Alice stays put while her twin Bella runs to a marker and back, and the sim shows the trip from either twin's frame. It lets you compare what the two clocks read when the twins meet again, the standard test of what a clock measures in special relativity.

What's on screen

• Alice (cyan) with her clock τ_A above her, in years.
• Bella (amber) with her clock τ_B. The small tag under each name gives that twin's speed in the current frame.
• Turnaround pylon and the label above the corridor: the distance from the start line to the marker.
• Status chip and banners: which frame you are in, what each twin is doing, and the moments of the turnaround and the reunion.
• Clock readings at reunion (sidebar): the two proper times, and their difference, for your settings.

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

Fixed: c = 1, so distances are in light-years and times in years. Bella runs at one constant speed on both legs and turns instantly. A run takes about 12 s at 1× speed, whatever Δt_A is.

The twins part at the start event and meet at the reunion event. Every observer agrees on these two events and on what each clock reads at the second one. A clock records its own proper time τ: the time that passes along its own worldline between them.

• Alice never leaves x = 0, so dx = 0 and her proper time is the time of her frame: τ_A = Δt_A.
• Bella moves at constant v on each leg, so dx = v dt and dτ = dt√(1 − v²) = dt/γ. Her two legs together last Δt_A in Alice's frame.

• On each straight leg the accounts are symmetric. Each twin measures the other's clock running slow by the same factor 1/γ.
• The number of inertial frames is not symmetric. Alice needs one, S, from start to reunion. Bella needs two: S′ on the way out and S″ on the way back, because she reverses at the marker.
• Which twin is moving depends on the frame. Which worldline is straight does not: Alice's is straight in every inertial frame, Bella's has a corner in every one.
• In Bella's frame she is at rest and the corridor slides past. The start line and the marker are at rest in Alice's frame, so Bella measures the distance between them contracted from L to L/γ, and she measures Alice's clock running slow by 1/γ.

Time dilation compares one clock with the other frame's time. It does not say which clock reads less when the clocks are brought together.

From τ_A = Δt_A and τ_B = Δt_A/γ, the difference between the two readings at the reunion is:

• γ depends on v alone, so at fixed v the gap is proportional to Δt_A.
• At fixed Δt_A the gap grows with v, but it stays below the whole trip: any v < c leaves τ_B > 0.

At the opening settings, Bella's outbound leg lasts 3.00 yr on her clock. In S′ Alice's clock runs slow by 1/γ, so it advances 3.00/γ = 1.80 yr. The return leg adds another 1.80 yr: 3.60 yr in all, yet Alice's clock reads 10.00 yr at the reunion.

• Switching from S′ to S″ changes which event on Alice's worldline Bella counts as simultaneous with her turn.
• Alice's clock does not skip anything. The jump is in Bella's bookkeeping: the clock face in her frame shows the reading she assigns to "now".
• The sim draws the leg-by-leg count, without the switch, as a dashed ghost clock beside Alice's clock.

• Between two events, the unaccelerated worldline records the greatest proper time. Alice's is that straight line. Any other path between the same events, Bella's included, records less (in flat spacetime).
• This is the reverse of distance in a plane, where the straight line is the shortest. The opposite signs of the time and space terms in ds² = −dt² + dx² make the difference.
• In this sim the turn is instantaneous and takes no time, so the gap builds up entirely along the straight legs. The turn selects Bella's path; it does not add to either clock.
• For the same Δt_A, a higher v makes Bella's path bend more, and τ_B falls.

• GPS. Satellite clocks move at about 3.9 km/s, which on its own makes them run slow by about 7.2 μs a day. Gravity at orbit height acts the other way and is larger (about 46 μs a day), so the system is built with both corrections.
• Cosmic-ray muons. A muon lives 2.2 μs on average, which is 660 m at nearly c. Made about 15 km up, with γ ≈ 20 it travels about 13 km on average, so many reach the ground. In the muon's frame it is the atmosphere that is contracted.
• Muon storage rings. At CERN, muons circulating at γ = 29.3 lasted 29.3 times longer than at rest, agreeing with the prediction to about 0.1%.

Key points

• τ_A = Δt_A and τ_B = Δt_A/γ, so the gap is Δt_A(1 − 1/γ), in units where c = 1.
• The gap is proportional to Δt_A and grows with v.
• Each leg is symmetric, but Alice uses one inertial frame and Bella uses two.
• The frame switch moves Bella's "now" on Alice's worldline: relabelling, not a tick of Alice's clock.

Common mix-ups

• "Each twin sees the other's clock slow, so the readings must match." That holds on each straight leg, but Bella also changes frames.
• "The acceleration makes Bella younger." It selects her path. The same turn on a trip twice as long gives twice the gap.
• "Alice really ages 6.40 yr in an instant." Her clock ticks steadily. What jumps is the reading Bella assigns to "now".

Things to try on your own

• Use the v slider to find the speed at which τ_B is half of τ_A, then check it with γ = 2.
• In Bella's frame set v = 0.95c and note the jump in Alice's clock at the frame switch. Compare it with v²Δt_A.
• Set v = 0.10c and compare the reunion readings with the 0.80c trip.