Part II – From Pythagoras to Einstein, via Hamilton
We saw before that the correct way to measure spacetime distances was to include the time with a negative sign.
s² = x² + y² + z² – c²t².
Or, as preferred by some physicists
s² = – x² – y² – z² + c²t².
This was first proposed by Einstein’s maths professor, Minkowski, and so is know as the Minkowski metric. The important thing about it is that space and time have opposite signs. With either of these measures, their invariance in different frames of reference allows us to derive all of Special Relativity. Incidentally, the step from “Special” to “General” Relativity involves adding cross terms: xy, zt, etc. These cross terms correspond to a curved spacetime, where the curvature is effectively gravity.
To get a clue as to why flat spacetime is this way, we have to go back to the mid 19th century, before Einstein was born. William Rowan Hamilton was an Irish mathematical physicist. We met him previously in our discussion of Quantum Field Theory. He made huge contributions to mathematics, optics and mechanics which remain influential to this day. But there was one problem that had vexed him for years.
By the mid-19th century, mathematicians, including Hamilton himself, had established a fairly complete understanding of “complex numbers”. To understand what these are, we have to look at the ordinary real numbers and consider them as points on a line. The normal arithmetical operations of addition and multiplication can be regarded as simply movements back and forth along this line.
Now we introduce a new number “i”. Multiplying a real number by i doesn’t move it along the real line any longer. Instead, it rotates the number by 90 degrees anti-clockwise. This gives us a whole new set of numbers, called “imaginary” numbers. Imaginary numbers are all products of a real number and the number i and sit at 90 degrees from the real line.
By combining a real number with an imaginary number, say 1+2i, we can represent any point in a plane. These are the “complex numbers”. They have a lot of fascinating properties, especially when you start doing calculus with them. I’d love to go off at a tangent here but it would take too long. Just take my word for it, they are really, really useful, not just in maths, but in physics and in electrical and mechanical engineering as well.
If we multiply i itself by i, then our rule says: take i and rotate it by 90 degrees anti-clockwise. This takes us to -1. In other words, i² = -1. i is the square root of -1.
Complex numbers transform problems in plane geometry into problems in algebra. Flushed with their success, mathematicians turned to the obvious extension. If we can have a number i that solves problems in plane geometry, let’s try a second version, call it j, that allows us to solve problems in three dimensions. Except this doesn’t work. Here’s why.
Suppose we set up the real axis and the imaginary i axis at right angles as before. Now we add a third axis, the j axis, perpendicular to both of them. Multiplying a real number by i rotates it onto the i axis. Multiplying a real number by j rotates it onto the j axis. So far, so good. But what happens when we multiply i by j? Let’s try some possibilities.
Suppose ij = 1?
We can multiply both sides by i to give iij = i.
But i² = -1.
So this gives -j = i.
Looking at our diagram this clearly isn’t correct.
What about ij = i?
This time we divide both sides by i and get j = 1.
Again that’s not right.
The same happens if ij = j.
Divide both sides by j and we get i = 1.
Maybe there’s some funny combination like ij = ai + bj + c. But this also quickly leads to a contradiction.
Hamilton puzzled for years on how to solve this. Then, on Monday 16 October 1843, Hamilton suddenly realised how to solve it. The problem wasn’t with i and j. The problem was with the real numbers. By representing one axis by real numbers and the other two by imaginary numbers, the real axis is being singled out as special. What he needed was to identify the three spatial axes with three different roots of -1: i, j and k.
He was so overwhelmed by the discovery that he scribbled their defining property onto a bridge on the Dublin canal he was walking along.
i² = j² = k² = ijk = -1.
But we still have to include the real numbers. The fact that i j and k all square to -1, which is a real number, means that they are also part of the algebra, but they clearly don’t represent a spatial dimension any more.
So, if we want to create an algebra that expresses relationships in more than one dimension of space, then two dimensions won’t work. Neither will three. We must have at least three dimensions whose unit vectors square to -1, and a separate fourth dimension whose unit vector squares to one.
Ring any bells?
s² = – x² – y² – z² + c²t²
Hamilton called the new 4-tuples of numbers, “Quaternions”. He rightly recognised the enormity of his discovery and equally rightly identified the real numbers with time. He spent most of the rest of his life reformulating physics in quaternion form. Their study became compulsory in the Dublin maths and physics departments.
So why are they almost unheard of today? Why wasn’t relativity invented 50 years earlier?
I’ll say a little bit more about quaternions in the comments and why they fell out of fashion. But the biggest problem with quaternions is that, unfortunately, nature doesn’t use them.
We’ll discuss what it does use in the next part.
Platitude of the Day 