What we will explain is why, for objects moving uniformly in a straight line, time runs slower. We’ll use no mathematical symbols, and won’t even need any pictures!
OK, here goes:
Before we begin, you have to accept two facts. The first is that if you are in uniform motion, all the laws of physics appear exactly the same as if you were standing still. Think about it: if you are in a smoothly moving airplane, you can pour coffee into a cup, etc., just as if you were sitting on the runway. In short: if you are in a windowless box, unless you are accelerating, going around a curve (which is the same thing), etc., you can’t tell if you’re moving or standing still. Agreed? This is called the Principle of Relativity. The first person to point it out (that we know of) was Galileo.
The second fact is that the speed of light through space is constant, for all observers. This is just a curious fact of the world, but it’s been shown to be true. If I’m standing still, and a beam of light whizzes past me, and you take off after it at a million miles an hour, we are still both going to measure the speed of the receding beam at about 186,282 miles per second — a value known as c. I won’t belabor this part — but, for example, if the speed of light weren’t constant, then light coming toward Earth from a faraway planet when it’s moving toward us in its orbit would travel toward us faster than light shining from the planet when it’s moving away, and we’d see everything all out of sync. For that not to be true, there’d have to be some medium (like air for sound waves) that held the velocity constant. But we know there isn’t, thanks to the Michelson-Morley experiment of 1887. Everybody, no matter how they are moving, always sees light moving at the same speed. That’s just the way it is.
The rest is really easy.
Imagine a special kind of clock. It consists of a mirror on the floor, a mirror on the ceiling, and a pulse of light that bounces endlessly back and forth between them. Call the time this takes one “tick”. Because the speed of light is constant, the length of the ticks are always exactly the same.
Now imagine mounting one of these in a railway car. Let’s also put a very accurate conventional clock on the wall of the car. They should run nicely in sync, because, after all, they are both very accurate clocks.
Now set the train in motion, whizzing by as you stand on the platform. (We’ll give the railway car a glass wall, so we can see in.) The train runs very smoothly and steadily. Inside the train is an observer (let’s call him Al) with his back to the window. The train runs so smoothly, he can’t even tell for sure if he’s moving at all.
For Al, each tick of the clock takes a constant time, which is the height of the car (i.e., the distance between the mirrors), divided by the speed of light.
But things are different for you, standing on the platform, watching the train go by. By the time the light travels the height of the car, the bottom mirror has moved down the track a little way (along with the rest of the train). And by the time it bounces back to the ceiling, the train has moved the same distance, again. This means that, as far as you are concerned, the light didn’t go straight up and down, but made a sort of zigzag down the track, which is a longer path. (To be specific, the length of each bounce is the hypotenuse of a right triangle that has the height of the car as one leg, and the distance traveled by the train as the other.)
Are we clear about this so far? Can you picture it?
What this means, then — because as far as you can see, the distance traveled by the light is longer on the moving train, and because the speed of light is always constant — is that each tick of the clock, as seen by you on the platform, takes longer than it does as seen by Al on the train. (Remember that Al doesn’t even know if he’s really moving or not, and so for him the light just seems to be bouncing straight up and down.) And if you think that this is just some jiggery-pokery involving weird light-clocks designed just to give this effect, remember that if the light-clock were to get out of sync with the wall clock, Al would notice, and would therefore be able to tell he was moving! But we already know he can’t possibly be able to tell in any way whether he’s moving or not, thanks to the Principle of Relativity.
So there it is: since time is what clocks measure, and since clocks tick slower on a moving train, then time runs slower on a moving train! Simple, right?
If you want to figure out just how much slower, it’s easy; it’s all just a bunch of right triangles, and a little algebra. You can do it. (Maybe I’ll do it for you in another post, but it’s really not hard at all.)
Last thing: that train gets shorter, too. If you want to understand that, just turn the clock so it’s lined up lengthwise down the train. I’ll leave that one to you.
So: want to have some fun? Teach a little kid how Special Relativity works. You’ll both feel great.