Special Relativity & Minkowski Diagram
Minkowski spacetime diagram for two frames: drag events and read their coordinates in both, with the light cone, lines of simultaneity, the twin paradox, and γ, time dilation, length contraction, Doppler shift and relativistic energy.
About this tool
Compare the same events in two inertial frames. S′ moves at signed speed β = v/c relative to S. Both diagram axes use light-seconds (ls); ct = 1 ls corresponds to one second. Equal horizontal and vertical scales keep light rays at 45°.
Click the diagram to add an event, drag to move it, or Alt+click/right-click to delete it. Select an event with its table button. “Edit events” also supports exact coordinates and keyboard operation. Up to eight events are available; coordinates outside the viewing window remain in the tables. The interval column classifies separation from the origin; the second table compares consecutive events.
The grey axes/grid belong to S, the blue ones to S′. Their lines of simultaneity differ. The primed grid uses integer coordinate spacing adapted to the viewing range and speed; it is not an ordinary Euclidean rotation. Calibration hyperbolae have invariant x² − ct² = ±k².
Scenarios regenerate when β or the axis range changes. Simultaneity compares two events at ct = 0. Time dilation follows a clock at rest in S′ until ct = R/2 in S; its own elapsed time is R/(2γ), so both ticks stay visible even at high γ. Length contraction compares rod endpoints at equal S time: L = L₀/γ. The twin travels out and back with an instantaneous reversal at ct = 0.45R and returns at 0.9R; its proper elapsed time is shorter than the inertial home clock’s.
The calculator uses γ = 1/√(1 − β²), Δt = γΔτ, L = L₀/γ, E = γmc², K = (γ−1)mc², |p| = γm|β|c and u/c = (β + u′/c)/(1 + βu′/c). Both speed inputs must have magnitude below one. Doppler outputs describe approaching/receding sources with speed |β|. The transverse wavelength γλ₀ uses a photon observed at 90° to the source velocity in the observer frame S. Momentum is shown as a magnitude.
Flat spacetime, one spatial dimension, ideal inertial motion and instantaneous twin turnaround. No gravity or finite acceleration model. Inputs and computations remain in your browser.