I've added this section to point out the difference between the *separation* between two points in space and two events in space-time.

Consider the triangle in figure 4.12. It could be a triangle in space or space-time.

In space, any side of the triangle will be shorter than the sum of the lengths of the other two sides and in particular, $a \lt b + c$.

If we are in space-time, then the lines are all world-lines so that the separations between the events are time-like.

Now, $OA$ is the *world-line* of a stationary observer, $OBA$ is (could be) the *world-line* of a moving observer, in which case it follows that $a \gt b + c$.

To see this clearly, figure 4.13 shows how we might compare $a$ with $b + c$ as *spatial* lengths.

First draw a circle of radius $b$ about $O$ (dashed blue) and see where it intersects $OA$, at $B'$.

Then translate the line $BA$ around this circle to $B'A'$ as indicated.

Finally, draw a circle of radius $c$ about $B'$ (dashed green) and see where it intersects $OA$, at $C$.

Clearly, the *length* $OC = b + c$ is greater than the *length* $OA = a$

We can do the same thing in space-time, but now we are dealing with proper time, as opposed to spatial distance, in which case we need to use hyperbolae to calibrate times, not circles. This means that we cannot trust our eyes when comparing proper times in a space-time diagram.

Figure 4.14 shows how we might compare the *proper times*, $a, b + c$.

First draw a hyperbola of radius $b$ about $O$ and see where it intersects $OA$, at $B'$.

Then translate $BA$ to $B'A'$.

Finally, draw a hyperbola of radius $c$ about $B'$ and see where it intersects $OA$, at $C$.

Clearly, the proper time $OC = b + c$ is less than the proper time $OA = a$.