General relativity and curvilinear coordinates

["Don't panic!"]

I have just been asked why we use curvilinear coordinate systems in general relativity. I replied that, from a heuristic point of view, space and time are relative, such that the way in which you measure them is dependent on the reference frame that you observe them in. This implies that coordinate systems change from point to point in spacetime, i.e. they are, in general, curvilinear coordinate systems. From a more mathematical point of view, spacetime is represented by a 4 dimensional manifold which is, in general, curved (physically this is caused by the presence of matter, although it is also possible for the spacetime to be intrinsically curved, even in vacuum). We wish to be able to describe such a manifold without embedding in some higher dimensional space (as after all, we have no a priori reason to believe that our universe is embedded in a higher dimensional space), and therefore we can only describe it in terms of local coordinate maps, in which we can construct locally invertible maps between Euclidean space $\mathbb{R}^{n}$ (which itself is most straightforwardly described by Cartesian coordinates) and the manifold (generally non-Euclidean). Thus the coordinate systems are necessarily curvilinear, as they can only locally describe the manifold, and will change as we move across the manifold.

Would this be an acceptable answer? Any feedback, improvements would be much appreciated.

 

[PeterDonis]

This looks ok, heuristically, but I think there is a simpler way to put it: Cartesian (non-curvilinear) coordinate systems only exist globally in a flat manifold, and spacetime in the presence of gravity is not flat. So we have to use curvilinear coordinate systems for spacetime in the presence of gravity because they're the only ones that exist.

 

["Don't panic!"]

Thanks for taking a look. Is it also fair to say that, in order to describe a curved space one requires a set of curvilinear coordinates defined locally. We can find a mapping from such a curvilinear coordinate system to local Cartesian coordinates. For example, if the manifold is a 2-sphere it can be described by a curvilinear coordinate system $(r,\theta)$, that can be locally mapped to a Cartesian coordinate system, such that we can describe the whole surface through a collection of coordinate patches from open sets of $(r,\theta)$ to Cartesian coordinates?!

 

[PeterDonis]

Is it also fair to say that, in order to describe a curved space one requires a set of curvilinear coordinates defined locally.

Any coordinates will be "defined locally"; there are no other kinds of coordinates.

 

["Don't panic!"]

Any coordinates will be "defined locally"; there are no other kinds of coordinates.

And is this why we consider curvilinear coordinates, as locally implies that that coordinates change from point to point?! (Sorry, I think I've gone and confused myself now)

 

[PeterDonis]

locally implies that that coordinates change from point to point

No, it doesn't. "Locally" just means that coordinates are assigned to points in the manifold, and nearby points get assigned nearby coordinate values, and within a small enough patch around a given point we can use flat coordinates, at least to a good approximation. But that will be true whether the manifold is globally flat or globally curved. Curvature is something that doesn't show up locally.

 

["Don't panic!"]

Curvature is something that doesn't show up locally.

I get that, but then why do we use curvilinear coordinates at all, couldn't we just use Cartesian coordinates locally? Or is the point that points on the manifold exist independently of the coordinates that we assign them, and so we are free to use whaf ever coordinates we like?

 

[PeterDonis]

why do we use curvilinear coordinates at all, couldn't we just use Cartesian coordinates locally?

Because many of the questions we would like to answer are not local questions. For example, if we want to describe the event horizon of a black hole, we can't do it using local coordinates; the horizon is a global feature of the spacetime geometry, not a local one. (It's the same problem we would have if we tried, for example, to describe the route an airliner takes from Los Angeles to Sydney; there's no way to do it using local Cartesian coordinates.)

 

["Don't panic!"]

So, for example, if we wanted to describe a path across the surface of a sphere we would need to do so using curvilinear coordinates (in this case spherical polar). I think what confuses me, is that coordinate systems on a manifold are usually only valid over a particular patch. Is the point that the patch that a particular coordinate system is valid over may be large enough that curvature needs to be taken into account? (Intuitively I can picture a coordinate grid "wrapped" around a sphere, for example)

 

[PeterDonis]

So, for example, if we wanted to describe a path across the surface of a sphere we would need to do so using curvilinear coordinates

Yes.

