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I'm not referring to the dimensional choice that makes $c=1$; rather I'm imagining something more about replacing all references that apparently involve velocities with the appropriate $\gamma$ factors or rapidity. In this description, the kinematic feature that maps to speed would then be defined on $[0,\infty)$, and we wouldn't have to deal with the question of "why is the speed of light finite, and has the value that it has?" $^*$

$^*$ Or, at least we wouldn't have to deal with it as a fundamental question about the universe.

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Are you thinking of something like rapidity? – Alfred Centauri Dec 11 '12 at 3:22
Yes, has anyone tried to express more physics than just special relativity in this representation? – Dave Dec 11 '12 at 4:18
there are a couple papers out there deriving the rules of relativity without making reference to the speed of light. Is that what you're looking for? – Mike Flynn Dec 11 '12 at 5:47
@MikeFlynn that might be what I'm getting at. – Dave Dec 11 '12 at 13:42
Rapidity ( works well in 1+1D (linear addition of rapidities, rapidity being the integral of proper acceleration, etc.), less so in higher dimensions. – Johannes Dec 11 '12 at 16:36

If you need equations without velocities, just with $\gamma$, just replace any $v$ by $c\sqrt{\gamma^2-1}/\gamma$ and you're done.

Incidentally, $\gamma$ isn't the only useful function of $v$ that takes values between $0$ and $\infty$. You could also use the rapidity $\eta={\rm arctanh}(v/c)$.

It's good to use $\gamma$ and $\eta$ and in various situations, it simplifies physics or makes it clearer. However, it's still true that $d\vec x / dx^0=\vec v$ and nothing else for the trajectory of a particle, so this basic defining equation of the slope of the trajectory takes the simplest form with $\vec v$! And this "actual velocity" is smaller than $c$ for all massive bodies: this basic statement of relativity is surely not just a convention and can't be hand-waved away by some change of variables. Only one quantity, $v$, deserves to be called the velocity.

Also note that $\gamma$ and even $\eta$, if defined in the easy way above isn't even a vector: it only knows about the magnitude of the speed, not its direction.

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In reference to your 3rd paragraph, I'm wondering if anyone has tried to write out kinmatics/dynamics in a way that would never require taking that derivative -- and thus never referring to speed. In reference to your 4th paragraph, can't we just decide to always formulate things in terms of momentum. – Dave Dec 11 '12 at 13:50
Dear Dave, you may express the same information differently but it's still possible to talk about $d\vec x/dx^0$ and it's a simple object. Your efforts to deny the existence of the concept of "velocity" is somewhat strange. Of course that you can avoid the concept, in various awkward ways, but why? As long as mechanics for particles with $x(t)$ functions follows from a theory, it's clearly natural to talk about the first derivatives $x'(t)=v(t)$. They're the velocities and relativity says that they're smaller than $c$. It's a true, non-vacuous statement. – Luboš Motl Dec 13 '12 at 7:53

Use a pure four-space notation $x^0 = ct$. Expressions such as $(1/c) (d/dt)$ transform to $(d/dx^0)$. For instance the typical $(v/c)$ terms transform to $(dx/dx^0)$.

For proper-time derivatives use $ds = cd\tau$.

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