Rotations in space-time

So far, I have discussed in detail two of the ingredients that govern the structure of our Universe: quantization and symmetry. Now we will consider the third, namely relativity. As it happens, relativity is actually associated with a symmetry, albeit a rather unusual one. Whether this is a coincidence or whether it indicates a deeper level of physical truth is not understood at present. As an introduction to the symmetry that produces relativity, let us return for a moment to rotations in space.

A symmetry group always leaves something unchanged; in the case of space rotations, the invariant is the distance between points. In three-dimensional space, we need three numbers (coordinates; say x, y and z) to describe the position of a point. The distance D is given by the Pythagoras recipe: D2 = x2 + y2 + z2. In a plane, two numbers (say x and y) suffice. Under rotations of the plane, D2 = x2 + y2 is an invariant.

Suppose that some sixteenth-century navigators in Amsterdam were to measure the latitudes and longitudes of two points on Earth, and from these calculate the distance between the points. Now let some Parisian navigators do the same. In olden times, people were every bit as chauvinistic about the location of their country as they are today, so naturally the Dutch used the meridian of Amsterdam, and the French used the Paris meridian for their observations.