Time Dilation: Light Clock

Time Dilation

Teach me

Speed

Guided Inquiry

1 · The light clock

Here’s our clock. A single photon bounces between two mirrors a height h apart inside a train moving at speed v, and one round trip is one tick. Observer A rides along with the mirrors, observer B stands on the platform, and light travels at c for both of them.

2 · Timing one tick

Let’s start with the obvious question. A and B time the same round trip of the same photon. What do you think — who records the longer time for it?

Now switch to the platform frame and follow one bounce with your eye.

3 · Where the difference comes from

So from the platform, B’s reading runs ahead of A’s. But why? What do you think produces that difference?

Follow one bounce from the platform and you’ll see it: the photon meets the mirror further down the track than it left.

4 · Raising the speed

Now let’s turn up the speed. At β = 0.10 one tick stretches by only 0.5 %. The speed slider goes all the way to β = 0.95. Make a guess — by roughly what factor does a tick stretch there?

Try it: drag β from 0.10 up to 0.95 and watch γ and the two clock readings.

5 · A friend says…

"A's clock runs slow in B's frame, so A's is the truly slow one."

Your friend sounds pretty sure. So let’s ride with A instead and look back at B’s clock. What do you think A finds?

One thing to bear in mind: the panel shows one comparison only, A’s clock against the platform’s synchronised clocks. This scene doesn’t draw A’s own row of clocks.

6 · Proper time

Let’s give that shorter reading its name. It’s the proper time Δτ — the time read by the one clock present at both events — and every other inertial frame records Δt = γ·Δτ.

And here’s a teaser for later: mutual time dilation between two inertial frames is symmetric, but the twin paradox isn’t — and only because one twin accelerates.

Listen

CONTROLS

Reference frame

Parameters

β = v/c 0.50

h (mirror separation) 1.00

Statistics

β = v/c 0.500

γ (Lorentz) 1.155

t_A (train clock) 0.000

t_B (platform clock) 0.000

θ (beam from vertical) 30.0°

L / L₀ (contraction) 0.866

Elapsed Time: t_A vs t_B

Same photon, two views: a zigzag from the platform, a straight bounce inside the train.

Invariant speed of light

Train frame A — clock at rest

Platform frame B — Pythagoras (half-trip)

Solve for the round-trip time

Lorentz factor (frame-invariant)

Proper time = invariant interval

The Physics

Time dilation · what this sim shows, the physics behind it, and every control

At a glance

One photon bouncing between two mirrors makes a clock, and this sim rides that clock on a moving train. It lets you read the clock from the train and from the platform, and compare what the two observers record for the same tick.

What you'll learn

How the time A and B record for one tick are related, and why.
How that relation depends on the train's speed β and on the mirror gap h.
What proper time is.
What each observer concludes about the other's clock.

One tick is one round trip of the photon between the mirrors.

What's on screen

Train frame (blue, observer A): the view that rides with the mirrors.
Platform frame (pink, observer B): the view from the track, with the train moving past. In 3D you pick one view at a time. In 2D both are drawn, one above the other.
Mirrors and photon: two mirrors a height h apart, and one photon bouncing between them.
Statistics and the elapsed-time graph: γ, the two clock readings t_A and t_B, the beam angle θ and the length ratio L/L₀.

See itLook at the clock from inside the train. Watch for: the photon bouncing between the two mirrors, one round trip per tick.

The controls

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

Control	What it changes	Range
2D / 3D	Switches between the 3D scene and the 2D side view.	2 views opens in 3D
Reference frame	In 3D, the camera: inside the carriage (Train frame) or on the platform (Platform frame). In 2D both frames are always drawn.	2 frames opens on Train
β = v/c	The train's speed as a fraction of the speed of light.	0–0.95 opens at 0.50
h (mirror separation)	The gap between the mirrors, and so how long one tick is.	0.5–2.5 opens at 1.00
Statistics	β, γ, the clocks t_A (train) and t_B (platform), the beam angle θ from vertical, and L/L₀. The clocks are running totals in the sim's own time unit (c = 1), not nanoseconds. Their rows are coloured pink (t_A) and blue (t_B), the opposite of the frame colours, so read them by label.	readout panel
Elapsed time graph	Both clock readings plotted against time.	graph
Pause / Play	Freezes or resumes the animation and both clocks.	top bar
Speed	Plays the animation faster or slower on screen. The clock ratio does not change.	0.25×–4× opens at 1×
Reset	Restarts the animation and both clocks from zero.	top bar

Fixed: the train moves at constant velocity along the track, and the mirrors are separated vertically, at right angles to the motion. In 2D the clocks also restart when the train leaves the panel.

The platform sees a longer path

Light moves at the same speed c for both observers. Let t_A and t_B be the times A and B record for one round trip. A rides with the mirrors, so the photon leaves and returns to the same spot on the train (Δx_A = 0). A's clock is at both events, so A reads the proper time:

Train frame A: one clock at both events.

