If I eliminate a lot of details and just sketch the general ideas, then a common way of presenting SR is this:
- Axiom 1: Clocks exist.
- Axiom 2: Light rays exist.
This is the approach followed in, e.g., Hawking and Ellis (p. 63, "Postulate (a) enables..."), and in Geroch's popularization General relativity from A to B.
There is a well known paper by Ehlers et al. which basically changes this to:
- Axiom 1b: Inertially moving massive particles exist.
- Axiom 2: Light rays exist.
In both of these treatments, an awkwardness exists because axiom 2 seems to imply two logically separate things at the same time: that we're not in Galilean relativity, and also that measuring rods exist.
Question: Can one instead replace these with the following?
- Axiom 1b: Inertially moving massive particles exist.
- Axiom 2b: Speedometers exist. That is, an observer can shoot a bullet at some fixed speed $v$ relative to themselves.
- Axiom 3: Anti-Galilean postulate. If we carry out Einstein synchronization using the bullets described in 2b, then synchronization produces different results for different observers.
My goal here is to separate out the Galilean statements from the SR statements, so that we can flip back and forth between the two systems just by negating or not negating one postulate.
Axiom 2b is similar to Euclid's axiom that all right angles are equal. It takes us out of the realm of purely affine geometry and implies that we essentially have an inner product space. Given some other reasonable assumptions not presented explicitly in this sketch, this also implies things like being able to determine whether a bullet that whizzes past you is going at $v$ (because we can compare world-lines to see if they're the same).
References
The geometry of free fall and light propagation Jürgen Ehlers, Felix A. E. Pirani, Alfred Schild General Relativity and Gravitation, https://doi.org/10.1007/s10714-012-1353-4 (available on Sci-Hub)