What it does
A spacetime (Minkowski) diagram draws events with position x across and time t up. With c = 1, light rays run at 45°. Someone moving at speed β has their own time axis (their worldline, tilted towards the light ray) and their own space axis (the line of events they call simultaneous, tilted the other way). This makes the strange parts of special relativity visible:
- Relativity of simultaneity: events on one horizontal line in one frame are not simultaneous in another.
- Time dilation and length contraction: the factor γ = 1/√(1 − β²).
- The interval s² = Δt² − Δx² is the same in every frame. Timelike pairs (s² > 0) can be cause and effect and always keep their order. Spacelike pairs (s² < 0) cannot, and observers can disagree on which came first.
In Events mode you type events in frame S; the tool gives their coordinates in frame S′, draws both diagrams, and for every pair gives the interval, its type, the proper time (or proper length), whether the order is kept or reversed in S′, and the speed of the frame in which the two events are simultaneous or at the same place.
In Twin paradox mode one twin stays home while the other travels at β to a distance D and back. The tool gives both ages, the jump in “now on Earth” when the traveller turns round, and how many of Earth’s yearly signals arrive on each leg (the relativistic Doppler effect), which settles the “paradox” without any acceleration argument.
How to use it
- Choose Events or Twin paradox, and the units (c = 1, so years go with light-years, seconds with light-seconds).
- Set the speed β with the box or the slider (negative means moving towards −x).
- Type events as name, t, x, one per line, or the twin’s turning-point distance D.
- Read the tables and diagrams; download the table as CSV or a report.
Worked example
The made-up events are O (0, 0), A (t = 2, x = 1), B (1, 3) and C (3, 3), seen from a frame moving at β = 0.5 (γ = 1.155). In S′, A is at t′ = 1.732, x′ = 0 and B is at t′ = −0.5774: B now happens before O, because O–B is spacelike (s² = −8). A and C, which are at different times in S, are simultaneous in S′. O–A is timelike with a proper time of 1.732 years, and O–C lies on a light ray.
For the twin paradox, set β = 0.8 and D = 4 light-years: the page shows Traveller ages 6 y, Earth twin 10 y. At the turnaround, “now on Earth” jumps from 1.8 to 8.2 years (6.4 years). The traveller receives 1 of Earth’s yearly signals on the way out and 9 on the way back.
Tips
- Slide β and watch the tilted S′ axes close in on the light ray: that is why nothing reaches c.
- To find a frame where two events are simultaneous, read the “special frame” column and type that β.
- Negative β tilts the axes the other way.
FAQ
Who is right about the order of spacelike events? Both observers. Neither event can influence the other, so their order has no physical meaning.
Why does the traveller age less if motion is relative? The traveller changes frames at the turnaround; the stay-at-home twin does not. The diagram shows the traveller’s line of simultaneity swinging across Earth’s worldline at that moment.
Is gravity included? No, this is special relativity in flat spacetime. For curved spacetime, see the Schwarzschild orbit tools.
Where the numbers come from
- Lorentz transformation, spacetime interval, relativity of simultaneity, relativistic Doppler effect and the twin paradox: standard results of special relativity, for example Taylor, E. F. and Wheeler, J. A., Spacetime Physics (2nd edition, 1992).
- Checked by fibub against SymPy (interval invariance and composition of boosts) and NumPy (example events, and a numerical integral of the traveller's proper time).
- Example events and trip are made up.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.