Light Rail

I have returned from my trip; big thanks to everyone who hosted me, and showed me a wonderful time.  I have a couple post ideas kicking around from my travels, but I thought I'd take a minute to give a quick primer on the special relativity issues I mentioned last time.

Special relativity rests on the idea that the speed of light is constant.  This may not seem like such a significant statement, but you have to consider that it means the speed is always constant.  Normally if you were chasing after something, all you need to do is go faster than it, and eventually you'd catch up.  However, if you tried that with light, no matter how fast you went, its speed would always appear to be the same amount faster than you.  This leads to some interesting effects.

Imagine we construct a clock that uses light as its timekeeper – it contains a hollow tube with a beam of light that travels down it, bouncing off each end; each bounce is a tick.  It might look something like this:

 

Now suppose we strap this clock to a rocket, and watch as it flies by.  It would look something like this:

 

The light is still going at the same speed, but it has to travel a greater distance, so the tick that we measured earlier has been made longer by the movement.  This is the relativistic effect known as time dilation.  Most other interesting effects follow from similar thought experiments.

One of the important restrictions of special relativity is that it only applies in an inertial frame, that is, a perspective which is moving at a constant velocity.  Technically, it doesn't quite apply to the situation I talked about last time, since my train had to accelerate up to speed, then slow down again, but we can adapt it as long as we're careful.

While on the train, clocks outside will appear to be running slower than my own, but since I accelerate before and after such observations, they are potentially invalid.  However, someone outside looking at my clock would be correct in noting that it appears slower than theirs.  Since the person outside stays still during my entire trip, their observations must be correct.  If the train takes a time t according to the stationary person, then they will see a time t/γ pass on my clock.  Plugging in the numbers from last time gives the 95 picosecond difference I mentioned.