What’s up with the bullets?

April 30, 2012

That is undoubtedly the most common question about Velocity Raptor. As it turns out, I was wrong. Read the above post for more info…

That is undoubtedly the most common question about Velocity Raptor. And, admittedly, it is counterintuitive. Extremely so. Even and especially for people with a relativity course under their belts. (When I was making the game, it took me a lot of double-checking to feel comfortable with it.) But… we have to listen to the math. We can’t do relativity by intuition alone.

I mention this explanation in the Relativity 101 page, but I think these pictures, and elaboration, could help make the situation clearer.

To understand the behavior of the bullets, we just need one starting point: Length Contraction. If a train is speeding past, a stationary observer would measure it to be shorter than the same train when stationary. And the length contracts in the direction of the motion (the train seems shorter from engine to caboose, not from top to bottom). Not a hard pill to swallow.

As a concrete example about the motion of bullets in the game, we turn to Level 7. Our intrepid hero (Velocity Raptor) finds herself in a room with a cannon. The cannon is firing bullets horizontally across the screen, whereupon they hit a target. VR also decides to run up (vertically upwards on the screen). So we have three frames of reference.

• That of the Raptor
• That of the Room, gun, and target
• That of the bullets

We’ll want to make sure to treat them each independently. For the purposes of this game, we are witnessing the world through the raptor’s frame of reference. So we need to figure out how (a) the room, and (b) the bullets will contract. Let’s start with the room.

Since the raptor is running upwards, to her the room seems to be going down. The direction of relative motion is vertical, so the Length Contraction happens vertically as well. The result is the room seems squished from top-to-bottom. Not a big surprise there.

In the exact same way, we can figure out how the bullets will behave. This time, there are two velocities to take into account. The raptor is running up, and the bullets are flying to the right. So in the view of the raptor, the bullets have a velocity that is down-and-to-the-right. Note that we are being extremely qualitative. Velocity addition in SR is more nuanced than in the standard Newtonian world, but all we care about is that the bullets seem (to the raptor) to be going down and to the right. For ease of our picture, imagine that we’ve chosen the velocities such that the final summed velocity points from one corner of the dotted-line-rectangle to the other.

Then we squish the bullets’ frame along its direction of motion. So in our example, we squish from one corner to the other. (That dotted rectangle around the bullets just helps us keep track of how the frame contracts. You can even pretend it is real, if you’d like… its addition doesn’t affect the results, just makes the picture clearer.)

Again, just a simple Length Contraction. Nothing special. So why did I pretend this was strange?

Ah yes, the final step. Let’s put all the pieces together. The raptor, room, and bullets of course exist together.

And there you have our result. The room is squished in one direction, the bullets in another, and thus the bullets are no longer sitting on a horizontal path across the screen. The cannon (where we know the bullets are starting) and the target (where we know the bullets are ending) lie along a perfectly horizontal line. But the bullets do not. They are sitting on a path aimed up and to the right. At an angle. So you won’t always find the bullets (seemingly) directly between the cannon and the target. In fact, depending on where the raptor stands, she may even see the bullets as emerging at a point distinct from the cannon… and disappearing at a point distinct from the target.

But remember, we didn’t do anything special. No magic, no tricks. Just boring old Length Contraction.

We have, in fact, rediscovered a known fact with our pictures. Two Lorentz boosts in non-parallel directions (aka combining the raptor’s motion and the bullets’ motion) together make not just another Lorentz boost, but also a rotation. Hence why the bullets don’t just seem squished, but also rotated.

Counterintuitive? Sure! But what good result in physics isn’t?

*Did you find a mistake in the steps? I’d love to hear about it. After all, in science, we must always reserve the right to be wrong 😉