Mass-Energy Equivalence and the Twin Paradox

Special relativity is not limited to the geometry of spacetime: it also links mass and energy through the most famous formula in physics, E = m c². This equation means that a mass m, even at rest, has a considerable rest energy, since c² is a gigantic number, about 9 × 10^16 in SI units. A tiny amount of mass converted into energy therefore releases an enormous amount of energy, which explains the power of nuclear reactions, where a tiny fraction of the initial mass of the nuclei is turned into radiated energy.

More generally, the total energy of a moving object is E = gamma × m c², and we recover E = m c² when the object is at rest, that is, when gamma = 1. This formula also shows why nothing with mass can ever reach the speed c: it would have to be given infinite energy, since gamma diverges as v tends to c.

Let us now turn to the famous twin paradox. One twin stays on Earth while the other travels to a distant star at v = 0.8 c, then comes back. On Earth, ten years pass between departure and return. The traveling twin, however, ages only ten divided by gamma, or about six years, because their proper clock runs more slowly as seen from Earth. The apparent paradox comes from wrongly believing the situation to be symmetric: each twin should see the other aging more slowly. The resolution lies in a real asymmetry: only the traveling twin changes inertial frame during the turnaround, undergoing an acceleration, whereas the twin who stayed on Earth remains in a single inertial frame from start to finish.

        ct
         ^
         *  (twins reunited)
         #\
         # \   <- traveling twin: bent path
         #  \
         #   *  (turnaround, change of frame)
         #  /
         # /
         #/
         *-------------------------> x
         O (common departure)

(# : vertical worldline of the twin who stayed on Earth, a single frame)
(the traveling twin follows a bent path: two distinct inertial frames)
(between the same two events, the inertial (straight) worldline has the longest
 proper time, even though it looks shorter on the diagram)

The diagram above illustrates the two worldlines: the vertical one of the twin who stayed on Earth, and the bent one of the traveling twin, whose change of direction in spacetime marks precisely the moment of the turnaround. Remember the misconception to avoid: it is not speed itself that creates the asymmetry, but the change of inertial frame experienced by the traveling twin alone.