What is the hierarchy of time: Δt, Δs, and Δτ?

[Bristlethorn]

Hi, all.
My textbook presents an hierarchy of the three types of time, but doesn't back it up or explain where it came from.

According to it,

Δt ≥ Δs ≥ Δτ

Where
- Δt is the coordinate time between two events as observed in an inertial frame,
- Δs is the spacetime interval between two events,
- Δτ is a proper time between two events.

Can anyone explain where this hierarchy comes from?
Thanks,

Bristlethorn

 

[Ibix]

You seem to be assuming a +--- signature and units such that c=1 (that is, $\Delta s^2=\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2$). I would have said that Δs and Δτ were the same thing, at least for timelike intervals, in that context. Perhaps there's some more context provided about what Δs and Δτ mean to your textbook? And which textbook, by the way?

But Δt is obviously greater than or equal to the other two from the definition of the interval - see the expression in brackets above.

 

[haushofer]

Proper time equals the spacetime interval between two events only when the proper time is measured by an inertial observer.

To see this hierarchy: write down the three expressions for the different intervals. What do you get?

 

[Ibix]

Proper time equals the spacetime interval between two events only when the proper time is measured by an inertial observer.

That's why I was asking for context. I interpreted the "proper time between two events" without qualification as the proper time along a straight line. You seem to be interpreting it as the proper time along an arbitrary path, in which case, I agree.

Maybe my terminology is letting me down.

 

[robphy]

Can you think of a physical example demonstrating the left inequality? the right inequality?

 

[PeterDonis]

My textbook

Which textbook?