Special Relativity: Starring Trains

Back by popular demand... another physicsy post! Well, the truth be told, the only person who requested another physics post was... well, me. But this post is exciting because it stars the classic subject of many a dear first year physics problem: trains! Yay trains :)

Let's say that you're a train, and you're having a really nice time with your tank engine friend, Thomas. You might think to yourself, I'd like to have more time with Thomas. And as a train, you've likely heard of special relativity, so you turn to Einstein to ask how to stretch out time.

Einstein will likely tell you to get moving. In fact, he'll say you should move so fast you're almost travelling at the speed of light. Why is this? Well, let's start with a seemingly unrelated experiment, and then we'll see what this means for you spending more time with Thomas.

The build up:
Muons are little particles who don't live very long. We can study muons in the lab and find that they only live for around 2.2 microseconds. So even if muons were travelling even as fast as light, they couldn't possibly travel further than 600 m before they die. But... muons are produced in the upper atmosphere (10 000m up), and some are still able to make it all the way to the earth's surface! How is this possible?

Einstein predicted incredible experiments like this one (even before anyone had tried it!) with his theory of special relativity. One of the things I love about special relativity is it's simple basis - just two postulates:

1.  Any reference frame (aka. perspective) is a good one as long as it's not an accelerating reference frame. Physics works on a moving train. Physics also works on the train station platform. To clarify this point: imagine you are in outerspace (you're a space-train perhaps). If you see another space-train getting closer, you can't tell if you're moving towards it or if it's moving towards you. And Einstein's point here is that it doesn't matter - both of you have equally valid reference frames. 

2. The speed of light is constant no matter what reference frame you observe it from. In our example above, the other space-train moved at a perhaps 120 km/h in your reference frame, but moved at 0 km/h in its own reference frame. Light isn't like a train: in every good reference frame, it travels 3x10^8 m/s (in a vacuum). 

A classic way of understanding some of the impacts of these two postulates is through the thought-experiement of a "light clock".  The clock in my grandma's living room ticks every time a pendulum completes a swing back and forth. This light clock we've imagined ticks every time a pulse of light travels from a bottom laser to a top mirror, bounces off the mirror and hits a receiver on the bottom.


Reference Frame A: Clock is not moving relative to you:
If you are standing beside the clock (not moving relative to the clock), then you observe the light pulse to travel straight up and down. Here's a quick sketch of the situation when you are in the same reference frame as the clock:

Let's do a quick calculation to see how long a "tick" is on this clock. To make the math easier, we'll pretend light travels pretty slow - say 1 m/s. And the height of the clock is 1m. So it takes 1s for the pulse to travel to the mirror and another second to get back down: a tick happens every 2 seconds.

Reference Frame B: Clock is moving relative to you:
Now let's say this clock is stuck to the side of a moving train, and you're standing on the platform watching the clock move past. Now, from your perspective, the pulse will appear to travel on an angled further distance to the mirror, as sketched here:

The key is that it's not some illusion you're seeing. Einstein's first postulate tells you that your reference frame is equally valid. So in this reference frame, you'll see the pulse travel further (say 1.5m to the mirror and another 1.5m down), so you calculate the time for light to go from the laser to the receiver (pretending that light travels 1 m/s) will be 3 seconds.

Wait! What is going on? When we looked at the clock from Reference Frame A, we observed it ticking every 2 seconds, but when we looked at the same clock from Reference Frame B, it ticked every 3 seconds! 


What we just happened upon is the first exciting consequence of Einstein's two postulates: time is relative! It ticks slower or faster depending on your perspective. But then, every five year old already knows that - when you are waiting for an ice cream cone, time ticks so slowly, but when you're eating the ice cream, time speeds up and the ice cream disappears so so fast. Okay, so that's not exactly special relativity. Trains make a more accurate example: From the perspective of a person who is standing "still" on a platform (we know now from postulate #1 that "still" is a very arbitrary word), the events on a nearby moving train are happening more slowly than the events happening on the "still" train station platform. 


So, if you want to have more time with Thomas the tank engine, you should drive your train as fast as you can... right?

The let down:
Unfortunately, succeeding in stretching out time hasn't actually given you what you'd hoped for. When time dilates, every clock - your heart beat, the pace of your conversation, even the rate at which you age - all these things "tick" in step with the new time. So unfortunately, from your perspective, it doesn't feel like you have any extra time with Thomas after all.

However, when you're chatting with Henry the green engine later on, he'll tell you that he thinks you spent quite a lot of time with Thomas. From the perspective of Henry, who is not moving inside his tunnel, he sees you and Thomas moving quite quickly, so he sees time tick slower for you. And he's rather green with envy.

The rest:
Time isn't the only thing that's relative! As we saw from our muon experiment, distance is also relative. As you can guess, the relativity of distance is very related to the relativity of time. And mass is relative too. Mass is a big deal right now with all the higgs celebrations. But for now, I'll pause this post to ask the most important next question: does the higgs boson like hugs?