Is Coordinate-Free Relativity the Key to Understanding General Relativity?

[Quchen]

Hey there,

Does anyone know a book that consequently uses coordinate-free expressions to develop general relativity? I've been looking for something for some time now, but everything I could find just briefly introduced the reader to concepts like exterior algebra, only to (almost) never use the concept again in the rest of the book (example: Einstein's general theory of relativity by Grøn and Hervik).
(I'd really love to see Einstein's field equations compressed to something small yet powerful like it's been done with Maxwell's equations, \mathrm dF=0\,; \mathrm d*F=4 \pi S)

Thanks in advance,
David

 

[Daverz]

I would suggest Gravitational Curvature by Frankel.

 

[Quchen]

Sounds good, especially because I've already got his other book, the geometry of physics. I've heard that Straumann wrote an excellent book too, but I cannot find it anywhere to have a look at the contents (at least not enough to spend 80 bucks on it).

 

[Goldbeetle]

https://www.amazon.com/dp/3642060137/?tag=pfamazon01-20

 

[JDoolin]

I've heard that concept of "coordinate free relativity" used here and there, but I'm confused on what they mean by that.

Can you describe things without any reference point?

Can you actually picture anything, or draw anything, without it having a position and size? No, I don't think so.

Here, picture a box. Imagine it in your mind. Imagine it in the vacuum of space without any other particle around it.

No, because the moment you pictured the box, there is something else in the scenario: Your eyes. The very act of picturing something invokes an origin. There is no way you can picture anything, without an origin (your eyes), and without picturning it at a certain distance away from you.

And if it is a certain distance away from you, you can define a coordinate system based on that.

Coordinate free relativity, if it means what it sounds like it means, then it is nonsense. So it probably means something else.

 

[WannabeNewton]

I've heard that concept of "coordinate free relativity" used here and there, but I'm confused on what they mean by that.

There is a level of abstraction involved, yes. Instead of defining tensors in the coordinate dependent way as quantities that transform according to T^{\alpha ...\beta }_{\mu ..\nu } = \frac{\partial x^{\alpha }}{\partial x^{\gamma }}...\frac{\partial x^{\beta }}{\partial x^{\delta }}\frac{\partial x^{\sigma }}{\partial x^{\mu }}...\frac{\partial x^{\lambda }}{\partial x^{\nu }}T^{\gamma... \delta }_{\sigma... \lambda } you define an (m, n) tensor as a multi - linear mapping of one - forms and vectors: \mathbf{T}:\underbrace{V^{*}\times ...\times V^{*}}_{n}\times \underbrace{V\times ...\times V}_{m} \mapsto \mathbb{R} Instead of giving things in terms of components you give things abstractly in terms of the tensor object itself. Differential forms are very useful in this context because they are formulated in a coordinate - free way.

 

[JDoolin]

There is a level of abstraction involved, yes. Instead of defining tensors in the coordinate dependent way as quantities that transform according to


you define an (m, n) tensor as a multi - linear mapping of one - forms and vectors: \mathbf{T}:\underbrace{V^{*}\times ...\times V^{*}}_{n}\times \underbrace{V\times ...\times V}_{m} \mapsto \mathbb{R} Instead of giving things in terms of components you give things abstractly in terms of the tensor object itself. Differential forms are very useful in this context because they are formulated in a coordinate - free way.

I'm not entirely clear on the notation.

T^{\alpha ...\beta }_{\mu ..\nu } = \frac{\partial x^{\alpha }}{\partial x^{\gamma }}...\frac{\partial x^{\beta }}{\partial x^{\delta }}\frac{\partial x^{\sigma }}{\partial x^{\mu }}...\frac{\partial x^{\lambda }}{\partial x^{\nu }}T^{\gamma... \delta }_{\sigma... \lambda }

... defines a whole lot of different functions, right? The thing on the left hand side
T^{\alpha ...\beta }_{\mu ..\nu } represents a function of \alpha, ..., \beta, \mu, ..., \nu, and produces a function. Am I right in assuming that you can replace \alpha, ..., \beta, \mu, ..., \nu with varibles? For nstance, \alpha,...,\beta might be (t,x,y,z) or (t,r,θ,φ) while \mu, ..., \nu would be (t',x',y',z') or (τ,ρ,Θ,Φ) or something along those lines...

(I should be taking into account the fact that there are coordinates on the right-side, too.)

But in the next step, you're taking

\mathbf{T}:\underbrace{V^{*}\times ...\times V^{*}}_{n}\times \underbrace{V\times ...\times V}_{m} \mapsto \mathbb{R}

Am I correct that each of the V's and V*'s represent some kind of class of vectors? So they are basically the inputs? The V*'s correspond to \alpha,...,\beta, while the V's correspond to \mu,...,\nu?

Then what is actually going on, is you're just notationally describing the math without referring to the coordinates. So the coordinates are actually still there; but they are just hidden, right?

I also notice that you have a bold-faced T over on the left (usually indicating a multi-dimensional quantity), and mapping to a real number on the right. Is that a typo?

In any case, I'm more interested in expanding out the first equation and understanding what it means. The second equation seems to be designed to shovel a bunch of complexity under the rug. I don't mind short-cuts, but I don't like learning the short-cut to an answer, when I don't even know what the question is.

