# Transforming Reference Frames in Space Time Diagrams

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?

• A concept that may be of some use: en.wikipedia.org/wiki/Rapidity – dmckee --- ex-moderator kitten Mar 1 '18 at 19:52
• Not sure what you're trying to calculate for the "magenta event". But if you want to work out everything graphically, I would suggest looking into learning how to plot up and interpret Minkowski diagrams. Also, I recommend the book "It's About Time: Understanding Einstein's Relativity" by N. David Mermin. – user93237 Mar 1 '18 at 20:36
• I'm trying to calculate the time of the magenta event, in the reference frame of the spaceship. I've edited the post to make that more clear. Thanks. – Mike Skocik Mar 1 '18 at 21:04

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: