Twin Paradox — One-Way Signal

Source

Lecture: 12 — Time dilation & the twin paradox
Simulation: One-way call / signal (no-return variant)
Concept: A moving twin's proper time runs slow — no turnaround needed; a light signal reveals the asymmetry cleanly.
Serves: Reading Minkowski diagrams · proper time along a worldline · light-cone invariance · Lorentz frame change.
Inquiry: What time does Alice's clock read when Bella's call arrives, and how does that compare to Bella's clock at emission?
Misconception: “Both twins age the same because motion is relative.” The one-way variant shows time dilation is real even without a turnaround.

Pedagogy principles applied

ID	Principle	How it shows up here
A1	One concept per sim	Just the one-way call: proper time along Bella's line vs. Alice's clock at reception.
A3	Formal tools OFF by default	Equations live behind ⚛︎ The Physics; the diagram is readable first.
C1	Every number earns its place	Only the three events' (x,ct), τ_B, τ_A, and γ are shown — each is read by a step.
D6	Scene evolves per step	Bella's line, E_C, and the light signal appear as the student advances.
H2	Misconceptions structural	The “both age the same” idea is encoded as a wrong choice; feedback only appears if the student picks it.
J2	Frame change is explicit	Selecting S′ relabels axes, redraws the grid, and rewrites every event coordinate via Lorentz.
J5	Light stays at 45°	The signal is 45° in both S and S′ — light-speed invariance is visible, not asserted.

Twin Paradox

Teach me

Speed

Minkowski diagram — Frame S (Alice's rest frame)

Alice

Bella

Light signal

Guided Inquiry

1 · A call from a receding ship

Here’s our setup. Alice stays on Earth at x = 0. Bella leaves the same event E₁ at constant speed and never turns back.

Somewhere along her worldline, at event E_C, Bella reads her own clock and sends that reading home as a light pulse. Both worldlines are straight: nobody accelerates anywhere in this scene. Keep that in mind as we go.

2 · Bella's clock at the call

Let’s start with the call itself. Bella recedes at v = 0.60. In Alice’s frame she calls at t_C = 5.00. What do you think Bella’s own clock reads at that event?

τ_B is live now — you can read it beside E_C and in Clock readings.

3 · When the call lands

Now follow the call home. The pulse leaves E_C at x = 3.00 and runs back to Alice at 45° — light speed, in every frame. What do you think Alice’s clock reads at the reception event E_R?

τ_A is live now, and E_R sits at ct = 8.00 on Alice’s worldline.

4 · Seeing versus ageing

Here’s where it gets interesting. A call reporting 4.00 lands when Alice’s clock reads 8.00 — a factor of 2.00. But in Alice’s frame, by what factor do you think Bella’s clock actually runs slow?

Try raising v/c: γ and the ratio τ_A/τ_B pull apart further.

5 · Bella's point of view

Now let’s be fair to Bella. She has the same right to call herself at rest; in her frame S′ it’s Alice who recedes at −v. Described in S′, whose clock do you think runs slow?

Switch the Frame to S′ and watch: E_R moves to ct′ = 10.00 while τ_A holds at 8.00.

6 · What a one-way call settles

Let’s step back and see what we’ve learned. The boost to S′ moves every coordinate — E_C to ct′ = 4.00, E_R to ct′ = 10.00 — yet both clock readings survive it: 4.00 at the call, 8.00 at reception. A proper time, after all, is what a carried clock records.

The steady factor between them, k = √((1+v)/(1−v)) = 2.00, depends only on v. And since nobody accelerates and the twins never meet again, nothing here crowns a younger twin — for that, you need a turnaround.

Listen

Frame

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Controls

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Bella's speed v/c 0.60

Call time t_C (Alice frame) 5.0

Tip: you can also drag E_C directly along Bella's worldline in the diagram.

Events (active frame)

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E₁ (x, ct)(0.00, 0.00)

E_C (x, ct)(3.00, 5.00)

E_R (x, ct)(0.00, 8.00)

Clock readings

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γ = 1/√(1−v²)1.250

τ_B (Bella at E_C)4.00

τ_A (Alice at E_R)8.00

τ_A / τ_B2.000

Proper times are frame-invariant — τ_B and τ_A do not change when you switch frames.