What advantage does this coordinate free notation give you? Could it be described in terms, maybe an Object Oriented Programming expert might understand? Like, maybe V*, V*, V*, V, V, V are the high-level objects, but the end user programmer doesn't need to care or worry about the exact nature of the functions and subroutines the computer uses to render the output. However, without those functions being in place, your computer is not actually going to do anything, right?

So effectively, isn't the coordinate free representation, kind of like saying

Show[AmazingRenderedOutput]

If someone else has gone through the work of creating the amazing rendered output, and set up the program with a syntax that will show it when you type that command, then you've really done something by typing that command. But it doesn't imply any understanding of what is really going on.

 

[WannabeNewton]

Am I right in assuming that you can replace \alpha, ..., \beta, \mu, ..., \nu with varibles? For nstance, \alpha,...,\beta might be (t,x,y,z) or (t,r,θ,φ) while \mu, ..., \nu would be (t',x',y',z') or (τ,ρ,Θ,Φ) or something along those lines...
(I should be taking into account the fact that there are coordinates on the right-side, too.)

Yes they are essentially just the components. The "..." is just to indicate there can be an arbitrary number of upper and lower indices.

But in the next step, you're taking

\mathbf{T}:\underbrace{V^{*}\times ...\times V^{*}}_{n}\times \underbrace{V\times ...\times V}_{m} \mapsto \mathbb{R}

Am I correct that each of the V's and V*'s represent some kind of class of vectors? So they are basically the inputs? The V*'s correspond to \alpha,...,\beta, while the V's correspond to \mu,...,\nu?

The n V^{*}'s correspond to n copies of a dual vector space and the m V's correspond to m copies of the associated vector space. For some point on a manifold, these would be the n copies of the cotangent space and the m copies of the tangent space respectively, and the tensor value of the tensor field at this point maps these copies of the cotangent and tangent space to the reals.

Then what is actually going on, is you're just notationally describing the math without referring to the coordinates. So the coordinates are actually still there; but they are just hidden, right?

I feel like we might end up misinterpreting each other's words here but no the coordinates are not "hidden". You only need coordinates to obtain the components of a tensor relative to that coordinate's basis. The geometric object itself, which is the tensor, exists regardless of any coordinate system.

I also notice that you have a bold-faced T over on the left (usually indicating a multi-dimensional quantity), and mapping to a real number on the right. Is that a typo?

I made it bold just to indicate that we are talking about the tensor object and not its components. But it does map to the reals.

What advantage does this coordinate free notation give you?

You don't rely on coordinates. It is more elegant and highlights the fact that the geometric object exists regardless of a coordinate system. Nature does not supply us with one right?

Could it be described in terms, maybe an Object Oriented Programming expert might understand? Like, maybe V*, V*, V*, V, V, V are the high-level objects, but the end user programmer doesn't need to care or worry about the exact nature of the functions and subroutines the computer uses to render the output. However, without those functions being in place, your computer is not actually going to do anything, right?

I am not a programmer by any means so I am sorry if I cannot help you much here but I can try in terms of Java: let's say you write an interface. You cannot really perform anything with the interface alone. What you can do is write classes that all implement this interface and these classes can now give that interface some functionality. Since these classes all implement this interface, they are all related by polymorphism. Even though the interface only has functionality when implemented by a class (or a bunch of classes) it still exists regardless of implementation, it has some abstract definition. Tensors are geometric objects that can be expressed in terms of components relative to a coordinate basis so that you can actually use them for calculations, and you can have a tensor expressed in many different coordinate systems that can be continuously mapped from one to the other, but the tensor itself exists regardless. I hope that helps but again, I'm sorry I couldn't help completely because I am not much of a programmer.

 

[JDoolin]

I think you've basically got the gist of my object oriented programming comparison.

Now, if I understand right, we have a general method that will deal with all manner of coordinate systems. (Cartesian, Cylindrical, Spherical, Parabolic, Paraboloidal, Oblate spheroidal, Prolate spheroidal, Ellipsoidal, Elliptic cylindrical, Toroidal, Bispherical, Bipolar cylindrical, Conical). The methods of General Relativity can operate without error on any well defined coordinate system, and that is fine.

I have no problem with the idea, but just with the words used; calling it "coordinate free." Then it makes it sound like we could actually possibly use General relativity on an undefined coordinate system...

I may be forcing an analogy here, but isn't it sort of like taking a word-processor, which can deal with hundreds of different fonts, and calling it "font-free." Certaninly, you can take away all the fonts and still have the word-processor, but it won't work. It's going to either fail to compile or have a runtime error where it doesn't show you anything you're typing. The word-processor shouldn't be called "font-free" but more "font-ready."

So is the implementation really "coordinate-free" in that it still works, even when there is NO coordinate systems, or is it a "coordinate-ready" methodology that can work on any given coordinates.

 

[WannabeNewton]

It is coordinate - ready in a sense. But you only need coordinates to do calculations. You can express the field equations in terms of the abstract tensor entities and not refer to a coordinate system at all. You are just using the abstract definitions of the tensors instead of how the tensor behaves in a certain coordinate system. It depends on what you mean by "works" because the coordinate - free approach does describe everything its just we can't really do any calculations unless we specify components relative to a basis.

 

[twofish-quant]

The second equation seems to be designed to shovel a bunch of complexity under the rug. I don't mind short-cuts, but I don't like learning the short-cut to an answer, when I don't even know what the question is.

It turns out that mathematically a lot of the complexity is unnecessary, and the reason people like coordinate-free descriptions is that it removes a lot of the unnecessary complexity.

