Just about done with special relativity. It is simply marvelous to see how everything follows from the constancy of the speed of light — time moving more slowly for a moving object (relative to an object standing still in its own frame of reference), a moving object shrinking (ditto), the increase in mass which occurs as an object begins to approach the speed of light, and how this leads to the equivalence of mass and energy. Special relativity is even sufficient to show how a gravitational field will bend light — although to really understand this, general relativity is required.
The one fly in the intellectual ointment is the Minkowski metric for the space time of special relativity. In all the sources I’ve been able to find, it appears ad hoc, or is defined analogously to the euclidean metric. I’d love to see an argument why this metric (time coordinates positive, space coordinates negative) must follow from the constancy of the speed of light. It is clear that the Minkowski metric is preserved under the hyperbolic transformation of space-time, but likely others are as well. Why this particular metric and not something else.
Consider the determinant function of an n by n matrix. It has a god awful mathematical form involving the sum of n ! terms. Yet all you need to get the (unique) formula are a few postulates — the determinant of the identity matrix is 1, the determinant is a linear function of its rows (or its columns), interchanging any two rows of the determinant reverses the sign of the determinant, etc. etc. This basically determines the (unique) formula of the determinant. I’d really like to see the Minkowski metric come out of something like that.
Can anyone out there shed light on this or give me a link?