Question

Spaceship MST3K is traveling to TRAPPIST-1 (a system of 7 Earth-sized planets 40 light-years away from Earth) at a speed u=0.8c. Assume that all frames are inertial, ignoring acceleration, and that Earth and TRAPPIST-1 do not move relative to each other.

a.) According to Earth, how long will the journey take?

b.) Earth sends a very important radio transmission to MST3k 10 years after their journey begins. According to Earth, when does this message arrive at TRAPPIST-1? Will the message reach MST3K before the ship arrives at TRAPPIST-1?

c.) According to MST3K, how long will the journey take?

d.) According to MST3K, when does the transmission arrive at TRAPPIST-1? Will the message reach MST3K before the ship arrives at TRAPPIST-1?

e.) According to MST3K, when was the transmission sent? How long did it take to arrive? Explain whether your answer violates faster-than-light travel.

Answer 1

a)According to earth the time taken by to complete the journey =,here the distance is between Earth and TRAPPIST-1=, and the speed is the speed of the satellite w.r.t earth =.
time taken then =, here the speed is measures in the units of speed of light and thus we can directly put the values as given without changing them in SI units,but as the distance was in light years then it means that the final answer will be in years as it's unit!!!

b)Radio transmission will travel at the speed of light this means that the radio signal sent at a distance of 40Ly will reach the planet in exactly 40 years.

c) Looking from the rest frame of the spaceship the planet is approaching at the speed of .8c this means that the distance between the spaceship and the planet will be contracted.
using the length contraction formula the contracted distance is give by
, here
,

,

So the journey this time would be calculated taking 24Ly as the distance to travel speed being the same.

d)According to satellite the distance bewteen earth and the TRAPPIST-1 is the contracted distance calculated above i.e. 24Ly.This means that the time taken by the transmissin to cover this distance in frame of the satellite is 24 years.The answer to whether it will reach before the ship arrives will be answered at last.

e)According to earth the transmission was sent 10years after the mission started but this time is the dilated time seeing from the rest frame of the satellite.
dilated time ;   is the proper time in the satellite's frame,
for this problem what we know is the dilated time
;

after the satellite took off.

Time taken by transmission to cover 24Ly(the contracted distance as calculated above) is simply 24 years

This means that the total time according to spaceship frame the transmission took to reach the TRAPPIST-1 is=,means it is going to reach at exactly the same time satellite is going to reach the planet.This is in accordance with the observation from the earth that the time would be exactly same (40years for signal to reach and 10years after it was sent so total time is actually 50 years) so there is no violation of anything as such.