Could it be described in terms, maybe an Object Oriented Programming expert might understand? Like, maybe V*, V*, V*, V, V, V are the high-level objects, but the end user programmer doesn't need to care or worry about the exact nature of the functions and subroutines the computer uses to render the output. However, without those functions being in place, your computer is not actually going to do anything, right?

Except that what you do is to give a broad description of your function, and once you know about the characteristics of that function, you do general stuff without knowing the details of what is in the inside of the function. If I know that the function is "const" then I can do things that I couldn't do if I didn't know.

This turns out to be important for compiler design.

If you really want your mind blown. Take a look at this wiki page.

http://en.wikipedia.org/wiki/Covariance_and_contravariance_(computer_science)

It turns out that you can use the same language to talk about C++ classes and vector spaces, and there is a branch of mathematics called category theory

So effectively, isn't the coordinate free representation, kind of like saying

Show[AmazingRenderedOutput]

If someone else has gone through the work of creating the amazing rendered output, and set up the program with a syntax that will show it when you type that command, then you've really done something by typing that command. But it doesn't imply any understanding of what is really going on.

But the cool thing is that you can make some statements about the without knowing what is going on. A lot of mathematics involves trying to figure out what the minimum description you need to say something useful.

To use an example. If I know that "dump as html" dumps out html, then I can take that output and pump it into another function that reads in html. I don't know what or how it dumps as html, and I don't care.

 

[twofish-quant]

Now, if I understand right, we have a general method that will deal with all manner of coordinate systems. (Cartesian, Cylindrical, Spherical, Parabolic, Paraboloidal, Oblate spheroidal, Prolate spheroidal, Ellipsoidal, Elliptic cylindrical, Toroidal, Bispherical, Bipolar cylindrical, Conical). The methods of General Relativity can operate without error on any well defined coordinate system, and that is fine.

Yup, and it goes a bit further. I draw a coordinate system on an apple. I change the coordinate system. Nothing really changed. Now I draw a coordinate system on a flat plane. Things really are different. So there is a lot of mathematics that tells you when you are really changing things and when you aren't.

To use a programming analogy. You can write different programs to get the same output, so how do you know if you have two programs that give the same output, if you don't. This gets at the very deep connection between general relativity and compiler design, because when you are writing a compiler, you are trying to translate one set of instruction to another set of instructions that does exactly the same thing, only faster.

One way of thinking about this is imagine a 1GB dimension space in which each axis can take on the value 0 or 1. A computer program is a point in this space. Compiling and running a program is a set of instructions that describe how the point moves through space.

Then it makes it sound like we could actually possibly use General relativity on an undefined coordinate system...

You don't need to define a coordinate system. You can say that space as a certain set of characteristics, and once you list the characteristics of that space, you end up with a set of coordinate systems.

I may be forcing an analogy here, but isn't it sort of like taking a word-processor, which can deal with hundreds of different fonts, and calling it "font-free."

Not really. Another analogy is that you could write a word processor with different languages, and you end up with the same output. The problem with using coordinate systems is that you can have different coordinate systems that describe the same space, and it's not obvious whether two spaces are the "same" or "different."

So is the implementation really "coordinate-free" in that it still works, even when there is NO coordinate systems, or is it a "coordinate-ready" methodology that can work on any given coordinates.

It works when there are no coordinate systems. You have to describe the space in some way, but you can do it without coordinate systems.

For example, if I say that I have a flat 2-d surface, I've just uniquely described the space. No coordinates. If I want to describe a cube, I describe a flat surface, I describe the way of taping the surfaces together. Again no coordinates.

 

[twofish-quant]

Can you describe things without any reference point?

It turns out that you can.

Can you actually picture anything, or draw anything, without it having a position and size? No, I don't think so.

It turns out also that you can.

And if it is a certain distance away from you, you can define a coordinate system based on that.

You can. But you don't have to. People have come up with a set of mathematical definitions that you can talk and think about spaces without using coordinates.

One way of thinking about how this works is imagine you are blind, and try to think about how you would describe an apple, and tell the difference between an apple and a grape to someone else that is blind. It's hard, but it can be done. If someone gives me an apple and a plate, they feel different, and an apple feels more like an orange than a plate does.

 

[JDoolin]

It turns out also that you can.

I cannot picture anything without a reference point. Tell me how you do it. How do you "picture" something without referencing your eyes?

You can. But you don't have to. People have come up with a set of mathematical definitions that you can talk and think about spaces without using coordinates.

One way of thinking about how this works is imagine you are blind, and try to think about how you would describe an apple, and tell the difference between an apple and a grape to someone else that is blind. It's hard, but it can be done. If someone gives me an apple and a plate, they feel different, and an apple feels more like an orange than a plate does.

Doesn't even a blind person have a sense of where the apple is? Doesn't the blind person have a sense of direction, that they can reach forward an take the apple? That the apple has a size and shape and location. Nature supplies to them a distance scale of "arm's lenghth" or an area scale of "about the size of my palm, from which they should naturally invoke a coordinate system.

Are you saying that if you are blind, all the world is just an amorphous reality with no direction, position, or scale? That's not what I have seen. When I see blind people, though, they seem highly aware of (and quite concerned about) the positions of things relative to themselves.

