Riemann tensor in 3d Cartesian coordinates

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Suppose we wish to use Cartesian coordinates for points on the surface of a sphere. Then all derivatives of the metric would vanish and so the Riemann curvature tensor would vanish. But it would give us a wrong result, namely that the space is not curved. So it means that if we want to get correct results we must necessarily use spherical coordinates in this case?

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Orodruin
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You are using Cartesian coordinates in an embedding space. This is not the same thing as describing the intrinsic properties of the surface and even if the metric in the embedding space is flat, the induced metric on the submanifold can be curved.

Nugatory
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First you have to precise about what you mean by "using Cartesian coordinates on the surface of a sphere". If you're just attaching ordinary x,y and z coordinates to points on the surface of a soccer ball on the table in front of you, then you're working in a three-dimensional flat space (the room you're in) that happens to contain a soccer ball. A straight line between any two points on the surface of the ball will pass through the interior of the ball; a two dimensional creature living on the two dimensional surface of the ball would interpret that straight line as a path through the mystical third dimension that connects the two points without going through any of the points in between.

The two-dimensional surface of the ball is curved, and it will be identically curved no matter what two-dimensional coordinates you use to label the points on its surface. Latitude and longitude are familiar and easy to work with, but there are others. Note that in this two dimensional world, the paths through the interior of the ball do not exist; a straight line between any two points is a great circle along the surface.

You are using Cartesian coordinates in an embedding space
If you're just attaching ordinary x,y and z coordinates to points on the surface of a soccer ball on the table in front of you, then you're working in a three-dimensional flat space (the room you're in) that happens to contain a soccer ball.
a two dimensional creature living on the two dimensional surface of the ball would interpret that straight line as a path through the mystical third dimension that connects the two points without going through any of the points in between.

Then two-dimensional surface of the ball is curved, and it will be identically curved no matter what two-dimensional coordinates you use to label the points on its surface
I see I think. Good example Nugatory.

So to get non zero components of the Riemann tensor we would have to make a displacement on the surface of the sphere? Using spherical coordinates it's easy, because one just have to set the radius constant. Using Cartesian coordinates I think it is more difficult to keep on a "straight line", because we would have to change three parametres, namely x, y and z. In this case, we would get non-zero derivatives of the metric. Is this right?

Nugatory
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Using Cartesian coordinates I think it is more difficult to keep on a "straight line", because we would have to change three parametres, namely x, y and z. In this case, we would get non-zero derivatives of the metric. Is this right?
The surface of the sphere is a two-dimensional surface, so you only get two coordinates (if you introduce a third, you are working with a three-dimensional space of which the points making up the surface of your sphere is an arbitrary subset). No matter what coordinates you use, you will find that there is no two-dimensional coordinate system in which the metric components are ##g_{ij}=\delta_{ij}## everywhere on the surface of the sphere - which is to say that surface is not flat and it cannot be properly described with two-dimensional Cartesian coordinates.

Of course "flat" can be a pretty good approximation, and then we can use Cartesian coordinates locally. For example, if you ask a Manhattanite for directions, you're likely to hear something like "three blocks uptown, two blocks crosstown" - that's Cartesian coordinates with axes aligned along the street grid. But formally what we're doing is defining a completely different two-dimensional surface, namely a plane that happens to be tangent to the surface of the earth at the point that we're standing and using Cartesian coordinates to give directions for motion in that plane, not on the surface of the earth.

No matter what coordinates you use, you will find that there is no two-dimensional coordinate system in which the metric components are gij=δijgij=δijg_{ij}=\delta_{ij} everywhere on the surface of the sphere - which is to say that surface is not flat and it cannot be properly described with two-dimensional Cartesian coordinates.
Could it be accepted as a definition of Cartesian coordinates? That is, Cartesian coordinates are the ones which describe flat spaces, no matter what their dimension is?
formally what we're doing is defining a completely different two-dimensional surface, namely a plane that happens to be tangent to the surface of the earth at the point that we're standing and using Cartesian coordinates to give directions for motion in that plane, not on the surface of the earth.
I see.

Orodruin
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Could it be accepted as a definition of Cartesian coordinates? That is, Cartesian coordinates are the ones which describe flat spaces, no matter what their dimension is?
No. Cartesian coordinates are coordinates on a Euclidean space. There is no necessity for a flat space to be Euclidean.

No. Cartesian coordinates are coordinates on a Euclidean space. There is no necessity for a flat space to be Euclidean.
Can you give me an example of a flat, non-Euclidean space in which we cannot use Cartesian coordinates?

Orodruin
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Can you give me an example of a flat, non-Euclidean space in which we cannot use Cartesian coordinates?
A flat torus, a flat Möbius strip, a cylinder, a flat Klein bottle, a cone with the apex removed, etc.

a cylinder
In what sense a cylinder is flat?

Orodruin
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In what sense a cylinder is flat?
The usual sense. A cylinder embedded in R3 has zero intrinsic curvature. You can cut it open and lay it flat on a table.

