Spaceship Duel

We're exploring simultaneity and trying to find what Einstein found more than a century ago! We will divide this in parts in the form of Q&A, asking relevant questions and attempting to answer them.

Simultaneity seems like a no brainer, but you might be surprised by the end of thoís scroll. Here's a general overview/introduction to how this will go. We have different lettered events happening in different times. Now our job in a nutshell is to compare these events in different frames of reference together, for example if the time between event X and event Y is the same for two different frames of reference (with difference velocities I should specify.). Don't worry if you're confused now, it will make more sense as we go.

Alright consider the following: We have two spaceships, at a distance L apart moving with the same velocity. The leftmost spaceship is always at origin in the spaceship FOR (frame of reference shortened), in other words x' = 0. Exactly in the middle we have a third spaceship at point M (Middle), and it stays in the middle all the time (has the same velocity as the other spaceships that's how it manages to stay in the middle if you were wondering).

For the fun part, both spaceships will shoot a laser beam simultaneously at each other (in S-FOR; spaceship frame of reference). We have 4 events then:

Event A: Spaceship A shoots the laser beam

Event B: Spaceship B shoots the laser beam

Event C: Spaceship A hit by Spaceship B's laser

Event D: Spaceship B hit by Spaceship A's laser

Here's a sketch of the system at various times.

Let there be two observers. Kylie on the spaceship right down the middle and Jack on the planet.

For Kylie: Events A & B happen at the same time and events C & D happen at the same time, that makes sense, she's moving with the same velocity as both the other two spacecrafts meaning that as far she's concerned spaceship A & B are stationary, as well as her! The planet is moving! When we put the problem in this conext it becomes trivial, 3 people standing in a line, the two on the ends throw a ball at each other with the same exact speed thrown at the same time, will both hit the person standing in the middle? Yes!

Now if there was no kid in between, the balls will just go on and hit the person it's headed to, AT THE SAME TIME.

That was the easier part, let's now look at the P-FOR, the planet frame of reference (remember, in the S-FOR the planet moved with velocity v). Were the lasers emitted simultaneously? Let's investigate. There's a neat trick that can make this easier to understand if we implement it at the start, since all 3 spaceships move with velocity v, can't we then just treat them as if they're all aboard a train? The answer is yes, and it's more intuitive than some spaceships and planets in space. I'll answer this quesiton by first looking at a diagram and then explain the diagram, but try to understand yourself or see if it makes some sense what is happening

Hmm first thing you notice, the rays don't reach Kylie at the same time. The reason is, kylie is moving (the whole train is) to the right, so she kinda goes to meet the beam before it comes to her fully, remember the kids who throw balls at a poor lad? Imagine the boy running towards one of the throwers all while both throwers are running with the exact same velocity, who hits him first? Ok maybe made it complex let's stick to the train. So if we know that the light beams hit Kylie at the same time, then the left most guy's laser must have been emitted first! To compensate for being "late". This specially made laser explodes on impact, who explodes first? In P-FOR it's the guy on the left, his motion is opposite to the beam coming at him, so he hits it first, it's vice versa for the guy on the right. It might seem like it's the guy on the right who blows up first because he didn't flash his laser first, but still the beam has a longer way to go as explained earlier.

Now that's weird. Are they shooting at the same time or not? The answer is they only shoot at the same time in a reference system such as Kylie's, otherwise as we saw with the fella on the planet he sees different things, to be exact he sees it this way

1. Left ship shoots
2. Right ship shoots
3. Left ship explodes
4. Right ship explodes

But for Kylie it's like this

1. Left and right ships shoots
2. Left and right ships explode

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I calculated the difference between the events and found that they were something very familiar, but before I say it let's talk a tiny bit about what I did.

I first wrote equations of the positions of the left ship, Kylie and the light beam eitted from the leftmost spaceship as a function of time, velocity, time and distance L. Then because we know that time T_m pthe position of kylie is the same as the position of the beam emitted from the leftmost spaceship we equate the expressions and obtain the following

\(t_A =t_M- \dfrac{L/2}{1-v}\)

Then we do the same with the left spaceship exploding and obtain another expression similar but not equal to the one we obtained above. We take the explosion time minus the beam emitting time, and find the duration it took, and we find that the duration it took for the observer on earth is not the same as the duration between the two events for Kylie, we found the ratio to be

\(\dfrac{1}{1-v^2}\)

which looks very similar to time dilation equation except this one is time dilation (also called lorentz transformation but we'll come back to this later) to the power of two (we know that because the square root is gone), so something must've been wrong that you could not possibly have known about. That is length contraction, we assumed that length will not contract, which it does. But that's a topic for another day, see you next time friend!