- #26

- 8,932

- 2,919

I'm afraid your answer doesn't make much sense. I can claim that ds^2 = dx^2 - dy^2 describes a shorter path than ds^2 = dx^2 + dy^2, but I have no justification for arbitrarily flipping the sign on one of my dimensions. So why is Minkowski able to get away with it with time? Is not time orthogonal in every way to space?

The length of a path is relative to a metric. For example, if [itex]x[/itex] and [itex]y[/itex] are Cartesian coordinates, then traveling [itex]\delta x[/itex] in the x-direction and [itex]\delta y[/itex] in the y-direction will put you at a distance away from your start of [itex]\delta s[/itex] where [itex]\delta s^2 = \delta x^2 + \delta y^2[/itex]. On the other hand, in polar coordinates [itex]r, \theta[/itex], the distance [itex]\delta s[/itex] is given by [itex]\delta s^2 = \delta r^2 + r^2 \delta \theta^2[/itex]. The general notion of a metric (for 2D space, for simplicity) is a tensor, that can be represented as four numbers: [itex]g_{11}, g_{12}, g_{21}, g_{22}[/itex], and the corresponding notion of "distance" is given by:

[itex]\delta s^2 = \sum_{i j} g_{i j} \delta x^i \delta x^j[/itex]

For any such metric [itex]g_{ij}[/itex], there is a corresponding notion of "distance" (or distance-squared, actually), and that gives rise to its own notion of "minimal" (or "extremal"; it's not necessarily minimal) path, that generalizes "shortest path" in Euclidean space. This generalized notion of a metric views distance and the corresponding notion of extremal path as something empirical, rather than something you can discover by pure logic alone. So empirically, it turns out that the metric [itex]ds^2 = (c dt)^2 - dx^2 - dy^2 - dz^2[/itex] is important in nature. If you move a clock from point [itex](x,y,z)[/itex] at time [itex]t[/itex] to point [itex](x+dx, y+dy, z+dz)[/itex] at time [itex]t+dt[/itex], then the time on the clock will advance by an amount [itex]ds/c= \sqrt{dt^2 - (dx/c)^2 - (dy/c)^2 - (dz/c)^2}[/itex]. That's an empirical fact. The Minkowsky metric is an especially convenient way to express this fact.