Did your researches with the slot at §14.14 yield the ‘important discovery’ I hinted at. Perhaps you should try again.

Revise as necessary on isovals–a minimum of §§14.1, 14.5 and 14.9. If you didn't ask yourself whether a branch of an isoval could be a world line – representing something moving on the ID universe – consider the question now.

A time-like isoval couldn't be a world line. For its slope is always shallower than 45°, and this would imply a causal influence travelling faster than light, which is impossible (§§10.10–12).

But a branch of a space-like isoval, being always steeper than 45°, could be a world line. Check with the slot that it represents something always moving slower than light. What sort of motion will it represent?

World lines of inertial observers are straight. So this curve must represent the motion of something that is non-inertial. What does ‘noninertial’ mean?

The meaning is in the definition of §5.2–which please reread. The test particle moves away.

But when there is no gravity–as we are assuming (§5.5)–an inertial observer has constant speed relative to any other inertial observer (§5.6). Therefore a non-inertial observer moves with changing speedhe is accelerated in the ordinary sense of the word. So this space-like isoval represents accelerated motion.

Note that acceleration has a quite different status from speed (§§1.1–3, 5.1–2). Speed is purely relative. But everybody can see whether an observer and his test particle stay together or move apart; and so all must agree whether he is accelerated or not.