embedding space if such exists - and we cannot calculate extrinsic curvature for the universe.

[ursaminortaur]

All I'm saying about the existence of an embedding space is that it cannot be ruled out based only on observations within the universe. I agree there is no evidence for it.

When it comes to curvature, if this exists then whether intrinsic or extrinsic, it can only do so if a higher dimension exists. You can't have curvature of 3D space without that regardless of whether you can measure it or not. Or at least I can't see how you can.

BoE

Mathematically intrinsic curvature is an internal property and doesn't depend upon the existence or properties of an embedding space. For the universe an embedded space can neither be ruled in or ruled out but is irrelevant scientifically because we have no way of knowing the properties of any embedding space if such exists - and we cannot calculate extrinsic curvature for the universe.

Just to add.

I think part of the problem may be that you are only used to thinking of curvature in terms of extrinsic curvature since that is what you see when looking at surfaces within our universe from the outside - so for instance you are used to seeing a cylinder as being curved whereas its intrinsic curvature is flat.