# Axiomatization of SR: can we replace light rays with timelike world-lines?

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)

• is not galilean statement simply saying that the light rays have infinite velocity instead of finite? anyway i dont understand what is axiomatic about your 3rd axiom. Seems to me just a conclusion. Also, could you point me to the passage in hawking,ellis you used for writing down your postulates? Seems to me that Axiom 1 is just consequence of Axiom 2, since you can construct clocks using light rays. Shouldnt there be some hint of relativity principle in those axioms? Commented Nov 12, 2019 at 11:28
• @Umaxo: Thanks for your comments! Seems to me that Axiom 1 is just consequence of Axiom 2, since you can construct clocks using light rays. No, you actually can't construct clocks using only light rays. If you only have light rays in your geometry, then spacetime has a higher symmetry, a symmetry under conformal transformations. Shouldnt there be some hint of relativity principle in those axioms? In this style of axiomatization for SR, some of the notions that are primary in Einstein's 1905 postulates become secondary, and vice versa.
– user4552
Commented Nov 12, 2019 at 21:08
• @Umaxo: anyway i dont understand what is axiomatic about your 3rd axiom. Seems to me just a conclusion. If you mean by this that you think it follows from the other two axioms, then I would be interested to see an argument to that effect. The other two axioms are designed to hold in Galilean relativity, so if I've achieved that design goal correctly, then it shouldn't be possible to prove axiom 3 from them.
– user4552
Commented Nov 12, 2019 at 21:10
• @Umaxo: is not galilean statement simply saying that the light rays have infinite velocity instead of finite? This might work. Seems awkward to me because then you can't talk about SR and Galilean relativity by just specifying different real values of a variable. Also awkward because the Galilean limit of Maxwell's equations actually is not as simple as taking $c\rightarrow\infty$. See physics.stackexchange.com/questions/30999/…
– user4552
Commented Nov 12, 2019 at 21:13
• I feel like the details that were being “eliminated” are in fact essential to selecting a class of “possible kinematics” that contain only two members: SR and Galilei. Then you distinguish between those with your axiom 3. But if you chose different postulates than those that you implicitly assume, there could be other kinematics that axiom 3 would fail to distinguish. For example, truncation of Lorentz transformations upto $c^{-2}$ terms would satisfy axiom 3, but such transformations would not form a group, so a group law must be one of those implicit principles. Commented Nov 13, 2019 at 4:35