 

[redrzewski]

Here's some free course notes that are primarily coords free:

http://docs.google.com/viewer?a=v&p...RvbWFpbnx3aW5pdHpraXxneDoxN2EyNzZjYmViODQ3ZWQ

If you don't trust that link, search for:
"General Relativity Winitzki"

 

[Dale]

No, because the moment you pictured the box, there is something else in the scenario: Your eyes. The very act of picturing something invokes an origin. There is no way you can picture anything, without an origin (your eyes), and without picturning it at a certain distance away from you.

And if it is a certain distance away from you, you can define a coordinate system based on that.

Relativity is fundamentally a geometric theory. You can do a lot of geometry without using coordinates. Even notions of angles and distances are not based on coordinates but are geometrical. In fact, generally you define coordinates based on the underlying geometry, not vice versa.

 

[JDoolin]

You can do a lot of geometry without using coordinates.

What do you mean?

 

[Ben Niehoff]

Now, if I understand right, we have a general method that will deal with all manner of coordinate systems. (Cartesian, Cylindrical, Spherical, Parabolic, Paraboloidal, Oblate spheroidal, Prolate spheroidal, Ellipsoidal, Elliptic cylindrical, Toroidal, Bispherical, Bipolar cylindrical, Conical). The methods of General Relativity can operate without error on any well defined coordinate system, and that is fine.

These are all orthogonal coordinate systems on flat space. Such coordinate systems do not exist on curved spaces! You need something even more general.

I have no problem with the idea, but just with the words used; calling it "coordinate free." Then it makes it sound like we could actually possibly use General relativity on an undefined coordinate system...

We can and we do. I solve Einstein's equations in d dimensions without using coordinates all the time.

So is the implementation really "coordinate-free" in that it still works, even when there is NO coordinate systems, or is it a "coordinate-ready" methodology that can work on any given coordinates.

It is coordinate-free. It is entirely possible to do computations in coordinate-free notation without ever making reference to any coordinate system.

 

[Dale]

What do you mean?

I mean basically all of Euclidean geometry can be done without coordinates. For example you can prove that the sum of the interior angles of a triangle is 180 deg, and many other things, all without ever using any coordinates.

 

[JDoolin]

I mean basically all of Euclidean geometry can be done without coordinates. For example you can prove that the sum of the interior angles of a triangle is 180 deg, and many other things, all without ever using any coordinates.

All of the proofs I see on that topic rely on geometric constructions on paper. But if you have drawn it on a piece of paper, (or even if you're just visualizing it as if it were drawn on a piece of paper), then there is a coordinate system.

Can you show me your proof? (or tell me where I can see a similar proof.)

 

[JDoolin]

I solve Einstein's equations in d dimensions without using coordinates all the time.

For practice, or some practical application?

 

[twofish-quant]

I cannot picture anything without a reference point. Tell me how you do it.

There are lots of tricks. What I do is to learn the basic math rules, and after a few months, I'm able to "feel" how the math works. The point is that the idea of doing geometry without coordinates has some firm mathematical foundations, and where or not it's easy to visual or not, geometry can be done without coordinates.

Doesn't even a blind person have a sense of where the apple is? Doesn't the blind person have a sense of direction, that they can reach forward an take the apple? That the apple has a size and shape and location. Nature supplies to them a distance scale of "arm's lenghth" or an area scale of "about the size of my palm, from which they should naturally invoke a coordinate system.

Perhaps, but they don't have to. There are coordinate free definitions of length and curvature.

Are you saying that if you are blind, all the world is just an amorphous reality with no direction, position, or scale? That's not what I have seen. When I see blind people, though, they seem highly aware of (and quite concerned about) the positions of things relative to themselves.

I'm saying that there are rigorous mathematical definitions that allow you do to geometry without coordinates, and thinking of feeling an apple without looking at it gives you an intuitive explanation for how those definitions work.

 

[twofish-quant]

For practice, or some practical application?

The practical application of coordinate free geometry is that you can make extremely general statements and proofs that apply without having to worry about the details.

Also in practice, in order to do any real GR calculation, you have to vastly reduce the number of coordinates so people use symmetry arguments to reduce the complexity of the problem before actually trying to solve it. One thing about coordinate free arguments is that imposing a coordinate system in curved space is not easy, so you have to start by figuring what coordinates you can and cannot use, and what the properties of those coordinates are.

 

[JDoolin]

But if you have drawn it on a piece of paper, (or even if you're just visualizing it as if it were drawn on a piece of paper), then there is a coordinate system.

Okay, I think I now see an error in my thinking.

Rendering the image on a computer screen does require a coordinate system, but drawing a picture on paper does not invoke a coordinate system. You can use whatever coordinate system you like to describe the image on the paper.

Using your eyes to view the world does give you an origin, but it does not require you to use a spherical or Cartesian coordinate system to describe things in the world, (though those are the most convenient.)

It would still be correct to say there is no way you can picture anything, or describe shape, location, or size, without an origin (your eyes), and without picturing it at a certain distance away from you. It's just that you aren't constrained to use any particular form of coordinates to do so.

 

[Dale]

But if you have drawn it on a piece of paper, (or even if you're just visualizing it as if it were drawn on a piece of paper), then there is a coordinate system.

This is not the case. If I simply draw a curve on a piece of paper which direction is x and where is the origin? You can do all sorts of things with that geometric figure without ever specifying an origin and a pair of basis vectors. For instance, you can draw tangent lines to a couple of points, you can calculate the angle between those tangent lines, you can determine the distance between those two points, you can determine the length of the curve between them, etc. All without ever using coordinates.

