Space and time become spacetime
If observers disagree about lengths, times and simultaneity, is anything absolute? In 1908 Hermann Minkowski found the answer by fusing the three dimensions of space with time into a single four-dimensional spacetime. In this view an event is a point (a where and a when), and the whole history of a particle is a curve through spacetime called a worldline. Different observers are just different, tilted ways of slicing the one spacetime into 'space now' and 'time'.
The one thing everyone agrees on
Spacetime restores an absolute. Between two events, observers disagree about the space gap \Delta x and the time gap \Delta t separately, but a particular combination — the spacetime interval — comes out identical for everyone. It plays the role in spacetime that distance plays on a map, except with a crucial minus sign that encodes the difference between space and time.
The invariant interval. Observers trade Δt against Δx as they change frames, but this combination is frame-independent — the true geometric 'separation' of two events.
The sign of (\Delta s)^2 sorts every pair of events into three classes. If (c\Delta t)^2 > (\Delta x)^2 the interval is timelike — a signal slower than light can join them, so one can cause the other. If it is smaller, the interval is spacelike — no signal could connect them and their time order is frame-dependent. If equal, they lie on a light ray. For a timelike pair, \Delta s/c is exactly the proper time a clock reads travelling between them.
The light cone and cause and effect
Because nothing outruns light, the 45° light lines through any event fence off a light cone: the future cone holds every event this one can still influence, the past cone every event that could have influenced it, and the vast 'elsewhere' outside the cone is causally off-limits. Relativity reshuffles the order of some events, but it never lets an effect precede its cause — the light cone is the guardian of causality.
The twin paradox, finally settled
Here is the puzzle that trips everyone. Twin A stays on Earth; twin B rockets to a star at 0.60c and returns. By our earlier trip, each leg is 10 years for Earth and 8 for the ship, so B comes home having aged 16 years while A has aged 20 — B is genuinely younger. The 'paradox' asks: motion is relative, so why can't B claim it was A who flew away, making A the younger one? The situation looks symmetric — but it is not.
The break in symmetry is the turnaround. Twin A rides a single inertial frame the whole time. Twin B must decelerate, stop and accelerate back at the star, physically switching frames — B feels the engines fire; A never feels a thing. Only B's worldline has a kink, and that is the objective difference. On the spacetime diagram A's worldline is the straight vertical line between departure and reunion, while B's is bent; the interval integrated along each path gives their ages directly.