Friday, August 03, 2018

Friday Science: The Euclidean metric (1.14.1a)

About every other Friday I want to move through another section of Peter Collier's A Most Incomprehensible Thing: Notes Toward a Very Gentle Introduction to the Mathematics of Relativity.

Here is my first post in this series.

1. The brilliant idea of Collier's book, one to which I have long subscribed, is that theory is most effectively learned on a "need to know" basis. There are many gifted "abstracticians" among us who do not need an answer to the question, "What do I need to know this for? When am I ever going to use this?" Indeed, I was one of them in high school and college. I still affirm those who are wired this way. Go for it!

But if the goal is actual learning, most of us best learn theory as we are engaged in practice. This was the guiding principle of Phase 1 of Wesley Seminary at Indiana Wesleyan University, of which I was a co-founder. Indeed, philosophically, I have adopted the epistemological stance of a pragmatist/nominalist. That is to say, the abstraction of theory in fact reduces to a useful game humans play in order to operate more effectively in the concrete world of realia. In other words, ideas are useful abstractions of reality.

I thus mock those who say, "You need to know the theory in order to do the practice effectively." Poppycock! The theory is abstracted from effective practice. Since I operate in the world of academia, you can imagine how often I whisper, "Numbnuts" under my breath. The number of virtual Platonists around me is a constant source of frustration.

2. So my previous post was in 4.1 of Collier's book, introducing the curvature of space. Section 2 moves on to Riemannian manifolds and "the metric." But to understand the "metric" of general relativity, Collier reaches back into the metric of special relativity (3.5), which reaches back into the Euclidean metric (1.14.1). So if I am to follow my principle of "theory on a need to know basis," I must now post on the Euclidean metric. Then in two weeks time, the Minkowski metric of general relativity (3.5).

The Euclidean metric
3. In high school geometry, we basically learned "Euclidean" geometry, named after the ancient Greek mathematician. So when we refer to Euclidean space, we mean the way space would behave if it were exactly like what we learned in high school.

In three dimensional Euclidean space, the Pythagorean theorem applies: l2 = a2 + b2 + c2.

Don't let the expanded form mess you up. For three dimensions, we've just added an extra square. We've used L for the hypotenuse for "line," meaning the "line" that results from these three components.

4. When we begin to talk about space in advanced physics, we have to make some modifications. For one, we begin to talk about incredibly small increments of space. In calculus (which I won't review here), we talk about "infinitesimals," meaning almost infinitely small increments.

So when we talk about dx or dy or dz we are talking about almost infinitely small increments on the x axis or y axis or z axis. These are called coordinate differentials. So now we might say that

dl2 = dx2 + dy2 + dz2

We can call this version of L the line element, a really small increment of the line that results from these components (the "resultant).

5. Now I'm no fan of matrices. But they are all over relativity and quantum mechanics. I don't know why they work and this frustrates me because 2018 Ken is not 1984 Ken. So here I will simply suspend my questions and present the "game" of matrices as it relates here.

A metric or metric tensor is a matrix that presents the coefficients of the differential equation above.

So the first 1 in the top left reflects that there is a 1 in front of dx2. The second one in the middle reflects that there is a 1 in front of the dy2. The third 1 in the bottom right indicates that there is a 1 in front of the dz2.

"Ours is not to question why. Ours is just to memorize or die."

gij (which should be in brackets, sorry) is a way of referring to a matrix. The matrix is g, and the elements of the matrix are in i rows and j columns. So the dy component is in the position 2, 2 (row 2, column 2).

6. To explain the lay out a little more, think of it this way:

dx    dy    dz

Here you can see that the first 1 is in the cross of dx, etc.

7. Collier goes on to formulate this matrix in terms of polar coordinates, but I don't need them to understand the matrices of general relativity yet, so I'm going to pass for the moment.