Relativity ★★☆ Moderate

🚂 Relativity of Simultaneity

Two lightning bolts strike the front and back of a moving train at the same time for a platform observer — yet the train observer sees them at different times. Einstein's train thought experiment, brought to life.

📍 Platform Frame
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🚂 Train Frame
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γ = 1.250 Train rest length L₀ = 160 px Contracted L = 128 px Train-frame delay Δt′ = —
Δt′ = γβL/c  |  front: t′ = −γβL/2c  |  back: t′ = +γβL/2c

What is the Relativity of Simultaneity?

Einstein's 1905 special relativity showed that "simultaneous" is not absolute — it depends on your reference frame. Two events that happen at the same time but different places in one frame are not simultaneous in any other inertial frame moving relative to the first.

In the platform frame: the train moves at speed β·c; both lightning strikes occur at t = 0 (same time). The platform observer M at the midpoint receives both light pulses simultaneously.

In the train frame: using the Lorentz transformation t′ = γ(t − βx/c), the front strike has t′ = −γβL/(2c) < 0 (already happened) and the back strike has t′ = +γβL/(2c) > 0 (not yet happened). The train observer M′ sees the front strike first.

The time gap is Δt′ = γβL/c = β²γL₀/c. For β = 0.6 and L₀ = 160 px, with c = 1 px/unit: Δt′ = 0.6 × 1.25 × 128 = 96 px-units.

About this simulation

This simulation visualises the relativity of simultaneity, one of the core consequences of Einstein's 1905 special relativity: two lightning strikes that hit the front and back of a moving train at the same instant in the platform frame are shown, side by side, arriving at different times in the train's own frame. Both panels run the identical physical event, animated frame-by-frame with finite light-signal propagation, so you can watch the same reality look different depending on who is measuring it. The train-frame timing gap is computed directly from the Lorentz transformation t′ = γ(t − βx/c).

🔬 What it shows

Two synchronised canvases model the same lightning strikes at the ends of a train: the Platform Frame (where the strikes occur simultaneously at t = 0 and observer M at the midpoint receives both light signals at once) and the Train Frame (where the same two events are transformed via t′ = γ(t − βx/c), giving the front strike a negative time and the back strike a positive time, so observer M′ sees the front strike first). The stats bar shows the Lorentz factor γ, the train's rest length L₀ = 160 px, its length-contracted value L = L₀/γ in the platform frame, and the train-frame time gap Δt′ = γβL/c = β²γL₀/c.

🎮 How to use

Drag the β = v/c slider (0.05 to 0.95) to change the train's speed as a fraction of light speed, then press Run to animate both frames together from the same starting instant. Watch the Platform Frame result flip to "SIMULTANEOUS" once both light circles reach observer M, while the Train Frame result reports "NOT SIMULTANEOUS — front first" as soon as the front light circle reaches M′ before the back one does. Increasing β raises γ and stretches Δt′, making the discrepancy between the two frames larger and easier to see; Reset returns both canvases to the pre-strike moment.

💡 Did you know?

Einstein introduced this exact train-and-lightning scenario in his popular 1917 book "Relativity: The Special and General Theory" to explain a result he had derived in his original 1905 paper "On the Electrodynamics of Moving Bodies." The key insight is that simultaneity is not a property of events themselves but of the reference frame used to describe them — there is no universal "now" that all observers share, only the invariant speed of light c that every observer measures the same regardless of their own motion.

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 reference frame. Two lightning strikes that occur simultaneously in the platform frame (at t = 0, at the front and back of the train) do not occur simultaneously in the train's own frame — the Lorentz transformation t′ = γ(t − βx/c) gives the front strike a different time coordinate than the back strike whenever the two events happen at different positions x.

Why does the train observer see the front strike first?

Applying t′ = γ(t − βx/c) to the two strikes (both at platform time t = 0, but at x = +L/2 and x = −L/2) gives the front event t′ = −γβL/(2c), a negative time, and the back event t′ = +γβL/(2c), a positive time. Since the front event's train-frame time comes before the back event's, the train observer M′ necessarily perceives the front lightning as happening first, even though nothing physically different occurred at the two ends of the train.

What do the β slider and the stats bar actually control?

The β = v/c slider sets the train's speed as a fraction of the speed of light (0.05 to 0.95). From β the simulation computes the Lorentz factor γ = 1/√(1 − β²), the length-contracted train length L = L₀/γ seen in the platform frame (where L₀ = 160 px is the train's rest length), and the train-frame time gap Δt′ = γβL/c between the two strikes. Pressing Run animates both canvases from the same starting instant using these computed values; Reset recomputes them and returns to the start.

Is the "SIMULTANEOUS" versus "NOT SIMULTANEOUS" result just a delay in when light reaches the eye?

No — it is a genuine disagreement about the time-ordering of the two strikes, not merely a visual delay. The simulation explicitly separates the strike events (which happen instantaneously in each frame's own coordinates) from the light signals traveling from them to the observers. The platform observer M receives both signals at once because M sits at the midpoint of two equal light-travel distances; the train observer M′ receives the front signal first because, in the train's own coordinates, the front strike genuinely occurred earlier in time, not merely closer in distance.

How does this connect to Einstein's original 1905 argument and the invariance of light speed?

Einstein built special relativity on the postulate that the speed of light c is the same for every inertial observer, regardless of the observer's or the source's motion. If simultaneity were absolute (as in Newtonian physics), combining that with a universal light speed would produce contradictions between frames. Einstein's 1905 paper resolved this by showing that simultaneity itself must be relative — each inertial frame has its own definition of "at the same time" — and the train-and-lightning example, which he later used in his 1917 popular book, is the standard way to make that abstract result concrete and visual.