Understanding curvature and dimensions

[TrickyDicky]

When explaining the concept of spacetime curvature many popular science books (but I've also seen it mentioned in some textbooks) recur to the difference between intrinsic and extrinsic curvature, and how in a 2-D world with ants or bugs that are two dimensional, they would not be able to detect or directly perceive in any way a extrinsic curvature (because it obviouly would demand a third dimension to detect it and then it wouldn't be a 2D world anymore) but if they were clever enough and had developed their math and analytical tools to a certain degree, they could, in principle, be able to say whether their world has intrinsic curvature and maybe they live in a curved surface. For instance by measuring big enough triangles and adding their angles, if the result was different than 180 degrees they could infer their world was the surface of a sphere (if they added up to more than 180º). But besides those analytical ways they wouldn't be able to directly or obviously observe the curvature of their world without having access to a higher dimension.


Now when applying this to our world, I find it hard to follow this same reasoning.
We are told that we live in a 4D spacetime that is intrinsically curved, so the basic difference with the previous scenario is that we are creatures confined to 4D instead of 2D, (a more subtle difference is that one of the 4 dimensions is temporal but I guess the 2D example works also if it was a 2-spacetime ) but this is not at first sight important for the distinction made between intrinsic and extrinsic curvature.
But there seems to be a difference between the flatland case and ours, we can actually perceive the curvature of our universe that is usually referred in GR as gravity without any difficulty, we observe the parabolic curves objects describe when shot from the ground, or the curved trajectories of the orbits of objects in space, etc, we don't need math to acknowledge the effects of the curvature of our 4-spacetime, but we were told that this only happens in the case of extrinsic curvature, that only extrinsic curvature should be obvious to our senses.
Funny, isn't it?

 

[SamirS]

First, these are examples used to visualize something that can't be visualized in our brain, the intrinsic curvature of spacetime. What we see, the parabolic path of thrown objects, orbits etc. is not the curvature of spacetime, it is the effect that the curvature of spacetime has on straight lines. We can use this movement to say there must be a curvature, but we still can't "see" it. It's a mathematical description that translates poorly into our lowly post-reptile brains.Now there actually isn't any difference to a flatland physicist measuring that the angles of his triangle add to a higher number than 180. He is measuring the effect of the intrinsic curvature of his flatland universe; however, just as the parabolic path of a thrown ball shows the curvature of spacetime indirectly, the angle of the triangle shows the curvature of flatland spacetime. It would be a mathematical construct for the flatland physicist, he could never actually directly "view" or "imagine" how curved space looks because his brain didn't evolve in a way that allows such a picture.

In short, parabolic paths and orbits etc. in 4D spacetime are the equivalent of the higher-than-180°-angle-triangle in 2D spacetime.

Which doesn't change the fact that it still is a mathematically developed concept that poorly translates into our visual imagination.

 

[zhermes]

That is a very good, and thorough analysis, but you're off a little in your interpretation of GR. The curvature of 4D spacetime in GR is still in those 4D---not requiring additional dimensions. The analogy is for 3D space which is bent into the 4th (time) direction.

For us, with perception largely constrained to 3D space (or you can think of it as spatial-perception perfectly constrained to 3D space), we cannot perceive spatial curvature into the 4th dimension; when in 'actuality' space-time curves together in a 4D framework.

Does that make sense?

 

[TrickyDicky]

That is a very good, and thorough analysis, but you're off a little in your interpretation of GR. The curvature of 4D spacetime in GR is still in those 4D---not requiring additional dimensions. The analogy is for 3D space which is bent into the 4th (time) direction.

But I have not said anything in my post about requiring another dimension. I said 4D curvature of our spacetime is intrinsic. Are you saying that what is really curved is the 3D-space and that we perceive it as extrinsic curvature from the fourth temporal dimension? But according to current cosmology 3D space is flat.

For us, with perception largely constrained to 3D space (or you can think of it as spatial-perception perfectly constrained to 3D space), we cannot perceive spatial curvature into the 4th dimension; when in 'actuality' space-time curves together in a 4D framework.

