From two postulates to one geometry
In Volume I you met special relativity as a set of surprising rules: moving clocks run slow, moving rulers shrink, and simultaneity is relative. Those rules all follow from the two postulates — the principle of relativity and the invariance of the speed of light. But learning relativity as a list of effects is like learning geometry as a list of triangle facts. This track does the opposite: we build the geometry first, and let the effects fall out as shadows of it.
The central move, due to Hermann Minkowski in 1908, is to stop thinking of space and time separately and fuse them into a single arena called spacetime. A point of spacetime is an event — a place and a time, like `here, now`. A particle traces a curve of events called its worldline. Different inertial observers slice this four-dimensional block into `space` and `time` differently, the way two people tilt a loaf of bread and cut different slices. Relativity is the study of what stays the same when you re-slice.
The invariant interval
Different observers disagree about the time \Delta t between two events and about the spatial distance \Delta r between them. Newton thought \Delta t was absolute; relativity says it is not. So what do observers agree on? The answer is the single most important object in this track: the invariant interval, which combines time and space with a crucial minus sign.
The invariant interval between two events. Every inertial observer computes the same value of s^2, even though each measures different \Delta t and \Delta x.
That minus sign is the whole story. In ordinary space, the distance \Delta r^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 is what rotations preserve — turn your head and the coordinates of a point change, but its distance from you does not. In spacetime, a change of inertial frame plays the role of a rotation, and s^2 is the quantity it preserves. Time enters with a $+$ and space with a $-$ (or the reverse, depending on convention), and that single sign difference is what makes time different from space while still living in the same geometry.
Causal structure and the light cone
Because s^2 can be positive, zero, or negative, spacetime separations come in three flavours, and the sign is frame-independent — it is a genuine physical fact about a pair of events.
The three causal classes. Timelike-separated events can be linked by a slower-than-light signal; lightlike ones only by a light ray; spacelike ones by nothing at all.
Draw time upward and space sideways: the set of all lightlike directions through an event forms the light cone. Events inside the future cone can be reached from here at or below light speed — you can cause them. Events inside the past cone could have caused you. Everything outside the cone is spacelike-separated: no signal can connect it to you, so no observer can even agree on whether it happened before or after `now`. This is why c is not just a speed but the causal speed limit of the universe.
Where this track goes
With the interval and the light cone in hand, the rest of the track is a guided tour of the machinery that makes covariant relativity so powerful. Guide 2 shows that a change of frame — a Lorentz boost — is literally a rotation in spacetime, just a hyperbolic one. Guide 3 introduces the four-vector, the natural object that transforms like the coordinates and whose length is automatically an invariant. Guide 4 builds the four-momentum and derives E=mc^2 and E^2=(pc)^2+(mc^2)^2 as pure geometry. Guide 5 climbs to rank-2 tensors, the electromagnetic field tensor and the stress-energy tensor, and points the way to general relativity.