[Q-Q]

What are you guys on about

>if A sees B at speed v, then B sees A at the same speed v

is directly derivable from the Lorentz velocity addition formula which is the result of the usual two postulates. Why would you want that as an extra assumption...

https://en.wikipedia.org/wiki/Velocity-addition_formula#Spec...

vAA = (vAB + vBA)/(1+(vAB*vBA/c^2) -> 0 = vAB + vBA

so velocity of A relative to B and B relative to A both have magnitude v.

[superposeur]

Of course, if you already assume special relativity then this is baked in: taking the inverse of the Lorentz transformation matrix effectively sends v to -v. And velocity addition for sure already assumes you have the Lorentz transformations.

The point is that in the derivation of relativity from elementary postulates, the “reflectivity of velocities” assumption is used. For instance the linked paper uses it between equations (12) and (13). All such derivations use it at some stage.

Intuitively, “reflectivity of velocities” is why time dilation leads to length contraction with same gamma factor.

I don’t think it follows from absence of a preferred frame since one could imagine some complicated group structure relating the velocities of the various observers relative to each other in such a way that all are on equal footing and yet v_AB is not exactly -v_BA.

As mentioned, this is certainly a reasonable assumption but who knows what whacky alternatives are out there? After all, in the different context of quantum mechanics, such “obvious” relationships as xp = px no longer hold true.