Lorentz factor

Events in S (Alice's frame)

Bella's proper time at E_C

Alice's clock at reception E_R

Lorentz boost (S → S′)

Light-cone invariance

The Physics

Twin paradox, one-way call · what this sim shows, the physics behind it, and every control

At a glance

Bella flies away from Alice at constant speed and never turns back. At one moment she reads her clock and calls home on a pulse of light, and the sim compares that reading with Alice's clock when the call arrives. It separates three things that are easy to run together: how a moving clock runs, how long light takes to cross the gap, and what each frame says about the other's clock.

What you'll learn

How a clock's reading depends on the worldline it follows through spacetime.
How to work out what Alice's clock reads when the call arrives.
How the ratio of the two readings is related to the Lorentz factor γ.
What changes when the same events are described from Bella's frame, S′.

A sketch of the diagram: Alice's worldline (cyan), Bella's (amber), and the call as a 45° light pulse from E_C to E_R.

What's on screen

Alice (cyan): at rest at x = 0, so her worldline is vertical.
Bella (amber): leaves E₁ at speed v. A dot climbs her line as the animation runs.
Light pulse (dashed): the call, a 45° ray from E_C to Alice. The faint dashed cross is the light cone of E₁.
Event badges E₁, E_C, E_R: (x, ct) coordinates, plus a clock reading τ at E_C and E_R.
Events and Clock readings panels: the coordinates in the active frame, then γ, τ_B, τ_A and τ_A/τ_B.

See itSend the call at the opening settings (v = 0.60, t_C = 5.0). Watch for: the amber dot climbing Bella's line, then the dashed pulse running from E_C back to Alice's line.

The controls

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

Control	What it changes	Range
Bella's speed v/c	How fast Bella recedes from Alice, as a fraction of c. It sets the lean of her worldline.	0.05–0.95 opens at 0.60
Call time t_C	When Bella makes the call, on Alice's clock in frame S. It moves E_C along Bella's worldline.	1–9 opens at 5.0
Drag E_C	Grab the E_C dot on the diagram and slide it along Bella's worldline. It moves the t_C slider too.	0.2–9.5
Frame	Redraws the diagram and the Events panel in Alice's rest frame (S) or in Bella's (S′).	S / S′ opens in S
Events	The (x, ct) coordinates of E₁, E_C and E_R in the active frame.	panel
Clock readings	γ, Bella's clock at E_C (τ_B), Alice's clock at E_R (τ_A), and τ_A/τ_B. During the Guided Inquiry the answers stay hidden until you commit a prediction.	panel
▶ Play / ⏸ Pause	Runs or freezes the animation: Bella climbs her worldline, then the pulse flies home. It loops.	top bar
Speed	How fast the animation plays on screen. The events do not move.	0.25×–4× opens at 1×
↻ Reset	Rewinds the animation to the start and pauses it.	top bar

Fixed: c = 1, so x and ct share one scale and light always runs at 45°. Bella moves at constant velocity and never turns back.

Proper time is the length of a worldline

A clock counts the proper time along its own worldline. Take c = 1, so ct and t are the same number. Alice sits at x = 0, so her clock reads the coordinate time Δt. Bella's clock reads

Proper time along a straight worldline at speed v.

Between E₁ and E_C, Alice's frame measures Δct = t_C and Δx = v t_C. In Δs² = −(Δct)² + Δx² the spatial part has the opposite sign to the time part, so it shrinks −Δs², and with it Δτ = √(−Δs²).
So for the same Δct, a tilted worldline records less proper time than a vertical one. Nothing is wrong with Bella's clock: it is reporting its own path.

Bella's leg at the opening settings. The teal curve is every event 4.00 of proper time from E₁.

As in the lectures, the interval is Δs² = −(cΔt)² + (Δx)², negative for timelike pairs, and the proper time is Δτ = √(−Δs²)/c.

See itKeep t_C fixed and change the lean of Bella's line. Watch for: τ_B changing while t_C stays put. The closer her line leans to the 45° ray, the further τ_B falls below t_C.

Light needs time to bring the call home

Bella sends the call at E_C, at x = v t_C. Light covers one unit of x per unit of ct in every frame, so the call is a 45° line in S and in S′. Its interval is zero:

The call is a null interval, the same in S and S′.

