Lorentz Transform & the Minkowski Diagram

Spacetime as one diagram

A Minkowski diagram plots position on one axis and time (scaled by c, so ct has units of length) on the other. A stationary object is a vertical line; a moving object is a tilted line called a worldline, whose slope encodes its speed; a light ray, moving at the one speed every observer agrees on, is always a line at exactly 45°. Everything special relativity says about time dilation, length contraction and simultaneity can be read directly off this single picture.

live demo · a grid of gridlines skewing under a Lorentz boost● LIVE

The transformation itself

Switching to a frame moving at speed v = βc relative to the first applies the Lorentz transformation to every event's coordinates:

x'  = γ * (x - β * ct)
ct' = γ * (ct - β * x)
γ = 1 / sqrt(1 - β²)          (the Lorentz factor)

Why the axes tilt instead of rotate

A Euclidean rotation turns both axes by the same angle in the same direction, keeping them perpendicular. A Lorentz boost instead tilts the space axis and the time axis towards each other, both by the same angle θ = arctan(β), so they close in on the 45° light line symmetrically from opposite sides but never cross it — the geometric statement that no observer's own worldline can ever reach the speed of light. What is preserved under a boost is not Euclidean distance but the spacetime interval (ct)² − x², and the curves of constant interval are hyperbolas, not circles; those hyperbolas are exactly what you need to correctly read equal time or equal length off the tilted, unevenly-scaled new axes.

Time dilation and length contraction, read off the picture

A moving clock's own one-second tick marks, plotted along its tilted worldline and mapped back to the original frame's horizontal (constant-t) lines, land further apart than one second — by a factor of exactly γ. That is time dilation, drawn rather than derived. Symmetrically, a rod at rest in the moving frame, measured by finding both of its ends at one single instant in the original frame, measures shorter by a factor of 1/γ than its rest length — length contraction, from the same diagram, just read along the other axis.

The relativity of simultaneity

Because each frame's own axis of "same time" is a different tilted line, two events that lie on one horizontal line in the original frame generally do not lie on a horizontal line in the boosted frame's own tilted coordinates — events simultaneous for one observer are not simultaneous for another moving relative to them, unless the two events happen at exactly the same place. This is not a measurement error or a delay in light reaching the observer; it is a genuine disagreement about which pairs of events count as "at the same time", and it is the reason causality is defined by the light cone rather than by any single frame's notion of simultaneity: only events inside each other's light cones can possibly affect one another, in every frame, without exception.

Frequently asked questions

Why do the moving frame's axes appear to squeeze toward the 45-degree light line?

The boost angle theta = arctan(beta) grows as beta approaches 1, so both the tilted space and time axes rotate closer to the 45-degree light line the faster the frame moves relative to the original one, but they can only ever approach it, never cross it, since that would require beta = 1, an observer moving at the speed of light.

What is the invariant hyperbola used for on the diagram?

A Lorentz boost does not preserve ordinary Euclidean distance on the page, so a ruler laid across the diagram gives the wrong answer for elapsed time or length in the tilted frame. The hyperbola of constant spacetime interval calibrates the tilted, unevenly spaced axes correctly, the same way a circle calibrates a rotated Euclidean axis.

Can two events be simultaneous for one observer but not for another?

Yes, unless the two events happen at exactly the same location. This relativity of simultaneity follows directly from each observer's line of constant time being tilted by a different amount on the Minkowski diagram, and it is a real disagreement about ordering, not merely a delay caused by the finite speed of light reaching either observer.

Try it live

Everything above runs in your browser — open Lorentz Transform and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

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The Lorentz Transform: Reading Relativity Off a Spacetime Diagram