Spacetime & Lorentz Boosts
Minkowski's 1908 insight: space and time are not separate — they are coordinates of a single four-dimensional spacetime. A change of reference frame is a Lorentz boost: a "rotation" in spacetime that tilts the coordinate axes toward the light cone. Drag the $\beta$ slider to boost into a moving frame and watch simultaneity break, clocks dilate, and rulers contract — all written in the geometry of the diagram. Gray: lab frame S. Blue: boosted x′-axis. Red: boosted ct′-axis. Yellow dashed: light cone (invariant in all frames).
Minkowski spacetime
In special relativity, every event is a point in four-dimensional spacetime with coordinates (t, x, y, z). On a Minkowski diagram we suppress two spatial dimensions and plot ct (time scaled by the speed of light) vertically and x horizontally. The worldline of any object is the curve it traces through this diagram as time advances. An object at rest traces a vertical line; an object moving at constant velocity traces a tilted line. No worldline can be tilted past 45° — that would require speed greater than c.
The Lorentz transformation
The transformation between reference frame S (at rest) and frame S′ (moving at velocity $v = \beta c$) is:
where $\gamma = 1/\sqrt{1-\beta^2}$ is the Lorentz factor. This is a hyperbolic rotation — the spacetime analogue of an ordinary rotation, except using the Minkowski metric ds² = −(ct)² + x² instead of the Euclidean metric. The invariant interval ds² is preserved by every Lorentz boost, just as Euclidean distance is preserved by ordinary rotations.
Reading the diagram
Axes: The x′-axis (blue) is the set of events simultaneous in S′ (ct′ = 0). The ct′-axis (red) is the worldline of the S′ origin (x′ = 0). Both axes tilt toward the light cone as $\beta$ increases — the light cone is the same in every frame.
Proper time ticks: Dots on worldlines mark unit intervals of proper time (time measured by a clock carried along the worldline). The dots on the moving observer's worldline are spaced further apart in lab time — this is time dilation. $\gamma$ dots of lab time contain only 1 dot of proper time.
Simultaneity (second scenario): Events A and B occur at the same lab time (same ct). They lie on the x-axis. But a boosted observer's "now" is the tilted x′-axis — and A and B are at different heights above this line: they are not simultaneous in S′. The mismatch grows with $\beta$ and with the spatial separation of the events.
Twin paradox (third scenario): The stay-at-home twin traces a vertical worldline; the traveler goes out at velocity $\beta$, turns around, and returns. The proper time on each worldline is its Minkowski "length" — which, due to the minus sign in the metric, is shorter for the longer geometric path. The traveler ages less: τ′ = 2T/γ versus $\tau$ = 2T for the stay-at-home. There is no paradox — the traveler's worldline has a kink (acceleration at the turnaround), breaking the symmetry.
The light cone and causality
The light cone (ct = ±x) is the boundary between causal and non-causal relationships. Events inside the forward light cone can be reached from the origin (they are causally connected). Events outside the light cone are spacelike-separated from the origin: no signal can connect them, and their time order is frame-dependent — different observers disagree on which one happened first. The invariant interval ds² = −(ct)² + x² is negative for timelike separations (inside the cone), zero for lightlike separations (on the cone), and positive for spacelike separations (outside the cone).