# B Space cannot be curved?

1. Aug 9, 2017

### PauloConstantino

I have always had this question, and I wonder if someone can explain to me if I am wrong about it.

In relativity they say space(time) can curve around massive bodies. Let's just consider space for a moment.

For example on the surface of a sphere, you tell someone to walk in a straight line, and after a while you can see that the path taken by the person is curved, because you can see that the person has not followed a straight line but has curved around the sphere. You can then see a straight line from yourself to the person, and this straight line does not follow the surface of the sphere.

Is this the sense in which space is curved? Because to me, this shows that space can only be curved in relation to an absolute "straight space". Otherwise you would never know that it is curved. There must be an absolute straight space geometry underlying the curved space.

Can anyone comment on this please

2. Aug 9, 2017

### A.T.

No, what you describe is extrinsic curvature, which would also happen if the person walks around a cylinder. But the sphere also has intrinsic curvature, which the cylinder doesn't. Intrinsic curvature can be determined without reference to a flat embedding space.

3. Aug 9, 2017

### atyy

The intrinsic curvature of the sphere can be seen by a person walking straight lines on it without an external reference. The person on the sphere must walk straight lines such that his path is a triangle. He will find that the angles of the triangle add up to more than 180°, indicating that his space is curved.

http://web.stanford.edu/~oas/SI/SRGR/notes/SRGRLect8_2012.pdf

4. Aug 9, 2017

### jbriggs444

There is a distinction to be made between intrinsic curvature and extrinsic curvature. The easy to visualize curvature is extrinsic. The physically meaningful curvature is intrinsic. Intrinsic curvature is what general relativity deals with.

Imagine a sheet of paper laid flat on a table. On this paper is a bug. The bug walks along the shortest path on the paper from point A to point B. We can all agree that the paper is flat and that the line is straight. Now roll up the paper into a tube. Again the bug walks along the shortest path from point A to point B. The bug will have walked across the exact same path on the paper (assuming that the shortest path does not cross the seam). From the point of view of the paper, the path is straight. From our external point of view, the paper is curved and the path follows the paper.

The above is an example of extrinsic curvature. We have a two dimensional space (the paper) and a larger three dimensional space (our ordinary three dimensional geometry) in which it is embedded. The curvature of the path depends on how we roll up the paper -- how we choose to do the embedding. For an inhabitant living on the surface of the paper using measurements made only on that surface, extrinsic curvature is not detectable.

Extrinsic curvature is a property of the embedding. The bug traces out the same path on the paper, whether it is rolled or unrolled. It is only from our external viewpoint that we can see any "curvature".

In the case of a sphere, things are not so simple. We cannot roll up a flat sheet of paper into a sphere without stretching or wrinkling it. If the bug on a sphere follows shortest paths and keeps careful track of distances, he can notice that the surface he lives on is not Euclidean. The interior angles of a triangle will sum to more than 180 degrees.

The above is an example of intrinsic curvature. For an inhabitant living on the surface of the paper using measurements made only on that surface, intrinsic curvature is detectable.

Here comes the hard part...

In the examples above, we talked about a two dimensional surface embedded within a three dimensional Euclidean space. That was just an aid to visualization. We can talk about the geometry of a two (or three or four or more) dimensional space without requiring that it be embedded in a higher dimensional space at all. A common way to do that is to imagine that the inhabitants of the space are able to measure distances. From any point in the space to any other point in the space, they can measure the distance between them. The distance measurement is the mathematical notion of a "metric". A space for which a metric exists is called a "metric space". Given a metric space, one can define intrinsic curvature in terms of the metric.

A standard way of handling a metric space in physics is to divide it up (if needed) into pieces and equip each piece with a Cartesian coordinate system. There are some rules about the boundaries and how to handle the seams between the pieces that we need not concern ourselves with. The result is a "manifold".

A surface of a sphere can be modeled as a two dimensional manifold. There is no embedding in three dimensional space and no meaningful notion of extrinsic curvature. There is still non-zero intrinsic curvature for this space.

Edit: Just noticed that this is an "A" level thread asking a "B" level question. This answer is at B level -- no equations.

5. Aug 9, 2017

### Parixit

The example of walking on sphere is just to understand why something that is curved may seem straight. The curvature of space is real and that is what we call Gravity. Near any massive body, space is curved towards center and hence any free object will move towards center, and we call it freely falling body.

6. Aug 9, 2017

### PauloConstantino

I think there's something wrong somewhere, probably in my thinking.

What I want to say is that it seems to me impossible for something to be curved unless it is curved in relation to absolute straightness. Do you know what I mean?

How can you tell something is curved at all? You need a reference point, that is, something which isn't curved at all. And so there must be an absolute straight space to compare curved space with, and so the real space must be straight.

