Relativistic Rocket Calculator

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 must 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 4.3 years would pass on Earth. 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, but over 2.5 million years pass on Earth — demonstrating how extreme time dilation becomes at near-light speeds.