Special Relativity
© 6 Nov 2012 Luther Tychonievich
Licensed under Creative Commons: CC BY-NC-ND 3.0
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How velocity can be relative and have a maximum value.


I wrote a couple months ago about general relativity, or at least how it models gravity. The other half of that theory is special relativity, or how it models speed.

For whatever reason, relativity almost always plots space and time “‍sideways‍” like this:

If one person was walking a one meter per second (a relaxed walking pace) to the right and another was ambling at half that speed the other direction, we might plot their positions like this:

Notice that the slopes of the lines correspond to speed: shallower lines represent faster-moving objects than do steeper lines.

The first part of classical relativity we’ll consider is the relativity of coordinate systems. The origin was arbitrary; we could just as easily have plotted these two people like this:

Because the origin doesn’t matter, we’ll usually leave off the tick marks and draw the plots passing through the origin for simplicity.

The next part of classical relativity states that velocity is also relative. We can choose any arbitrary velocity and call it “‍still‍”, adjusting other velocities to ensure that distances remain the same.

In normal life we usually call the nearest part of the surface of the earth “‍still,‍” which makes sense since it is the biggest thing around.

One way to think about the classical relativity of velocity is that just like we can move the whole plot around as long as we move all of it the same amount, so too we can move all the velocities around so long as well move all of them the same amount along a horizontal line.

That concludes my introduction to classical relativity.

Special relativity picks up where classical left off by first postulating “‍there is a maximum speed.‍” A maximum speed doesn’t even make sense in classical relativity: speeds are slopes and you can shift things to make slopes as close to flat (i.e., as fast) as you want. And classical relativity seems to work in daily life. How can there be a maximum speed?

Einstein’s solution to this was to take that line that you can move velocities around on and replace it with a hyperbola instead.

The depicted curve is not quite a hyperbola; from here on out the images are only approximately correct.

Hyperbolas have the nice feature that their steepest slope is bounded. Pick a hyperbola whose steepest slope is 299,792,458 meters per second and we get to the constant (and maximal) speed of light we observe.

So what does this mean happens to speeds as we slide them around the hyperbola? For speeds not too close to the speed of light, it looks much like classical relativity:

But as speeds get closer to the speed of light two speeds that are just as far apart on the hyperbola have very similar-looking slopes (and hence speeds) to us.

That is about all there is to special relativity, besides a little bit of math to define distance-along-the-hyperbola in a more correct way than the Euclidean distance I used in my pictures. But from this simple idea comes… a lot. Once you accept this model of velocity you find that distance and duration are both relative to the speed of the observer. Perhaps I’ll go into how some of these come about in a later post.

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