Special Theory of Relativity

I must warn you from the outset, that if you're unwilling to go into some detail about this, you will never understand it.

The key to understanding special relativity is to first realize that, while spacetime is a unified whole as a geometric entity, it is not Euclidean -- it is Minkowskian [pron, "Min•CAUF•skee•en"].

What does that mean? Well, in Euclidean geometry, when you impose Cartesian coordinates, you can do translations and rotations without changing any geometric properties of figures -- that is, those kinds of transformations and only those kinds, preserve all distances and all angles. This can be shown to follow directly from a quantity called, the metric of space -- in 3 dimensions, it is

ds² = dx² + dy² + dz²

which follows from the Pythagorean Theorem, with ds being the distance between the endpoints of the segment (dx,dy,dz). When you change coordinates by translating and rotating, dx, dy, and dz will, in general, all change, but ds will not. That metric is called a Euclidean metric.

In special relativity, we introduce the time coordinate, convert it to the same units as the spatial coordinates by multiplying it by c. We call a spacetime "point" an "event," and the segment connecting two events an "interval," and find that the metric is now

ds² = -dt² + dx² + dy² + dz²
or
dτ² = -ds² = dt² - dx² - dy² - dz²

depending on which one of those RHS's is non-negative. That metric is called Minkowskian.
If the first one is positive, then ds is the proper length between the two events, and the interval is said to be "spacelike."
If the second one is positive, then dτ is the proper time between the two events, and the interval is said to be "timelike."
If they're both 0, then the interval is said to be "lightlike," and ds = dτ = 0.