Let's say a spaceship leaves Earth and is travelling at relativistic speeds. The spaceship's commander observes a supernova that took place somewhere ahead of the spaceship and then sends a signal to Earth to inform it of the supernova. On Earth, what will happen first, the reception of the signal or the observation of the supernova?
Common sense tells us that the light from the supernova will reach Earth first but doing the calculations tells us the opposite. How can this be explained?
Here is the full enunciation of the problem + solving:
The commander of a spaceship which is moving away from Earth on the x-axis with speed v=0.8c observes a supernova at the following spatio-temporal coordinates:
x'=6.2*10^8 ls (light-seconds)
measured from the spaceship's reference frame and reports the event to Earth.
What are the spatio-temporal coordinates of the supernova from the Earth's reference frame? Also, will Earth receive the information from the commander before or after the astronomers on Earth observe the supernova?
Applying the Lorentz transformations yields:
x=(x' - v*t')/sqrt(1-v^2/c^2)=2.3*10^8 ls
t=(t' - v/c^2*x')/sqrt(1-v^2/c^2)=1.733*10^8 s
The results match those of my teacher's although I find the results pretty strange. x < x' although it should be vice-versa because the commander of the spaceship sees space contracted. Also t < t', but since time flows slower on Earth from the reference frame of the ship, it should be vice-versa. My guess is that my teacher got this wrong by misusing the Lorenz transformations.