Proper Time Interval: Exploring its Definition on Wikipedia

In summary, proper time is the time measured by a clock moving along an arbitrary time-like world line.
  • #1
Kashmir
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Wikipedia article on proper time

"Given this differential expression for ##\tau##, the proper time interval is defined as
##
\Delta \tau=\int_P d \tau=\int \frac{d s}{c} .
##
Here ##P## is the worldline from some initial event to some final event with the ordering of the events fixed by the requirement that the final event occurs later according to the clock than the initial event."

In the integral
##
\Delta \tau=\int_P d \tau=\int \frac{d s}{c} ## , is the worldline a line on the time axis of the frame in which the clock is at rest? Also I can calculate the integral in another frame in which the clock is moving, however the world line will have spacetime coordinates different than the rest frame ,however due to invariance of ##ds## we get the same value of proper time as we have in the rest frame. Am I correct?
 
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  • #2
Kashmir said:
is the worldline a line on the time axis of the frame in which the clock is at rest?
No, it's any timelike line. In general you would write $$\begin{eqnarray*}
ds&=&\sqrt{c^2dt^2-dx^2}\\
&=&\sqrt{c^2-\left(\frac{dx}{dt}\right)^2}dt
&=&\sqrt{c^2-v^2(t)}dt
\end{eqnarray*}$$where ##v(t)## is the velocity of the worldline in whatever coordinates you're using.
Kashmir said:
Also I can calculate the integral in another frame in which the clock is moving, however the world line will have spacetime coordinates different than the rest frame ,however due to invariance of we get the same value of proper time as we have in the rest frame.
Yes.
 
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  • #3
Ibix said:
No, it's any timelike line. In general you would write $$\begin{eqnarray*}
ds&=&\sqrt{c^2dt^2-dx^2}\\
&=&\sqrt{c^2-\left(\frac{dx}{dt}\right)^2}dt
&=&\sqrt{c^2-v^2(t)}dt
\end{eqnarray*}$$where ##v(t)## is the velocity of the worldline in whatever coordinates you're using.
It is not any timelike line in the rest frame, but it is also not necessarily the time axis. It is a line parallel to the time axis.
 
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  • #4
Kashmir said:
is the worldline a line on the time axis of the frame in which the clock is at rest?
##P## is the worldline of any clock in any state of motion in any reference frame in any spacetime.

Kashmir said:
Also I can calculate the integral in another frame in which the clock is moving, however the world line will have spacetime coordinates different than the rest frame ,however due to invariance of ds we get the same value of proper time as we have in the rest frame.
Yes. You can change the reference frame as you like, including non-inertial frames, the value will not change.
 
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  • #5
Orodruin said:
It is not any timelike line in the rest frame, but it is also not necessarily the time axis. It is a line parallel to the time axis.
The formula above is not required to be evaluated in the clock’s rest frame
 
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  • #6
Dale said:
The formula above is not required to be evaluated in the clock’s rest frame
No, but it is what the OP asked about:
Kashmir said:
is the worldline a line on the time axis of the frame in which the clock is at rest?
 
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  • #7
Orodruin said:
It is not any timelike line in the rest frame
Ah yes, sorry. @Kashmir - the formula you quoted is completely general and works for any ##P##, whether the clock on that worldline is at rest/moving/accelerating/whatever. However, if the clock is inertial and in its rest frame then the worldline ##P## is parallel to the time axis, although it need not be the axis itself.
 
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  • #8
Orodruin said:
No, but it is what the OP asked about:
Yes, they were asking if P were a worldline on the time axis. That formula is valid for any timelike worldlines, not just ones on the time axis or parallel to it
 
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  • #9
Dale said:
Yes, they were asking if P were a worldline on the time axis.
Not quite - the original post says (my emphasis)
Kashmir said:
In the integral
##
\Delta \tau=\int_P d \tau=\int \frac{d s}{c} ## , is the worldline a line on the time axis of the frame in which the clock is at rest?
...to which the answer is no, but it's a line parallel to the axis, at least if we understand that ##P## is the worldline of the clock.
 
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  • #10
Given a time-like line you can always construct local rest frames by Fermi-Walker transporting a tetrade along this world line.

Proper time is the time measured by a clock moving along an arbitrary time-like world line, and of course you don't need to construct the local rest frames to calculate it since it's an invariant/scalar. It's always given by
$$\mathrm{d} \tau =\sqrt{g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}} \mathrm{d} \lambda/c,$$
where ##\lambda## is an arbitrary parameter for the time-like world line (I use the +--- signature for the pseudo-metric).
 
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Related to Proper Time Interval: Exploring its Definition on Wikipedia

1. What is the definition of Proper Time Interval?

The Proper Time Interval is a concept in physics that measures the duration between two events in the same frame of reference. It is also known as the proper time or the invariant time.

2. How is Proper Time Interval calculated?

The Proper Time Interval is calculated using the equation Δτ = √(Δt^2 - Δx^2), where Δt is the time interval between the events and Δx is the distance between the events in the same frame of reference.

3. What is the significance of Proper Time Interval in physics?

The Proper Time Interval is significant because it is an invariant quantity, meaning it is the same for all observers regardless of their relative motion. It is also a fundamental concept in the theory of relativity and is used to calculate time dilation.

4. Can Proper Time Interval be negative?

No, the Proper Time Interval cannot be negative. It is always a positive value, as it represents the duration between two events.

5. How is Proper Time Interval different from Coordinate Time?

Proper Time Interval and Coordinate Time are two different ways of measuring time in physics. Proper Time Interval is based on the duration between two events in the same frame of reference, while Coordinate Time is based on the time recorded by a clock at a specific location. They can be related through the equation Δt = γΔτ, where γ is the Lorentz factor.

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