Light Clock — Geometric Derivation of Time Dilation

Source

Lecture: 11 — Relative length and time
Simulation: Simulation 2 — Light clock & geometric derivation of time dilation (lecture-demo mode)
Concept: The time-dilation factor γ emerges geometrically from applying the Pythagorean theorem to the diagonal light path inside a moving light clock.
Serves: Derive time dilation (via the equivalent light-clock argument); explain the frame dependence of measured time intervals; distinguish proper time (invariant, one clock present at two events) from coordinate-dependent time.
Inquiry: Rest-frame bounce → moving-frame diagonal → Pythagoras → t = γ t₀, built up in five lecturer-driven additive stages while the animation and v-slider stay live.
Misconception: "Moving clocks run slow — so to find which frame measures the proper time, just figure out which one is moving." Neither frame is uniquely "the moving one"; the proper time is recorded by the single clock present at both events.

Pedagogy principles applied

ID	Principle	How it shows up here
A1	One concept	Every element serves the single light-clock derivation of γ.
A2	Declutter	Only h, v, t₀, t and γ are shown — no β, no length contraction, no second clock.
A3/A4	Formal layer optional	Equation chain is OFF by default; revealed at stage 5 or in the ⚛︎ The Physics sheet.
B1	Real terms	"mirror", "light pulse", "rest-frame round-trip time" — no invented jargon.
B3	Symbols defined	Each of h, t₀, t, γ, v carries its definition on first use; t₀ is visually distinct.
C1	Numbers referenced	v, γ, t₀, t are all read across the staged build; speed selector hidden.
C2	Numbers in context	"v = 0.60 c (lab frame)", "t = 1.25 t₀", "t₀ = 2h/c (proper time, clock at rest)".
D1	Visualization dominant	The two-clock canvas is the largest element; panels are secondary.
D3	Entities labelled	Mirrors, light pulse, and the three triangle legs labelled adjacent to themselves.
D5	No clipping	Triangle and clocks rescale uniformly so nothing clips at v = 0, 0.5, 0.9, 0.95 c.
D7	Playable from load	Animation runs and the v-slider works immediately; stages are additive emphasis only.
D8	Labels not narration	Canvas carries symbols; explanatory prose lives in the side panels.
E1	Faithful animation	Pulse moves at c in both views; the moving round trip takes exactly γ× longer.
E3	No decoration	One pulse disc, mirrors, geometric lines — no glow trails or flares.
G2	Invariants labelled	h and c are invariant; t₀ is labelled the proper time.
H1	Misconception structural	Not a standing card — a collapsed Discussion prompt and a wrong prediction choice.
H3	Persona states it	The friend's claim is voiced; the rebuttal reframes proper time as event co-location.
J1	Consistent notation	γ, t₀, t, h, v, c used throughout with no silent substitution.
J3	Correct model	γ recomputed from v each frame; (ct/2)²=h²+(vt/2)² → t = γ t₀.

Light Clock

Teach me

Speed

Scroll to zoom · double-click to reset

Guided Inquiry

1 · A clock made of light

Let’s meet the clocks. Each one is a pair of mirrors held a fixed distance h apart, with a single light pulse bouncing between them; one round trip is one tick, and a clock at rest ticks in t₀ = 2h/c. Alice’s clock stands still, while Bella’s identical clock is carried past to the right at speed v. Keep in mind that everything on screen is drawn as Alice sees it.

2 · The pulse's path

Let’s set Bella’s clock moving, to the right at v = 0.60 c. In Alice’s frame, what path do you think the pulse traces as it goes from the bottom mirror up to the top one and back?

Now press ▶ Play and watch the pulse trace out its path.

3 · How long the trip takes

So that slanted path is longer than the straight 2h Alice’s own pulse covers. Here’s the key question: in Alice’s frame, how long does one round trip of Bella’s pulse take, compared with t₀?

Keep an eye on the γ and t readouts — they’re live, and t is Bella’s tick as measured in Alice’s frame.

4 · Across the motion

Before we build the triangle, there’s one thing to check. Lengths along the direction of motion do contract. The construction treats Bella’s mirror separation — measured across the motion — as the same h Alice has. What do you think: does that separation contract as well?

And there’s the triangle, drawn for you now; its vertical leg is that unchanged h.

5 · Raising the speed

Now let’s turn up the speed. At v = 0.60 c one tick of Bella’s clock takes 1.25 t₀ in Alice’s frame. Make a guess — at v = 0.90 c, how long does one tick take?

Try it yourself — drag v toward c: γ reads 1.25 at 0.60 c, 2.29 at 0.90 c and 3.20 at 0.95 c.

6 · Pythagoras, and the name for it

Now let’s put it all together. Half a trip is a right triangle with legs h and vt/2 and hypotenuse ct/2, so (ct/2)² = h² + (vt/2)², and solving for t gives t = (2h/c)/√(1 − v²/c²) = γ t₀. That factor γ is the Lorentz factor, and t₀ — read by the one clock present at both the emission and the return — is the proper time. You’ve just derived time dilation with nothing more than Pythagoras.

