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Spacetime diagram: the maths

Special relativity in one space dimension: Lorentz transformations between inertial observers. Computed live.

Equations

t′ = γ (t − vx/c²), x′ = γ (x − vt)

The Lorentz transformation: when and where an event happens for an observer moving at speed v.

T_Earth = 2D / v, T_traveller = T_Earth / γ

The twin paradox: a round trip to a star D light-years away at speed v (turnaround treated as instant).

L = L₀ / γ

Length contraction: a pole of rest length L₀ moving at speed v is shorter by γ in your frame.

Constants

c

1 light-year per year

units chosen so c = 1

Worked examples ✓ checked on every change

Twins: 0.8 c to a star 4 light-years away — difference in ageing	4.00 years	Earth twin 10 years, traveller 6
Pole (5) and barn (4): fits in the barn's frame at 0.7 c?	yes	L = 5/γ = 3.57 < 4
…and at 0.5 c?	no	L = 5/γ = 4.33 > 4

The app's test suite puts the lab in each setup, reads the lab's own result, and fails if it strays from these values.

What's simplified

• One space dimension.
• The traveller's turnaround is instant; a real ship would take time to decelerate and return (the 1 g voyage lab does that).

Where it breaks

Only for flat spacetime: no gravity. With gravity, use general relativity.

Sources: H. Minkowski (1908); E. F. Taylor & J. A. Wheeler, Spacetime Physics (1992).