Coordinate-free relativity

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

Your point has nothing to do with coordinate systems.

DaleSpam

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
...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.

DaleSpam

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.

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

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

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.

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?

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

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.

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".

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

DaleSpam

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

JDoolin

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

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.

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.

DaleSpam

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.

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.

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

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.

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 [itex]distance = \left |x_b-x_a \right |[/itex], 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.

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.

That's correct. You could even choose some wierd 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.

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

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.

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.

DaleSpam

Since you seem to be aware that the distance is completely independent of choice of origin then I don't see why you think it must be defined at all. I made the measurement without defining a coordinate system and choosing an origin.

I never said it did. But the fact is that making a measurement with a chosen unit of length does not constrain nor inform your choice of coordinates in any way.

You seem to be under the misapprehension that a coordinate system is the same as making a measurement that results in a number. That is incorrect, all measurements are invariant under arbitrary coordinate transforms. So the existence of a measurement does not require nor inform you as to any coordinate system. Nature does not come equipped with a required set of coordinates, regardless of what measurements you may make.

You should know the definition of a coordinate system by now. Can you unambiguously define a unique coordinate system from the fact that the distance between A and B is 6"?

You don't need a coordinate system for any of those measurements.

I (almost*) agree, but that is irrelevant. The "from" point need not be the origin and the scale to mark off the distance need not correspond to any coordinates.

*The almost is that in order to measure the distance between A and B you do not need to identify one as "from" and the other as "to". The measurement of distance is invariant under that choice.