 

[twofish-quant]

Also imagine a sponge. If you try to imagine putting a coordinate system on a surface of a sponge in which you have to take into account every little nook and cranny, you will go insane, and you will find that you *can't* create a mapping from the surface of a sponge to a global coordinate system. Now imagine, a thousand black holes.

What you can do is to magnify the sponge and create a local "flat" coordinate system for a tiny piece of the sponge, and the tape a thousand of those coordinate systems together to describe the sponge. The description of the sponge then becomes a description of how you "tape" the local systems together.

Once you do that, you realize that you don't need the local coordinates at all. All you need is a description of how to "tape" different pieces of the sponge together.

One other way of thinking about how coordinate-free geometry works and why it is useful is to think about LEGO's. I want to tell you how to build a gas station out of LEGO's, and I *could* give you a bunch of X,Y,Z coordinates for each part of the gas station, but you'll go insane. Or I can tell you to connect piece one to piece two. Connect that piece to another piece etc. etc. By specifying how two pieces connect, you can build complex structures that don't involve coordinates.

In GR, the "LEGO's" are things called tetrads.

 

[twofish-quant]

It would still be correct to say there is no way you can picture anything, or describe shape, location, or size, without an origin (your eyes), and without picturing it at a certain distance away from you.

Mathematically you can.

One problem that people have with GR is that people are trying to fit it into their intuition of how three-space works. For example, the questions of "where did the big bang happen" or "what does curved space curve into" are questions because people are trying to picture thing using the rules that work for things they see everyday.

The important thing is that those rules don't apply. Space doesn't need to curve into anything. The big bang was everywhere and nowhere. Mathematically, you can describe spaces that aren't obviously connected to your daily visualization, and learning how to visualize those things is why math is hard.

One of the hard part of doing the math is to "let go" of your intuition and learn a new set of rules for how space works.

 

[JDoolin]

This is not the case. If I simply draw a curve on a piece of paper which direction is x and where is the origin? You can do all sorts of things with that geometric figure without ever specifying an origin and a pair of basis vectors. For instance, you can draw tangent lines to a couple of points, you can calculate the angle between those tangent lines, you can determine the distance between those two points, you can determine the length of the curve between them, etc. All without ever using coordinates.

Well, in any case, I stand by my correction in my previous post. Putting the figure on paper doesn't invoke a coordinate system, but observing the figure with your eyes, taking a photograph, or modeling the image on a computer screen all do invoke a coordinate system, or at least project the figure onto a fixed coordinate system.

Besides which, angles are observer dependent anyway, so you can't actually unambiguously state what any angle is without deciding on the velocity of the reference frame you're using.

 

[JDoolin]

Also imagine a sponge. If you try to imagine putting a coordinate system on a surface of a sponge in which you have to take into account every little nook and cranny, you will go insane, and you will find that you *can't* create a mapping from the surface of a sponge to a global coordinate system.

Well, if I'm permitted to use three dimensions, which I think is fair, since the sponge obviously occupies three dimensions, then there is no real difficulty, is there?

Maybe if you went with a Klein Bottle or something like that, I would agree with you; I might go insane with that one. But not with a sponge.

 

[Dale]

Putting the figure on paper doesn't invoke a coordinate system, but observing the figure with your eyes, taking a photograph, or modeling the image on a computer screen all do invoke a coordinate system, or at least project the figure onto a fixed coordinate system.

The only thing which invokes a coordinate system is actually defining a coordinate system.

By observing a figure with my eyes, what coordinate system have I defined? Where is the origin, is it in my center of mass, or is it in my right eye, or my left eye, or somewhere between them, does it matter if I am right or left eye dominant? Am I implying spherical, or cylindrical, or Cartesian, or some other arbitrary coordinate system? Which way are the axes oriented? Is the coordinate system right-handed or left-handed? Does my handedness make a difference? Is it orthonormal? None of this is specified, therefore you have not invoked a coordinate system.

A coordinate system is a 1-to-1 differentiable mapping from points in the manifold to points in R(n). Looking doesn't uniquely define such a mapping.

 

[JDoolin]

Besides which, angles are observer dependent anyway, so you can't actually unambiguously state what any angle is without deciding on the velocity of the reference frame you're using.

I went ahead and made an animation so that you can see what I mean.

The angle between the two marked paths is 90 degrees, but from the perspective of the dot in the middle, the angle between the particle paths is 180 degrees.

I'm still curious about your argument here:

If I simply draw a curve on a piece of paper which direction is x and where is the origin? You can do all sorts of things with that geometric figure without ever specifying an origin and a pair of basis vectors. For instance, you can draw tangent lines to a couple of points, you can calculate the angle between those tangent lines, you can determine the distance between those two points, you can determine the length of the curve between them, etc. All without ever using coordinates.

When you draw lines on that paper, you are implicitly using the reference frame of the paper. You may not be specifying the origin, or the direction of the unit vectors, but you are specifying the inertial reference frame where you've decided that the angle measurement will be made. Specifically, you've decided that you're going to accelerate your protractor until it matches the speed of the paper, and then you'll do your angle measurement there.

 

[JDoolin]

I was also reminded of this video, at about 2:10, people drawing straight up-and-down lines on a piece of paper passing by;

 

[Dale]

When you draw lines on that paper, you are implicitly using the reference frame of the paper. You may not be specifying the origin, or the direction of the unit vectors, but you are specifying the inertial reference frame where you've decided that the angle measurement will be made.

