⏱️ Relative Simultaneity
Two lightning bolts strike the ends of a moving train simultaneously in the platform frame. Watch both reference frames side by side: the platform observer sees both flashes at once; the train observer does not. Adjust β = v/c and see the Lorentz-transformed flash times shift.
Train Speed
Presets
Relativistic Values
Flash Times (S′)
x′ = γ(x − vt)
γ = 1/√(1−β²)
Δt′ = γvL/c²
About Relative Simultaneity
The Thought Experiment
Einstein (1905) imagined a train moving at speed v past a platform. Two lightning bolts strike the front and rear of the train at the same moment according to the platform observer (placed at the midpoint of the train). Both light pulses travel at c in every frame. The train observer, moving toward the front flash, encounters it before the rear flash — and concludes the front bolt struck first. Neither observer is wrong; simultaneity is frame-dependent.
Lorentz Transformation
The platform frame S assigns coordinates (t, x) to events. The train frame S′ moves at velocity v relative to S. Event coordinates transform as t′ = γ(t−vx/c²), x′ = γ(x−vt). For two events with the same t but different x, t′ differs: Δt′ = −γvΔx/c². The impossibility of causal paradoxes is preserved because |Δx/Δt| > c for spacelike-separated simultaneous events; no signal can connect them.
Spacelike Separation
The two lightning events are spacelike separated: the spacetime interval s² = c²Δt²−Δx² < 0. For spacelike intervals, the time ordering is observer-dependent — different inertial frames can disagree on which event came first, or even find them simultaneous. This does not violate causality because no causal influence (travelling ≤ c) can connect the two events.
Physical Implications
Relative simultaneity is not a measurement artefact; it reflects the geometry of Minkowski spacetime. Consequences include the relativity of length (Lorentz contraction) and time (time dilation). GPS satellites must correct for both special-relativistic time dilation (−7 μs/day due to orbital speed) and general-relativistic gravitational blueshift (+45 μs/day) to maintain metre-level positioning accuracy.