Theory of relativity

The twin paradox

After a round trip, the traveler can be younger than the stay-at-home twin because they follow different worldlines and the traveler changes frames while turning around.

Model the traveler as moving at speed v over distance D, turning around, then returning at the same speed. Earth-frame time is 2D/v; the traveler accumulates less proper time on both legs.

\Delta t_{\mathrm{Terre}}=\frac{2D}{v},\qquad \Delta\tau_{\mathrm{voyageur}}=\frac{2D}{v}\sqrt{1−\frac{v^2}{c^2}}

Definition: Definition

Proper time is accumulated along each person’s worldline. The stay-at-home twin is approximately inertial; the traveler accelerates and changes inertial frames at turnaround, so their situations are not symmetric.

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Physical meaning

The acceleration can be brief. The point is not simply “who moves,” but that the proper-time integrals along the two worldlines differ. An instantaneous turnaround idealization ignores propulsion and gravitational details.

Example: Worked example

A traveler goes to a point 4 light-years away at 0.8c and immediately returns. How long passes on Earth and for the traveler?

Solution

Earth: $2D/v=8/0.8=10$ years. Since γ=5/3, the traveler experiences $10/γ=6$ years, neglecting turnaround time.

Example: Proper time on a round trip

An astronaut travels to a star $4$ light-years from Earth at $0.8c$ , turns around quickly, and returns. Neglect acceleration time; compare elapsed Earth time with the astronaut’s clock.

Solution

Earth measures $2(4)/0.8=10$ years. Since $γ=1/\sqrt{1-0.8^2}=5/3$ , the traveler’s total proper time is $10/γ=6$ years. The difference occurs because the worldlines joining the same meetings have different proper lengths.

A person’s elapsed time is the length of their worldline between meetings in the Lorentzian metric: $\tau=\int\sqrt{1-v^2/c^2}\,dt$ in the Earth frame. Between the same reunion events, the stay-at-home twin follows nearly one inertial path, while the traveler follows two velocity segments and changes frame at turnaround. Turnaround acceleration can be brief; the decisive comparison is the proper-time integral along each complete worldline.

Neglecting acceleration time merely simplifies the calculation; the age difference depends on the complete worldlines, not on an instantaneous turnaround.

Quick check

In the round-trip model, who changes inertial frames at turnaround?

Which quantity is invariant under Lorentz transformations?

References

Edwin F. Taylor, John Archibald Wheeler (1992). Spacetime Physics
Robert Resnick (1968). Introduction to Special Relativity