(in this case spherical polar).

Or Mercator, or stereographic, or...

The point being that "curvilinear coordinates" is a very general term; for any manifold, there will be lots of different possible curvilinear coordinates that could be chosen.

coordinate systems on a manifold are usually only valid over a particular patch. Is the point that the patch that a particular coordinate system is valid over may be large enough that curvature needs to be taken into account?

Yes. For example, spherical polar coordinates on a sphere are valid everywhere except the poles.

 

["Don't panic!"]

Great, I think that's started to clear things up for me.
Is it correct to say that, in general, curvilinear coordinates are required to describe points, curves etc on a manifold, as if one used Cartesian only one would only ever be able to describe very small patches on the manifold and would be unable to describe paths across the manifold (which would be pretty useless).

 

[PeterDonis]

Is it correct to say that, in general, curvilinear coordinates are required to describe points, curves etc on a manifold, as if one used Cartesian only one would only ever be able to describe very small patches on the manifold and would be unable to describe paths across the manifold (which would be pretty useless).

To describe curves and other geometric objects that cover more than a small local patch of the manifold, yes. Points, strictly speaking, do not fall in this category; you can always describe a single point and a small local patch around it with local Cartesian coordinates.

 

["Don't panic!"]

Ah, ok. So we could choose to use local Cartesian coordinates to describe small patches, but in general this wouldn't be useful as we would be unable to describe curves, vector fields etc. that are defined over several coordinate patches.

 

[PAllen]

Ah, ok. So we could choose to use local Cartesian coordinates to describe small patches, but in general this wouldn't be useful as we would be unable to describe curves, vector fields etc. that are defined over several coordinate patches.

Well, even here trying to use lots of small Cartesian patches, you run into the fact that none of them can be precisely Cartesian if the manifold is curved. Curvature is non-zero at a single point. In rough analogy with simple calculus, the deviation between a tangent and a curve gets smaller the closer you get to the point of tangency; yet the second derivative does not approach zero, and has a specific value at every point.

 

["Don't panic!"]

Well, even here trying to use lots of small Cartesian patches, you run into the fact that none of them can be precisely Cartesian if the manifold is curved. Curvature is non-zero at a single point. In rough analogy with simple calculus, the deviation between a tangent and a curve gets smaller the closer you get to the point of tangency; yet the second derivative does not approach zero, and has a specific value at every point.

Would it be correct to say that as Cartesian coordinates describe a (hyper) plane, then one could run into the situation where the manifold has intrinsic curvature at each particular point and so one could not construct a Cartesian coordinate patch around any point as one would "move off" the manifold Such that the coordinate system doesn't accurately describe the points on the manifold at that point (as per your analogy, the points in the Cartesian coordinate system would be in a hyperplane tangent to a point on the manifold and would deviate from the actual description of points in the patch it's trying to describe)?

I (maybe incorrectly) visualise it as follows. Locally, a manifold can be mapped to Euclidean space. Such a manifold will be curved in general (even locally) and so in order to accurately describe points on the manifold in a given coordinate patch in terms of coordinates in Euclidean space we require a coordinate system whose coordinate lines curve (such that they recreate the curvature of the manifold in that patch). For example, with a sphere we can choose a hemisphere as a coordinate patch and "wrap" our coordinate system over the hemisphere such that it recreates the curvature as we move between different points on the hemisphere. This can be achieved by using, for example, spherical polar coordinates which describe the coordinates on a spherically curved surface in Euclidean space.

As an aside, when people talk of gravity as causing coordinate acceleration is it meant that because gravity is the manifestation of curved spacetime (due to matter content) in a region, when we describe this region in a coordinate system we will need to use curvilinear coordinates which will change as we move in that region, hence give the illusion that objects are accelerating relative to this coordinate system?