Listen

Controls

▾

v (Bella's speed in Alice's frame)

v = 0.60 c γ = 1.25

t = 1.25 t₀

Construction stage

Discussion

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"A friend says: 'Moving clocks run slow — so to find which frame measures the proper time, just figure out which one is moving.'"

From the simulation: is either frame uniquely "the moving one"?

Neither frame is uniquely "the moving one." Each frame sees the other's clock as moving — the first postulate guarantees no frame is special.

The proper time t₀ is measured by the clock that is present at both events (the same clock records the pulse leaving and returning). It's not about which frame "is moving" — it's about which single clock is at both events.

Pythagoras on the light triangle (one half-trip)

Solve for t

Lorentz factor

Live values

Symbols

h = mirror separation (frame-invariant)
c = light speed (invariant, same in every frame)
t₀ = 2h/c = Alice's round-trip time — the proper time of the at-rest clock (one clock, two events)
t = Bella's round-trip time as observed in Alice's frame
v = Bella's speed in Alice's frame (v > 0 → rightward)
γ = 1/√(1 − v²/c²) ≥ 1 for 0 ≤ v < c (frame-invariant — both frames agree on γ)

The Physics

The light clock and time dilation · what this sim shows, the physics behind it, and every control

At a glance

A light clock is one pulse of light bouncing between two mirrors, the simplest clock whose rate can be worked out from first principles. This sim sets an identical clock moving past one at rest, then builds the geometry that relates their ticks.

What you'll learn

How a light clock ties the length of a tick to the constancy of c.
How the light's path in Alice's frame compares with its path in Bella's.
How Bella's tick, timed by Alice, depends on v, and how to compute it.
What proper time is, and which clock reads it.

Two identical clocks, mirrors a distance h apart. Everything on the canvas is drawn as Alice sees it.

What's on screen

Alice's clock (cyan mirrors, left): at rest, with its pulse and the tick time t₀ = 2h/c underneath.
Bella's clock (amber mirrors): the same clock moving right at v, restarting when it reaches the edge.
Bella's pulse path (rose): traced as the pulse moves; the Construction stage sets how much is drawn.
Grid: h is always two squares; as v rises the whole drawing, grid included, scales down to fit.
Readouts (Controls panel): v, the Lorentz factor γ, and t, one tick of Bella's clock as Alice times it, in units of t₀.

See itRun the two clocks side by side at v = 0.60 c. Watch for: Alice's cyan clock on the left, Bella's amber clock moving right, one pulse in each.

The controls

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

Control	What it changes	Range
v slider	Bella's speed in Alice's frame, in units of c. The clocks restart when it moves.	0–0.95 c opens at 0.60 c
Readouts	The live values of v, γ (the Lorentz factor, derived in section 05) and t, one tick of Bella's clock in Alice's frame. The Guided Inquiry hides γ and t until you have answered the matching card.	display
Construction stage	How much geometry is drawn. Rest clock: both clocks and the pulse trails. Diagonal: Bella's pulse path built up trip after trip. Triangle: the first round trip closed into a triangle. Labels: its three sides named. Pythagoras: the equation written on the canvas.	5 stages starts at Rest clock
Scroll zoom (on the canvas)	Scroll down to zoom the drawing out, so more ticks fit before Bella's clock reaches the edge. Zooming restarts the clocks; double-click restores the default.	0.22×–1×
Discussion	A common claim about proper time, with a Reveal rebuttal button. It opens on the last Guided Inquiry card and is always open in Free Exploration.	panel
⏸ Pause / ▶ Play	Freezes both pulses mid-flight, or sets them going again.	top bar
Speed	Plays the animation faster or slower on screen. v, γ and t do not change.	0.25×–4× opens at 1×
↻ Reset	Returns v to 0.60 c, the stage to Rest clock and the zoom to default, and restarts the mode you are in.	top bar

Fixed: one tick of Alice's clock lasts 2.2 s on screen at 1× speed. The mirror gap h is the same for both clocks at every v.

Why the moving pulse leans

Bella carries her clock past Alice at speed v. Each pulse leaves the bottom mirror and has to reach the top mirror directly above it.

In Bella's frame her mirrors are at rest, so the pulse goes straight up and straight back. One tick takes t₀ = 2h/c.
In Alice's frame the top mirror moves on to the right while the light is in flight. To meet it, the pulse must travel on a slant, so each half-trip is a diagonal and the whole trip is a zig-zag.
Both descriptions are correct. They are the same two events, emission and arrival, described in two frames.
The speed of light is c in Alice's frame too (the second postulate). The pulse's velocity there has a sideways part v, the mirrors' own speed, and an upward part √(c² − v²).

See itRun both clocks at v = 0.60 c. Watch for: Alice's pulse running straight up and down, while the rose trail of Bella's pulse zig-zags to the right.

The light triangle

Take one half-trip in Alice's frame, from emission to the top mirror. Call the whole round trip t, as Alice times it. This half-trip lasts t/2 and has three sides.

