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⏱️ 2D Lorentz Transform

Interactive 2D Minkowski diagram: watch two events that are simultaneous in one frame become non-simultaneous after a real Lorentz transformation. Adjustable velocity, live x', ct' readout.

Special Relativity2DModerate60 FPS📱 Mobile-adapted⇄ 3D version

Relativity of Simultaneity, in Real Coordinates

This diagram tracks two independent events, A and B, at the same time on the same S-frame spacetime plot used by the classic Minkowski diagram, but here both events can be moved freely and the frame S′ axes are drawn at the exact tilt angle θ = arctan(β) that special relativity requires — not a decorative skew.

x′ = γ(x − βct) — space coordinate in the moving frame
ct′ = γ(ct − βx) — time coordinate in the moving frame
γ = 1/√(1−β²) — Lorentz factor

What to Explore

Press "Sync B" to force ct_A = ct_B — the two events are now simultaneous in S (Δct = 0)
Raise β — Δct′ in S′ moves away from zero even though Δct in S stays exactly zero: the events are no longer simultaneous for the moving observer
Move A and B to the same x — simultaneity survives the boost (a purely time-like separation transforms without mixing when Δx = 0)
Spread A and B far apart in x with a small β — even a modest velocity produces a large Δct′, because the effect scales with β·Δx

About this simulation

This 2D Minkowski diagram places two independent events, A and B, on a spacetime plot and applies the exact Lorentz transformation to compute their coordinates in a second inertial frame S′ moving at velocity v = βc. Because the transformation mixes space and time, two events that occur at the same instant in S (Δct = 0) generally do not occur at the same instant in S′ (Δct′ ≠ 0) — the famous relativity of simultaneity, shown here with live numbers rather than a thought experiment alone.

🔬 What it shows

A spacetime plane (x horizontal, ct vertical, light-seconds). The S′ axes are drawn at the true angle θ = arctan(β) toward the 45° light cone, with tick marks calibrated to the Minkowski metric — not a cosmetic skew. Every number in the side panel, including x′, ct′, Δct and Δct′, comes directly from x′ = γ(x − βct), ct′ = γ(ct − βx).

🎮 How to use

Drag the β slider to set the S′ frame's speed. Move event A and event B independently with their x and ct sliders, or press "Sync B" to force them simultaneous in S. Watch the verdict box: it reports whether A and B are simultaneous in S, in S′, in both (only possible when Δx = 0), or in neither.

💡 Did you know?

The size of the simultaneity break, Δct′ = γ(Δct − βΔx), depends on Δx: two co-located events (same x) stay simultaneous in every frame, but the farther apart they are in space, the larger the time gap a moving observer measures between them — even at everyday speeds, just too small to notice.

Frequently asked questions

What is the relativity of simultaneity?

It is the special-relativity result that whether two events happen "at the same time" depends on the observer's frame of motion. Events simultaneous in one inertial frame are, in general, not simultaneous in a frame moving relative to the first. This simulation demonstrates it directly: set Δct = 0 in frame S, then increase β and watch Δct′ in frame S′ move away from zero.

Why does moving events apart in x change the effect?

The exact relation is Δct′ = γ(Δct − βΔx). When Δx = 0 (the events share the same position), Δct′ = γΔct, so simultaneity (Δct = 0) is preserved. As Δx grows, the βΔx term grows too, so even a small velocity β can produce a large Δct′ once the events are far apart in space.

Are the S′ axes just a visual skew, or real geometry?

They are the real geometry. The x′ axis is the locus of ct′ = 0, which solving ct′ = γ(ct − βx) = 0 gives ct = βx — a line through the origin with slope β, i.e. angle arctan(β) from the x-axis. The ct′ axis is the locus x′ = 0, giving x = βct, the mirror-image tilt from the ct-axis. Both axes tilt by the same angle toward the light cone, which is why light stays at 45° in every frame.

What does the "Sync B" button do?

It copies event A's ct value onto event B, so the two events become exactly simultaneous in frame S (Δct = 0) while keeping their different x positions. This is the standard starting condition for demonstrating relativity of simultaneity: two spatially separated events, one instant, in one frame.

Is this the same simulation as the 3D Lorentz Transform?

No. The 3D version renders a single event's projection through a WebGL shader. This 2D version tracks two independent, freely positioned events on a plain canvas so you can directly compare Δct in S against Δct′ in S′ and read a live simultaneity verdict — a different mechanic built around the same underlying Lorentz transformation.

⚙ Under the hood

2D Minkowski spacetime diagram with two independent events. Applies the exact Lorentz transformation x'=γ(x-βct), ct'=γ(ct-βx) to draw the moving frame's real tilted axes at the true angle arctan(β), and shows live how two events simultaneous in one frame genuinely stop being simultaneous in the other.

relativitylorentz-transformminkowskisimultaneityspecial-relativityspacetimetime-dilationinertial-frames

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install