 

[martinbn]

This looks ok, heuristically, but I think there is a simpler way to put it: Cartesian (non-curvilinear) coordinate systems only exist globally in a flat manifold, and spacetime in the presence of gravity is not flat. So we have to use curvilinear coordinate systems for spacetime in the presence of gravity because they're the only ones that exist.

This depends on what one calls Cartesian coordinates. For example if the manifold is $\mathbb R^3$, then are the usual coordinates $(x, y, z)$ Cartesian? I'd guess that most people would say yes, and that they are global. Is this manifold flat? Well, that is not a meaningfull question until a metric is specified (or just a connection). It can be flat (with the usual structure) or it could be non-flat.

 

[stevendaryl]

This depends on what one calls Cartesian coordinates. For example if the manifold is $\mathbb R^3$, then are the usual coordinates $(x, y, z)$ Cartesian? I'd guess that most people would say yes, and that they are global. Is this manifold flat? Well, that is not a meaningfull question until a metric is specified (or just a connection). It can be flat (with the usual structure) or it could be non-flat.

I would say that coordinates are Cartesian only if the metric (or line element) is $ds^2 = dx^2 + dy^2 + dz^2$. So you can't have Cartesian coordinates on a curved space, even on a tiny patch.

 

[PeterDonis]

Is this manifold flat? Well, that is not a meaningfull question until a metric is specified (or just a connection).

Just to clarify, when I said "flat manifold", I meant a manifold with a flat metric (zero Riemann curvature tensor) defined on it.

 

[PeterDonis]

Would it be correct to say that as Cartesian coordinates describe a (hyper) plane, then one could run into the situation where the manifold has intrinsic curvature at each particular point and so one could not construct a Cartesian coordinate patch around any point as one would "move off" the manifold

Not really. The problem is not that you can't assign Cartesian coordinates to points on the manifold. The problem is that the actual metric on the manifold is not the flat Cartesian metric. So if you use the Cartesian coordinates, with their metric, to compute physical quantities, you will get incorrect answers.

in order to accurately describe points on the manifold in a given coordinate patch in terms of coordinates in Euclidean space we require a coordinate system whose coordinate lines curve (such that they recreate the curvature of the manifold in that patch).

Not exactly. The coordinate lines seem "curved" only because you are visualizing them in a higher dimensional flat space--for example, great circles on the Earth seem curved to you because you are visualizing them as being embedded in flat 3-dimensional space. But with respect to the intrinsic metric of the Earth's surface, great circles are not curved; they are straight lines (geodesics). So coordinate lines do not have to be curved in any intrinsic sense in order to cover a curved manifold. (Note that they can be curved--lines of latitude on the Earth, except for the equator, are not geodesics, even though lines of longitude are. But they do not have to be curved.)

when people talk of gravity as causing coordinate acceleration is it meant that because gravity is the manifestation of curved spacetime (due to matter content) in a region, when we describe this region in a coordinate system we will need to use curvilinear coordinates which will change as we move in that region, hence give the illusion that objects are accelerating relative to this coordinate system?

It is true that any coordinates in which gravity causes coordinate acceleration will be curvilinear, yes. But the main thing the term "coordinate acceleration" is emphasizing is the physical fact that objects moving solely under gravity are in free fall; they feel zero proper acceleration. So any talk of "acceleration due to gravity" can only refer to coordinate acceleration. This is important because coordinate acceleration is (obviously) dependent on the coordinates you choose, whereas proper acceleration is an invariant.

 

["Don't panic!"]

Not really. The problem is not that you can't assign Cartesian coordinates to points on the manifold. The problem is that the actual metric on the manifold is not the flat Cartesian metric. So if you use the Cartesian coordinates, with their metric, to compute physical quantities, you will get incorrect answers.

I think part of my confusion arises because we tend to relate Cartesian coordinates to more general coordinates. For example, in a polar coordinate system $(r,\theta)$ we can map to Cartesian coordinates via $$x=r\cos (\theta),\qquad y=r\sin (\theta)$$ Sorry, I've got myself into a real muddle here, I thought I understood this all before, but now having started to try and learn differential geometry I'm having severe doubts about my understanding!

