Minkowski Diagram

Spacetime diagram · Frame SHorizontal: x · Vertical: ct

Event · x3.00

Event · ct3.80

Boosted · x′1.27

Boosted · ct′2.66

Lorentz transformation: x′ = γ(x − βct) · ct′ = γ(ct − βx). The light cone stays fixed for all inertial frames.

Events & trajectories

Keep several worldlines and events on the same spacetime diagram.

Add a spacetime event

Label Color Position · x Time · ct

Light cone from this event

Click an event in the list or on the diagram to select it. Drag to reposition.

Add a particle trajectory

Start position · x₀ Start time · ct₀ Velocity · v/c Duration · Δ(ct) Color

Show dashed trajectory preview

Worldline: x = x₀ + (v/c)[ct − ct₀]. Values |v/c| > 1 depict spacelike, nonphysical particle tracks.

Diagram uses c = 1. Hold Shift and drag the canvas to pan; use +/− to zoom. The original cyan event E remains available.

Reading a Minkowski diagram

What it is

A Minkowski diagram is a drawing of spacetime. Position, x, runs across the page. Time runs up the page as ct, so that with c = 1 a year and a light-year are the same length. Light travels on the dashed lines, at 45 degrees.

A point is an event: one place at one time. A line is a worldline, the history of one object. An object at rest is a vertical line. Motion tilts that line toward the light. A worldline with mass stays inside the light lines.

How to read it

The pale axes are x and ct for an observer at rest. The violet axes are x′ and ct′ for an observer moving at speed β along x. Change β and the violet axes tilt. The same event then has two pairs of coordinates, and the numbers under the diagram give both.

The gold dashed lines are light through the origin. The opening above the origin is its future. The opening below is its past. An event inside either opening can be joined to the origin by a signal slower than light. An event outside cannot.

What it is for

The diagram shows a Lorentz transformation directly. The tilted axes are the moving observer’s coordinates. Drag an event and both frames update together. The interval between two events, (cΔt)² − (Δx)², does not change from one inertial frame to another. Time dilation, length contraction, and disagreements about which events are simultaneous are different readings of that interval.

Time-like and space-like

These diagrams were made in the tool above, with the speed set to rest. The moving axes then lie on the stationary axes, and the light lines are at 45 degrees. A is the origin. E is the other event. The colored line joins them.

Minkowski diagram at rest. Event A is at the origin and event E is at x = 2, ct = 4, inside the future light cone. A green line between them is steeper than the light line. — **Time-like.** A is at x = 0, ct = 0, and E is at x = 2, ct = 4. The time part is larger: 4² − 2² = 12. E is inside the future light cone, and the green line is steeper than the light line. A traveler at 0.5c goes from A to E. Every inertial observer agrees that A happens first.

Minkowski diagram at rest. Event A is at the origin and event E is at x = 4, ct = 2, outside the light cone. A pink line between them is flatter than the light line. — **Space-like.** E is at x = 4, ct = 2. The distance is larger: 2² − 4² = −12. E is outside the light cone, and the line is flatter than the light line. The gap would take a speed of 2c. No signal can cross it, and other frames can disagree about which event happens first.

Minkowski diagram at rest. Event A is at the origin and event E is at x = 3, ct = 3, on the light line. A pale line between them lies along the light cone. — **Light-like.** E is at x = 3, ct = 3. The two parts are equal: 3² − 3² = 0, so E lies on the light line. Light from A reaches E. Nothing slower does.