Relativity ★★☆ Moderate

⬦ Minkowski Spacetime Diagram

Drag the velocity slider to Lorentz-boost the moving frame. See world lines tilt, the light cone stay fixed at 45°, and simultaneity lines rotate.

Rest frame (S) axes Moving frame (S′) axes Light cone (c = 1) Simultaneity lines World lines
γ = 1.000 Time dilation: Δτ′ = Δt/1.000 Length contraction: L′ = L/1.000
β = 0.00  |  γ = 1/(√1−β²) = 1.000

Minkowski Spacetime

In a Minkowski diagram the vertical axis is time (ct) and the horizontal axis is space (x). An object at rest traces a vertical world line; a moving object tilts toward the light cone. The light cone (45° lines) is frame-independent — light always travels at c.

A Lorentz boost at velocity β = v/c rotates both the ct′ axis and the x′ axis symmetrically around the light cone. Events that appear simultaneous in S (same ct) are not simultaneous in S′ (same ct′), and vice versa.

About the Minkowski Spacetime Diagram

A Minkowski diagram plots time on the vertical axis (ct) and space on the horizontal axis (x), so an event is a single point and an object's history is a world line. This simulation draws the rest frame S, the light cone fixed at 45 degrees where c = 1, and a second frame S' related by a Lorentz boost, illustrating the geometric structure of Einstein's special relativity.

The velocity slider sets the dimensionless speed beta = v/c, which tilts the ct' and x' axes symmetrically toward the light cone and rotates the red lines of simultaneity. The readouts show the Lorentz factor gamma = 1/sqrt(1 - beta squared) together with the resulting time dilation and length contraction. This geometry underpins GPS timing, particle accelerators and the relativistic limit on signalling speed.

Frequently Asked Questions

What is a Minkowski spacetime diagram?

It is a two-dimensional graph that combines one space dimension and time into a single picture of spacetime. Time, scaled as ct, runs vertically and position x runs horizontally, so a point is an event and a line traces an object's motion through time.

What does the beta slider control?

The slider sets beta = v/c, the speed of the second observer as a fraction of light speed, ranging from -0.95 to 0.95. Moving it applies a Lorentz boost: the primed axes tilt, the simultaneity lines rotate, and the gamma, dilation and contraction figures update.

Why does the light cone stay at 45 degrees?

Because the speed of light is the same in every inertial frame. With units chosen so c = 1, a light ray satisfies ct = x, which is a 45-degree line. Since all observers measure the same c, that yellow cone never tilts when you change beta.

What is the Lorentz factor gamma?

Gamma equals 1/sqrt(1 - beta squared). It governs how strongly time dilation and length contraction act. At beta = 0 it is 1, and it grows without bound as beta approaches 1, which is why nothing with mass can reach the speed of light.

Why are the S' axes not perpendicular?

A Lorentz boost is a hyperbolic rotation, not an ordinary rotation. The ct' axis acquires slope 1/beta and the x' axis slope beta, so both axes close in symmetrically on the light cone. They only appear non-perpendicular because the diagram is drawn in the rest frame's coordinates.

What do the simultaneity lines mean?

The red dashed lines are events the moving observer judges to happen at the same time, that is constant ct'. Because they tilt with beta, two events simultaneous in S are generally not simultaneous in S', which is the relativity of simultaneity made visual.

How are time dilation and length contraction shown?

The stats bar reports that a moving clock's proper time interval is divided by gamma and a moving object's length is divided by gamma. Both effects come from projecting onto the tilted primed axes, so a larger beta gives a larger gamma and stronger distortion.

What do the Add and Clear World Line buttons do?

Add World Line draws an extra orange line for a hypothetical object with a random starting position and velocity, letting you compare several trajectories at once. Clear Lines removes those extra world lines, and Reset returns beta to zero and clears everything.

Is this simulation physically accurate?

Yes, within its idealisation. It uses the exact Lorentz transformation for a 1+1 dimensional spacetime, the correct axis slopes, and the standard gamma factor. It omits the second and third space dimensions and any acceleration, so it models inertial frames in flat spacetime only.

How does this relate to the spacetime interval?

The quantity s squared = (ct) squared - x squared is invariant under a Lorentz boost, so all observers agree on it even though they disagree on time and distance separately. This invariant hyperbolic geometry is exactly why the primed axes shear rather than rotate.

Where is this geometry used in the real world?

Minkowski diagrams clarify causality, GPS satellite clock corrections, and the behaviour of fast particles in accelerators. They also explain why faster-than-light signalling would break the ordering of cause and effect, a key constraint in modern physics.