The twin paradox

The setup

In the twin paradox, one twin stays on Earth while the other travels at high speed to a distant star and returns. When they reunite, the traveling twin is younger. This seems paradoxical because each twin might claim the other was moving and therefore the other's clock ran slow.

The resolution is that the twins do not play symmetric roles. The Earth twin remains approximately in one inertial frame, while the traveling twin changes inertial frames during turnaround.

Proper time along worldlines

Each twin's aging is the proper time along that twin's worldline:

$au=int d au=int sqrt{1-v(t)^2/c^2},dt$

The stay-at-home twin follows a nearly straight inertial worldline between departure and reunion. The traveling twin follows a bent worldline: outbound, turnaround, and inbound.

In Minkowski spacetime, the straight inertial path between two timelike-separated events maximizes proper time. Therefore the traveling twin ages less.

Simple calculation

Suppose the traveler moves at speed $v$ for coordinate time $T/2$ outbound and $T/2$ inbound in Earth's frame, ignoring the short acceleration time. The Earth twin ages

$Delta t_E=T$

The traveler ages

Delta au_T=rac{T}{gamma}

where

gamma=rac{1}{sqrt{1-v^2/c^2}}

Since $gamma>1$, the traveler is younger by

T-rac{T}{gamma}

Role of acceleration

Acceleration marks the change of frames, but the age difference is not caused only by acceleration as a local clock-slowing effect. The main issue is the total spacetime path. Acceleration allows the traveling twin to leave one inertial frame and return, making reunion possible.

Even if the acceleration period is brief, the age difference can be large if the outbound and inbound speeds are high and the trip is long.

Relativity of simultaneity explanation

During the outbound leg, the traveler says Earth's clocks run slow. During the inbound leg, the traveler also says Earth's clocks run slow. But at turnaround, the traveler's definition of simultaneity changes dramatically. The slice of distant Earth time considered simultaneous jumps forward.

This simultaneity shift accounts for why the traveler still predicts the Earth twin will be older at reunion.

Experimental reality

The twin paradox is not merely fictional. Particle lifetimes, atomic clocks on airplanes, and satellite clocks all confirm path-dependent proper time. Different worldlines between events accumulate different amounts of proper time.

No contradiction

The apparent paradox comes from applying inertial-frame symmetry to a situation that is not symmetric. The traveler changes frames; the Earth twin does not in the simplified version. More deeply, the proper time integral along each worldline is different.

The big idea

The twin paradox is resolved by spacetime geometry. Aging is proper time along a worldline. The traveling twin follows a non-inertial, shorter-proper-time path between departure and reunion, while the stay-at-home twin follows a longer-proper-time path. The result is real, measurable, and non-contradictory.