Coordinate-free relativity

Ben Niehoff

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

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

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

DaleSpam

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

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

For practice, or some practical application?

twofish-quant

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.

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

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

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

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.

DaleSpam

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

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

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

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.

DaleSpam

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

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:

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.

Attached Thumbnails

 

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;

http://www.youtube.com/watch?v=igOWR_-BXJU

DaleSpam

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.