My background is in mathematics, and I am well versed in differential geometry and general relativity. I have not, however, studied special relativity as much, and frankly view Minkowski space as a vacuum solution of GR. So one thing that eludes me are some elegant ways to move from the classical postulates of SR to viewing it differential geometrically as dynamics on Minkowski spacetime.
ex. How does one arrive to the conclusion of SR being described by Minkowski spacetime from statements like "the laws of physics have Poincaré invariance" and "the speed of light is constant in all inertial frames of reference."
As well discussed in this post, for me an inertial frame of reference is
an isometry of $(M,g)$ and $(R^4,η)$ where $η$ is the standard Minkowski metric. Note that this is different from a general chart because charts are not required to be isometries, and indeed maps geodesics in M to straight lines in R4
And the Lorentz group is
subgroup of GL(4) that leaves the pseudo-inner product on that tangent space invariant.
But how do we get these succinct mathematical characterizations of the underlying concepts of SR from the original postulates? This post, gives an answer in terms of the isometry groups, however I don't understand how the physics fits in with the mathematics.
Namely, I would appreciate the physics of this clarified
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.
What is meant by accommodating? Are we referring to symmetries of physical laws, their solution spaces, frames?