I would like to think that I understand how 4-velocity is defined in special relativity. It makes sense to me that one takes a differential segment of the world line ##(dt, d\vec x)## and translates it into a frame in which its spatial component disappears, leaving only ##(d\tau, 0) = (dt/\gamma, 0)##, where ##d\tau## is referred to as the "proper time." One then divides the differential world line segment by the proper time, and the 4-velocity pops out:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\vec U = {1 \over d\tau}(dt, d\vec x) = {\gamma \over dt}(dt, d\vec x) = \gamma(1, \vec v)

[/tex]

The magnitude of the 4-velocity is then frame invariant, massless particles can't have 4-velocities, etc..., etc...

What I don't understand is this: The 4-velocity involves the terms ##dt/d\tau##, ##dx/d\tau##, ##dy/d\tau##, ##dz/d\tau##. Now, these are derivatives. Every other time I've seen a derivative anywhere in calculus or physics it has been the case that the thing being differentiated is a function that can be parameterized by the specified variable. Yet in this case I can't make sense of the idea that the world line is parameterized the proper time ##\tau##. To begin with, the proper time for a given differential world line segment is only well defined in the momentarily co-moving reference frame (MCRF). Every differential world line segment will have a different MCRF, and this makes the proper time feel ephemeral, strange, and not globally persistent enough to parameterize the world line as a whole.

Could anyone help clear up my confusion here? Should I even be trying to think of the world lines as parameterized entities? And if not, then how can we possibly take the derivatives of an entity that isn't even a function (in the sense that functions are nothing if not "things which are parameterized").

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# 4-Velocity and World Line Parameterization

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