Why is "incorrect" parametrize a Timelike world-line using the time "t" of some frame S?

Question

In some relativity books ( Carrol, Schutz, Rindler) is common to treat the worldline of a massive particle as a parametric curve $x^\mu (\lambda)\ $ of spacetime, with $\lambda$ being a real parameter. In these books, the usual parameter used is the proper time $\ \tau \ $ of the particle. Why is unusual to parametrize the worldline of a particle with respect the time parameter $t$ of some inertial frame S (wich could not be the rest frame of the particle)? If using the "t" as parameter leads to an inconsistence or a algebraic/conceptual problem, please explain to me with some example, etc

Answer 1

It is conceivable to parametrise the worldline of a particle with the time t of some inertial (or non-inertial) frame B. After all, an observer at rest in B can observe the particle and record its position as a function of time t, thus providing the parametrisation.

However, if you consider a general situation, what frame B would you use? It would be an arbitrary, artificial choice, of "some" frame B that has nothing to do with the problem. Choosing the time in the rest frame of the particle (i.e. the proper time of the particle), which rest frame need not be inertial, is more natural. The rest frame of the particle is "there", because the particle is. The frame does not need to be introduced artificially, as above frame B would have to.

Furthermore, when it comes to derivatives, the derivative of the worldline with respect to the proper time is the 4-velocity of the particle, and a 4-vector. If you consider instead the derivative with respect to the time t of some frame B, then as long as the frame is fixed, the derivative is also a 4-vector (all observers, in all frames, agree with respect to the time that passes in frame B, although the time in their own respective frames is different). From the notation, however, this would be less evident, possibly more awkward, and definitely more error prone.