12.3: Worldlines and Proper Time

The trajectory followed by a particle through space and time is commonly called a worldline in the theory of relativity. In general, a worldline does not have to be a straight line in a spacetime diagram - you can of course speed up and slow down (relative to a stationary observer) as you please. Any clock you bring with you will, from your point of view, keep on ticking at the same rate as always. For the stationary observer its rate will however depend on your speed, and if you change direction or accelerate, that will have an effect as well. We call the time recorded by a clock in the comoving reference frame of a particle the particle's proper time, usually denoted by \(\tau\). We can calculate the proper time with respect to a stationary observer by chopping up the particle's trajectory into small pieces. In each of the pieces, the particle's velocity will be constant, and we can calculate the time dilation as we did before, then we'll integrate over the pieces to get the total proper time. To see how this is done, consider a part of the trajectory between \(t=t_i\) and \(t=t_{i+1}\), in which the particle travels from \(x_i\) to \(x_{i+1}\) as measured by a stationary observer. That observer then calculates the particles speed in this interval to be

\[
v_i=\frac{x_{i+1}-x_i}{t_{i+1}-t_i}
\]

We now define the interval of proper time as the length of the invariant interval \(\Delta s_i\) corresponding to the displacement divided by the speed of light, which gives:

\[
\begin{aligned}
\Delta \tau_i & =\frac{\Delta s_i}{c}=\frac{1}{c} \sqrt{c^2\left(t_{i+1}-t_i\right)^2-\left(x_{i+1}-x_i\right)^2} \\
& =\left(t_{i+1}-t_i\right) \sqrt{1-\frac{\left(x_{i+1}-x_i\right)^2}{c^2\left(t_{i+1}-t_i\right)^2}}=\Delta t_i \sqrt{1-\frac{v_i^2}{c^2}}=\frac{\Delta t_i}{\gamma\left(v_i\right)}
\end{aligned}
\]

\(\Delta \tau_i\) is the time interval as measured on a comoving clock. Unsurprisingly, it is related to \(\Delta t_i\) through the time dilation factor \(\gamma\left(\nu_i\right)\). To calculate the total time which has passed on the comoving clock, we simply sum over all discrete intervals \(i\). In the limit where the length of the interval becomes infinitesimally short, that sum becomes an integral, and we can calculate the proper time by:

\[
\Delta \tau=\int_{t_{\mathrm{a}}}^{t_{\mathrm{b}}} \sqrt{1-\frac{v(t)^2}{c^2}} \mathrm{~d} t
\]

where \(\left[t_{\mathrm{a}}, t_{\mathrm{b}}\right]\) is the time interval as measured by the stationary observer. Note that equation (\(\PageIndex{3}\)) holds for any kind of motion, not just motion at constant velocity.