Don’t Stop Believing (in Special Relativity)

The study of physics is pretty well known for being hidden behind expensive college courses, fancy experiments, and intentionally over-complicated jargon. But the mysteries of the universe belong to you just as much as to any tenured professor at whatever ivy league institution. In fact, there’s an entire category of experiment that we can do right now, without any advanced math or funding from the National Science Foundation. These “thought experiments” are great for getting an intuitive feel for some of the most bizarre theories we’ve ever come up with.

I am hoping to write a series of these, tentatively titled “Alice and Bob, Demons and Cats,” where I will summarize some of my favorite thought experiments. Some of them might be a bit longer than others… To start off, let’s take a look at Albert’s favorite train. This one’s got it all, rail travel, lightning, angst, and time paradoxes. Let’s get into it:

Alice was just a small town girl, living in a lonely world. Just how lonely, however, she had yet to learn. So she took a midnight train going anywhere. Bob was a city boy, born and raised in south Detroit. He, however, did not take a midnight train, but decided to stand by the tracks and watch them roll by on a stormy evening. Maybe one day he would go, but not tonight…

Albert Einstein, who was clearly the inspiration for Journey’s greatest hit, spent lots of time imagining something like this. But his goal was not late night karaoke, he instead aimed to dramatically change the concept of time using a simple thought experiment. But before we take a look at Alice, Bob, and the train, we should choose what assumptions we are going to make about how the universe works.

One of the amazing things about special relativity, the theory that explains this experiment, is that it all comes from two simple assumptions. They are:

  1. Everyone has the same laws of physics, no matter how fast they happen to be going.*
  2. The speed of light always appears the same to you, no matter who you are or how fast you’re going in any direction. That speed is usually written as “c” for some reason.

*Unless you are accelerating, either speeding up or slowing down. Turns our acceleration requires a different, “generalized” theory of relativity, creatively named “general relativity”. That one definitely deserves its own post.

The first one is a pretty safe assumption, and it makes sense. One of the goals of physics is to have general laws that describe things all across the universe under the same umbrella. I should be able to use the same laws of physics in Boston as you see wherever you are when you read this. If that one isn’t true, then what are we even doing? Who would know anything? It would be chaos! Total chaos! Let’s sidestep all that and say that 1. Is true.

The second one is a bit weirder, and it’s what makes all the difference. Albert wrote about what it would be like to chase a beam of light, asking what it would look like? If you chase ordinary things like a train or bus that you just missed, it looks to you like they slow down. If you pass cars on the highway, they look to you like they are moving backwards, when you actually know that you’re both moving fast, you’re just going a bit faster. Albert is suggesting that if we chase a beam of light, no matter how fast you chase it, it still looks like it’s moving at the speed of light. You could never pass a beam of light on the highway, no matter how much you speed up. The point here is that light doesn’t work like normal physical objects (your buses and trains and all). A bit strange, yes, but it has been tested many times. In fact, it was first proven in a famous experiment, generally known as the most famous “failed” experiment of all time. So for our purposes here, let’s accept this strange little idea and see where it takes us.

So now that we have the assumptions, what’s the setup for this thought experiment? Let’s go back to our slightly modified Don’t Stop Believing scenario for a moment. We have Alice zooming along on the train, and Bob hanging out by the train tracks on a stormy night. At the exact moment Alice passes Bob, he sees a flash of lightning, illuminating her face behind the train’s window! It looks to him like lightning struck the ground a ways down the tracks (in the direction the train was coming from) and the light from the strike reached both him and Alice at the exact moment that Alice passed him. This situation is illustrated in the first figure below. Very dramatic, Bob thinks, I wonder if Alice saw that striking (hah) coincidence of timing! What are the odds? Not good, as it turns out.

Picture 1: Bob’s perspective as he sees the light from the lightning flash (made up of little particles of light called photons) illuminate both him and Alice through the train window at the same moment.
Image 2: The moment when the lightning first strikes, though neither Bob nor Alice have seen it yet! The lightning photons (little bits of light) have just started to travel from the bolt of lightning towards Alice and Bob. They’re both still in the dark.

We know that the light took some time to travel from the place it struck to reach Bob, so let’s rewind to the moment exact moment the lightning hit the ground, before it has had a chance to travel to either Alice or Bob. This situation is illustrated in the second picture. At the moment the lightning strikes, Alice’s position on the train is closer to the spot where the lightning hits. So now we have rays of light traveling from the charred crater on the ground towards both Alice and Bob (well, in all directions, but we mostly care about Alice and Bob). Let’s think about when Alice would see the flash of light. To do that, we have to put ourselves in her shoes, and her shoes are traveling very fast on the train. So let’s move to Alice’s “frame of reference” of the train.

Picture 3: Alice’s perspective as she sees the flash of lightning, and it illuminates her face. Strangely, this happens before she has passed Bob’s place on the embankment, because she started closer to the light, and she sees it travel the same speed relative to Bob as it does relative to her. Alice is illuminated, but Bob is not.

This is where we have to remember our weird assumption about the speed of light: it appears the same to all observers, regardless of how fast they’re going themselves. So we have light traveling at the same speed towards two different observers, one of whom is closer. The light has to reach Alice first, right? Well it does! At least from Alice’s perspective (Illustrated in the third figure). She sees light moving towards both her and Bob at the same speed. She is closer, so the light reaches her first, then it reaches Bob later.  

But let’s go back to thinking about Bob’s perspective. He sees the light traveling towards them both at the same speed. Alice is initially closer to the blast, but she is moving towards him. To Bob, it looks like Alice is able to outrun the light (we know this isn’t how it can look from Alice’s point of view though, so it doesn’t violate assumption 2). That’s why he sees the light reach them both at the same time, exactly when they pass each other (look again at the first picture)! So we have a troubling conflict in our timeline: Bob sees the light reach them both at the same time, but Alice sees the light reach her first? Who is right?

Nobody is right. Or Alice and Bob are both right. We’ve just re-discovered one of the many troubling paradoxical predictions of Special Relativity, which is generally called the “relativity of simultaneity.” We can reconcile this apparent logical paradox by giving up on our understanding of different events being simultaneous. It turns out that the simultaneity of different events of any kind is completely dependent on how fast you are going. Bob can see two things happen at the same time, while Alice sees them happen at different times. Time, as it turns out, is entirely relative. The way you experience of the passage of time and the relationship between consecutive events is unique to your perspective. It’s a lonely thought, or a very cool one, probably both.

But don’t get too spooked, you can’t even get close to observing this in your everyday life. Turns out that the speeds we travel at in our real lives are so small relative to the speed of light that phenomena like this one are unrecognizably small. If you wanted to actually test this scenario in real life, Alice’s train would have to be traveling way faster than is possible with any vehicle humans have ever built. Alternatively, you could test relativity with less inspiration from Journey and use a type of precision measuring technique interferometry, which is how the theory has been tested in reality. I should also be writing a post soon about interferometry in general, because I think it’s one of the most amazing types of measurement we’ve ever come up with.

So this is just one of the many strange thought experiments we can do with special relativity to unravel the concept of time. Stay tuned for the next post in this series, about another of my favorite thought experiments: the twin paradox.

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