The question I'm doing states that the nearest star to earth is 4 light years away, as measured on earth and a spaceship can get there in 5 years, as measured by an observer on earth. It asks how long the pilot would measure the journey to take and how far from earth the pilot would measure the star to be.
Using $t= \gamma t_0 $ I have (correctly) calculated the time measured by the pilot to be 3 years. This makes sense as I have taken $t_0$ to be the time measured by the pilot (stationary frame) and $t$ to be the time measured on earth.
However, using the same principle with the length contraction formula $l= \frac{l_0}{ \gamma} $, taking $l_0$ as the pilot's distance and $l$ as the distance measured on earth I get $l_0=4 *(5/3)=6.66… $ which is wrong as it is not a contraction. The right answer is obtained by switching the roles of $l$ and $l_0$ around but I don't understand why my initial thought process was wrong? Also, switching the roles seems inconsistent with how I'm using the time dilation formula.
Also, as a side question, I've heard stuff along the lines of "you're always safer using the Lorentz equations instead of the simple time dilation, length contraction equations." – I'm wondering how I'd use the Lorentz equations in this case. I think I successfully got the time answer using the Lorentz time equation but again, had trouble on how to even use the $x$ coordinate Lorentz equation for the second part of this question.
Many thanks for any answers!
Best Answer
So the question is: Why is $l_{o}$, the proper length, measured in the Earth's frame of reference and not the pilot's? $l_{o}$ is the length of something measured in the frame of reference in which it is at rest. On Earth we see the Earth and the star at rest. The pilot sees them as moving.