Does that make sense?

I didn't actually understand this bit.

 

[TrickyDicky]

First, these are examples used to visualize something that can't be visualized in our brain, the intrinsic curvature of spacetime. What we see, the parabolic path of thrown objects, orbits etc. is not the curvature of spacetime, it is the effect that the curvature of spacetime has on straight lines. We can use this movement to say there must be a curvature, but we still can't "see" it. It's a mathematical description that translates poorly into our lowly post-reptile brains.


Now there actually isn't any difference to a flatland physicist measuring that the angles of his triangle add to a higher number than 180. He is measuring the effect of the intrinsic curvature of his flatland universe; however, just as the parabolic path of a thrown ball shows the curvature of spacetime indirectly, the angle of the triangle shows the curvature of flatland spacetime. It would be a mathematical construct for the flatland physicist, he could never actually directly "view" or "imagine" how curved space looks because his brain didn't evolve in a way that allows such a picture.

In short, parabolic paths and orbits etc. in 4D spacetime are the equivalent of the higher-than-180°-angle-triangle in 2D spacetime.

Which doesn't change the fact that it still is a mathematically developed concept that poorly translates into our visual imagination.

Why do you say that our brains can't visualize parabolic paths? Mine can. Basically you are saying that intrinsic and extrinsic curvature are the same thing. They aren't. See the WP page on curvature.

 

[SamirS]

Why do you say that our brains can't visualize parabolic paths? Mine can. Basically you are saying that intrinsic and extrinsic curvature are the same thing. They aren't. See the WP page on curvature.

Well because I don't say that. You can visualize a parabolic path, of course, and this path is caused by curvature, but what you can't visualize the curvature itself.

The 2D physicist can visualize the triangle with the strange angles, and this is caused by curvature, but he too can't visualize the curvature itself.

What you are basically trying is to make up a picture of how spacetime curvature "looks". However, this makes no sense because we can only perceive its effects, nothing more. You are mistaking an analogy that was created to help people who do not have the required math skills to understand, in a very simple sense, how gravity may work. It's just like the rubber sheet analogy - it is not accurate, and it is a popular science representation, but it is not science nor what curvature "looks like".

 

[TrickyDicky]

Well because I don't say that. You can visualize a parabolic path, of course, and this path is caused by curvature, but what you can't visualize the curvature itself.

The 2D physicist can visualize the triangle with the strange angles, and this is caused by curvature, but he too can't visualize the curvature itself.

What you are basically trying is to make up a picture of how spacetime curvature "looks". However, this makes no sense because we can only perceive its effects, nothing more. You are mistaking an analogy that was created to help people who do not have the required math skills to understand, in a very simple sense, how gravity may work. It's just like the rubber sheet analogy - it is not accurate, and it is a popular science representation, but it is not science nor what curvature "looks like".

I'm actually talking about two different types of curvature that are not analogies to help people without the required math skills but that are actually used by mathematicians, it doesn't have anything to do with the rubber sheet analogy, although actually the concept can be explained in lay terms.

You are blurring that distinction completely. I don't even know what you mean by "you can't visualize the curvature itself". When you see a basket ball, in what sense can't you visualize its curvature? Well if you do visualize it is because it is an example of intrinsic 2D-curvature seen from a higher dimension (3D space). Intrinsic curvature can always be visualized from an embedding higher dimension. If you were a 2D bug living in the basketball surface, then you wouldn't be able to visualize it, but you still would be able to measure it, using Riemannian geometry tools.
Can you visualize the curvature in the surface of a cylinder? You should be able. That is an example of "only" extrinsic curvature, because it can't be neither visualized nor measured from a 2D POV, you can only do it from a higher embedding dimension.

