# Exploring the Relativistic Invariance of Proper Time

• bernhard.rothenstein
In summary, special relativity teaches us that proper time intervals between two events are invariants.

#### bernhard.rothenstein

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

bernhard.rothenstein said:
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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"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.

daniel_i_l said:
"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.

daniel_i_l said:
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.)

bernhard.rothenstein said:
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?

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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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.

## 1. What is relativistic invariance of proper time?

The relativistic invariance of proper time refers to the concept that the passage of time is relative and depends on the observer's frame of reference. This means that the measurement of time can vary depending on an observer's relative speed and gravitational forces.

## 2. Why is it important to explore the relativistic invariance of proper time?

Understanding the relativistic invariance of proper time is crucial in the field of physics, as it helps explain the behavior of objects moving at high speeds and in strong gravitational fields. It also plays a significant role in the theories of relativity and has practical applications in fields such as space travel and GPS technology.

## 3. How is proper time different from coordinate time?

Proper time is the time measured by an observer in a specific frame of reference, while coordinate time is the time measured by an observer in a different frame of reference. Proper time takes into account the effects of relative speed and gravitational forces, while coordinate time does not.

## 4. Can the concept of relativistic invariance of proper time be observed in everyday life?

Yes, the effects of relativistic invariance of proper time can be observed in everyday life, although they are usually very small. For example, GPS satellites have to account for the difference in time due to their high speeds in orbit, or else they would not be accurate.

## 5. How is the concept of relativistic invariance of proper time tested and verified?

The concept of relativistic invariance of proper time has been extensively tested and verified through experiments, such as the Hafele-Keating experiment and the Pound-Rebka experiment. These experiments involve measuring the difference in time between two clocks, one on Earth and one in motion, and comparing it to the predictions of the theory of relativity.