The Twin Paradox — Proper Time

Source

Collection: Wiki SR simulations (v1) — SuperSTEM special-relativity engine.
Concept: Proper time is the time a clock reads along its own worldline; the straight (inertial) worldline accumulates the most proper time, resolving the twin paradox.
Serves: Shows two twins' worldlines on a Minkowski diagram: Sam's straight line and Tara's bent round trip. As a marker advances in coordinate time, each clock accumulates dτ = dt/γ along its path, so the reunited twins read different ages.
Inquiry: Who ages more? At what rate does the moving clock tick? What physical fact breaks the symmetry between the twins?
Misconception: That the traveler ages more, or that the twins stay the same age because “motion is relative” — ignoring that only Tara's turnaround switches inertial frames.

Pedagogy principles applied

ID	Principle	How it shows up here
H1	Misconception is structural, not a text card	The “Tara is older” and “same age” errors are wrong prediction choices; the rebuttal appears only if that choice is picked.
D6/D7	Guided inquiry reveals in stages	Five sidebar cards run fundamental → full: predict → observe the run → verify the 1/γ rate → uncover the turnaround asymmetry → resolve.
A2	Fewest objects	Opens with two worldlines and a turnaround marker; no vestigial readouts that no step reads.
C	Every number earns its place	β, γ, coordinate time and each twin's proper-time age are read or compared by a step; the age difference is the quantity the activity is about.
Physics honesty	Animate only true physics	Proper time accumulates as a marker advances in coordinate time along timelike worldlines (honest). Velocity β and distance d are static sliders — never auto-swept, since SR frames are inertial.

The Twin Paradox

Speed

Spacetime diagrampress Play to run the round trip

Sam (stays home — straight worldline) Tara (traveler — bent worldline) turnaround (frame switch)

Guided Inquiry

1 · Predict — who ages more?

Sam stays home; Tara flies to a distant star and returns. Both born the same day. When they reunite at the same place, who has aged more?

2 · Run the round trip

Press ▶ in the top bar. A marker sweeps up in coordinate time while each twin's clock accumulates its own proper time along its worldline. Watch the two ages diverge.

3 · What did the clocks read?

At the reunion, compare the two proper times. Along the whole trip, Tara's clock ran at what rate compared with Sam's coordinate time?

4 · Why isn't it symmetric?

“Motion is relative” — so why is there a definite answer at all? What physically distinguishes the twins?

5 · The longest proper time wins

Coordinate time between depart and reunion is the same for both (it's read off Sam's frame). What differs is the proper time along each path: dτ² = dt² − dx²/c². The bent worldline logs less, so Tara returns younger. Turn on ∑ Formal for the algebra, or drag the sliders to explore. Hit Finish to tuck this away.

Controls

▾

Turnaround distance d = 4.0 light-years

1 ly8 ly

Travel speed β = v/c = 0.60 (Sam's frame)

0.10c0.95c

Readouts

▾

β = v/c—

γ (time-dilation)—

coord. time (Sam's frame)—

Sam's age (proper time)—

Tara's age (proper time)—

age difference—

Lorentz factor

Sam's proper time (straight — longest)

Tara's proper time

The stay-at-home worldline is a straight (inertial) line — and in relativity the straight worldline has the LONGEST proper time. Tara's worldline is bent: her engine burn at the turnaround switches her reference frame. That turnaround is the physical asymmetry that resolves the “paradox” — the two twins are not interchangeable.