B sees the mirrors move on while the light is in flight. In half a trip the light covers the hypotenuse of a right triangle: one leg is the gap h, the other is how far the train moves in that half trip.

Platform frame B, half a trip.

The hypotenuse is longer than the vertical leg h. At the same speed c, light needs more time for it, so t_B > t_A.
This is not signal delay. Each observer reads an event on a clock right beside it.

See itWatch the same photon from the platform. Watch for: a zigzag path in the bottom panel against a straight bounce in the top, and the platform clock t_B running ahead of the train clock t_A.

Each tick stretches by γ

Solving the triangle gives the platform's time for one tick. The gap h cancels, so the ratio is set by β = v/c alone:

γ is the Lorentz factor, and γ ≥ 1 for every speed.

β h

These sliders move the sim too.

Speed v/c	Lorentz factor	One tick is longer on the platform by
0.10	1.005	0.5 %
0.50	1.155	15 %
0.95	3.20	220 %

The growth is far from linear: negligible at everyday speeds, then steep as β approaches 1. The sim stops at β = 0.95.

See itPush the train to β = 0.95. Watch for: γ reading 3.20, and the platform reading t_B growing about 3.2 times as fast as the train reading t_A.

Only the speed sets the stretch

The gap h sets the length of a tick, t_A = 2h/c, but it cancels in t_B/t_A. A bigger gap makes both observers' ticks longer by the same factor.
The clock itself does not matter. Any clock at rest on the train keeps step with the light clock. If it did not, A could tell the train is moving, which the principle of relativity forbids.
The sim also reports two geometric readings. The beam leans from vertical by θ = arcsin β in B's frame (the triangle below). L/L₀ = 1/γ is the carriage length that B measures, against its rest length L₀. That is a separate effect, which needs both ends read at the same instant in B's frame.

The triangle gives sin θ = β.

See itRun β = 0.80 with h = 0.5, then with h = 2.5. Watch for: the photon bouncing five times more slowly in the second run, while the clock ratio t_B/t_A reads the same in both.

Each frame finds the other's clock slow

From A's frame, B is the one moving at speed v, so A finds B's clocks slow by the same γ. Both statements hold. They do not contradict each other, because each compares different pairs of events, and the two frames disagree about which events are simultaneous. Take a clock at rest on the train (x′ = 0). The Lorentz transformation then gives Δt = γΔt′:

The primed frame is the train, so Δt′ = t_A and Δt = t_B.

Both frames do agree on the interval between the two events. The interval is s² = −c²Δt² + Δx². For the two ends of a tick it is −c²t_A² in A's frame, and −c²t_B² + v²t_B² in B's. Equating them gives t_B = γ t_A again.

Here c Δτ = √(−s²), which for clock A (Δx = 0) is c t_A.

The sim draws one comparison only: A's clock against the platform's clocks. It cannot show B's clock as A reads it, so this section has no See-it.

Where it matters

Cosmic-ray muons. A muon lives 2.2 μs at rest. At β = 0.998 (γ = 15.8) the ground sees it last about 35 μs, so it flies about 10 km on average instead of 0.66 km. In this idealised picture about a quarter of the muons made 15 km up reach the ground, against about 1 in 10¹⁰ without time dilation.
Particle accelerators. Muons stored in a ring at γ ≈ 29 last about 64 μs, against 2.2 μs at rest.
GPS. A satellite's orbital speed alone makes its clock lose about 7 μs a day. A separate, larger effect from gravity pushes the other way, and the system corrects for both.

Go deeper: the muon told in both frames

One journey, two accounts: a stretched lifetime on the ground, a contracted atmosphere on board.

In the muon's own frame the atmosphere is contracted by the same γ, so the 15 km trip is 0.95 km long and takes about 3.2 μs. Both accounts give the same physical outcome.

Summary

In one line. The platform records one tick of the train's light clock as γ = 1/√(1 − β²) times the train's own proper time, because the light covers a longer path at the same speed c.

Key points

A's clock is beside both events, so it reads the proper time t_A = 2h/c, the shortest reading.
The platform records t_B = γ t_A, where γ depends on β alone.
Each observer finds the other's clock slow by the same γ.

Common mix-ups

"The train's clock is slow because its mechanism runs down." Every clock on the train keeps step with the light clock. What differs is B's measured light path.
"The delay is light taking time to reach B's eye." Each reading is taken at a clock beside the event, so signal delay is already removed.
"A's clock is the truly slow one." A finds B's clock slow by the same γ. No frame is privileged.

Things to try on your own

Set β = 0.80 and predict the platform clock when the train clock reads 6.000. Then check it in the Statistics panel.
Work out the β at which γ = 1.5, set the slider close to it, and read γ.