Question

You are the astrogator on a starship that has just left a space station one hour ago heading toward galactic East at half light speed. The captain receives orders listing the coordinates of a rendezvous with another ship as (1 light-hour North, 2 light-hours East). The coordinates, of course, are measured relative to the space station, and the captain wants them translated into coordinates relative to your starship. What do you tell him?

Answer 1

Here, the velocity is: v = c/2 traveling east

1 light hour = distance light travels in one hour with speed v = c

Since the starship is traveling east, the north-bearing direction will not change since it is perpendicular to the direction of motion.

distance covered by the starship in one hour = 0.5 light hour (since v = c/2)

so, the remaining distance is 2 - 0.5 = 1.5 light hour

this distance will suffer Lorentz contraction due to the relativistic speed of the starship.

therefore, the east-bearing coordinate will be: $L=\sqrt{1-\frac{v^2}{c^2}}L_o = 0.866 (1.5) = 1.3 \text{ light hours}$

the final coordinates measured relative to the starship is (1 light-hour North, 1.3 light hour East).