Measured Against Reality

Tuesday, May 29, 2007

Why Special Relativity is Obvious

I’ve been meaning to write this post about why special relativity is obvious for a while, and since the new Microsoft Word comes with equation writers, I’ve got an excuse to finally do it.

Special Relativity is a favorite whipping boy among cranks. You’ll see attempts to disprove it all the time (I got one in my inbox a few weeks ago). It also has a reputation as being difficult to understand. I hope to address both of those with this post.

I was in my electrodynamics lecture, and the professor was going through the derivation of wave solutions to Maxwell’s equations in a vacuum (if you’re interested, the derivation is here, with H in place of B, for some weird reason). The final product is below (which is where Word’s new equations came in handy).



Which is immediately recognizable as a wave traveling with speed c. This is pretty easily derived for waves in matter too (with D and H replacing E and B respectively. D and H are simply the electric and magnetic fields in matter, and for most materials end up being just a factor different, which leads to a different propagation velocity).

So what? Everyone knows that electromagnetic waves exist. And what does this have to do with relativity?

Well, for those not familiar with it, special relativity has two postulates, the rest of the theory rest on their back. The first is that there exist inertial reference frames, and the laws of physics are the same in all inertial frames (a frame is more or less just a coordinate system with a clock (so you can measure position and time), and an inertial frame is defined as one where the laws of physics hold, so the second part is something of a tautology). This postulate is hard to argue with, you’d essentially be saying that physics is invalid.

So if a frame is inertial (like say, the one you’re in while you’re sitting at your computer reading this), then any frame moving with a constant velocity relative to it is also an inertial frame. How do we know that? Because if you’re moving with constant velocity in a closed train (meaning you can’t see the outside), there’s no way to tell that you’re moving. You might not believe this at first, but most of your cues that you’re moving come from tiny accelerations, you really can’t tell when you’re moving with constant velocity; it’s impossible, which means that it’s an inertial frame.

The second postulate is more interesting, and that is that the speed of light is the same in all inertial frames. But this is obvious: I just showed above that Maxwell’s equations have waves that propagate with speed c, and since Maxwell’s equations are a law of physics they’re true in all frames. That means that an observer in any inertial frame must see light moving at speed c, no matter how they’re moving relative to anything else.

I think we should really speak of the postulate of relativity, because the second comes naturally from the first, and it might clear up why this wacky theory has to be correct (and is totally obvious).

Of course, the consequences of this aren’t obvious at all: Time Dilation, Lorentz Contraction, the relativity of simultaneity, the weird addition of velocities, and everyone’s favorite: the equivalence of mass and energy. Those are all truly bizarre consequences, but they have to be true, and they’re Einstein’s brilliant insight. I’m not going in to them now, the Wikipedia articles should satisfy the curious , but they have to be true given the two postulates.

Or, at least, they have to be true if Maxwell’s equations and the rest of physics is correct. If Maxwell were wrong pretty much no modern technology would work, rendering that highly implausible. So unless you think that inertial frames don’t exist, you must think that special relativity is correct. I hope I’ve made it clear exactly why this is true.

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3 Comments:

  • "with H in place of B, for some weird reason"?

    And you passed e&m?

    By Anonymous Anonymous, at 11:12 PM, May 29, 2007  

  • Anonymous, it's weird because they're mixing E and H. Every E&M book I have ever seen uses E and B to be the fields in vacuum and D and H to be in matter. Mixing them is never really done. I realize that H's ties to free current make it more useful in certain applications, but I've never seen them mixed before. That's after reading both Griffiths and Purcell, which are pretty much the textbooks for undergrad E&M.

    For example, Griffiths's Maxwell's equations in matter use all four, but only E&B and D&H appear together, E and H are never mixed.

    So now I ask, you passed E&M?

    By Blogger Stupac2, at 11:22 PM, May 29, 2007  

  • More interesting is how magnetism collapses into "electrostatics" when you put relativity into the equations - i.e. magnetism is just a relativistic correction applied to electrostatics.

    By Anonymous Anonymous, at 9:18 PM, May 30, 2007  

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