Coordinate Time
0.0
Proper Time (Rocket)
0.0
Velocity
0.00c
Distance Traveled
0.0 ly
Rocket Parameters
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The Relativistic Rocket Problem
A rocket under constant proper acceleration (what the astronauts feel) follows a hyperbolic worldline in spacetime. The velocity asymptotically approaches the speed of light, but never reaches it.
Notice how the rocket ages more slowly than clocks back on Earth due to time dilation, allowing distant journeys within a human lifetime for the travelers.
Key Physics
Worldline (c=1):
x(τ) = (1/α)[cosh(ατ) - 1]
t(τ) = (1/α)sinh(ατ)
Velocity:
v(τ) = tanh(ατ)
Rindler horizon:
x = -1/α (events behind this line can never be seen by the rocket)
x(τ) = (1/α)[cosh(ατ) - 1]
t(τ) = (1/α)sinh(ατ)
Velocity:
v(τ) = tanh(ατ)
Rindler horizon:
x = -1/α (events behind this line can never be seen by the rocket)
Journey Example
At 1g acceleration (α ≈ 1 c/year), a rocket can reach:
- Alpha Centauri (4.4 ly): ~3.5 years proper time
- Galactic center (26,000 ly): ~20 years proper time
- Andromeda (2.5 M ly): ~28 years proper time
But Earth ages much more!