The sim's Labels stage at v = 0.60 c. The lengths are worked out in the next section.

h, vertical. The mirror gap is the same in both frames: lengths across the motion do not change. If they did, two equal rings sliding past each other would each have to pass inside the other.
vt/2, horizontal. How far the mirrors travel in time t/2.
ct/2, the slant. How far the light travels in that time, since its speed is c in Alice's frame.
Motion is horizontal and h is vertical, so the corner between them is a right angle.

See itDraw the labelled triangle at v = 0.60 c. Watch for: the dashed vertical leg h, as tall as the gap in Alice's clock, with the right-angle mark at the corner.

Why the tick is longer

Pythagoras on the triangle ties the three sides together:

One half-trip, in Alice's frame.

Solving for t, the time of one tick of Bella's clock as Alice measures it:

The mirror gap h cancels, so the result depends on v alone.

The light's path, ct/2, is longer than h, and its speed is the same c. A longer path at the same speed takes longer, so t is greater than t₀.
Nothing inside Bella's clock has changed. The longer tick comes from the constancy of c.

See itShow the Pythagoras stage at v = 0.60 c. Watch for: the overlay writing (ct/2)² = h² + (vt/2)², and its γ and t matching the readouts.

How the stretch grows with speed

The Lorentz factor depends only on v, and both frames agree on its value:

This slider moves the sim too.

γ is not proportional to v. The root closes on zero as v approaches c, so γ is 1.25 at 0.60 c, 2.29 at 0.90 c and 3.20 at 0.95 c, the sim's top speed.
At low speed the effect is small. For v much less than c, γ − 1 is about v²/2c²: at 0.1 c a tick is longer by only 0.5%.

See itSpeed Bella up to v = 0.90 c. Watch for: the Tick readout rising to 2.29 t₀ (it was 1.25 t₀ at 0.60 c), and Bella's pulse path leaning much further.

Proper time, and what each frame measures

Emission and return are two events. The interval between them has the same value in every frame. The interval is s² = −c²Δt² + Δx².

The same result, from the invariant interval.

Proper time is the time on a single clock present at both events. Bella's clock is at both, so she reads t₀. Alice needs two separated clocks to time the same events, and gets t.
The Lorentz transformation gives the same result, with Δt Alice's time and Δt′ Bella's: Δt = γ Δt′.
Any clock carried with Bella must agree with her light clock. If it did not, she could detect her own uniform motion.
The effect is reciprocal. Bella times Alice's clock as moving and finds Alice's ticks longer by the same factor: each frame times the other's clock as slow. The sim draws only Alice's view.

See itOpen the Discussion panel and reveal its rebuttal. Watch for: the rebuttal saying that proper time belongs to the one clock present at both events.

Where it matters

Cosmic-ray muons: both arrows are mean flights, measured in the ground frame.

Cosmic-ray muons. A muon's mean lifetime at rest is 2.2 μs, which is 0.66 km of flight at nearly c. In this simple estimate (made about 15 km up, γ = 20), the ground frame gives a mean flight of 13.2 km, so about 32% reach the ground. Without dilation, only about 10⁻¹⁰ would.
GPS. At 3.9 km/s a satellite clock loses about 7 μs a day to this effect alone. Gravity pushes the other way by about 45 μs, and the net is corrected in the satellite's clock.
Everyday speeds. A clock on an airliner at 250 m/s falls behind by only about 11 μs per year of continuous flight. Atomic clocks flown round the world (Hafele and Keating, 1971) showed the effect, together with gravity's.

Go deeper: the muon in its own frame

In the muon's frame its clock runs normally and its mean flight is still 0.66 km. The 15 km of air is contracted instead, to 15/20 = 0.75 km.
Both frames give the same survival fraction, e^−0.75/0.66 ≈ 32%. Time dilation and length contraction describe the same events from two frames.

Summary

In one line. Light travels at c in every frame, so a longer path means a longer tick: Alice times Bella's tick as t = γ t₀, with t₀ = 2h/c.

Key points

Bella's pulse goes straight up and down in her frame, and on a slant in Alice's.
The triangle h, vt/2, ct/2 gives t = γ t₀, with γ = 1/√(1 − v²/c²).
γ is 1 at rest and grows ever faster near c: 1.25, 2.29 and 3.20 at 0.60 c, 0.90 c and 0.95 c.
t₀ is a proper time, read by the one clock present at both events. The effect is reciprocal.

Common mix-ups

"The frame that is moving is the one that measures proper time." Either frame can be called moving. Proper time belongs to the single clock present at both events.
"Each frame finds the other's clock slow, so they contradict." They compare different pairs of events, so there is no contradiction.
"The mirror gap h shrinks too." Lengths change only along the motion. h is across it, so it is unchanged.

Things to try on your own

Set v = 0.80 c, read t from the Tick readout, and check it against 1/√(1 − 0.8²).
At v = 0.60 c, Alice times 10 s between two ticks of Bella's clock. Work out what Bella's own clock reads between them.
At v = 0.95 c, step through the five stages and compare the base vt/2 with the height h.