So far we have collected a bag of relativistic surprises one at a time: moving clocks run slow, moving rulers shrink, distant "nows" fall apart. Each came from its own clever thought experiment. But physics prefers one master key to a ring of separate ones — a single rule that, given an event's coordinates in your frame, hands back its coordinates in mine. That rule is the Lorentz transformation, and from it time dilation, length contraction, and the relativity of simultaneity all tumble out as effortless corollaries.

The old, common-sense way of relating two frames is the Galilean transformation: if my frame slides along yours at speed v, then x' = x - vt and — crucially — t' = t, one universal time for all. That extra assumption, t' = t, is exactly the one the constancy of light forbids. The Lorentz transformation is what replaces it, mixing space and time together so that everyone agrees on the speed of light even while they disagree on lengths and durations.

The transformation itself

Let frame S' move at constant speed v along the x-axis of frame S, their origins coinciding at t = t' = 0. An event with coordinates (t, x) in S has coordinates (t', x') in S' given by:

The single new ingredient over Galileo is the t' = \gamma(t - vx/c^2) line. That -\,vx/c^2 term is the mathematical face of the relativity of simultaneity: two events at the same time t in S but at different places x get different times t' in S'. Time now depends on where you are, not just when.

Everything we already found, in one line each

The power of the transformation is that our earlier results are no longer separate miracles — they are one-step consequences.

Time dilation. Watch a single clock sitting at rest at a fixed place in S, so between two of its ticks \Delta x = 0. The transformation gives \Delta t' = \gamma\left(\Delta t - v\,\Delta x / c^2\right) = \gamma\,\Delta t. The moving observer records the longer time — a clock at rest ticks the proper (shortest) time.

Length contraction. To measure a moving rod you note both ends at the same instant in your frame, so \Delta t = 0. Using the inverse transform on the rod's rest length \Delta x' = L_0 gives L_0 = \gamma\,\Delta x, i.e. the measured length \Delta x = L_0/\gamma — shorter, exactly as before.

Relativity of simultaneity. Two events simultaneous in S (\Delta t = 0) but separated by \Delta x get \Delta t' = -\gamma\,v\,\Delta x/c^2 \ne 0 — not simultaneous in S'. Three famous effects, one equation.

The invariant interval: what everyone does agree on

If observers disagree about times and lengths, is anything absolute? Yes — and it is the deep reward of the whole theory. Just as rotating your graph paper changes the x and y of a point but never its distance from the origin \sqrt{x^2 + y^2}, a Lorentz transformation changes t and x but never the combination

Notice the crucial minus sign: spacetime distance is not the ordinary Pythagorean sum but a difference of the time and space parts. That single sign is the whole geometry of relativity. Because s^2 is invariant, the light cone (s^2 = 0) looks the same to everyone — which is just the second postulate wearing a geometric costume.

See the interval stay put

Here is a single event, marked in the rest frame at (x, ct) = (1, 2) (in units where c = 1). Drag the velocity slider to view it from frames moving at different \beta = v/c. The event's coordinates (x', ct') change — it slides left and right, up and down — but it is trapped on one hyperbola c^2 t'^2 - x'^2 = s^2, drawn as the curve. The live readout of c^2 t'^2 - x'^2 never budges. The dashed 45^\circ lines are the light cone; the hyperbola hugs but never crosses them.

Worked examples

Example 1 — transforming an event. In units where c = 1 (distances in light-seconds, times in seconds), an event happens at x = 4, t = 5. A frame moves at \beta = 0.6, so \gamma = 1/\sqrt{1-0.36} = 1.25. Then

The event sits at (x', t') = (1.25,\ 3.25) in the moving frame.

Example 2 — checking the interval. Use the same event's separation from the origin. In S: s^2 = t^2 - x^2 = 25 - 16 = 9. In S': s^2 = t'^2 - x'^2 = 3.25^2 - 1.25^2 = 10.5625 - 1.5625 = 9. Identical — the interval is invariant, just as promised.

Example 3 — time dilation from the transform. A clock rides along in S' at its origin (\Delta x' = 0) and ticks off \Delta t' = 1\ \text{s}. Using the inverse transform, \Delta t = \gamma(\Delta t' + v\,\Delta x'/c^2) = \gamma\,\Delta t'. At \beta = 0.6, \Delta t = 1.25\ \text{s} passes in S — the moving clock runs slow, recovered in one line.