Sure, but that is not a coordinate system. Remember, a coordinate system is a 1-to-1 differentiable mapping between an open subset of the manifold and an open subset of R(n). Until you do that you do not have a coordinate system.

 

[Ben Niehoff]

As Dale is hinting, a reference frame and a coordinate system are not the same thing.

A coordinate system is a map from an open subset of manifold into R^n.

A reference frame is a collection of n linearly-independent vectors at a single point.

A local inertial frame is the GR analogue of an orthonormal frame: it is a collection of n mutually orthonormal vectors at a single point.

 

[JDoolin]

Well, regardless of the definitions of manifolds, reference frames, and coordinate system, let me just reiterate my point:

There is no way you can picture anything, or describe shape, location, or size, without an origin (your eyes), and without picturing it at a certain distance away from you.

 

[Ben Niehoff]

Well, regardless of the definitions of manifolds, reference frames, and coordinate system, let me just reiterate my point:

There is no way you can picture anything, or describe shape, location, or size, without an origin (your eyes), and without picturing it at a certain distance away from you.

Your point has nothing to do with coordinate systems.

 

[Dale]

There is no way you can picture anything, or describe shape, location, or size, without an origin (your eyes), and without picturing it at a certain distance away from you.

I agree completely. In fact, that further emphasizes the idea that a lot of geometry can be done without coordinates.

 

[JDoolin]

Ben says "A coordinate system is a map from an open subset of manifold into R^n."
What are examples of open subsets of manifolds? Are the chalkboard or the computer monitor not fair examples of subsets of manifolds?
What are examples of R^n? If you describe the tension in each contractible muscle in your arm, is that not an example of R^n? If you describe the direction of an image in front of your face, wouldn't the natural inclination be to describe this either in terms of left, right, up and down, forward and backward; either in a rectangular or spherical coordinate system?
Is there any way to describe distance without invoking some kind of numerical measure. (I can acknowledge that a dog may or may not invoke numbers in estimating distances, but if not, he also cannot communicate to other dogs where something is. On the other hand, bees are known to communicate quantitatively about distant locations.)

Let me try some questions and see if you can answer them without invoking any kind of coordinate system.

• How big is the screen you're looking at?
• How far away is it?
• Which direction is it from you?
• When you write or draw a picture on a chalkboard, what position do you hold your shoulder, your elbow, your wrist?
• How far away do you stand from the chalkboard?

What is it exactly about this statement

There is no way you can picture anything, or describe shape, location, or size, without an origin (your eyes), and without picturing it at a certain distance away from you.

...that says it does NOT involve a coordinate system?

 

[JDoolin]

Upon mentioning bees in my last post, it occurred to me that another way of describing things is with landmark based geometry. Instead of stating a distance, you just say from this landmark go to that landmark.

You need not mention direction, distance, shape or size.

 

[lurflurf]

JDoolin Coordinates are almost like a religion to you. The point is not "Is it possible to use Coordinates?" but "Is it helpful to use Coordinates?". If we do decide to use Coordinates we must decide which ones to use and how many. A reasonable answer is none to both. Even when possible, using coordinates is not always worth the trouble they cause. Often even a pro coordinate zealot will be say things like

"Imagine that we have some coordinates, but we do not know anything about them, but they are there really they are, they are really messed up, but that is okay, we love them anyway, they are really complicated, we do not know how to get any numbers, if we had numbers it would not help because there would be so many and there would be so much error and the calculations would be so impossible, that is okay though we are not going to use them anyway, also these coordinates require that we embed the object we are working with into a much more complicated object which might be impossible..."

Why would we want to introduce coordinates if (in a particular situation) we are not going to use them and they are not helpful? At best we have wasted time, and introduced needless complications.

 

[Monocles]

Maybe something that would help understanding is to emphasize the point that, in general, if you are given two different sets of coordinates of some objects (which may or may not be the same object), it is extremely difficult to tell if the sets of coordinates are describing the same object. So, we describe objects in a coordinate-free fashion so that we never have to worry about that problem.

Using a programming analogy, this is (I believe) an equivalent problem to the halting problem, since the halting problem is equivalent to the word problem for groups, and group presentations are 'coordinates for groups'.

For examples of how mathematicians think about things that have no concept of size, distance, etc. consider topological spaces. None of these notions exist until you define a metric.

 

[Ben Niehoff]

JDoolin, the concepts of "distance" and "angle" do not require the use of coordinate systems at all.

Think back to the classical Greek geometry you did in high-school. Suppose I have a triangle ABC composed of three lines AB, AC, and BC. There is an unambiguous notion of the angle A that exists independently of any coordinate system. There is an unambiguous notion of the distance AB that exists independently of any coordinate system. In fact, there is no need to use coordinate systems at all; relying on a simple set of axioms, one can derive all geometrical facts using only the pure geometrical concepts of distances and angles.

It was Descartes who invented (or reinvented) the notion of "analytic geometry": that is, marking the points of the triangle ABC by some coordinate system and then using the coordinate system to derive facts about the triangle ABC. This method makes some geometrical proofs more straightforward, but it is certainly not necessary to use a coordinate system, and in many cases it adds unneeded complexity.

For example, try to show that the so-called "conic sections" are actually sections of a cone. Using coordinate systems, this is an algebraic nightmare. Using pure geometry, there is an elegant trick.

 

[Dale]

What are examples of open subsets of manifolds? Are the chalkboard or the computer monitor not fair examples of subsets of manifolds?