Is the point that if we use a curvilinear coordinate system then we can treat it as Cartesian (in it's own system, i.e. set up a coordinate grid, for example, in polar coordinates there would be an $r$ -coordinate line and a $\theta$ - coordinate line) if we then map to an actual Cartesian coordinate system though, and represent the curvilinear coordinate system in a Cartesian coordinate system the coordinates lines of the curvilinear coordinate system will appear curved?!

 

[PeterDonis]

Is the point that if we use a curvilinear coordinate system then we can treat it as Cartesian (in it's own system, i.e. set up a coordinate grid, for example, in polar coordinates there would be an $r$ -coordinate line and a $\theta$ - coordinate line)

Having a coordinate grid does not mean you are treating the coordinates as Cartesian. Any coordinate chart will have a coordinate grid.

if we then map to an actual Cartesian coordinate system though, and represent the curvilinear coordinate system in a Cartesian coordinate system the coordinates lines of the curvilinear coordinate system will appear curved?!

They might or they might not. Polar coordinates on a plane are a bad example because the plane is flat, so you can set up global Cartesian coordinates. For a better example, imagine setting up a local Cartesian coordinate grid centered on the point on the Earth where the equator and the prime meridian intersect. In this local Cartesian coordinate grid (which will only cover a small patch of the Earth's surface centered on the chosen point), the equator and the prime meridian will be straight lines (because they are both geodesics, and geodesics will appear as straight lines in a local Cartesian grid).

 

["Don't panic!"]

Is the point that a manifold locally "looks like" (is homeomorphic to) $\mathbb{R}^{n}$ (and thus Euclidean), however one can only use Cartesian coordinates if the metric defined on the manifold is locally flat (i.e. the metric is locally Euclidean, $ds^{2}=dx^{2}+dy^{2}+\cdots$). As such, if the metric is something different and describes a curved geometry, then although we can map the manifold (locally) to $\mathbb{R}^{n}$, the manifold will not be locally flat and we are unable to use Cartesian coordinates to represent the points on the manifold (in the patch that we are considering) in Euclidean space, instead we must use more general (curvilinear) coordinate systems to represent points on the manifold which we can map locally to Cartesian coordinates via coordinate transformations and thus represent the points in Euclidean space (sorry this may not be worded very well)?

In other words, as a manifold is generally non-Euclidean, the only way to represent it locally in Euclidean space is via curvilinear coordinates that we can locally map to Cartesian.

 

[Mentz114]

@"Don't panic!"
There is a mapping from the manifold into itself called a tetrad field which transforms the global metric ( curved or otherwise) into a local set of orthogonal axes with Minkowski metric. This space has a limited range of validity which gets smaller with increasing curvature. This local space is what a material body traveling on a curve in spacetime can construct at every point of the curve. A tetrad field exists for all timelike curves thanks to the equivalence principle.

 

[pervect]

The main point I think is that coordinates in and of themselves are just labels. You can use whatever sort of coordinates you like and are suited to the problem - polar, hyperbolic, elliptical, whatever. As such, they have no direct physical significance as they are arbitrary labels.

There are geometric and physical features of the geometry that are independent of the coordinates, for instance the distance between the same two points will have a definite value, regardless of what coordinate choices you make.

You can regard the metric as a tool for converting from coordinates to distances, though it has other features.

As far as needing curvilinear coordinates goes, it turns out on curved manifolds (consider the surface of the Earth as an example), you can't assign coordinates such that a change in coordinate represents a constant change in distance everywhere like you can on flat manifolds (consider a plane).

If you can assign coordinates such that a change in a coordinate always represents the same distance, then your metric is a unit diagonal matrix, and the space described by the metric is flat (a formal proof involves knowing a bit about how intrinsic curvature is defined, something that hopefully you'll get to eventually).

