This is real special-relativistic time dilation applied to a hypothetical traversable wormhole of proper throat length D (in light-seconds, so v is a plain fraction of c). The traveler's velocity follows a trapezoidal profile: a linear ramp up to cruise speed β over ramp time ta, a constant-β cruise, then a symmetric ramp down. At every instant, the Lorentz factor is
γ(β) = 1 / √(1 − β²)
and the traveler's proper time accumulates slower than the outside observer's coordinate time by exactly that factor:
dτ = dt · √(1 − β(t)²) = dt / γ(t)
The sim integrates this live, frame by frame, as the ship crosses the throat. For the ramp phases (linear v(t) = β·t/ta) the integral has a closed form used to cross-check the live totals:
τ_ramp = t_a · [ ½√(1−β²) + arcsin(β)/(2β) ]
If the throat is too short for the ship to finish ramping up before reaching the midpoint, the profile automatically collapses to a triangle (accelerate straight into decelerate, no cruise) so the ramp distance β·ta never exceeds D.
- Tunnel view — the ship's position along the throat, its instantaneous speed gauge, and both clocks.
- β / γ graph — the velocity profile and Lorentz factor over the whole trip; γ diverges as β → 1.
- Clock divergence — traveler proper time τ plotted against observer coordinate time t; the growing gap between the two lines is the time-dilation effect, not a decorative animation.