Ibix
A cylinder has no intrinsic curvature. Draw a triangle on a piece of paper and roll it into a cylinder. The angles still add to 180. Circles still have circumference ##2\pi r## if the radius is measured in the surface of the cylinder. Etcetera. It's pretty much the canonical simple example of a space with extrinsic curvature but no intrinsic curvature.

JorisL
Orodruin
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Also, in general, a metric need not be induced by an embedding.

DrGreg
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In what sense a cylinder is flat?
You can tightly wrap a flat piece of paper round it without stretching, tearing or crumpling. So its "intrinsic" local structure (as a manifold) is no different than a flat piece of paper's. Of course they have different "extrinsic" properties (which aren't part of the manifold, but of the embedding in 3D Euclidean space) and different global properties, but flatness is a local intrinsic property.

Nugatory
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In which sense a cylinder is flat?
Take a sheet of ordinary paper. It will pass any test for local flatness you try: the interior angles of triangles will add to 180 degrees, the pythagorean theorem works, there is a coordinate system in which the metric components are ##g_{ij}=\delta_{ij}##. Roll the sheet up into tube and it will form the surface of a cylinder, but it will still have all those flatness properties. The easiest way to see this is to imagine that you drew some geometric shape on the piece of paper before you rolled it up - the marks on the sheet of paper don't move around on the surface so their geometric relationships don't change.

The surface of a sphere is fundamentally different because there's no way of forming the sheet of paper into a sphere without stretching the paper (which doesn't work because paper doesn't stretch - better to use a rubber sheet instead) and altering the geometric relationships between nearby points.

This would be a good time to google search for the difference between "intrinsic curvature" and "extrinsic curvature" if you are not already familiar with those terms.

Orodruin
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It is official. This thread is now a test of how many different ways one can say "make a paper roll".

davidge and Nugatory
@Orodruin , @DrGreg , @Nugatory and @Ibix, thanks for explaining what is the meaning of intrinsic and extrinsic curvature.

No matter what coordinates you use, you will find that there is no two-dimensional coordinate system in which the metric components are ##g_{ij} = g_{ji} = \delta_{ij}## everywhere on the surface of the sphere
Trying to understand the meaning of this, I have tried the following:

Suppose we want to calculate the distance between two points on a sphere, where one point is on the pole and the other is on the equator of the sphere. We can set our coordinate axes such that one coordinate, say z, is always zero.

The infinitesimal Euclidean distance is ##ds^2 = dx^2 + dy^2##, but ##x^2 + y^2 = r^2## in this case, and so ##ds^2 = \frac{x^2}{y^2}dx^2 + \frac{y^2}{x^2}dy^2##. (Is this right?) It seems clear that the metric will have first, second, etc... derivatives. Now it seems that that space (surface of the sphere) has a curvature, although I have not worked on the components of the Riemann tensor. (There will be any non-zero component?)

Nugatory
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Suppose we want to calculate the distance between two points on a sphere, where one point is on the pole and the other is on the equator of the sphere. We can set our coordinate axes such that one coordinate, say z, is always zero.
This picture makes no sense as a way to assign coordinates to the two-dimensional surface of the sphere because there are points on all three coordinate axes that do not lie on the surface of the sphere - so whatever set of points are being labeled by those coordinates, it's not the set of points that corresponds to the surface of the sphere. Furthermore, you have three coordinates not two - taking ##z=0## reduces the number of coordinates by one, but the surface that is being described by the two remaining coordinates is not the surface of the sphere, it's a plane that intersects the sphere.

One way or another you need two coordinates, not three, that cover the surface of the sphere without picking up any points outside the sphere (informally, this means that the coordinate axes lie on the surface of the sphere). Latitude and longitude would work, as would stereographic coordinates. You can calculate the value of the components of the metric tensor for the surface of the sphere in your chosen coordinate system by using the concept that @Orodruin mentioned above: it's an induced metric once you've decide to think of the two-dimensional sphere as something embedded in Euclidean three-dimension space.

Ok
You can calculate the value of the components of the metric tensor for the surface of the sphere in your chosen coordinate system by using the concept that @Orodruin mentioned above: it's an induced metric once you've decide to think of the two-dimensional sphere as something embedded in Euclidean three-dimension space.
How would this induced metric look like in terms of Cartesian coordinates?

Nugatory
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How would this induced metric look like in terms of Cartesian coordinates?
You mean the Cartesian coordinates of the three-dimensional Euclidean space in which the sphere is embedded?
Suppose you go with latitude ##\phi## and longitude ##\theta## as your coordinates on the surface of the sphere. We have (although wise people will check my algebra):
##x=R\cos\phi\sin\theta##
##y=R\cos\phi\cos\theta##
##z=R\sin\phi##
relating the ##\theta## and ##\phi## coordinates of points on the two-dimensional surface of the sphere to the x,y, and z coordinates of points in the three-dimensional Euclidean space in which the sphere is embedded. Some algebra and the formula in the wikipedia article I linked will see you home from there.

