# Proper time

bernhard.rothenstein
do you kinow a simple and convincing argument for the fact that proper time is a relativistic invariant?

## Answers and Replies

nakurusil
do you kinow a simple and convincing argument for the fact that proper time is a relativistic invariant?

Have a look here

In the case of inertial motion proper time reduces to:
$$c\Delta \tau=\sqrt(c^2\Delta t^2-\Delta x^2)$$

Since $$c^2\Delta t^2-\Delta x^2$$ is frame invariant , it follows that in the case of inertial motion proper time is also invariant.

The above proof was facilitated by using the fact that $$dx/dt=constant$$. What happens in the case of $$dx/dt$$ not constant? Since the integrand used in the definition of $$\tau$$ varies with time, we cannot use the reasoning seen above anymore. Most likely, in the general case, proper time is no longer an invariant.

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Gold Member
"Proper time" between 2 events is the time that is measured by an observer moving from event1 to event2. How could different observers disagree on what some other observer measured? (they can all measure different times between the events but they all agree on what one "chosen" observer measured)
Also, all observers agree on the ST interval between events. the proper time is equal to this interval.

Homework Helper
Gold Member
"Proper time" between 2 events is the time that is measured by an observer moving from event1 to event2. How could different observers disagree on what some other observer measured? (they can all measure different times between the events but they all agree on what one "chosen" observer measured)

To amplify daniel_i_l's comment:

Of course, elapsed proper-time is a timelike-path-dependent quantity between two timelike-related events. Geometrically speaking, the elapsed proper time between two events is the spacetime arc-length of the given timelike curve [which all observers will agree upon]. Proper time requires the specification of a timelike-curve.

Also, all observers agree on the ST interval between events. the proper time is equal to this interval.

More correctly, the interval is a measure of the largest proper time between the two events [in a nice enough region of spacetime]. (The clock effect.)

The time associated with the "spacetime-interval between two timelike events" is a specialized case where the events are infinitesimally close and a geodesic path is taken. In Minkowski space, this definition can be extended to distant timelike-related events joined by a straight inertial worldline. This time for this spacetime-interval is the largest proper-time among all other proper-times along timelike-curves joining those events. (The Euclidean analogue is that the length of the straight-line joining two points is the shortest among all other curves joining those points.)

do you kinow a simple and convincing argument for the fact that proper time is a relativistic invariant?

As daniel_i_l suggests above,
probably the simplest argument just relies on
asking what that clock-owner measured, more specifically: count of the number of ticks his clock registered. No one can dispute that!

What you then might ask is how his clock was constructed and how it registers the ticks, i.e. at what events do those ticks occur?

bernhard.rothenstein
As daniel_i_l suggests above,
probably the simplest argument just relies on
asking what that clock-owner measured, more specifically: count of the number of ticks his clock registered. No one can dispute that!

What you then might ask is how his clock was constructed and how it registers the ticks, i.e. at what events do those ticks occur?
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Please confirm if I am right interpreting your help.
You and I we are in relative uniform motion. We have identical wristwatches and let tau be the period of the two clocks as measured by me and by you respectively. Both clocks read a zero time when we are located at the same point in space. We make the convention to measure N ticks of our clocks and so the time intervals Ntau. Special relativity teaches us that counted numbers are invariants and what we measure under such conditions are proper time intervals having for both of us the same magnitude.:rofl:
Thanks you and to all participants on my thread.