An open subset specifically excludes the border, so if you draw a line and say everything inside the line (but not including the line) then that is an open subset of the manifold of the surface of the chalkboard or monitor.

What are examples of R^n?

For a 2D manifold like a chalkboard it would be R^2, i.e. pairs of real numbers (x,y) or (r,theta) or ...

Let me try some questions and see if you can answer them without invoking any kind of coordinate system.

• How big is the screen you're looking at?
• How far away is it?
• Which direction is it from you?
• When you write or draw a picture on a chalkboard, what position do you hold your shoulder, your elbow, your wrist?
• How far away do you stand from the chalkboard?

1) ~23" on the diagonal
2) ~19" from the monitor to the tip of my nose
3) The center of the monitor is straight ahead and at a ~100º angle from vertical
4) I move them all over
5) ~15" away

Note that none of the above required the specification of a coordinate system.

 

[JDoolin]

JDoolin, the concepts of "distance" and "angle" do not require the use of coordinate systems at all.

Think back to the classical Greek geometry you did in high-school. Suppose I have a triangle ABC composed of three lines AB, AC, and BC. There is an unambiguous notion of the angle A that exists independently of any coordinate system.

Go back to my post 31 and see if you can say without question whether the angle between the paths is 90 degrees or 180 degrees.

There is an unambiguous notion of the distance AB that exists independently of any coordinate system.

And what notion of distance would that be? The cartesian distance? The space-time-interval? The arc length of a geodesic? The arc-length of the null path in the reference frame of the nearest gravitational well (which is, of course zero)? Which null path would you choose to use? Which unambiguous notion of distance are you using?

In fact, there is no need to use coordinate systems at all; relying on a simple set of axioms, one can derive all geometrical facts using only the pure geometrical concepts of distances and angles.

It was Descartes who invented (or reinvented) the notion of "analytic geometry": that is, marking the points of the triangle ABC by some coordinate system and then using the coordinate system to derive facts about the triangle ABC. This method makes some geometrical proofs more straightforward, but it is certainly not necessary to use a coordinate system, and in many cases it adds unneeded complexity.

For example, try to show that the so-called "conic sections" are actually sections of a cone. Using coordinate systems, this is an algebraic nightmare. Using pure geometry, there is an elegant trick.

Well, I will acknowledge that there are surely some interesting things you can do with geometry without defining the locations of points. Like sewing instructions... You can take the corners of a rectangle and sew the ends together to make a mobius strip.

But once you have decided that you are using ANGLES and DISTANCES to describe the location of landmarks and features, you have implicitly defined a coordinate system. I don't know what you mean by "pure" geometric structures. But I can say that you need to look a little deeper for the "impurities" and ambiguities that really do exist in these lofty concepts.

 

[Ben Niehoff]

And what notion of distance would that be? The cartesian distance? The space-time-interval? The arc length of a geodesic? The arc-length of the null path in the reference frame of the nearest gravitational well (which is, of course zero)? Which null path would you choose to use? Which unambiguous notion of distance are you using?

We're talking about ordinary, Euclidean plane geometry here, so I don't see why you're going on about spacetime intervals. There is no time dimension involved. There is a triangle ABC formed by three lines AB, AC, and BC. The line AB has a length we can measure by holding a ruler up against it. The angle A can be measured by holding a protractor up against it. We can rotate and translate the paper in any way we like; the length of the line AB and the measure of the angle A are invariant.

Well, I will acknowledge that there are surely some interesting things you can do with geometry without defining the locations of points. Like sewing instructions... You can take the corners of a rectangle and sew the ends together to make a mobius strip.

The "sewing instructions" thing you've described is topology, not geometry. Topology studies how spaces are connected and how different spaces can be mapped into each other.

Geometry studies what happens once you define a notion of "distance" and "angle".

But once you have decided that you are using ANGLES and DISTANCES to describe the location of landmarks and features, you have implicitly defined a coordinate system.

No, I haven't. Why do you think so?

 

[Dale]

But once you have decided that you are using ANGLES and DISTANCES to describe the location of landmarks and features, you have implicitly defined a coordinate system.

This is not correct. See my above reply to your list of questions. I never defined a coordinate system.

 

[JDoolin]

Let me try some questions and see if you can answer them without invoking any kind of coordinate system.
How big is the screen you're looking at?
How far away is it?
Which direction is it from you?
When you write or draw a picture on a chalkboard, what position do you hold your shoulder, your elbow, your wrist?
How far away do you stand from the chalkboard?

1) ~23" on the diagonal
2) ~19" from the monitor to the tip of my nose
3) The center of the monitor is straight ahead and at a ~100º angle from vertical
4) I move them all over
5) ~15" away

Note that none of the above required the specification of a coordinate system.

See my above reply to your list of questions. I never defined a coordinate system.

To the contrary, almost all of your answers specify coordinate systems.

Your first answer maps an open subset of the space in your room to R^1.
Your second answer maps an open subset of the space in your room to R^1.
The third answer is a projection of a vertical plane in your room onto [0,360).
Your fourth answer only succeeds in avoiding a coordinate system by failing to be specific.
Your fifth answer invokes three dimensions, since you are probably describing the average distance between the surface of the chalkboard and the surface of your body.

 

[JDoolin]

We're talking about ordinary, Euclidean plane geometry here, so I don't see why you're going on about spacetime intervals.

If you are using Euclidean plane geometry (i.e. nothing is moving; nothing has any relative velocity), then I would have to agree that angle and distance have unambiguous meanings.

