Transforming Reference Frames in Space Time Diagrams

Question

I'm really rusty on my special relativity, and could use some help on this problem. (Doing for fun, not homework.) Thanks.

Here's a space-time diagram with c=1 units:

Yellow Lines: Light Cones

Blue & Red Vertical Lines: Planets (at x = 0 & 0.5)

Cyan Line: Space Ship moving at 0.9

Green & Magenta X's: Events that happen on each planet (t = 0.10 & 0.15)

Green & Magenta Lines: The light traveling from the event

I'm trying to transform this diagram into one that is in the reference frame of the space ship, but I'm struggling with my Lorentz boosts. I know the length should contract between the planets:

$$ L' = L/\gamma $$

I know I should be using the Lorentz equations

$$ t' = \gamma (t - vx) \\ x' = \gamma (x - vt) $$

but when I try these, I get weird answers. For example, when I try to calculate the time of the magenta event in the reference frame of the spaceship:

$$ t' = \frac{1}{1 - 0.9^2}(0.15 - 0.9 * 0.5) = -0.69 $$

which is really far in the past, as far as these events are concerned.

Can someone help me with how to transform this? Specifically:

1. Am I doing the contraction right between the planets?

2. Should the planets be seen moving away at a speed of 0.9, or does that change?

3. How do I calculate the timing of the events in the new coordinate frame where the spaceship is at rest?

Answer 1

OK, so I figured it out. My original answer was right, I just didn't accept it because it seemed weird, but relativity is pretty weird.

The magenta event DOES happen with a negative time in the spaceship's proper (inertial) frame. The reason for this is that very little time passes for the ship, because it's moving so fast, so the magenta event happens long ago.

Here are the updated space time diagrams:

To answer my questions:

1) Yes, I did the contraction right. This arises naturally from the Lorentz equations, and is redundant, so I don't have to do them. However, while it is valid to view the space between the planets as a rod of a set distance, I cannot assume simultaneity between the two ends of the rod.

2) Yes, the planets are observed from the spaceship as moving away at a speed of 0.9.

3) I calculated the times right with the Lorentz equations. Plugging in the values for distance and time gives the right answers.

Thanks to everyone who took the time to read this.

Side note: This is why FTL communication is not possible. Imagine the green event happened, and the blue planet alerted the red planet of the event (which is the magenta event). The spaceship would then receive word of the magenta event before the green event happened. The spaceship could then send a FTL signal to the blue planet, which would arrive before the green event took place.