Deriving the Minkowski metric from homogeneity of spacetime and the isotropy of space

Question

In this wikipedia page, it says that one can derive the spacetime interval between 2 arbitrary events from the second postulate of special relativity, together with the homogeneity of spacetime and the isotropy of space.

The "second postulate" as numbered in the reference is this the invariance of $c$ stated like this:

As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity $c$ that is independent of the state of motion of the emitting body. Or: the speed of light in free space has the same value $c$ in all inertial frames of reference.

How exactly is this done?

Answer 1

The Wikipedia page (at least as it stands when I read it today) says that "it follows" from the assumptions stated in the question that the spacetime interval between two events 1 and 2 is $$ c^2 (t_1 - t_2)^2 - (x_1 - x_2)^2 - (y_1 - y_2)^2 - (z_1 -z_2)^2$$

It's not really a derivation. The second postulate (that the speed of light is constant) is required to make the $c^2$ factor on the first term meaningful in light of the homogeneity and isotropy assumed in this case, which together imply that the form of the interval should be the same throughout space. The rest of the expression is the obvious form for a metric under the assumptions (e.g. treats all directions the same).