What's the cleanest/quickest way to go between Einstein's postulates [1] of
- Relativity: Physical laws are the same in all inertial reference frames.
- Constant speed of light: "… light is always propagated in empty space with a definite speed $c$ which is independent of the state of motion of the emitting body."
to Minkowski's idea [2] that space and time are united into a 4D spacetime with the indefinite metric
$ds^2 = \vec{dx^2} – c^2 dt^2$.
Related to the question of what is the best derivation of the correspondence are:
Is the correspondence 1:1? (Does the correspondence go both ways?)
and are there any hidden/extra assumptions?
Edit
Marek's answer is really good (I suggest you read it and its references now!), but not quite what I was thinking of.
I'm looking for an answer (or a reference) that shows the correspondence using only/mainly simple algebra and geometry. An argument that a smart high school graduate would be able to understand.
Best Answer
I will first describe the naive correspondence that is assumed in usual literature and then I will say why it's wrong (addressing your last question about hidden assumptions) :)
The postulate of relativity would be completely empty if the inertial frames weren't somehow specified. So here there is already hidden an implicit assumption that we are talking only about rotations and translations (which imply that the universe is isotropic and homogenous), boosts and combinations of these. From classical physics we know there are two possible groups that could accomodate these symmetries: the Gallilean group and the Poincaré group (there is a catch here I mentioned; I'll describe it at the end of the post). Constancy of speed of light then implies that the group of automorphisms must be the Poincaré group and consequently, the geometry must be Minkowskian.
[Sidenote: how to obtain geometry from a group? You look at its biggest normal subgroup and factor by it; what you're left with is a homogeneous space that is acted upon by the original group. Examples: $E(2)$ (symmetries of the Euclidean plane) has the group of (improper) rotations $O(2)$ as the normal subgroup and $E(2) / O(2)$ gives ${\mathbb R}^2$. Similarly $O(1,3) \ltimes {\mathbb R}^4 / O(1,3)$ gives us Minkowski space.]
The converse direction is trivial because it's easy to check that the Minkowski space satisfies both of Einstein postulates.
Now to address the catch: there are actually not two but eight kinematical groups that describe isotropic and uniform universes and are also consistent with quantum mechanics. They have classified in the Bacry, Lévy-Leblond. The relations among them is described in the Dyson's Missed opportunities (p. 9). E.g., there is a group that has absolute space (instead of absolute time that we have in classical physics) but this is ruled out by the postulate of constant speed of light. In fact, only two groups remain after Einstein's postulate have been taken into account: besides the Poincaré group, we have the group of symmetries of the de Sitter space (and in terms of the above geometric program it is $O(1,4) / O(1,3)$).
Actually, one could also drop the above mentioned restriction to groups that make sense in quantum mechanics and then we could also have an anti de Sitter space ($O(2,3) / O(1,3)$). In fact, this shouldn't be surprising as general relativity is a natural generalization of the special relativity so that the Einstein's postulates are actually weak enough that they describe maximally symmetric Lorentzian manifolds (which probably wasn't what Einstein intented originally).