Length contraction

Alright, maybe you found the discussions about simultaneity and time dilation a little difficult to picture. In that case, you’ll probably appreciate the third consequence of relativity a little more. Length contraction simply claims:

Moving objects are shortened.

Creepy, I know. To be a little more specific, moving objects are shortened in the direction of motion. So if a Maserati MC12 goes zipping by you at near light speed, its front bumper will be closer to its back bumper, but its height and width will stay the same. Just as with simultaneity and time dilation, I want to remind you that this is not some trick of observation: it’s not just that the car looks shorter, it’s that it actually is shorter (in the reference frame of the ground). While we’re on the subject, take a guess: by what factor do you predict the length changes?

Okay, well, I hate to keep you in suspense. If you guessed that the factor was γ, just like for time dilation, then you have a devious mind. But you’re also correct. If the length of an object at rest is L’, then the length that someone on the ground measures is L = L’/γ. If you don’t believe it, you can check out my proof:

https://mywebspace.wisc.edu/mweinberg/web/LengthContractionProof.pdf

So, huh. That’s weird. Why γ? Why not, I dunno, 3γ? Or some entirely different function of velocity? If you read the proof, you’ll see that the way γ crops up is quite different from the way it came about in time dilation, so what gives? Actually, it’s no coincidence that the factor by which time is increased is exactly the same as the factor by which length is decreased.

You’ve probably just started studying vectors in physics, but of course you know that they’re basically just a magnitude and a direction. If I want to, I’m free to rotate a vector however I like (by multiplying by something called a “rotation matrix”). Of course, the rotation can’t possibly change the magnitude of the vector, just which way it’s pointing. The secret is, special relativity is just a very abstract kind of rotation: instead of rotating a normal three-dimensional vector, say, out of the x-direction and into the y-direction, it rotates a four-dimensional vector out of space entirely, and into time. The magnitude of the “spacetime” vector doesn’t change, which is why the amount that length decreases has to exactly match the amount that time increases.

Of course, the guy driving the Maserati doesn’t think his car is shortened. To measure it, he’d have to use some kind of measuring stick, but all his measuring sticks are also shortened by the same factor! As a result, no matter how carefully he measures, he’s never going to agree that it’s not its normal length.

In fact, just as with time dilation, remember that as far as he’s concerned, he’s stationary, and it’s the road that’s moving underneath him. Actually, if he sees anyone on the ground whip out a measuring stick, he’s going to think they’re the ones that are shortened. So this raises a similar problem to the one in the time dilation section: if Al is the guy in the car and Bob is a scientist on the ground, Al says Bob’s measuring sticks are short, and Bob says Al’s measuring sticks are short. Which one is right? Write me a comment and let me know what you think. In the meantime, here are a couple of questions about length contraction to entertain you. (Remember, if you get stuck, ask me questions; I’m always happy to help out.)

https://mywebspace.wisc.edu/mweinberg/web/LengthContraction.pdf