Part I – From Pythagoras to Einstein
Several times over the years, I’ve tried to convince people that Pythagoras’ Theorem was one of the most profound things that they learned at school. If I’m honest, I don’t think I’ve ever succeeded in convincing anybody. But, ever the optimist, I thought I’d try again, this time on my blog.
Let’s start with a statement of Pythagoras’ Theorem: the square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides. The hypotenuse is just the longest side, the one opposite the right angle. You’re probably so familiar with this that it doesn’t strike you as being at all remarkable.
In the diagram: r² = a² + b². There are dozens of proofs of this. I give a very simple one in the comments.
There are a number of important properties of this that are worth pointing out. First, no matter how you shift or rotate the triangle, the length of the hypotenuse remains the same. As of course it must. Mathematicians say that the result is invariant under translations and rotations.
Second when you rotate the triangle in a cartesian coordinate system, a measurement that was a pure “x” value before, becomes a mixture of x and y values afterwards. Coordinates can get mixed up, even though the structure of the physical object remains the same.
Extending this to three dimensions is trivial. We just take the 2d triangle result and add a third dimension and apply exactly the same rule again. So now r² = a² + b² + c².
Einstein realised that, in order to consistently understand experiments in optics and the theory of electromagnetism, it wasn’t enough to specify a position. You had to include WHEN something happened, as well as WHERE. But it was his old maths professor, Hermann Minkowski, who realised that this involved a geometrical interpretation. It was this geometrical interpretation that set Einstein off on a geometrical theory of gravity, resulting many years later in the General Theory of Relativity.
So how do you measure a “distance” in spacetime? Simple, you just extend Pythagoras’ Theorem from three dimensions to four. But there are two fascinating twists.
The first is that you can’t just add a time to a length.
3 secs + 4 metres makes no sense.
You have to convert a time into a length by multiplying it by a speed first.
3 secs x 2 metres per sec + 4 metres = 6 metres + 4 metres.
So we need a speed to multiply all our times by. Let’s give this speed a symbol. Let’s choose “c”. And let’s give it a name. Let’s call it “the speed of light”.
The second twist is even more important though. We don’t add time, we SUBTRACT it. So our formula for measuring distance in spacetime, traditionally given the symbol “s”, is:
s² = x² + y² + z² – c²t²
How do we know this? Experiment. Requiring this measure to be invariant in different frames of reference, recovers all the properties of Special Relativity. And special Relativity has been so thoroughly tested experimentally that it’s as close to a certainty as you’re ever likely to get in science. I’ll derive a couple of these consequences in the comments.
Just as before, a rotation in space can mix up spacial coordinates. Now however, we can rotate between space and time. This corresponds to a change in speed. So a pure “x” coordinate, for example, can become a mixture of “x” and “t” coordinates by changing from a stationary to a moving frame.
As an aside, c²t² is usually much larger than the other three lengths. So some physicists define the signs the other way round.
s² = c²t² – (x² + y² + z²)
Unfortunately, these two conventions exist side by side. Most particle physicists use one convention and most cosmologists use the other. I can never remember which way round each prefers, and besides, it’s not an unbreakable rule in either group. So every time you pick up a textbook or a paper on either subject, you have to check which convention the author is using. The really important thing though, is that space and time use different signs.
But for me, the interesting question is not how to construct the correct spacetime measure. It’s Why? Why does nature have a 4 dimensional spacetime? Why does time have a different sign from the other three dimensions?
Stay tuned for the next episode.
Platitude of the Day 