However, I think it is also interesting (if we are talking about relativity) to consider objects that are moving.

There is no time dimension involved. There is a triangle ABC formed by three lines AB, AC, and BC. The line AB has a length we can measure by holding a ruler up against it. The angle A can be measured by holding a protractor up against it. We can rotate and translate the paper in any way we like; the length of the line AB and the measure of the angle A are invariant.

If you are constraining yourself to talking about Euclidean plane geometry, lying on a stationary page, then all you are saying is correct. But, again, the thread is about coordinate free "relativity" so my question in post 44 about post 31 is still valid in the larger context.

 

[Dale]

To the contrary, almost all of your answers specify coordinate systems.

No, not one of them did.

Your first answer maps an open subset of the space in your room to R^1.
Your second answer maps an open subset of the space in your room to R^1.

The space in my room is 3D, so a coordinate system in my room maps open subsets of the space in my room to open subsets of R^3.

However, for the sake of argument, even considering a 1D embedded manifold in my room (so that we can map open subsets of the manifold to open subsets of R^1), a measure of the distance between two points does not establish a coordinate system.

First, the measure of distance is invariant under shifts of the origin. Is the origin on me or is it on the monitor or on some other point elsewhere? Second, the measure of distance is invariant under reversals of the basis vector. Do coordinates increase from me to the monitor or from the monitor to me? Third, the measure of distance in inches does not preclude the use of a coordinate system using other units. Do the coordinates change at a rate of one coordinate per inch or one coordinate per meter? Fourth, the measure of distance does not indicate if the coordinate system is uniform. Do the coordinates change at a linearly decreasing rate as a function of distance?

A measure of distance simply does not establish a mapping to R^n. There are many unspecified details. By telling you the distance from me to the monitor is 19" you cannot tell me unambiguously what is the coordinate position for me nor what is the coordinate for the monitor nor what are the coordinates for each point between us. Nothing less constitutes a coordinate system.

The third answer is a projection of the horizonal plane in your room onto [0,360).
Your fourth answer only succeeds in avoiding a coordinate system by failing to be specific.
Your fifth answer invokes three dimensions, since you are probably describing the average distance between the surface of the chalkboard and the surface of your body.

Similarly with all of these. None specify a coordinate system because all of them leave a huge variety of details unspecified. Do you understand the difference between measuring a distance and specifying a coordinate system?

 

[JDoolin]

No, not one of them did.

The space in my room is 3D, so a coordinate system in my room maps open subsets of the space in my room to open subsets of R^3.

There is a path from on corner of your screen to the other corner of your screen. I suppose it may not actually be an OPEN subset of your room, since it is only one-dimensional. In the other two dimensions, you might call it a closed subset, since a set containing only one point is a closed set.

I believe some use the word "clopen" to describe such subsets as a line or a plane through space.

However, for the sake of argument, even considering a 1D embedded manifold in my room (so that we can map open subsets of the manifold to open subsets of R^1), a measure of the distance between two points does not establish a coordinate system.

First, the measure of distance is invariant under shifts of the origin. Is the origin on me or is it on the monitor or on some other point elsewhere?

If you are finding the distance from point A to point B, then your origin is at point A. But if you use the distance formula distance = \left |x_b-x_a \right |, then x_b and x_a must be defined from some other point (the origin).

Even if you use an unnumbered ruler to measure the distance, you still must determine the "from" point, and by doing so, you have defined an origin.

Second, the measure of distance is invariant under reversals of the basis vector. Do coordinates increase from me to the monitor or from the monitor to me?

This would depend on where you place your oigin (where you are measuring from.)

Yes, you can arbitrarily designate your origin, but that does not mean you can make the measurement without choosing an origin at all.

Third, the measure of distance in inches does not preclude the use of a coordinate system using other units. Do the coordinates change at a rate of one coordinate per inch or one coordinate per meter?

That's correct. You could even choose some weird logarithmic scale if you wanted. But again, similarly, you can arbitrarily designate any unit length you wish, but that does not mean you can make the measurement without choosing a unit length at all.

Fourth, the measure of distance does not indicate if the coordinate system is uniform. Do the coordinates change at a linearly decreasing rate as a function of distance?

Again, that all depends on your choice of how to define your unit size.

A measure of distance simply does not establish a mapping to R^n. There are many unspecified details. By telling you the distance from me to the monitor is 19" you cannot tell me unambiguously what is the coordinate position for me nor what is the coordinate for the monitor nor what are the coordinates for each point between us. Nothing less constitutes a coordinate system.

Nothing less? I disagree.

A coordinate system need only have dimension high enough to measure whatever quantities you are interested in. If I ask you for the diagonal length of your computer screen, we only need a one dimensional coordinate system. If I ask you the dimensions of the computer screen, we need a two-dimensional coordinate system. If I ask you where your left eye is in relation to the computer screen, we need a three-dimensional coordinate system. If I allow for the fact that your eyes may be moving at a relative velocity wih the computer screen, and I ask the same question, we need a four-dimensional coordinate system.

Similarly with all of these. None specify a coordinate system because all of them leave a huge variety of details unspecified. Do you understand the difference between measuring a distance and specifying a coordinate system?

I cannot imagine any way to measure a distance without specifying a "from" point and defining some scale with which to mark off the distance.

I think you could define a coordinate system without determining any distances, (so I see there is a difference), but I don't think you can go the other way, and determine a distance without defining, at least, a one-dimensional coordinate system.