Two twins leave the same event and meet again at the same later event, each carrying a clock. This sim draws both journeys on a spacetime diagram and counts each clock's proper time, so you can compare what the two clocks read when the twins meet.

What's on screen

• Alice's worldline (blue): a straight line up x = 0.
• Bella's worldline (red): out at speed v to the turnaround Eₜ, then back to E₂.
• "Now" line (dashed amber): sweeps up once, through the events that are simultaneous in the frame shown. A dot marks where it crosses each worldline, with a τ counter: that twin's proper time so far.
• Readings panel: the totals, and the numbers for the frame you are viewing.

Press ⌖ on a row to see where that control sits in the sim.

Fixed: T = 10 yr of Alice's time from E₁ to E₂, and c = 1 (years and light-years). One sweep takes about 11 s.

A worldline is a twin's path through spacetime. A clock carried along it records the proper time τ. With c = 1, a straight stretch that covers Δt in time and Δx in distance has interval Δs² = −Δt² + Δx² and proper time Δτ = √(−Δs²). A bent path is a sum of straight pieces; a smooth one is an integral:

Alice stays at x = 0, so Δx = 0 and her clock simply counts the coordinate time. From E₁ to E₂ that is T = 10 yr.

The interval is Δs² = −c²Δt² + Δx², so a timelike pair has Δs² < 0 and its proper time is Δτ = √(−Δs²)/c.

Each of Bella's legs lasts Δt = T/2 and covers Δx = vT/2, so each is worth (T/2)√(1 − v²) of proper time. Together:

Her path looks longer on the page, but in Δs² = −Δt² + Δx² sideways distance pushes Δs² towards zero, and Δτ = √(−Δs²) falls with it. The faster she goes, the further out Eₜ swings and the less her clock records.

The gap grows slowly at first and then quickly. At v = 0.10c it is only 0.05 yr (about 18 days); at v = 0.80c it is a full 4.00 yr.

Between two events, the straight, inertial worldline records the greatest proper time. Any bend between them lowers it.

• Alice is inertial from E₁ to E₂: one straight line, one inertial frame (S).
• Bella needs two inertial frames, S′ on the way out and S″ on the way back, and she switches between them at Eₜ.
• The bend belongs to her worldline, not to a frame. Redraw the picture in S, S′ or S″ and the corner at Eₜ is still there.
• The turnaround is instantaneous in this sim, and T√(1 − v²) has no acceleration term in it. The result comes from the shape of the path.

The sim's two boosts (c = 1) relabel every event:

• In S′, Bella is at rest on the way out and Alice moves at −v. Bella's return leg is faster there, with speed w, still below c:
• The reunion E₂ lands at t′ = γT, which is 12.50 yr at v = 0.60c, at x′ = −γvT. That is a coordinate time in S′, not the reading of either twin's clock.
• Alice's clock still reads τ_A = 10.00 yr and Bella's τ_B = 8.00 yr (at v = 0.60c). A boost preserves Δs² = −Δt² + Δx² along every piece of every worldline, so each total keeps its value.

In One-way + signal mode Bella never turns around. At E_send (t = T/2 in S) she radios Alice, and the light signal reaches Alice at E_recv. At v = 0.60c:

Ask what Alice's clock read at the moment Bella signalled. In S the answer is 5.00 yr. In S′ it is (T/2)/γ² = 3.20 yr. The two frames draw different "now" lines through E_send, and they meet Alice's worldline at different events. There is no second shared event, so there is no frame-free answer.

• Cosmic-ray muons. At rest a muon lasts 2.2 μs on average, enough to travel about 0.66 km. Take one made 15 km up at 0.994c: the trip takes about 50 μs by Earth clocks but only 5.5 μs by its own, about 2.5 lifetimes. That is why a useful fraction reach the ground.
• GPS. The satellites move at about 3.9 km/s, so speed alone slows their clocks by about 7 μs a day. Gravity, which this sim does not model, speeds them up by about 46 μs a day. Without a correction for both, position errors would grow by about 11 km a day.
• Muon storage rings. At CERN in 1977, muons circling at γ ≈ 29 lived about 64 μs instead of 2.2 μs. They were accelerating all the time, yet the lifetime matched the prediction from their speed alone.

Key points

• Proper time is Δτ = √(−Δs²), with Δs² = −Δt² + Δx² and c = 1, added up along the worldline.
• τ_B = T/γ falls as v rises, though her drawn path gets longer.
• Only Bella's worldline bends. A boost changes t and x but not τ, and "at the same time" depends on the frame.

Common mix-ups

• "Bella sees Alice moving, so the two twins are symmetric." Alice's worldline is straight in every frame. Bella's bend never disappears, and only she changes inertial frame.
• "A longer path on the page means more time for Bella." That is the Euclidean instinct. In Δs² = −Δt² + Δx² the spatial step has the opposite sign to the time step, so it lowers the proper time Δτ = √(−Δs²).
• "The acceleration at the turnaround is what slows her clock." The result T√(1 − v²) contains no acceleration. It comes from the shape of her path.

Things to try on your own

• Set v = 0.80c and check τ_B in the Readings panel against T√(1 − 0.80²).
• View S′ at v = 0.50c and compare Bella's return speed on the diagram with 2v/(1 + v²).
• In One-way mode at v = 0.30c, work out τ_B and the arrival time before you read them off.