My Favorite Theory

Scientific theories are cool and complex beasts: you observe some data or think about previous results, formulate an idea about how the world behaves, and then test out how well that idea holds up in the presence of more observation and further development.

My favorite theory is, hands-down, Theory of Special Relativity. It’s not my favorite because of its far-outside-common-experience implications, or because of mathematical obscurities, or because of its attachment to the great celebrity-scientist Albert Einstein, or even because of its amazing practical applications (like nuclear power and lasers and GPS) – cool though all those things may be. It’s my favorite theory because it springs from just a couple simple ideas, and I can derive its wild and wonky implications straight from those ideas using nothing more than basic geometry. It’s a testament to the power of the “thought experiment,” and a wonderful demonstration of how a few brilliant ideas can lead to extraordinary outcomes!

The Theory of Special Relativity basically boils down to just two postulates:

  1. The laws of physics are the same in all non-accelerating reference frames.
  2. The speed of light is the same when measured in any reference frame.

That’s it! Now, check this out: I’m going to derive in a few lines the famous relativistic effect known as time dilation.

Suppose I go screaming by you in a cartoon rocket ship while you stand bewildered on the ground. My rocket’s velocity v is a substantial fraction of the speed of light c. Because it’s just that awesome.

Whoosh!

Inside my rocket ship I have a special type of clock. It’s like a pendulum clock, but it works with light. In a pendulum clock, the pendulum swings through a certain arc in a certain time. In my clock, a laser bounces a pulse of light off a mirror a certain well-measured distance away, and a detector right next to the laser picks up the light pulse. (The laser and detector are so close together that the light basically retraces its steps back from the mirror.) A timer hooked up to the whole thing tracks the amount of time between the laser firing and the detector registering the light.

My light clock

Here’s a closer look at the clock and how it works:

I know that the distance from the laser to the mirror is d, so the beam has to travel a distance 2d every time it fires. I also know that the speed of light is c, so the total time the light beam takes to travel this distance is t’ = 2d/c. That’s one tick of the clock, as I measure it.

Now suppose the clock is right near the porthole on my rocket ship, so that you can see it as I whiz past. You see the entire rocket traveling with speed v to the right, so in a time t the rocket moves a distance vt. And you see the light beam travel along a slightly different path than I do:

Your view of the clock

Why do you see the light travel along this angled path? Why, the first postulate of Special Relativity is the reason! The laws of physics have to be the same for both of us. I look at the laser and see that it has zero horizontal velocity (because we’re both standing on the cartoon rocket deck), so the beam just goes straight up and down. But you look at the laser and see it zooming along with horizontal velocity v, so the light the laser shoots out picks up that additional velocity.

Let’s look at that beam path carefully for a minute, and add some math – don’t worry, nothing too scary! Just the Pythagorean Theorem, to figure out the distance the light beam had to travel.

Okie-dokie. Sounds great, but here’s the thing: Special Relativity Postulate #2 says that the speed of light in vacuum is constant as measured by all observers. So how long do you measure it takes the laser beam to travel this path?

Now, hold on here – I measured one tick of the clock as t’ = 2d/c. You measure it as t’ = 2d/sqrt(c2v2). But because of postulate #1, we know that we are describing the same physics! Let me write t in terms of t’ for comparison.

This gamma quantity is kind of a funny thing, and it shows up all over relativity. Since v always has to be smaller than c, then (1 – v2/c2) is always less than one and gamma = 1/sqrt(1 – v2/c2) is always greater than one. That little fact means that t will always be greater than t’. That’s relativistic time dilation! Put in simple terms, if I am traveling very fast with respect to you, then the time of one tick of my clock seems longer for you than it does for me. In fact, since gamma depends on my rocket ship’s velocity, the effect gets more and more pronounced the faster I go, getting towards infinity as I get closer to the speed of light:

Gamma as a function of speed

It’s not just my light clock that gets stretched out in this way. All clocks, and all processes that involve time are subject to time dilation! (You could figure that out by the same method that I just did, if you carefully track the paths of light beams.)

Eventually, what I think is one second on my clock will be a year according to you. If I go faster still, I could get one second on my clock to be a century or a millennium to you! This phenomenon is one reason why we know that nothing can travel faster than light: because if my rocket ship could go at light speed, then time dilation would stretch things out such that (according to you) an infinite amount of time would elapse if I go anywhere!

The coolest thing about all this, to me, is that Einstein came up with these ideas through careful consideration of “thought experiments:” what if we could ride along with light beams? what if I zoom by you on a relativistic rocket? He formulated his postulates carefully, and he fleshed out their implications carefully – but the derivations themselves are wonderfully simple and easy to follow. The physics that result, though…crazy!

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