My take on dimensions, numbers, and general relativity

Sounds like basically the four dimensions used in GR are all mathematically pretty much the same on a fundamental level. It's not like the fourth dimension of the model space in GR (which is R^4) acts much different from the first three. The metric that gives distance between two points in space-time, however, does do something different for the fourth dimension than it does for the first three. The square of distance infinitesimal element ds^2 is, by the transfer principle of logic, in addition to Pythagoras' theorem, is not merely

dx^2 + dy^2 + dz^2 + dt^2

I think that it's dx^2 + dy^2 + dz^2 - k * dt^2

for some value of k (I have no idea what k means physically). However, it didn't have to be the fourth one we write with a -k in front; it could have been the first one or the third one. What's necessary is that exactly three of the dimensions have positive metric but exactly one of the four must be multiplied by some negative number.

I don't think that GR is able to answer the question why 3 must have positive sign and 1 must have negative sign.

So, in that small technical sense of how points' distances between each other, the time dimension is only distinguishable as something who contributes negatively to distance squared. And if you remember your complex analysis, supposing there is more timelike distance than spacelike distance between two points, that means the distance itself will be a non-real, complex number.

The distance for two points only differing in the first three coordinates is never non-real. If the two points are the same point in 3-space, but if they aren't the same then they differ in the time-coordinate; in that case, the distance between them is imaginary. This tells us that the concept of measure/distance is dependent upon a sense in which space and time do not work the same exact way, the way, say, dimension 2 and dimension 1 work.

But let me say a word on that foul word, imaginary. Imaginary numbers get a bad rap. First of all, imaginary is a word describing contrast with what are known as real numbers. The real numbers are visualized and thought of as a line, well, points on a line. ALL imaginary numbers are are the numbers that are visualized and thought of as points in a PLANE.

Once you define how to add, subtract, divide, and multiply two complex numbers (i.e., basically imaginary numbers), the plane becomes a field just like the real numbers are. The complex numbers just have one more dimension than the real numbers; then not surprisingly, the complex numbers are specified by two coordinates.

I think complex numbers get a bad rap because of their unfortunate appellation. We owe the quirky, psychological descriptors labels for types of numbers to the Greeks. These numbers have colorful descriptions like irrational, transcendental, real, imaginary, negative, etc, etc.. The way the story was told to me it seems the Greeks basically couldn't fathom irrational numbers as ever possibly "actually" existing. The first way to need irrational numbers is when trying to deduce width and length from area. If a square can have 4 units of area and a 3-square can have 9 units of area, then why can't there be a square having 5 units of area? This needs the square root of 5 to be invoked.

At any rate, the complex numbers are essentially not a whole lot different from irrational numbers in a sense. Note that these numbers can solve the area problem when the area is negative the same way irrational numbers can solve the area problem when the area is not a perfect square number.

Oh yeah, perfect, that is another heavily-human word.