Am I incorrect? If so why ? How can you prove that absolute space is curved?

I am a maths graduate so you can use equations if you like.

7. Aug 9, 2017

8. Aug 9, 2017

### Staff: Mentor

Yes, we know exactly what you mean. That is why several of us have already pointed out the difference between intrinsic and extrinsic curvature. You need to stop just repeating your question and start responding to the answers.

This is extrinsic curvature, and it is not the kind of curvature we use in GR. We use intrinsic curvature which does not require a external reference.

Take a piece of paper, put it on your desk and draw a triangle on it. The sum of the angles is 180, indicating that the paper is intrinsically flat. Now, roll the paper into a cylinder. Extrinsically, the paper is now curved, but the angles still add up to 180 so intrinsically it is still flat.

Now draw a triangle on a sphere. The sum of the angles is greater than 180, indicating that the surface of the sphere is intrinsically curved. There is no need to go outside the surface of the sphere to know that it is curved.

Last edited: Aug 9, 2017
9. Aug 9, 2017

### Staff: Mentor

An easy way to test for curvature is to choose three points and stretch strings between them to form a triangle, then measure the three interior angles of the triangle. If they add up to something different than 180 degrees then you know that you're in a curved space.
This does depend on a particular definition of "straight": a taut string, which is the shortest path between two points, is considered to be "straight"; and indeed this is pretty much an informal way of stating what the formal mathematical definition of "straight" means.

The two-dimensional space that is made up of all the points on the surface of a sphere is a curved space, and the string/triangle test will find the curvature without ever involving any points that are not part of that space. In that example, the points making up the two-dimensional space are a subset of the points that make up the larger three-dimensional space (we would say that two-dimensional space is "embedded" in the larger three-dimensional space), and the string/triangle test will show that that space is flat. But that's a different space, one that includes more points than the two dimensional surface But we don't need access to that larger flat space to measure that the two-dimensional space is curved.

In our four-dimensional spacetime we can "draw" straight lines by shining laser beams and tracing the paths (called "worldlines") of objects moving inertially, and their behavior allows us to measure the curvature even though that spacetime is not embedded in a larger five-dimensional space.

Last edited: Aug 9, 2017
10. Aug 9, 2017

### Janus

Staff Emeritus
Think of it this way, in curved space, the rules of geometry are different than they they are in flat space. In flat space, if you join three straight lines to form a triangle, the sum of the interior angles will be 180 degrees. In curved space, joining three straight lines can result in interior angle sums of less or greater than 180 degrees. You don't need an absolute flat space reference to compare this to, you just measure the interior angles. You don't need "real" flat space to exist except as one of infinite possible values of curvature that could exist.

11. Aug 9, 2017

### Staff: Mentor

Another example that demonstrates the intrinsic curvature of a spherical surface: Consider two airplanes that start out flying from the Equator, heading straight north (i.e. parallel to each other) along different longitude lines. They maintain their respective headings without change. As they approach the North Pole, they come closer and closer to each other, contrary to what we know about parallel lines on a flat surface.

12. Aug 9, 2017

### Ibix

It's easy to visualise a curved surface embedded in a Euclidean volume. As others have noted, though, this gives you two distinct ways to define "curved". One ("extrinsic" curvature) is by comparison to the geometry of the volume in which the surface is embedded. The other ("intrinsic" curvature) is by reference to the geometric properties of lines and angles in the surface itself.

However, there is no need to visualise the surface. You can simply describe it as a set of points, and define rules for the distance between points. This is what we do in relativity. You can't have the notion of extrinsic curvature with this definition, but you get intrinsic curvature. And it doesn't seem to matter. We never need the extrinsic definition for any physics.

Is this abstract definition "actually" a curved space? Who knows? Who cares? We describe it as a curved spacetime by analogy to what it would look like if it were embedded in something. But we need no such assumption.

13. Aug 9, 2017

### pervect

Staff Emeritus
If you are a math graduate, try reading about non-euclidean geometry. The wolfram article seems a good place to start, http://mathworld.wolfram.com/Non-EuclideanGeometry.html

If you view geometry from a mathematical perspective, it is convenient to start with Euclidean geometry. This puts your undefined intuitive notion of "an absolute straight space" which doesn't have any formal mathematical meaning that I'm aware of, into a well-accepted and rigorous mathematical formulation.

Then all we are saying is that mathematically consistent geometries exist that are not Euclidean. That's the mathematical perspective. Another element of the mathematical perspective may be useful here is to talk about what elements of the postulates that mathematically define Euclidean geometry need to be modified to have a non-Euclidean geometry? The short answer is that we need to modify the parallel postulate.

