Problem — No signals faster than light

Problem

No signals faster than light. Two events separated in space can appear simultaneous in one frame and not in another. Why does this relativity of simultaneity not allow sending signals faster than light? Distinguish spacelike and timelike intervals.

Solution

The spacetime invariant. Between two events one defines the interval

$\Delta s^2 = c^2\,\Delta t^2 - \Delta x^2,$

which is the same in all inertial frames. Its sign classifies the pair of events and does not depend on the observer.

Spacelike intervals ($\Delta s^2 < 0$, i.e. $|\Delta x| > c\,|\Delta t|$). These are events too far apart in space to be connected by a signal at speed $\le c$: they lie outside the light cone. For these events the temporal order is relative: changing frame, one may precede the other or be simultaneous. But precisely because no light signal or slower one can connect them, this ambiguity does not carry information: there is no cause preceding an effect.

Timelike intervals ($\Delta s^2 > 0$, i.e. $c\,|\Delta t| > |\Delta x|$). These are events connectable by a signal at speed $\le c$: they lie inside the light cone. For these, the temporal order is invariant: if in one frame the cause precedes the effect, it does so in all frames. Causality is preserved.

Why no signals faster than $c$. A superluminal signal would connect two events separated by a spacelike interval. But then there would exist a frame in which the effect precedes the cause: one could, in principle, receive an answer before having sent the question, generating causal paradoxes. This is relativity’s prohibition:

$\ev{\text{causal order is invariant only for timelike intervals}}$

The relativity of simultaneity only concerns spacelike events, which cannot communicate: no paradox, no FTL signal.

Links

Topics: Relatività ristretta Concepts: Relatività della simultaneità · Invariante spazio-temporale Skills: Costruzione di diagrammi di Minkowski