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

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In summary: David has a post on his blog about Coordinate-free Relativity. He explains that it is a way of thinking about space that is independent of coordinates. It is a way of thinking about space that is based on the idea of curved space.
  • #1
Quchen
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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, [itex]\mathrm dF=0\,;[/itex] [itex]\mathrm d*F=4 \pi S[/itex])

Thanks in advance,
David
 
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  • #2
I would suggest Gravitational Curvature by Frankel.
 
  • #3
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).
 
  • #5
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.
 
  • #6
JDoolin said:
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 [tex]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 }[/tex] you define an (m, n) tensor as a multi - linear mapping of one - forms and vectors: [tex]\mathbf{T}:\underbrace{V^{*}\times ...\times V^{*}}_{n}\times \underbrace{V\times ...\times V}_{m} \mapsto \mathbb{R}[/tex] 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.
 
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  • #7
WannabeNewton said:
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: [tex]\mathbf{T}:\underbrace{V^{*}\times ...\times V^{*}}_{n}\times \underbrace{V\times ...\times V}_{m} \mapsto \mathbb{R}[/tex] 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.

[tex]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 }[/tex]

... defines a whole lot of different functions, right? The thing on the left hand side
[tex]T^{\alpha ...\beta }_{\mu ..\nu }[/tex] 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

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

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

Code:
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.
 
  • #8
JDoolin said:
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

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

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 [itex]V^{*}[/itex]'s correspond to n copies of a dual vector space and the m [itex]V[/itex]'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.
 
  • #9
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.
 
  • #10
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.
 
  • #11
JDoolin said:
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

Code:
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.
 
  • #12
JDoolin said:
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.
 
  • #13
JDoolin said:
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.
 
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  • #14
twofish-quant said:
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.
 
  • #16
JDoolin said:
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.
 
  • #17
DaleSpam said:
You can do a lot of geometry without using coordinates.

What do you mean?
 
  • #18
JDoolin said:
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.
 
  • #19
JDoolin said:
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.
 
  • #20
DaleSpam said:
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.)
 
  • #21
Ben Niehoff said:
I solve Einstein's equations in d dimensions without using coordinates all the time.

For practice, or some practical application?
 
  • #22
JDoolin said:
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.
 
  • #23
JDoolin said:
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.
 
  • #24
JDoolin said:
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.
 
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  • #25
JDoolin said:
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.
 
  • #26
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.
 
  • #27
JDoolin said:
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.
 
  • #28
DaleSpam said:
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.
 
  • #29
twofish-quant said:
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.
 
  • #30
JDoolin said:
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.
 
  • #31
JDoolin said:
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.

attachment.gif


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:

DaleSpam said:
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.
 

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  • #32
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;

 
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  • #33
JDoolin said:
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.
 
  • #34
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.
 
  • #35
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.
 
<h2>1. What is Coordinate-Free Relativity?</h2><p>Coordinate-Free Relativity is a mathematical framework that describes the principles of General Relativity without relying on a specific coordinate system. It is based on the idea that the laws of physics should be independent of the observer's chosen coordinates.</p><h2>2. How is Coordinate-Free Relativity related to General Relativity?</h2><p>Coordinate-Free Relativity is a more general and abstract approach to understanding General Relativity. It provides a deeper understanding of the underlying principles and allows for a more elegant and concise formulation of the theory. However, it is still based on the same physical laws and predictions as General Relativity.</p><h2>3. Why is Coordinate-Free Relativity important?</h2><p>Coordinate-Free Relativity is important because it allows for a more intuitive and geometric understanding of General Relativity. It also allows for a more unified and elegant description of the theory, which can lead to new insights and predictions.</p><h2>4. Are there any limitations to Coordinate-Free Relativity?</h2><p>Like any mathematical framework, Coordinate-Free Relativity has its limitations. It is not always the most practical approach for solving specific problems and can be more complex to work with compared to traditional coordinate-based methods. However, it is still a valuable tool for understanding the fundamental principles of General Relativity.</p><h2>5. How can Coordinate-Free Relativity be applied in real-world scenarios?</h2><p>Coordinate-Free Relativity is primarily used in theoretical and mathematical physics, but it has applications in various fields such as cosmology, astrophysics, and even engineering. It can help in understanding the behavior of objects in extreme gravitational fields, such as black holes, and can also aid in the development of new technologies, such as GPS systems.</p>

1. What is Coordinate-Free Relativity?

Coordinate-Free Relativity is a mathematical framework that describes the principles of General Relativity without relying on a specific coordinate system. It is based on the idea that the laws of physics should be independent of the observer's chosen coordinates.

2. How is Coordinate-Free Relativity related to General Relativity?

Coordinate-Free Relativity is a more general and abstract approach to understanding General Relativity. It provides a deeper understanding of the underlying principles and allows for a more elegant and concise formulation of the theory. However, it is still based on the same physical laws and predictions as General Relativity.

3. Why is Coordinate-Free Relativity important?

Coordinate-Free Relativity is important because it allows for a more intuitive and geometric understanding of General Relativity. It also allows for a more unified and elegant description of the theory, which can lead to new insights and predictions.

4. Are there any limitations to Coordinate-Free Relativity?

Like any mathematical framework, Coordinate-Free Relativity has its limitations. It is not always the most practical approach for solving specific problems and can be more complex to work with compared to traditional coordinate-based methods. However, it is still a valuable tool for understanding the fundamental principles of General Relativity.

5. How can Coordinate-Free Relativity be applied in real-world scenarios?

Coordinate-Free Relativity is primarily used in theoretical and mathematical physics, but it has applications in various fields such as cosmology, astrophysics, and even engineering. It can help in understanding the behavior of objects in extreme gravitational fields, such as black holes, and can also aid in the development of new technologies, such as GPS systems.

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