The Length of World Line: Prerequisite for Proper Time

[GR191511]

What is the prerequisite of "the length of world line equals proper time"?C=i?orC=1?If metric is-+++:
$ds^2=-c^2d\tau^2\Rightarrow whenC=i,s=\tau$
If metric is +---:
$ds^2=c^2d\tau^2\Rightarrow whenC=1,s=\tau$
So,which one?

 

[Dale]

The prerequisite is that the object have mass. The scaling factor is unimportant, it is just an arbitrary choice of units and sign convention.

 

[Ibix]

What is the prerequisite of "the length of world line equals proper time"?

The worldline must be timelike everywhere, or else proper time is not defined for it (equivalent to Dale's comment about mass).

The value of $c$ is just a choice of ratio between units of timelike and spacelike displacements, so is unimportant. It's similar to putting a meter rule and a yardstick end to end - the total length is the sum of the components' lengths regardless of the units I used to specify them. Experienced people convert units without prompting - inserting $c$ as needed is just a novel example of something you would take in stride in other contexts.

Otherwise, if I were being precise I'd say that $d\tau=\sqrt{|ds^2|}$. I think that saying that proper time is the length of a worldline is an imprecise analogy. I use it either with people who don't understand Minkowski geometry at all (B-level twin paradox threads) or with people who understand it well enough to know I'm being sloppy and will insert modulus signs as needed. Either way I'd probably scare-quote it.

 

[GR191511]

The prerequisite is that the object have mass. The scaling factor is unimportant, it is just an arbitrary choice of units and sign convention.

Thanks

 

[GR191511]

The worldline must be timelike everywhere, or else proper time is not defined for it (equivalent to Dale's comment about mass).

The value of $c$ is just a choice of ratio between units of timelike and spacelike displacements, so is unimportant. It's similar to putting a meter rule and a yardstick end to end - the total length is the sum of the components' lengths regardless of the units I used to specify them. Experienced people convert units without prompting - inserting $c$ as needed is just a novel example of something you would take in stride in other contexts.

Otherwise, if I were being precise I'd say that $d\tau=\sqrt{|ds^2|}$. I think that saying that proper time is the length of a worldline is an imprecise analogy. I use it either with people who don't understand Minkowski geometry at all (B-level twin paradox threads) or with people who understand it well enough to know I'm being sloppy and will insert modulus signs as needed. Either way I'd probably scare-quote it.

Thank you！ I got it