# Physical Motivation for Four-Velocity definition

The first definition transforms as a four-vector: $$\dfrac{dx^{'\mu}}{d \tau} = \Lambda^{\mu}{}_{\nu} \dfrac{dx^{\nu}}{d \tau}$$.

The second definition transforms not quite as a four-vector: $$\dfrac{dx^{'\mu}}{d t'} = \dfrac{dt}{dt'} \Lambda^{\mu}{}_{\nu} \dfrac{dx^{\nu}}{d t}$$.

This makes sense, since in the first definition you divide the differentials of a four-vector (which themselves also transform as a four-vector) by a scalar (invariant under the Lorentz group).

I would like to point out conceptual problems. We would like the 4-velocity to somehow characterize the movement of an object through spacetime. Conceptually it makes sense to demand, that such quantity should depend only on the quantities that have direct relation to that movement. So bringing some random observer's time that has nothing to do with the movement of the object in would be conceptually weird decision. It makes sense to define 4-velocity as tangent vector to the objects worldline, because this mathematical entity is directly connected with it and thus also with objects movement. Of course, we need some parametrization of the worldline, which would be ideally natural to the worldline/movement itself and does not depend on any external quantities. Since in spacetime, every object has its own clocks, this curve is naturally parametrized by clock of the object itself, that is - by its proper time.

Note, that in this way, you don't need to talk about Lorentz group at all. When I first learned about 4-velocity, the decision to use proper time in the derivative felt to me as random decision just to make some Lorentz 4-vector. But it actually has deeper geometrical reasons, as I tried to explain.