A clock carried along a path through spacetime reads the proper time τ of that path. This sim sends two travellers from one event to another along different worldlines, so you can compare their clocks and see what a clock's reading depends on.

What's on screen

• Blue line A: the straight worldline from E₁ to E₂.
• Amber line B: a worldline with one kink at the amber dot, which you can drag.
• Violet dots E₁ and E₂: the two shared events. They can be dragged too.
• Dashed lines: the light cone of E₁. A clock's worldline has to stay steeper than these.
• Proper time panel (sidebar): the clock reading τ of each worldline.

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

Fixed: c = 1, so Δ(ct) and Δx share one unit. One run lasts 6 s at 1×.

An event is a point (x, ct), and a worldline is the path a traveller takes through events. A clock carried along one straight segment reads

The signature is (−, +): the time term is negative, so a timelike interval has Δs² < 0.

A clock can only follow a segment that stays steeper than the light cone, |Δx| < Δ(ct). Then Δs² is negative and Δτ is real. With Δx = 0 the clock stays at one place and Δτ = Δ(ct).

Hold Δ(ct) fixed and widen Δx. Δx² enters Δs² with a plus sign, so Δs² rises towards 0 and Δτ = √(−Δs²) falls. On paper the slanted segment is longer than the vertical one; on the clock it is shorter.

• In S a segment with Δx/Δ(ct) = v/c has Δτ = Δ(ct)/γ, the familiar time dilation: S measures the moving clock running slow by γ.
• B's legs in the opening layout have v = 0.60c, so γ = 1.25 and Δτ = 2.50 / 1.25 = 2.00.
• With the kink at height ct = 2.5, moving it to x = 0, 1.50 and 2.40 gives each leg Δτ = 2.50, 2.00 and 0.70.
• As |Δx| approaches Δ(ct), Δτ goes to 0. Light itself has Δτ = 0.

A bent worldline is a chain of straight segments, and its clock reads the sum of their Δτ.

This is the triangle inequality turned round: in Δs² the space steps carry the opposite sign to the time steps, so two sides of a spacetime triangle sum to less than the third. In the opening layout τ_A = 5.00, while B reads 2.00 + 2.00 = 4.00. A second bend costs more: the sim's worldline C reads 2.88.

Put the kink at x = 0 and B lies along A, so the readings agree.

A second observer S′ moves along x at speed v relative to S, with β = v/c and γ = 1/√(1 − β²). The sim redraws every worldline in S′ using the boost below, with c = 1.

The boost mixes the time and space steps, so Δt′ changes with β. It also keeps Δs² = −Δ(ct)² + Δx² fixed, so Δτ = √(−Δs²) does not change.

• At β = 0.50 (γ = 1.155), B's legs take Δt′ = 2.02 and 3.75 in place of 2.50 and 2.50, yet each still reads Δτ = 2.00.
• A's Δt′ is 5.77 in S′, and its clock still reads 5.00. In the opening layout A is at rest in S, so τ_A happens to equal S's coordinate time there.
• Every observer sees the same kink in B's worldline. A and B cannot be swapped by changing frame.

• Cosmic-ray muons. A muon's own clock gives it a lifetime of about 2.2 μs. At 0.999c, γ = 22.4, so in the ground frame it lasts about 49 μs and travels roughly 15 km. Without dilation that distance would be under 1 km.
• GPS. Satellite clocks lose about 7 μs a day to their motion and gain about 45 μs a day from weaker gravity, a net 38 μs a day. Left uncorrected, that would put positions off by roughly 11 km a day.
• Flown clocks. In 1971, Hafele and Keating flew atomic clocks around the world on airliners. Compared with a clock that stayed home, they differed by tens to hundreds of nanoseconds, as motion and gravity together predict.
• The twin paradox. This is the same picture: the travelling twin's worldline is bent between two meetings, so that twin's clock records less.

Key points

• Along one segment, Δs² = −Δ(ct)² + Δx² and Δτ = √(−Δs²), with c = 1.
• At fixed Δ(ct), a wider Δx gives a smaller Δτ.
• A worldline's total is the sum of its segments' Δτ, and a bent path between two events records less than the straight one.
• Boosting to S′ changes Δt′ but not Δτ.

Common mix-ups

• "The path through space that is longer on the diagram must take more time." In Δs² the space step has the opposite sign to the time step, so the longer-looking path records less.
• "B ages less because B was the one moving." Motion is relative. What separates the travellers is straight versus bent.
• "Δτ depends on which frame you compute it in." It does not. Only the coordinate time Δt′ changes under a boost.

Things to try on your own

• Drag E₂ to (2, 5) with B's kink at (1.5, 2.5), and read both clocks. Check τ_A = √(−Δs²) = √21 ≈ 4.58 by hand.
• Switch to S′ and set v/c = −0.50. Compare B's two Δt′ with those at +0.50.
• Show worldline C and drag W₂ and W₃ onto A's line. Watch τ_C climb towards τ_A.