The actual sort of geometry you'd want to study for General relativity is called "differential geometry", and be warned it's not quite as simple as some of what I'm presenting. It is something that can be studied, it is not something that I can present cleanly in a short post.

Mathematically we can talk about embed a non-Euclidean geometries in an Euclidean geometry of higher dimension, but this isn't always convenien, even though I believe it's possible. . I'm not really familiar with the embedding theorems, but Google finds "Whitney Embedding Theorem" https://en.wikipedia.org/wiki/Whitney_embedding_theorem. The point here is you are saying that such an embedding is "required", when what you should be saying is probably more along the lines of "I am so used to Euclidean geometry that I want to leverage off of it as much as possible rather than learn something new". Unfortunately for you, attempting to leverage off your existing knowledge of Euclidean geometry turns out not to be very productive, so you need to learn something new. If for no other reason than that's what the literature does, so if you want to take advantage of the literature, you need to study what it says.

The sub-point is that it's defintiely NOT required to study non-Euclidean geometry as some embedding in a space of higher dimension. Because something is possible doesn't imply that it's required, or even that it's necessarily a good idea.

14. Aug 10, 2017

### Orodruin

Staff Emeritus
Just to add that the argument with measuring the angles of a triangle applies to metric spaces. You can define (intrinsic) curvature without any reference to a metric in the form of a curvature tensor. All you need is a notion of what it means for a vector field to be parallel along a curve, i.e., an affine connection. A space is then flat if the parallel transport of any vector around any closed curve that homotopic to a point returns the same vector. (Or, equivalently, the curvature tensor is zero.)

15. Aug 10, 2017

### bahamagreen

I'm sympathetic to the OP's frustration.

The "triangle on the surface of a sphere" is not a triangle with straight lines between the vertices; that triangle has curved edges that follow the curve of the sphere's surface. Those lines may be shortest paths through the curved surface, but they are not straight. Three points that make the vertices of the "triangle" on the surface of a sphere may be connected through the interior of the sphere with straight lines.

Similarly, the "parallel" latitudinal paths going north from the equator are not straight lines and not parallel; they are curved like the Earth's surface. Lines that are tangent to the Earth's surface only touch at one place and don't bend to form a contact path on the surface.

None of the explanations provided are getting past distinguishing intrinsic from extrinsic curvature beyond calling an apparently curved path a straight path by ignoring or denying embedding and calling the shortest path through a curved space a straight line.

The examples have obviously curved things claiming to be straight. The OP was accused of repeating the question, but who is repeating the same answers without demonstrating justification how embedding can be removed from consideration?

16. Aug 10, 2017

### Staff: Mentor

Yes, they are, because in a curved manifold, the definition of "straight" is not the one you are using. The correct definition of "straight" is "geodesic"--on a 2-sphere, for example, geodesics are great circles, and drawing a triangle on a 2-sphere is done using 3 segments of great circles.

Why is this the correct definition of "straight"? Because it is the correct generalization of the Euclidean definition to cases where you do not have any Euclidean space available at all. In the case of the Earth, yes, we know the Earth's surface is a 2-sphere embedded in a 3-dimensional space, which (at least as best we can tell) is Euclidean. But in the general case, there will not be any such embedding available, and any manifold has to be analyzed solely in terms of its intrinsic properties for a correct understanding of curvature and geometry.

No, they are using the correct definition of "straight" in the general case where there is no embedding in a Euclidean space to fall back on.

17. Aug 10, 2017

### Staff: Mentor

But there is no absolute "straight space" in the general case. For example, we have no evidence of any higher dimensional "straight space" (the correct term would be "Euclidean", or one of its synonyms in the case of a spacetime) in which our universe is embedded. So the only way we have to analyze the curvature of the spacetime of our universe, including the curvature of any "spaces" included in it, is to do so using intrinsic properties that don't rely on knowledge of any absolute "straight space", since we have no such knowledge.

18. Aug 10, 2017

### A.T.

They have extrinsic curvature w.r.t the embedding 3D-space, but no curvature w.r.t the embedding 2D surface of the sphere, which we are inserted in.

That's the definition of intrinsic curvature.

19. Aug 10, 2017

### Ibix

Why do you think you need to consider embedding? The formal process of describing a manifold simply starts with a set of points and adds rules for how they are related. The end result is some maths that also happens to be useful for describing curved sub-spaces of a Euclidean manifold, but that doesn't mean it is in any way reliant on such an embedding.
I know what you mean. But, ultimately, he's confusing "this maths does describe curved manifolds embedded in higher dimensional spaces" with "this maths can only describe curved manifolds embedded in higher dimensional spaces". None of the physics we see needs an embedding, and that is our justification for dropping it as an assumption.

20. Aug 10, 2017

### bahamagreen

Thanks, those seem like better approaches to graspable answers... curious what the OP might think.