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You mean the Cartesian coordinates of the three-dimensional Euclidean space in which the sphere is embedded?
Yes
Suppose you go with latitude ϕϕ\phi and longitude θθ\theta as your coordinates on the surface of the sphere. We have (although wise people will check my algebra):
x=Rcosϕsinθx=Rcos⁡ϕsin⁡θx=R\cos\phi\sin\theta
y=Rcosϕcosθy=Rcos⁡ϕcos⁡θy=R\cos\phi\cos\theta
z=Rsinϕz=Rsin⁡ϕz=R\sin\phi
relating the θθ\theta and ϕϕ\phi coordinates of points on the two-dimensional surface of the sphere to the x,y, and z coordinates of points in the three-dimensional Euclidean space in which the sphere is embedded. Some algebra and the formula in the wikipedia article I linked will see you home from there.
Yea, in this case we would obtain the components in terms of ##r, \theta## and ##\phi##. I would like to get them in terms of ##x,y## and ##z## instead.

Orodruin
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Yes

Yea, in this case we would obtain the components in terms of ##r, \theta## and ##\phi##. I would like to get them in terms of ##x,y## and ##z## instead.
You are missing the point. The radius ##r## is not a coordinate on the sphere. The sphere is two-dimensional and is described with only two coordinates.

The radius rrr is not a coordinate on the sphere
I'm sorry for including the ##r## into above. So how can we obtain the components in terms of ##x,y## instead of ##\theta, \phi##?

Orodruin
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I'm sorry for including the ##r## into above. So how can we obtain the components in terms of ##x,y## instead of ##\theta, \phi##?
Start from the line element and the relation ##ds^2 = dx^2 + dy^2 + dz^2## and the expression of z in terms of x and y ##z = \sqrt{R^2 -x^2-y^2}## to find ##dz##.

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pervect
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@Orodruin , @DrGreg , @Nugatory and @Ibix, thanks for explaining what is the meaning of intrinsic and extrinsic curvature.

Trying to understand the meaning of this, I have tried the following:

Suppose we want to calculate the distance between two points on a sphere, where one point is on the pole and the other is on the equator of the sphere. We can set our coordinate axes such that one coordinate, say z, is always zero.

The infinitesimal Euclidean distance is ##ds^2 = dx^2 + dy^2##, but ##x^2 + y^2 = r^2## in this case, and so ##ds^2 = \frac{x^2}{y^2}dx^2 + \frac{y^2}{x^2}dy^2##. (Is this right?) It seems clear that the metric will have first, second, etc... derivatives. Now it seems that that space (surface of the sphere) has a curvature, although I have not worked on the components of the Riemann tensor. (There will be any non-zero component?)
You might find articles on the Mercator projection helpful. If you look at a map of the globe, the most common map is the Mercator projection, though there are others. If you read about about cartography, which is less advanced than GR but seems applicable to what you're struggling with, you can get a better idea of how we create a 2d map of the 2d surface of the Earth - and the formulas we use to get distances from the map coordinates, which would basically be the metric.

Using x,y, and z as coordinates doesn't really work, because not all coordinate values are constrained to lie on the sphere. You could conceivably use x and y as coordinates on a hemisphere, but you won't have the amount of reference material available to you that you will have if you go with lattitude, longitude, and the mercator projection technique.

If you did use x and y as coordinates, I can quickly sketch out what I think you should get - which doesn't look particularly close to what you wrote earlier. You want to compute ##ds^2 = dx^2 + dy^2 + dz^2## with the constraint equation that defines z(x,y) = ##\sqrt{R^2 - x^2 - y^2}##

Using the chain rule, we can write

$$dz = \frac{\partial z(x,y)}{\partial x}dx + \frac{\partial z(x,y)}{\partial y}dy$$

So we would get

$$ds^2 = dx^2 + dy^2 + dz^2 = dx^2 + dy^2 + \left( \frac{\partial z(x,y)}{\partial x}dx + \frac{\partial z(x,y)}{\partial y}dy \right)^2$$

And if you expand it out you'll get a line element in terms of dx and dy. Conceptually, you'd be mapping the earth by a projection process, but instead of the mercator projection process I suggested earlier, you'd map points on half the sphere to a unit circle on the x,y plane. So you'd really need two maps to cover an entire sphere.

The mercator projection technique isn't that much more invovled, is a bit more standard, and maps all points on the globe except for the north and south poles on one map, so it's probably a preferable technique. I don't have any good references for it, alas - you can try the wiki, it does have an article, but I'm not sure how indepth it really is.

In any event, you'll wind up with 2 coordinates for every point on the 2d manifold that's the surface of the globe, which is the correct noumber of coordinates. And you'll find that there is some formula that will give you the distance between two points , but it won't be the pythagorean formula that you seem to want it to be, it will be some other more complex formula.

.

Nugatory