Problem — Which twin diagram

Problem

Which diagram? Two twins: one stays on Earth, the other sets off on a trip at $v = 0{,}8c$, returning after a proper time $\tau$. Which spacetime diagram correctly represents the situation from Earth’s point of view? (Epstein 2002)

Solution

What the diagram must show (as seen from Earth).

• The twin who stays on Earth has constant spatial coordinate $x = 0$: their world line is the horizontal line along the $t$ axis (in blue). ✓ (the same in all three)
• The travelling twin moves away at constant speed $v = 0{,}8c$, then turns around and comes back: their world line is a “triangle”-shaped broken line that leaves and returns to the Earth twin’s axis.

Decisive physical constraint: the slope. The traveller’s world line represents motion with $v = 0{,}8c < c$. Its inclination relative to the time axis must be that of $0{,}8c$: less steep than the light cone ($v = c$) but with a well-defined, realistic slope.

• Diagram A: intermediate slope, consistent with $v = 0{,}8c$ (the apex reaches an $x$ compatible with the speed and flight time). ✓ plausible candidate — but let’s compare.
• Diagram C: the apex is too low, corresponding to too small a speed (the traveller moves away too little): it does not represent $0{,}8c$.
• Diagram B: the broken line has the correct opening for $v = 0{,}8c$ — the traveller reaches the correct distance in the round-trip time, with a slope smaller than that of light. It is the only one geometrically consistent with $v = 0{,}8c$ and with returning to the same point.

Answer.

$\ev{\text{Diagram B}}$

Only diagram B has the traveller’s world line with the correct inclination (speed $0{,}8c$, sub-luminal) leaving from the origin and returning to the Earth twin’s world line, closing the round trip.

Links

Topics: Relatività ristretta Concepts: Dilatazione dei tempi Skills: Costruzione di diagrammi di Minkowski