Thursday, April 10, 2008

Why is Relativity Necessary?

Anyone who reads science fiction probably knows that something strange happens when objects move at high speeds relative to each other. A space traveller can take a one year journey in a high speed ship, only to return and find out that hundreds of years have passed for the rest of humanity. More realistically, one of two synchronized clocks can be taken up into a fast moving plane and, on its return, found to read earlier than the one that remained stationary.

How can we possibly believe that time slows down for objects/beings moving with respect to us? Even stranger, to the moving being, WE are the ones moving, and they would view our time as slowed. Well, to my mind, all this strangeness springs quite naturally from a single strange starting point: the speed of light is constant.

Imagine I clap my hands, and you can measure the speed of the soundwave as it passes you by. If you were to turn and run away from me at 5m/s, then the speed you measure would of course be reduced by exactly 5m/s. Moving away from the source of the sound seems to slow the speed of the sound wave. Well, with light waves, this is not the case! Say I flick on a lightbulb instead of clapping my hands, and you measure the speed of the passing light wave using a stopwatch and a meter stick (no comments on the quick reflexes required please!) You could stand still, run away at 5m/s or fly away in a jet at half light speed. In all cases, you'll measure the speed of light to be exactly the same! (this of course is straightforward - if not easy - to verify experimentally)

Here's an example to show how this ruins our intuition about time. Imagine the bulb is flicked on, and we have two observers, one (call him A) is stationary with respect to the bulb, and the other (call him B) is flying away at half light speed. By the time A sees the wave travel 1km from the bulb, B has of course travelled half a km, so for B the wave has only moved half of one km. Yet, when each measures the speed of the wave, they get the same result, though B only saw it move half as far!

Something has to give in this contradictory scenario, and it turns out to be time (actually, distance as well, but let's not get too complicated...). The reason A and B can agree on the speed of light, but disagree on it's distance travelled is that they also disagree on how much time has passed.

Does this make anyone else's heart beat quickly?!

It turns out that some creative high school trigonometry can give you the exact formula to determine at what rate B's clock is moving as observed by A (slower), but as the textbooks say, I'll leave that as an excersize :)


Rachael said...

Does B start moving at the same time the bulb is turned on? If she does, then by the time she travels half a km shouldn't the light have passed her and extend another 1/2 km ahead of her?

So B should agree with A, that the wave has traveled 1km for each of them although B's place relative to the wave is different than A's. This all seems perfectly usual.

What doesn't seem usual -- is that I think what you're saying is that it doesn't matter how fast your moving or where you are when you measure the speed of light, you'll always measure the same speed *without* correcting for your own speed and location relative to the starting point. This is extremely weird.

We'd expect measures of the speed of sound to be identical if we made these corrections, but we'd be quite surprised if we made identical measurements without the corrections!

I mean this all as a big question that I'm not sure how to ask in fewer words.

Chris said...

The weird thing about relativity is that everyone's "own speed" is zero! I could equally well have set up the problem with A (holding the light bulb) passing B in space at relative speeds of half light speed. Both are gliding along free of gravity, so both A and B feels that they are motionless. A flicks on the buld the moment they pass one another. Which one has to do the correcting now?

The answer is neither, but still, if they both hold out a meter stick and time the light pulse as it passes them, they will both find that the speed reads the same, with neither one correcting for any speed!

"The speed of light is constant" means NEVER having to correct for one's own speed when measuring, and still finding the same value.

Chris said...

If that seems crazy, it is! But never the less, it is true.