Final answer:
The Earth-bound observer measures the proper length between Earth and Alpha Centauri. The astronaut experiences length contraction, observing a shorter distance. As a result, the proper length is measured by the Earth-bound observer, while the shortest length is perceived by the astronaut.
Step-by-step explanation:
When a spaceship travels from Earth to Alpha Centauri at speeds close to the speed of light, two different measurements of distance are obtained due to the effects of special relativity. The Earth-bound observer measures the proper length, which is the actual distance between Earth and Alpha Centauri when both points are stationary relative to the observer. This distance remains constant and unaffected by the high velocities involved in the journey.
On the other hand, the passengers aboard the spacecraft experience length contraction, a phenomenon that occurs at relativistic speeds, meaning the distance between Earth and Alpha Centauri appears shorter than the proper length. Due to the relativistic effects, the astronauts perceive a contracted distance, which allows them to traverse the space between the stars in what they measure as their proper time without exceeding the speed of light.
The conclusion is that the Earth-bound observer measures the proper length, whereas the astronaut measures the shortest length due to length contraction.