It remains to convince yourself that it is necessary and sufficient for coordinate changes to represent distances directly that the geometry is "Caartesian", I suppose, that this statement (or the equivalent statement about a flat metric) are equivalent.

 

[PeterDonis]

Is the point that a manifold locally "looks like" (is homeomorphic to) $\mathbb{R}^{n}$ (and thus Euclidean), however one can only use Cartesian coordinates if the metric defined on the manifold is locally flat (i.e. the metric is locally Euclidean, $ds^{2}=dx^{2}+dy^{2}+\cdots$)

You seem to be confused about the meaning of the word "Euclidean". Usually, as I understand it, it means both that the manifold is $\mathbb{R}^{n}$ and that the metric is flat.

Also, the usual definition of "manifold" requires it to be locally Euclidean in the above sense; the usual term for this is "Riemannian". If we extend consideration to manifolds like spacetime, which have a metric that is not positive definite, those are referred to as "pseudo-Riemannian" manifolds, and they are locally Minkowskian (or Lorentzian), not Euclidean, since their metric is locally flat in the spacetime sense (i.e., locally the Minkowski metric).

 

["Don't panic!"]

You seem to be confused about the meaning of the word "Euclidean". Usually, as I understand it, it means both that the manifold is Rn\mathbb{R}^{n} and that the metric is flat.

That's what I thought originally too, but then if we are mapping the manifold locally to Euclidean space I don't see why we couldn't just use Cartesian coordinates in the first place and forgo curvilinear coordinates altogether (apart from the fact that they may simplify the description)?

Sorry, I feel like I'm going round in circles a bit.

As far as needing curvilinear coordinates goes, it turns out on curved manifolds (consider the surface of the Earth as an example), you can't assign coordinates such that a change in coordinate represents a constant change in distance everywhere like you can on flat manifolds (consider a plane).

So do we need curvilinear coordinates so that we can account for a non constant change in measurement of distance (i.e. a non constant change in the metric)? I'm struggling to convince myself why curvilinear coordinates need to be used alternatively to Cartesian. I understand that curvilinear coordinates are useful for describing curved surfaces etc. as they change as we change position on the surface and so make it easy to describe points on such a surface, but I can't see why one would definitely want to use them over Cartesian?

Is the point that we are free to choose Cartesian coordinates in a particular patch, however, when we transition to another coordinate patch (having started with Cartesian coordinates), the coordinates in this new patch will not be Cartesian, in general, and therefore we will end up with a different set of coordinates. Thus we need the notion of more general, curvilinear, coordinate systems such that we can describe all of the coordinate patches on the manifold?!

 

[stevendaryl]

That's what I thought originally too, but then if we are mapping the manifold locally to Euclidean space I don't see why we couldn't just use Cartesian coordinates in the first place and forgo curvilinear coordinates altogether (apart from the fact that they may simplify the description)?

If your space is nonflat, then there ARE no Cartesian coordinates. At best, there are locally Cartesian coordinates, which means that they are approximately Cartesian in a tiny neighborhood. By "approximately Cartesian", I just mean that the error in computing distances within that neighborhood using a Cartesian metric is negligible (although negligible depends on your purposes).

No matter how small the patch, the coordinate system within that patch is not going to be Cartesian unless your space is flat.

 

["Don't panic!"]

If your space is nonflat, then there ARE no Cartesian coordinates. At best, there are locally Cartesian coordinates, which means that they are approximately Cartesian in a tiny neighborhood.

By this, is Cartesian coordinate system reserved for a global coordinate system?

Is the point that we are free to choose local Cartesian coordinates to represent coordinates in a particular patch, however, when we transition to another coordinate patch (having started with local Cartesian coordinates), the coordinates in this new patch will not be Cartesian, in general, and therefore we will end up with a different set of coordinates. Thus we need the notion of more general, curvilinear, coordinate systems such that we can describe all of the coordinate patches on the manifold?!

