Faster than light?

Imagine that all points of a directed straight line l' in K' are of black color. Let P'0 be a point on l', and let every point P' of l', just by coincidence, switch its color from black to red at coordinate time t'P' = d'(P'0, P') / w', where d' is a signed distance and w' is an arbitrary positive constant. The resulting speed at which the redness property, i.e. the red ray (or the tip of the red ray, in point-like terms), propagates along l' is w'. Curiously, w' can be greater than c, and the formulas for velocity addition remain valid even in that case.

Using the notations of the previous section:

(Theorem) w' > c if and only if w > c.

If w'x' = -c2 / v, then Δt = 0 for any Δx, which means that wx (and thus w too) is "infinite", at least in the sense that the red color appears at the very same moment along a whole straight line in K. This seems to violate the continuity assumption of trajectories.

If w'x' < -c2 / v, then Δt < 0 for any Δt' > 0, which means that the red ray travels "backward in time", in the sense that the points of l' are becoming red in the opposite order when viewed from K. This seems to violate causality at first sight.

Note: whenever w'x' < -c, there always exists a v < c such that w'x' < -c2 / v holds, from which it follows that no signal can travel faster than light.

However, it's important to emphasize that in these examples neither rigid objects nor signals, but only a state of space, made up of independent point-like properties, is propagating. A red ray that is faster than light cannot be "sent", it can only arise either due to coincidence or pre-arrangement. Nevertheless, the phenomenon is well-defined and can be considered when speculating about hypothetical particles moving faster than light.

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