Four-Vectors

What to observe

1. Drag the boost slider and watch the amber arrow. It never moves. Only the indigo and teal axes swing, so only the shadows the arrow casts on them change. The arrow is the physics, the components are just a reading taken by one observer.
2. The two shadows trade against each other: as one grows the other grows too, yet the invariant bar at the bottom right stays frozen. That is because the length is built with a minus sign, s² = (ct)² − x², and the growth in one term is exactly cancelled by the other.
3. Follow the hollow marker. Every observer in the universe reads a different component pair, and every one of those pairs lands somewhere on thesame hyperbola. Rotations in ordinary space move points on a circle, boosts move them on a hyperbola: that single swap is all of special relativity.
4. Push the arrow across the dashed 45° cone. Inside the cone the invariant ispositive (timelike, and its square root is the proper time a clock would log). Outside it turns negative (spacelike, a proper length, and no observer can be present at both ends).
5. Switch to Four-velocity. Slide the particle to any speed you like: the arrow stretches and tilts, but its length is pinned at exactly1 (that is, c). Everything moves through spacetime at the speed of light, and speeding up through space only trades time for space.
6. Switch to Four-momentum and drag the mass to zero. The arrow liesflat on the light cone, the invariant collapses to nothing, and E² − p²c² = 0 becomes E = pc. That is a photon, and it is the same picture as everything else, just with zero length.
7. Keep the mass fixed and raise the energy. The tip climbs the same hyperbola, and the gap between E and pc closes without ever shutting. The rest mass is the arrow's length, and no amount of energy can change it.