In the case of gravity, we can perceive it without the aid of Riemannian geometry (and of course we can model it with the tools of differential geometry), various possibilities are open, the curvature that we visualize: is the intrinsic curvature of 3D space from a 4D POV? Or is it the intrinsic 4D spacetime curvature seen from a fifth dimension? Or even another possibility, is it an extrinsic curvature of flat 3D space from 4D spacetime (expanding space wrt time universe)? This last seems to be a possible interpretation of the L-CDM model.
But still the most common interpretation from GR is that gravity is the intrinsic 4D curvature of spacetime, which is hard to reconcile with something that we can easily see and feel from our 4D perspective.
When we see the elliptic path of a planet or the parabolic path of a stone thrown from the ground, we are perceiving a curved shape in time, just like when we see the curved shape of an apple, and these lead one to think that the observed curvature is not a intrinsic 4D-curvature in one case or 3D intrinsic curvature in the case of the apple.

 

[zhermes]

But there seems to be a difference between the flatland case and ours, we can actually perceive the curvature of our universe that is usually referred in GR as gravity without any difficulty, we observe the parabolic curves objects describe when shot from the ground, or the curved trajectories of the orbits of objects in space, etc, we don't need math to acknowledge the effects of the curvature of our 4-spacetime, but we were told that this only happens in the case of extrinsic curvature, that only extrinsic curvature should be obvious to our senses.

What I was trying to say was that these are actually examples of extrinsic curvature: e.g. parabolic trajectories---extrinsic curvature embedded in 3D space (or 4D space-time). This we can see when we throw a rock, or fly a rocket-ship. They are caused by intrinsic curvature in 4D space-time, which we cannot see, but need mathematics to understand.

 

[TrickyDicky]

What I was trying to say was that these are actually examples of extrinsic curvature: e.g. parabolic trajectories---extrinsic curvature embedded in 3D space (or 4D space-time). This we can see when we throw a rock, or fly a rocket-ship.

This I can follow.

They are caused by intrinsic curvature in 4D space-time, which we cannot see, but need mathematics to understand.

Do you mean that 4D spacetime intrinsic curvature causes the extrinsic curvature that we see?
This would mean if we take it to the 2D example that the intrinsic curvature of a surface can cause the extrinsic curvature of a circle drawn on this surface that 2D ants can perfectly see.
It makes little sense to me, but I'm no expert.

 

[zhermes]

Do you mean that 4D spacetime intrinsic curvature causes the extrinsic curvature that we see?
This would mean if we take it to the 2D example that the intrinsic curvature of a surface can cause the extrinsic curvature of a circle drawn on this surface that 2D ants can perfectly see.
It makes little sense to me, but I'm no expert.

Yeah, its a pretty strong statement, but I stand by it. Its the basis of the GR interpretation of gravity (it wouldn't be the case in a quantum gravity---not necessarily at least). The 4D, intrinsic curvature of spacetime is what causes massive bodies to feel gravity, which causes them to move in extrinsically 'curved' paths that are readily apparent. That was the underlying insight of Einstein, I suppose.

 

[TrickyDicky]

Yeah, its a pretty strong statement, but I stand by it. Its the basis of the GR interpretation of gravity (it wouldn't be the case in a quantum gravity---not necessarily at least). The 4D, intrinsic curvature of spacetime is what causes massive bodies to feel gravity, which causes them to move in extrinsically 'curved' paths that are readily apparent. That was the underlying insight of Einstein, I suppose.

I agree that seems to be the usual interpretation, but is it accurate mathematically? That is my doubt.

 

[zhermes]

I agree that seems to be the usual interpretation, but is it accurate mathematically? That is my doubt.

Absolutely. Maybe I don't understand, why might it not be accurate?

 

[TrickyDicky]

Absolutely. Maybe I don't understand, why might it not be accurate?

This is an example for 2D gaussian curvature, just translate it to 4D spacetime curvature
From Wolfram mathworld: http://mathworld.wolfram.com/IntrinsicCurvature.html

"Intrinsic Curvature: A curvature such as Gaussian curvature which is detectable to the "inhabitants" of a surface and not just outside observers. An extrinsic curvature, on the other hand, is not detectable to someone who can't study the three-dimensional space surrounding the surface on which he resides."