A spaceship is heading towards earth at 0.99c, when it is 1*10^9 metres away from the earth as measured by an earthly observer, the spaceship ejects a life pod. Is the distance between the earth and the spaceship at the instance the life pod is ejected the same for both the spaceship pilot and earthly observer?
Intuitively I think that as the situation is identical from both observer's frame of reference with the other heading toward them at 0.99c, the measured distance would also be the same. However, when I tried to solve this with Lorentz transforms, I failed miserably.
I tried to first align the frames of reference at: time(earth) = time(spaceship) = 0s
from this, I can find the time at which the life pod was ejected in earth's frame of reference by solving: timeEarth(ejection) = -(1*10^9)/(0.99c) = -3.38 seconds, which means the spaceship ejects the life pod 3.38 seconds before it passes the earth, from the earth's frame of reference.
I then assign spacetime coordinates for this event in the earth's frame of reference and let everything occur 1 dimensionally along an x-axis, so the event occurs at (110^9 m, -3.38 s) in the earth's frame of reference where the position of the space ship is 110^9 metres to the right. If I then apply Lorentz transforms to find the spacetime coordinates of the event from the spaceship's frame of reference:
Lorentz factor (γ) = 1 / sqrt( 1 - (0.99c / c)^2 ) = 7.089
timeSpaceship(ejection) = γ * ( -3.38 - (0.99c / c^2) * 1* 10^9) = -47.45 seconds
PositionSpaceship(ejection) = γ * (1*10^9 - 0.99c * -3.38) = 1.42 * 10^10 metres
This result clearly looks wrong, because imagining the frame of reference of the spaceship which remains still, the life pod must be ejected at the spaceship's position, and I would expect PositionSpaceship(ejection) = 0, since time was synchronised to zero when the spaceship passes the earth and so the spaceship would remain at x=0 in its frame of reference, while the earth moved toward the spaceship from the left at 0.99c.
I'm sure I've just made a mistake in using Lorentz transforms, but any help would be greatly appreciated.