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⏱️ Lorentz Transform — Minkowski Diagram

Interactive spacetime diagram showing two inertial frames S and S'. Drag event A, adjust velocity β and watch coordinates transform via x′ = γ(x − βct), ct′ = γ(ct − βx).

Frame S′ velocity

γ (Lorentz factor) 1.250

Event A (drag dot)

Coordinates

S: (x, ct) –
S′: (x′, ct′) –
s² = c²t²−x² –
Interval type –
■ S frame   ■ S′ frame
╌ Light cone (x = ±ct)
Dashes = coordinate projections

Lorentz Transformation

The Lorentz transformation relates spacetime coordinates between two inertial frames S and S′, where S′ moves at velocity v = βc relative to S:

Reading the Minkowski Diagram

What to Explore

Spacetime Interval

The quantity s² = c²t² − x² (or −s² = x² − c²t²) is Lorentz-invariant — the same in all inertial frames. If s² > 0 the interval is timelike (causal connection possible), if s² = 0 it is lightlike, if s² < 0 it is spacelike (no causal connection, time order can be reversed).

About this simulation

This interactive Minkowski spacetime diagram shows how the coordinates of a single event A transform between two inertial frames S and S' when S' moves at velocity v = βc. Adjusting the velocity tilts the orange S' axes toward the 45° light cone, while the applet computes the new coordinates using the Lorentz transformation x′ = γ(x − βct), ct′ = γ(ct − βx). It is a vivid way to grasp time dilation, length contraction and the relativity of simultaneity.

🔬 What it shows

A 2D spacetime (x horizontal, ct vertical) measured in light-seconds. The blue grid and axes are frame S; the orange grid and axes are frame S', which the code derives by tilting both axes by the same angle toward the yellow light cone (x = ±ct). Event A's coordinates are computed live with x′ = γ(x − βct), ct′ = γ(ct − βx) and γ = 1/√(1−β²).

🎮 How to use

Drag the β = v/c slider (0 to 0.98) to set the S' frame speed and watch γ update. Drag event A directly on the canvas, or use the x and ct sliders (−4 to 4 ls). The side panel reads out the S coordinates, the transformed S' coordinates, the invariant interval s² = c²t² − x², and whether the interval is timelike, lightlike or spacelike.

💡 Did you know?

The S and S' axes always make equal angles with the light cone, so a light ray stays at 45° in every frame — the geometric expression of the constant speed of light. This is why the diagram is drawn with Minkowski rather than Euclidean unit intervals.

Frequently asked questions

What is a Lorentz transformation?

It is the set of equations in special relativity that converts the space and time coordinates of an event from one inertial frame to another moving at constant velocity. In this simulation it uses x′ = γ(x − βct) and ct′ = γ(ct − βx), where β = v/c and γ = 1/√(1−β²). Unlike the everyday Galilean transformation, it mixes space and time together.

What does the velocity slider β actually do?

β is the speed of frame S' as a fraction of the speed of light, ranging here from 0 to 0.98. Increasing β raises the Lorentz factor γ and tilts the orange S' axes symmetrically toward the 45° light cone. At β = 0 the two frames coincide; as β approaches 1, γ grows without bound and the axes squeeze together near the light cone.

What is the spacetime interval shown in the panel?

The quantity s² = c²t² − x² is the spacetime interval. Its value is the same in every inertial frame, which is why it is called Lorentz-invariant. If s² is positive the interval is timelike (a cause-and-effect link is possible), if it is zero it is lightlike, and if it is negative it is spacelike, meaning no signal can connect the two events.

Why can the time order of events sometimes reverse?

For a spacelike interval (s² less than zero), event A lies outside the light cone of the origin. Because no signal can travel between such events, different frames legitimately disagree on which came first. The simulation lets you place A in this region and increase β to see the transformed ct′ flip sign, demonstrating the relativity of simultaneity.

Are the calculations physically accurate?

Yes. The applet applies the exact special-relativity Lorentz transformation with no approximations, so the displayed S' coordinates, the γ-factor and the invariant interval are quantitatively correct for the chosen β. Distances are expressed in light-seconds so that c = 1 and the light cone sits at a clean 45 degrees, but the underlying physics is unchanged.