 

[PeterDonis]

if we are mapping the manifold locally to Euclidean space I don't see why we couldn't just use Cartesian coordinates in the first place and forgo curvilinear coordinates altogether (apart from the fact that they may simplify the description)?

Because, as we said before, if all you use are local Cartesian coordinates on local patches of the manifold, you cannot describe geometric features that do not fit in a single local patch. The only way to describe such features using coordinates at all is to use coordinates that cover a large enough region of spacetime to include the feature. In a curved spacetime, such coordinates must be curvilinear. (Note that you can, if you want, describe geometric features without using coordinates at all. But then this whole discussion would be moot.)

 

[Mentz114]

By this, is Cartesian coordinate system reserved for a global coordinate system?

Is the point that we are free to choose local Cartesian coordinates to represent coordinates in a particular patch, however, when we transition to another coordinate patch (having started with local Cartesian coordinates), the coordinates in this new patch will not be Cartesian, in general, and therefore we will end up with a different set of coordinates. Thus we need the notion of more general, curvilinear, coordinate systems such that we can describe all of the coordinate patches on the manifold?!

My post above describes how in the context of GR one can construct these 'flat' patches. It is important that the equivalence principle guarantees the existence of such local patches.

 

[stevendaryl]

Is the point that we are free to choose local Cartesian coordinates to represent coordinates in a particular patch, however, when we transition to another coordinate patch (having started with local Cartesian coordinates), the coordinates in this new patch will not be Cartesian, in general, and therefore we will end up with a different set of coordinates. Thus we need the notion of more general, curvilinear, coordinate systems such that we can describe all of the coordinate patches on the manifold?!

Even within a small patch, you can't have Cartesian coordinates if the space is curved. As I said, at best you can have coordinates that are approximately Cartesian. The criterion of being "locally Cartesian" is only true at a single point. If you get away from that point, the coordinates will not longer be locally Cartesian.

 

["Don't panic!"]

Sorry to be going on so much about this guys, really appreciate you putting up with me. I guess I'm struggling with the fact that a manifold locally maps to Euclidean space, so how is it not possible to use Cartesian coordinates? Is it that the manifold itself is non-Euclidean and so even locally it won't be flat (apart from at a single point) and so curvilinear coordinates need to be used to take into account this deviation from flat space as one "moves away" from that point?

 

[PeterDonis]

I guess I'm struggling with the fact that a manifold locally maps to Euclidean space, so how is it not possible to use Cartesian coordinates?

It is possible to use local Cartesian coordinates, because the manifold locally maps to (a small patch of) Euclidean space. (Actually, this is only an approximation, as others have pointed out.) But it is not possible to use global Cartesian coordinates, because the manifold does not globally map to Euclidean space.

Please read the above carefully; you are confusing yourself by being inconsistent about including the words I bolded above. Being careful to always put them in where needed should help resolve your confusion.

 

[George Jones]

As others have pointed out, consider the surface of the Earth.

Can you use 2-dimensional Cartesian coordinates that cover the entire surface of the Earth? Can you lay out a rectangular grid locally of a few city blocks by a few city blocks?

 

["Don't panic!"]

It is possible to use local Cartesian coordinates, because the manifold locally maps to (a small patch of) Euclidean space. (Actually, this is only an approximation, as others have pointed out.) But it is not possible to use global Cartesian coordinates, because the manifold does not globally map to Euclidean space.

Ok, so is this anywhere near to being correct?
So as a manifold locally maps to a small patch of Euclidean space (why is this an approximation? I thought that a manifold was covered by a set of coordinate charts, each of which are homeomorphisms from small local patches on the manifold to small patches of Euclidean space, e.g. $\phi :U\subset M\rightarrow V\subset\mathbb{R}^{n}$), a particular local subset of the manifold can be represented by local Cartesian coordinates. However, it is not possible to use global Cartesian coordinates, and thus when one transitions to a neighbouring coordinate patch on the manifold, the coordinates describing this neighbouring patch will not be Cartesian (as the manifold is curved and so the transition maps between the two patches will not be constant maps).