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

+ 0 like - 0 dislike
543 views

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

recategorized Feb 7, 2020

+ 0 like - 0 dislike

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.

answered Feb 6, 2020 by anonymous

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverfl$\varnothing$wThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.