Special Relativity & Minkowski Diagram

Everything here happens in one space dimension with c = 1: distances are light-seconds and time is plotted as ct, also in light-seconds. In those units light always travels at 45°, a boost is a hyperbolic rotation, and the diagram stays readable at any speed.

The diagram. The black axes are the rest frame S. The blue axes belong to frame S′, moving at speed β to the right. The ct′ axis is the worldline of the S′ origin (x = βct) and the x′ axis is the set of events S′ calls simultaneous with the origin (ct = βx). Both tilt toward the light cone by the same angle arctan β, which is why nothing can be tilted past 45°: that would mean overtaking light.

Interaction: click empty space to place an event, drag it to move it, Alt+click or right-click to delete it. Click a table row to select an event; the dashed lines through the selected event show which other events S calls simultaneous with it (grey) and which S′ does (blue). Those two lines disagree — that is the relativity of simultaneity, and it is the root of almost every apparent paradox.

Scenarios

• Simultaneity — two events at the same time in S, at equal distances left and right. Read the ct′ column: in S′ they happen at different times, and the order flips if you reverse β.
• Time dilation — a clock at rest in S′ ticks off ct′ = R/2 of its own time. In S the same tick happens at ct = γ·R/2: the moving clock runs slow by exactly γ.
• Length contraction — a rod of rest length L₀ sits at rest in S′; its two ends trace the green worldlines. S measures both ends at the same time (ct = 0) and gets L₀/γ.
• Twin paradox — the traveller leaves, turns around at ct = R/2 and comes home. Their worldline is longer on the page but shorter in proper time: in spacetime the straight (inertial) path has the most elapsed time, the opposite of ordinary geometry.

Overlays: the calibration hyperbolae x² − ct² = ±k² connect points that are one unit from the origin in every frame. They show why the tilted S′ grid is not simply a squashed copy of the S grid — its unit lengths are stretched along the axes.

The interval s² = Δx² − Δct² is the same in all frames. Negative means timelike (cause can reach effect, and √−s² is the proper time a clock reads between them), positive means spacelike (no signal can connect them, and their order depends on the frame), zero means lightlike.

Formulas. γ = 1/√(1 − β²); x′ = γ(x − βct), ct′ = γ(ct − βx); Δt = γΔτ; L = L₀/γ; longitudinal Doppler λ = λ₀√((1 ± β)/(1 ∓ β)); transverse Doppler λ = γλ₀; E = γmc², p = γmβc, and rapidity φ = artanh β, which unlike velocity simply adds.

Limitations: one spatial dimension, flat spacetime, inertial frames only. Acceleration appears only as an instantaneous turnaround, and gravity is out of scope — that needs general relativity.