Is proper time an intrinsic value of Minkowski space?

Question

What is proper time? Is it a part of Minkowski space (that is a mere spacetime interval)? Or is it an intrinsic characteristic of massive particles (a sort of "aging")? Example: In the following Minkowski diagram,

• A is the worldline of a particle moving with the observer's reference frame.

• B is the worldline of another particle with proper time between two events = 4 (space distance x= 3, time interval t = 5, proper time τ = 4, according to the equation $$ \tau^2 = \delta t^2 - \delta x^2 $$)

• C is just 2 events without any worldline between, and if a particle would travel through both events, its proper time would be 4

Is there proper time in C even if no particle is travelling there? Or is in C only hypothetic proper time in case a particle is travelling this path?

Answer 1

Ok, before we fill up the comment section with this, I will write this as an answer:

Proper time $\tau$ along a path $\gamma$ is

$$ \tau := \int_\gamma \sqrt{\mathrm{d}x^\mu\mathrm{d}x_\mu}$$

and a clock moving along $\gamma$ will have $\tau$ as its elapsed time at the end of the path. Yet, the definition of proper time $\tau$ involves such clocks not more than the definition of voltage involves voltmeters. You would not say "There is no voltage" because you have nothing to measure it with, and you would not say "There are no lengths" just because you have no ruler. And just as well you would not say "There is no proper time" just because there is no clock travelling along that path.

Answer 2

As stated by Moonraker your point C is the same as saying there is no space at all thereby any definition of 2 points makes no sense at all.

Now to the question "what is proper time?"

"It is the time which is measured in the rest frame of an observer passing through two events in space-time"

This will depend on the kind of movement the observer experiences thereby you cannot just take the direct distance in space-time as done in B but you have to integrate over all small space-time segments. You can do that because the space-time is an invariant under lorentz transformations so that you can always transform into a system in which your you are at rest so that you only move along the time axis.(Only allowed locally if you take acceleration into account)

Answer 3

The proper time between two space-time points is very, very like the ordinary distance between two points in ordinary space. In fact, in the case that the two points are at the same time in a given frame the proper time between them is just (minus) the distance between them divided by c, the speed of light. It does not require a clock to be real, any more than distance requires a ruler