You should spend some time doing calculations in the “hypercomplex plane”. That is, something like the complex plane, but where you’ve taken the real number line and added a a new value u such that u² = 1 instead of the value i such that i² = –1.
Try to work out what perpendicular numbers look like in this space, what “circles” look like (hint: like hyperbolae), what the tangents to those circles look like, what kind of metrical relationships arbitrary triangles or other shapes have, and so on.
If you play with this simple 2-dimensional number system for a while, you’ll come to understand the nature of special relativity much better.
* * *
The displacement between two points in spacetime can be either “spacelike” in which case there is some inertial reference frame in which the two points are simultaneous, or it can be “timelike” in which case there is some inertial reference frame where the two points are at separate times, unmoving in space, or the displacement vector can be “lightlike”, which is a boundary between the two – the two points are always in every inertial reference frame separated by some speed-of-light separation.
Try to work out what perpendicular numbers look like in this space, what “circles” look like (hint: like hyperbolae), what the tangents to those circles look like, what kind of metrical relationships arbitrary triangles or other shapes have, and so on.
If you play with this simple 2-dimensional number system for a while, you’ll come to understand the nature of special relativity much better.
* * *
The displacement between two points in spacetime can be either “spacelike” in which case there is some inertial reference frame in which the two points are simultaneous, or it can be “timelike” in which case there is some inertial reference frame where the two points are at separate times, unmoving in space, or the displacement vector can be “lightlike”, which is a boundary between the two – the two points are always in every inertial reference frame separated by some speed-of-light separation.