Relativistic Rocket Calculator: Ship Time vs Earth Time

The relativistic rocket is a thought experiment and practical engineering concept: a spacecraft that accelerates continuously at constant proper acceleration for an interstellar voyage. At speeds approaching the speed of light, Newtonian physics breaks down. Einstein’s special relativity has to be used instead.

Why Relativity Matters at High Speed: At low speeds, time passes the same for everyone. At relativistic speeds, a moving clock ticks slower (time dilation). A ship traveling to Proxima Centauri at 1g acceleration would arrive in about 3.5 years of ship time, but about 5.9 years would pass on Earth. Light itself takes 4.24 years to cover that distance, so a ship starting from rest has to take longer than that back home. At higher accelerations the gap widens dramatically.

The Brachistochrone Mode (accelerate to midpoint, flip, decelerate): This is the fastest way to arrive at rest at the destination. The ship accelerates for the first half, flips 180°, then decelerates for the second half.

Key Relativistic Equations (constant proper acceleration a, distance d): Using the standard formulas derived from hyperbolic motion in special relativity:

For one leg of distance d_half = d/2:

• Ship proper time (τ): τ_half = (c/a) × acosh(a × d_half / c² + 1)
• Earth coordinate time (t): t_half = (c/a) × √((a × d_half / c² + 1)² − 1)
• Peak velocity: v_max = c × tanh(a × τ_half / c)
• Peak Lorentz factor: γ_max = cosh(a × τ_half / c)

Total ship time = 2 × τ_half, total Earth time = 2 × t_half.

For fly-by mode (accelerate only, no deceleration):

• τ = (c/a) × acosh(a × d / c² + 1)
• t_earth = (c/a) × √((a × d / c² + 1)² − 1)

Constants used:

• c = 299,792,458 m/s (speed of light)
• g = 9.80665 m/s² (standard gravity)
• 1 light-year = 9.461 × 10¹⁵ m

Reference Destinations:

At 1g acceleration in brachistochrone mode, a trip to Andromeda takes about 28 years of ship time, while over 2.5 million years pass on Earth. That is how extreme the dilation gets at near-light speeds.

The part the arithmetic hides: fuel

The times above are real physics and they are also completely out of reach, for a reason that has nothing to do with relativity being hard. Holding 1g for years means carrying the reaction mass to produce it, and the rocket equation in its relativistic form is brutal about that.

For a perfect photon rocket, one that turns fuel into a beam of light with no losses at all, the mass you start with divided by the mass you arrive with is e raised to the power of (a × τ ÷ c), where τ is the ship’s own elapsed time under thrust.

Proxima Centauri at 1g, arriving at rest, needs a mass ratio near 39 to 1. Andromeda at 1g needs about 7 trillion to 1, so a 1,000-tonne ship would set off carrying roughly 7 × 10¹⁸ kg of perfect antimatter fuel, which is asteroid territory. Any real drive is far worse than perfect, and the ratio is an exponential, so “far worse” compounds fast.

None of that makes the time dilation wrong. It just means the ship is the science fiction, not the clock. The page reports the mass ratio alongside every answer so the two stay in the same view.