Proper Time and Time Dilation

Suppose we have a particle moving with a velocity v in a given coordinate system $K$ . In a time $dt$ (in that system) it moves $d{\bf x} = {\bf v}dt$ . Then its invariant infinitesimal interval is (ds)^2 = (c dt)^2 - dx^2 = c^2 dt^2 (1 - &beta#beta;^2). In the particular frame where the particle is at rest ($dx' = 0$ ) we define the proper time to be d&tau#tau;= dt' so that (ds)^2 = c^2 (d&tau#tau;)^2 . Thus the proper time is just the time experienced by the particle in its own rest frame.

From the relations above, it is easy to see that d &tau#tau;= dt 1 - &beta#beta;^2(t) = dt&gamma#gamma;(t) and to find the interval between two events on some world line it is necessary to integrate: t_2 - t_1 & = & &int#int;_&tau#tau;_1^&tau#tau;_2 d&tau#tau;1 - &beta#beta;^2(&tau#tau;)
& = & &int#int;_&tau#tau;_1^&tau#tau;_2 &gamma#gamma;(&tau#tau;) d&tau#tau;. If $\beta$ is constant (so the frames are inertial) then we get the usual time dilation &Delta#Delta;t = &gamma#gamma;&Delta#Delta;&tau#tau; or &Delta#Delta;&tau#tau;= &Delta#Delta;t&gamma#gamma; but this is not true if the particle is accelerating. Applying it without thought leads to the ``twin paradox''. However, the full integral relations will be valid even if the two particles are accelerating (so that $\beta(\tau)$ ). You will need to evaluate these relations to solve the twin paradox for one of your homework problems.

Finally, I want to note (without discussing it further at this time) that proper time dilation leads to a relativistic correction to the usual doppler shift. Or should I say that the non-relativistic doppler shift is just a low velocity limit of the correct, time dilated result.

Now that we have some notion of what an infinitesimal time interval is, we could go ahead and try to defince 4-dimensional generalizations of momentum and energy. First, however, we will learn how velocities Lorentz transform.