On Monday, May 23, 2022 at 1:02:57 PM UTC-7, patdolan wrote:
> > > Velocity Postulate:
> >
> > In physics, in terms of any operationally-defined system of coordinates x,t, and for any given trajectory x=f(t), at any given event, the quantity dx/dt is defined as the "velocity" of that trajectory at that event. This is the definition of the word "velocity", it is not a postulate.
> >
> > > [spacetime version...
> >
> > There is no separate "spacetime version" distinct from the standard historical definition of the word "velocity", as given above.
> >
> > > ... there can exist at most one instantaneous velocity magnitude ||v|| shared
> > > between any pair of observers.
> > That doesn't make sense. Velocity is as defined above, dx/dt in terms of any specified system of coordinates. There are infinitely many systems of coordinates, and entities have infinitely many different velocities, depending on which system of coordinates you are referring to. What you might be trying to say is that for two standard systems of inertial coordinates S and S', with aligned space axes, objects at rest in S' have velocity v in terms of S, and objects at rest in S have velocity -v in terms of S'. But these are by no means the only possible coordinate systems that can be defined, and hence those objects have any other velocities in terms of many other coordinate systems. It is necessary always to specify the coordinate system when referring to a velocity.
>
> To have infinitely many coordinate systems is tantamount to having infinitely many observers.
Nope, you are mistakenly conflating "observers" with coordinate systems, one of the most common newbie mistakes. Every object is at rest in terms of infinitely many coordinate systems, with distinct temporal foliations, so there is no unique system of coordinates in which an object is at rest.
> In this problem I have limited the ... coordinate systems to only two, [the two objects]
> co-moving coordinate systems.
I already spoonfed you your unexamined stipulations, pointing out that you are (unwittingly) referring to the standard inertial coordinate systems, S and S', in which the two object are respectively at rest.
> Given a velocity value |v| shared between a pair of observers...
Translation: Given two systems of standard inertia-based coordinates S and S' with mutual velocity v...
> ...each observer must be able to calculate that velocity value from the standpoint
> of his own FoR...
It's pointless to talk in terms of "what an observer must be able to calculate", because an observer could be someone like you, who can't calculate his way out of a paper bag. The propositions of physics are objective facts, not assertions about anyone's ability to calculate things. What you need to say is that given the velocity of an object in terms of one coordinate system, and given the relationship between that coordinate system and some other coordinate system, the velocity of the object in terms of that other system is fully specified.
> and from the standpoint of the other observer's FoR. Furthermore, all four of these velocity values must be identical.
Four? Again, a given object at a given event has a specific velocity in terms of any specified system of coordinates. So far you're just talking about two systems of coordinates, and you seem to be tacitly assuming they are inertial coordinate systems, which we can call S and S', and each object has a velocity in terms of those two coordinate systems (not four). As noted above, an object with dx/dt = 0 has d'x/dt' = -v, and an object with d'x/dt'=0 has dx/dt = v. If those are the four velocities you are talking about, they are not identical, two are zero and one is v and the other is -v.
> Relativity corollary: All velocities shared between all pairs of observers must be less than the velocity of light.
No, that isn't a corollary, and it isn't the relativity principle. It's a complete non-sequitur from what you have said.
> Stan now calculates what Townes will say Stan's velocity is according to Townes' co-moving coordinate system.
Again, if the "co-moving coordinate systems" you're referring to are standard inertia-based coordinate systems S and S' in which the two objects are at rest, then the object at rest in S' has velocity v in terms of S, and the object at rest in S has velocity -v in terms of S'. These are objective facts.
Of course, we could refer to other systems of coordinates... for example, we could use the time coordinate from S and the space coordinates of S', or vice versa, and the resulting hybrid coordinate systems would have different velocities, but these are not standard inertial coordinate systems. Again, there are infinitely many coordinate systems (in fact, there are infinitely many in which a given object is at rest), and every object has infinitely many velocities, depending on which system of coordinates you are referring to. So nothing that you are saying makes any sense. Now do you understand?