The Twin Paradox
One of the most famous puzzles in special relativity: two twins start together. One stays home, the other rockets away and returns. When they reunite, the traveling twin has aged less. This isn't symmetric — there's no paradox once you understand worldlines and proper time.
What You're Seeing
The Stay-at-Home Twin (Red)
This twin's worldline is a straight vertical line in spacetime. They're at rest, moving only through time. Their path through spacetime is the longest possible timelike path between the departure and reunion events.
The Traveling Twin (Blue)
This twin's worldline has three segments: outbound journey, turnaround (acceleration), and return journey. The path through spacetime is bent, creating a "shorter" worldline in the sense of proper time — they experience less elapsed time.
The Turnaround "Kink"
This is crucial! The traveling twin must accelerate to turn around. This acceleration breaks the symmetry between the twins. The stay-at-home twin never accelerates (stays in an inertial frame the whole time), but the traveler does. That's why they age differently.
Why Is There No Paradox?
The "paradox" comes from naively applying time dilation both ways: "Each twin sees the other's clock running slow, so how can one be younger?" The resolution is that the twins' situations are not symmetric. Special relativity applies in inertial frames, but the traveling twin changes frames during turnaround.
During the acceleration phase, the traveling twin is in a non-inertial frame where different rules apply. From the traveler's perspective during turnaround, the stay-at-home twin's clock appears to jump forward dramatically. This "jump" accounts for all the missing time.
The Mathematics
Proper time along a worldline is given by integrating the spacetime interval:
τ = ∫ √(1 - v²) dt
For the stay-at-home twin, v=0 always, so τ = t (they age normally with coordinate time). For the traveling twin, v≠0 for most of the journey, so the integral gives a smaller value. The faster you go, the less proper time elapses along your worldline.
In spacetime geometry, straight worldlines have the maximum proper time between two events (this is opposite to spatial geometry, where straight lines are shortest!). Any detour through spacetime reduces proper time — you age less.
Things to Try
Increase the Speed: Drag the velocity sliders higher. Watch how the age difference grows. At β=0.9, the effect is dramatic — the traveler ages only about half as much!
Longer Journey: Increase the distance to turnaround. Even at moderate speeds, a long journey creates significant age differences. This is how interstellar travel might work — you could reach distant stars within a human lifetime (your proper time), even if centuries pass on Earth.
Asymmetric Return: Make the outbound and return velocities different. Notice that what matters is the total proper time, not the symmetry of the journey. A slow outbound + fast return gives the same result as fast outbound + slow return if the total speeds average the same.
The Turnaround Duration: Shorten the acceleration phase to nearly instant. The age difference remains the same! The acceleration itself doesn't cause the aging difference — it's the time spent traveling at high velocity. Acceleration just breaks the symmetry.
Very Fast Turnaround: Set turnaround to 0.1 years at β=0.9. You get a sharp kink. This is the "instantaneous turnaround" approximation often used in textbooks. Real accelerations would be gentler, but the result is the same.
Real-World Examples
Muons in the Atmosphere
Muons created by cosmic rays high in the atmosphere travel at ~0.99c. They have a half-life of 2.2 microseconds, so classically they should decay before reaching the ground. But time dilation means they experience much less than 2.2 μs (in their frame), so many reach the surface. This has been measured countless times.
Particle Accelerators
Unstable particles in accelerators live much longer than their rest-frame half-lives. A pion with a 26 ns half-life can circle the Large Hadron Collider for microseconds because, in the lab frame, its clock runs slow.
GPS Satellites
GPS satellites orbit at ~14,000 km/h. They're moving slower than ground clocks (time dilation makes them age slower by ~7 μs/day), but they're also higher in Earth's gravitational field (general relativistic effect makes them age faster by ~45 μs/day). The net effect is ~38 μs/day fast, which GPS systems must correct.
Common Misconceptions
"It's just perspective"
No! When the twins reunite, there's a real, physical difference. The traveler has fewer wrinkles, has read fewer books, has experienced less subjective time. All observers agree on who aged less.
"The traveler is in a gravitational field"
Sort of, but this is a red herring in special relativity. You can understand the twin paradox entirely within SR by noting that the traveler changes inertial frames. Acceleration creates pseudo-gravitational effects, but the aging difference exists even for arbitrarily gentle accelerations.
"Both twins see each other's clocks run slow"
True while both are in inertial frames! But during turnaround, the traveler's frame changes, and the stay-at-home twin's clock appears to "catch up" and overtake. The symmetry is broken by acceleration.
Philosophical Implications
The twin paradox demolishes the Newtonian idea of absolute time. There is no universal clock ticking away in the background. Each observer carries their own clock — their proper time — and different paths through spacetime accumulate different amounts of proper time.
This means "now" is not just observer-dependent for distant events (relativity of simultaneity) but also that the amount of time that passes depends on your worldline. Time is not a universal parameter but a property of paths through spacetime.
In a very real sense, spacetime is the fundamental reality, and our division of it into "space" and "time" is frame-dependent bookkeeping. The traveling twin doesn't age less because time slows down — they age less because their worldline through spacetime is geometrically shorter (in the proper-time sense).