Interpretation of Negative Time in Minkowski Diagram

[techsingularity2042]

Homework Statement

A spacecraft is moving at a speed v = 0.8 c with respect to a space station whose proper length is 500 m. Two light bulbs A and B are placed at each end of the station.

The frames are such that when clocks in both frames show zero, the origins of the two frames coincide.

In the frame of reference of the station, the two light bulbs are turned on simultaneously.
According to an observer in the spacecraft, which light bulb is turned on first and what is the time interval between the two lights turning on?

Relevant Equations

Lorentz transformation

I want to explain my thoughts longer, but since my English is terrible, I will try to keep it as short as possible.

This is the Minkowski diagram for the question above. ct-axis is the worldline of the space station, and ct'-axis is the worldline of the spacecraft.
What I am confused about is how ct' of B is negative.

How could it have happened in the past in the frame of the spacecraft if it occurred when t=0 if both their clocks have been set to zero when their origins coincide? Doesn't that mean before their origins even coincided, the event B already happened in the frame of the spacecraft, whereas in the station's frame, it occurs when their origins coincide?

Thank you in advance.

 

[Ibix]

Doesn't that mean before their origins even coincided, the event B already happened in the frame of the spacecraft, whereas in the station's frame, it occurs when their origins coincide?

Yes. But there is no consequence to the ordering of spacelike-separated events. If you pick a person and a time on their wristwatch, all frames will agree whether he can see any lamp lit, although if he can't see both they may disagree about whether either is lit.

This is a major difference from pre-relativistic theories. If one event is in the future lightcone of the other, their order is agreed by all frames. If neither event is in the future lightcone of the other, some frames will say they happened in one order, some in the other order, and some at the same time.