Alice is at rest in S, so her clock reading at E_R is the coordinate time of E_R: when the call left, plus the time light needs to cross the gap.

Alice's clock at the moment the call arrives.

Key idea. With c = 1, the call arrives later than it left by exactly the x-distance it has to cover. The faster Bella recedes, the bigger that gap.

See itSend a call at v = 0.80, t_C = 4.0. Watch for: the pulse crossing the gap from E_C to Alice's line at 45°, so E_R sits above E_C by as much as E_C sits to the right of Alice.

What Alice sees is not how fast Bella ages

Compare the two readings the call joins: τ_A at reception and τ_B at emission. Their ratio is the factor k.

k is the Doppler factor. It depends on v only.

Time dilation, γ, is only one factor in k. The other, 1 + v, is the growing light-travel delay.
If Bella sent a call every 1.00 of her own time, Alice would receive them every k of hers, because each call starts from further away.
So the ratio of the two readings is how the call looks. γ is how fast Bella's clock runs.

v/c t_C

These sliders move the sim too.

See itCompare γ with τ_A/τ_B at v = 0.30 and then v = 0.90. Watch for: τ_A/τ_B staying above γ, and the gap between them widening as v grows.

Each frame finds the other's clock slow

In S′, Bella is at rest and Alice recedes at −v. The boost gives the coordinates of every event there:

The boost the Frame buttons apply.

It changes where the events sit, not what a clock reads at an event: τ_B and τ_A are the same in S and S′. What differs is which events each frame calls simultaneous.

Opening settings, drawn in S. Each frame's "now" through E_C meets Alice's worldline at a different event.

In S, Alice reads 5.00 when Bella reads 4.00. Bella's clock is slow.
In S′, Alice reads 3.20 when Bella reads 4.00. Alice's clock is slow, by the same γ.
The two "now" lines meet two different Alice events, so both statements hold. The twins never meet again, so the sim cannot name a younger one. A reunion needs a turnaround.

See itSwitch to Bella's frame, S′. Watch for: E_R in the card showing S′ coordinates, with τ_B and τ_A keeping the values they had in S.

Where it matters

Cosmic-ray muons. A muon's mean life of 2.2 μs would carry it about 660 m. At γ ≈ 22 its mean decay length in the ground frame is roughly 15 km, so many muons made 15 km up still reach sea level.
GPS. A satellite's speed alone makes its clock run about 7 μs a day slow. Gravity pulls the other way and wins, and the clocks are pre-adjusted. Without the corrections, position errors would grow by kilometres a day.
Doppler shifts. Light from a source receding along the line of sight arrives stretched by k = √((1+v)/(1−v)), the same k as the call, and spectroscopy of receding sources uses it.

Go deeper: why a round trip does settle it

Between two fixed events, the straight worldline has the most proper time. A twin who turns around follows a bent worldline between the same two events, so she records less.
The turnaround does not cause the difference. It is what lets the twins meet again, so their readings can be compared at one event.

Summary

In one line. A clock reads the proper time of its own worldline, and the call carries that reading across a gap that light needs time to cross. The ratio of the two readings is the Doppler factor k, not the dilation factor γ.

Key points

τ_B = t_C/γ: the proper time of Bella's tilted worldline is less than t_C.
τ_A = t_C(1 + v): the time of emission plus the light-travel time.
τ_A/τ_B = k = γ(1 + v) = √((1+v)/(1−v)). It depends on v only, not on t_C.
A boost moves coordinates, not clock readings. A one-way call names no younger twin.

Common mix-ups

"The ratio of the two readings is the time dilation factor." It is k = γ(1 + v). The dilation is γ alone, and the factor 1 + v is the light-travel delay.
"Each frame says the other's clock is slow, so one of them must be wrong." They compare different pairs of events, so both hold.
"Switching to S′ changes the clock readings." Only the coordinates move. τ_B and τ_A read the same.

Things to try on your own

Set v = 0.95 and t_C = 9, then check in Clock readings that τ_A/τ_B equals γ(1 + v).
Drag E_C along Bella's line and note which readings change and which do not.
Set v = 0.05. Compare how far τ_B sits below t_C with how far τ_A sits above it.