Suppose we have a particle moving with a velocity **v** in a given
coordinate system
. In a time
(in that system) it moves
. Then its **invariant infinitesimal interval** is
(ds)^2 = (c dt)^2 - d**x**^2 = c^2 dt^2 (1 - &beta#beta;^2).
In the *particular* frame where the particle is at *rest* (
)
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
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
). 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.