Minggu, 21 Juli 2013

Relativity with Pictures

I don't like relativity. It's not my territory, and I'm not that comfortable with it. The last time I wrote about relativity was when I did e=mc-squared back here But some questions came up last week, and I've been working on it. Maybe you know I don't like working with equations; I need to be able to draw pictures. So here's what I've come up with.

We like to work in a one-dimensional universe...that is, just one spatial dimension and the time dimension. Here is a graph of our space, with units chosen so that the speed of light (shown as the blue arrow) is one:

The first thing we do is we put a train into the picture, and we start with a train standing still. It looks like this:

Now we ask the question: what if the train is moving? Let's give it something like one quarter the speed of light. Then obviously the world-line should look something like this:

The horrifying thing about relativity is that this picture is completely wrong. It's actually correct for what we call "Gallilean Relativity". But the universe we live in turns out to be different. Here is what the world-line looks like for an actual moving train:

Yes, this is it. There are a number of disturbing observations we need to make. First of all, notice that I haven't just drawn in the world-lines representing the front-end and the back-end of the train. I've drawn in the whole train to show it moving along. And I've tilted the moving train with respect to the stationary train. What does this mean?

The x-t axis is drawn from the perspective of a stationary observer, who could be inside or outside the stationary train. Points along the t-axis are stationary points: that is, they represent the same position at different times. But the other side of the coin is that points along the x-axis represent simultaneous points: that is, they pare points at the same time but at different locations. It goes both ways.

What Relativity says is that physics must have the same laws whether you are a moving observer or a stationary observer. So now we have to consider what the world looks like for the man inside the moving train. (Although I've drawn four cars, we should really think of it as one long tunnel-like car, where you can see clear from on end to another, but you can't see outside. Relativity asks what happens if the man inside doesn't know if he's stationary or if he's moving.

For this observer, the back end of the train (and the front end) are both stationary points. But we already figured out for a stationary observer that the world-line of a stationary point runs parallel to the t-axis. So for the man inside the train, the time axis must follow the world-line from the back of the train. It is tilted.

There is nothing so unusual about this: it is exactly the same thing in traditional Gallilean relativity. It doesn't mean time is means points along the time-axis are stationary points. It's very straightforward.

But I've drawn the train as being tilted with respect to the x-axis. This is something entirely different and it is brand-new with relativity. It says that in addition to the time-axis being tilted, the space axis is also tilted. But we already figured out that for a stationary observer, points along a line parallel to the space axis are simultaneous points. So if the space axis really is tilted, then it tells us that the way I have drawn, from the point of view of the man inside the train, simultaneous "snapshots" of the location of the whole train.

We're soon going to see why it has to be this way; but before we do, there's just a couple more important observations to make about my drawing. If you look carefully, you'll see that the world-lines of the front end of the train do not intersect along the x-axis, but in fact just a little bit to the right. This is not an accident: I've drawn it that way on purpose. And furthermore, if you look at the intersection of those world-lines, neither is it simultaneous with the back of the train passing through the origin. Although you can't see it from my drawing, the location of the front of the train at the moment the back end passes through the origin is actually above the horizontal world line by approxamately the same amount as it is below the world line when it crosses the x-axis - something like this:

The time axis has shifted...that already happened in Gallilean relativity, and it just meant that position is relative. But here the space-axis also shifts...and it means that simultaneity is relative. In fact, we will eventually see that the space axis tilts exactly the same amount as a the time axis - at least, it does when we choose our units so that the speed of light is unity (c=1).

I've told you as best I can what the world looks like in relativity. When we return, I'll try to explain why it looks that way.

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