Exoplanet Travel Planner — Relativistic Journey Time Calculator

Formula

How this is calculated

Special relativity predicts that a clock moving at speed v relative to a stationary frame runs slower by a factor of γ = 1 / √(1 − β²), where β = v/c is the speed as a fraction of the speed of light. This effect is called time dilation. For an interstellar journey of distance d at speed v, the time measured by Earth (coordinate time) is t = d/v. The crew aboard the ship ages by only τ = t / γ = t × √(1 − β²), which can be dramatically shorter than t at speeds close to c.

At the same time, the ship crew measures the distance to the destination as length-contracted to d_contracted = d / γ, which is consistent with their perceived travel time: at their contracted distance d/γ travelling at speed v they arrive in time τ = (d/γ)/v = d/(v·γ) = t/γ. This is not a paradox — the two frames genuinely disagree on distance and time, and both are self-consistent.

This calculator does not model acceleration phases (which would require integrating the hyperbolic trajectory) — it assumes the ship is already at cruise speed for the entire journey. Real missions would include significant acceleration and deceleration legs; the actual proper time would be shorter than predicted here if a constant-g acceleration profile were used instead. The kinetic energy per kg in units of mc² is (γ − 1), illustrating why propulsion requirements grow extremely steeply near the speed of light.