Schrödinger's Cormorant

2.7 Slimming Advice

Another weird consequence of special relativity: distances are contracted for a moving observer, compared to those measured by a static one.

You and Nefertiti settle back in your deck chairs, feeling rather pleased with yourselves for resolving the mystery of the muons. Cormorant, meanwhile, drops magnificently down from his exalted altitude, makes a couple of swift passes up and down the beach, comes in to land and immediately inserts a fly into the ointment. “OK but put yourself in the muons’ shoes”, he challenges, “how do we make it work in their frame of reference?” While Nefertiti is scooping the insect out of her medication, let’s explore what he means:

If you are one of the muons, you are at the origin of your own reference frame, so you’re not moving and therefore your time is not dilated. So from your point of view, the problem remains that the sizeable proportion of you and your colleagues who survive the journey from Cormorant’s height to the beach seems to imply that you’re travelling faster than light. Now while muons may quite like the “live fast, die young” image, they know in their heart of hearts that they can’t break the fundamental laws of the universe. So what’s going on here? Could it be that the relative movement of the different frames of reference affects not just time but distance?

Question 2-11

If you worked through this question, you should have derived this equation:

II-ii

This is called the length contraction equation: it describes quantitatively how the length of something, if it is measured by a moving observer, is contracted compared to the length measured in the rest frame of that thing.

Note that, so far, we have only considered lengths in the direction of travel of the moving reference frame. We will consider what happens in other directions a bit later. So the meaning of this equation can be stated as: a length will be contracted in the direction of motion, in the reference frame of a moving observer compared to a static observer.

Having figured it out, you are about to explain it to your playmates while sharing some doughnuts, when Cormorant takes a look at you and blurts out: “you don’t look as slim as you did earlier when I was flying past. Just saying.” Nefertiti disagrees. .....

Question 2-12

So special relativity means that it's not just time that's affected by the movement of an observer, it's distance too. With this in mind, it's worth briefly revisiting our starting point, at the beginning of this chapter:

Question 2-13

As with time dilation, this effect is normally far too small to see. You can see that by looking at equation II-ii, above: unless u is approaching the speed of light, √(1 – u2/c2) is very close to 1, in which case, for practical purposes, D0 = D1.

Next you’re going to start to wonder: do they shrink in all directions, or just in the direction of travel? Nefertiti proposes an experiment. She will mount a sharp stick at a fixed height on her bike. With you and the bike both stationary, she will prod you with the stick, leaving a neat bruise on your torso. Then, having marked this as bruise number 1, you’ll repeat the experiment but this time it will be a ride-by prodding. You’ll stay where you are but this time, Nefertiti will get you as she is riding past at high speed. Let’s explore what happens:

Question 2-14

The Gedankenexperiment in Question 2-14 has, hopefully, convinced you that length contraction cannot happen in the perpendicular direction. It occurs only in the direction of relative motion.

This is why, in our original time dilation investigation (figures II-v and II-vi), our setup was designed to make sure that relative movement of the reference frames was perpendicular to the direction of flight of the sardines / light pulse, so that only time would be affected by the motion and not the length of the flight path.

At this point we have established the basics of what happens to time and distance in special relativity. In the next chapter we’ll use these ideas to explore the implications for momentum and kinetic energy.

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