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by Bruce Linnell |
Transformations and/or coordinate systems can be categorized a few different ways. (1) Orthogonal and non-orthogonal (2) linear and non-linearand (3)observer dependent, and description dependent.
Orthogonal and Non-Orthogonalareis a descriptions more of coordinate systems, rather than the transformations between them. A coordinate system makes it so you can locate an object by giving two or more numbers. For instance,if I usea standardCartesian Coordinate system, I might designate point (1,2) as a point 1 unit to the right of the origin, and2 units up.The directions (left-right) and the directions (up-down) are perpendicular to each other.
Similarly, I might choose to give the point (1,45^o) in a polar coordinate system. This would be 1 unit away from the orgin, but at an angle of 45 degrees to the right. This is still an orthogonal coordinate system, because the angular direction is effectively perpendicular to the radial direction.
On the other hand, I could also use a coordinate system where we use a system of north/south streets, which are crossed diagonally by a system of north-east/south-west streets. In this case, you can use vectors in the directions of the two streets to determine a location, or you can use vectors in directions perpendicular to the two streets to determine a loction. The vectors parallel to the directions are called "contravariant" vectors, while the vectors perpendicular to the directions are called "covariant" vectors.
In the case of orthogonal coordinate systems, covariant and contravariant vectors defined in this way are identical.
The polar coordinate system is a "nonlinear" coordinate system. If you select a region of "constant r" meaning you want a set of points where r is the same, then you get not a line but a circle. If you choose a region of "constant θ" though, you will have a line, because you are asking for all the points that have the same angle from the origin.
On the other hand, Cartesian coordinate systems, a region of constant x, or a region of constant y, will both give you lines, so these are linear coordinate systems.
There are two types of coordinate transformations: Those that change the actual locations of things, and those that simply change the descriptions of the locations of things. Special Relativity tends to focus on the types of transformations that are Observer Dependent, while General Relativity tends to focus on the sorts o transformations that are Description Dependent.
Consider a situation where three balls move away from each other. The left-hand-ball and the right-hand-ball move directly away from the central ball. The angle between thepaths of the balls is 180 degrees, since they are moving directly away from each other. However, another observer passing by above will see an entirely different situation:
Here the angle between the paths of the two balls is 90 degrees. This is a situation where by changing the point-of-view of the observer, we have made real changes in the positions and paths of the objects.
For instance, if I say that my table is two feet in front of me, but you say the table is 10 feet to your left, have we changed the description of locations of things, or have we changed the location? You could argue that you've only changed the description of the location of the thing. One of the descriptions is reative to me. The other description is relative to you. But by changing the observer, you're changing the location... Not just the description of the location.
Some might try to take from this that now "location" is some kind of arbitrary or meaningless concept. But quite the contrary: Location is "Observer Dependent." That is not the same thing as arbitrary. Arbitrary means you have some kind of choice in setting the parameters, a decision that can be made on a whim. Observer dependence is not something you can decide on a whim, because as an observer, you have limits in your freedom of motion.
On the other hand, you may say that the chair is 3 feet in front of me and 3 feet to my right. But I say the chair is 45o to my right, and 4.2 feet away from me. Now we have a situation where we are both describing the same location, but using a different description. The first description 3 feet in front, and 3 feet to the right, is using rectangular coordinates, while the second description 4.2 feet, and 45 degrees, is using polar coordinates.
In general these sorts of transformations which just change the descriptions, are nonlinear transformations.
Rotation:
Notice there is exactly one parameter in the 2X2 matrix, Θ. This is the relative angle between the observer and the top of the smiley face.
Horizontal Stretch
Here the single parameter inside the 2x2 matrix is s, representing the horizontal scale factor.
Vertical Stretch:
Now s represents the vertical scale factor.
Skew: (Galilean Transformation in space-time)
Here, the v represents a slanting coefficient. When this is 1, the slant is a 45 degree angle. At 0, there is no slant, and the smiley face looks normal.
Taffy Stretch: (The x component stretches while the y component contracts, and vice versa.)
Here, the image is horizontally stretched by a factor, s, and simultaneously vertically contracted by the same factor, 1/s.
LT Smiley: One diagonal component stretches while the other contracts, and vice-versa; This is the Lorentz Transformation
I switched variables on you from (x',y') because I wanted to show this is the Lorentz Transformation (and because I was tired of typing equations). Two features of the Lorentz Transformation are that the slopes of the diagonal lines x = c t and x = - c t do not change, so that means the speed of light doesn't change. The other feature is that in the middle, the transformation looks exactly like the skew transformation.
Polar to Rectangular:
Rectangular to Polar:
So a couple of obvious differences between the linear and nonlinear transformations should be noted. The first, notice that when doing a linear transformation, we start with (x,t) and end with (x',t') whereas with the nonliear transformations we are switching back and forth between (x,y) and (r,θ). This is because with the linear transformation, the coordinates themselves are not changing definitions, but rather, the objets are changing their locations. On the other hand, wih the nonlinear transformations, the objects are NOT changing their positions, but the coordinates are changing definitions. Instead of using (x,y) we'll use (r,θ) and describe exactly the same situation with different numbers.
Another big difference, you might notice is that all of the linear transformations can be expressed with matrices, while the nonlinear transformations cannot be. However by using a bit of differentiation, (the chain rule) we can get something that looks close to the matrix form.
We'll take our polar-to-rectangular equations:
and differentiate both equations:
And then this differentalequation is expressible in a matrix form:
However, unlike the linear transformations above, the quantities inside the 2X2 matrix are functions of the coordinates r and θ. That means that the nature of the transformation changes from point-to-point.
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by Bruce Linnell |