Proper Time Interval: Exploring its Definition on Wikipedia

[Kashmir]

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?

 

[Ibix]

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.

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.

 

[Orodruin]

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.

 

[Dale]

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.

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.

 

[Dale]

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

 

[Orodruin]

The formula above is not required to be evaluated in the clock’s rest frame

No, but it is what the OP asked about:

is the worldline a line on the time axis of the frame in which the clock is at rest?

 

[Ibix]

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.

 

[Dale]

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

 

[Ibix]

Yes, they were asking if P were a worldline on the time axis.

Not quite - the original post says (my emphasis)

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.

 

[vanhees71]

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).