Relativity That ClicksEpisode 2 / γ — a moving clock runs slow, set only by β

We folded speed into β. Next, time itself — from a single principle, "c is the same for everyone," time dilation follows

γ — a moving clock runs slow, set only by β A fast-moving clock ticks slowly. How much it slows is a single function set only by β, γ = 1/√(1−β²).
There is one seed — "the speed of light c is the same as seen by anyone." After that, it comes out from the Pythagorean theorem alone.

Tools you'll need: the Pythagorean theorem, β from Episode 1, and "c is the same for everyone" This episode's function: γ = 1/√(1−β²)

In Episode 1 we folded speed into the unitless $\beta=v/c$. What moves in this episode is time itself. A fast-moving clock ticks more slowly than a stationary one — the famous "time dilation." It seems like magic, but there is only one seed to it — the speed of light $c$ is the same value as seen by every observer (the very thing we called "the core" in Episode 1's honest line). Grant that one point, and the rest comes tumbling out from nothing more than the middle-school Pythagorean theorem: a single function that expresses how much the clock slows — the Lorentz factor $\gamma=1/\sqrt{1-\beta^2}$. Time dilation, length contraction, and $E=mc^2$ from the next episode onward — this $\gamma$ takes all of it on. Let's derive it with our own hands.

01A moving clock runs slow — but why?

Hearing "when it moves, time slows down," you might think the clock's mechanism is thrown off by the shaking. Wrong. Whatever the clock's mechanism, time itself slows. To grasp the reason by the shortest path, we use the simplest possible clock — the light clock. Because its workings are determined by the speed of light $c$ alone, the principle "c is the same for everyone" acts on it directly.

02The light clock — starting from "c is the same for everyone"

The light clock is this. Between two mirrors held a distance $L$ apart, light bounces up and down. One round trip (strictly, one one-way trip) counts as "one tick." For a stationary light clock, the light just goes straight up and down — the one-way time is $\Delta t_0=L/c$. This is the clock's own intrinsic tick.

Now, run this light clock sideways at speed $v$. To us watching from the ground, as the light goes up it is also carried sideways with the clock, so it appears to travel a diagonal path. A diagonal is longer than straight up and down. And yet — the speed of light is the same $c$ whether it is running or not (this is the principle). If you cover a longer distance at the same speed, it takes more time. So one tick $\Delta t$ seen from the ground is longer than the clock's own $\Delta t_0$ — this is what "a moving clock runs slow" really is.

03Pythagoras gives us γ

All we do is draw a triangle. During one tick of time $\Delta t$ seen from the ground — the light travels $c\,\Delta t$ diagonally (hypotenuse), the clock moves $v\,\Delta t$ sideways (base), and the mirror spacing stays $L=c\,\Delta t_0$ (height). It's a right triangle, so, by Pythagoras:

Try it — hypotenuse² = base² + height² $$(c\,\Delta t)^2=(v\,\Delta t)^2+(c\,\Delta t_0)^2$$

Rearrange both sides and solve for $\Delta t$ ($\beta=v/c$)

$$(c^2-v^2)\Delta t^2=c^2\Delta t_0^2\ \Rightarrow\ \Delta t=\frac{\Delta t_0}{\sqrt{1-v^2/c^2}}=\frac{\Delta t_0}{\sqrt{1-\beta^2}}$$

There it is. Ground time $\Delta t$ is $1/\sqrt{1-\beta^2}$ times the clock's proper time $\Delta t_0$. This multiplier is the one that has a name.

This episode's function — the Lorentz factor γ

$$\gamma=\frac{1}{\sqrt{1-\beta^2}}\qquad\Rightarrow\qquad \Delta t=\gamma\,\Delta t_0$$

$\gamma$ is a pure function set only by the unitless $\beta$. A moving clock's own one tick $\Delta t_0$ appears stretched by a factor of $\gamma$ on the ground — that is, a moving clock ticks slow by a factor of $1/\gamma$. $\gamma\ge1$, and it goes to infinity as $\beta\to1$.

04Let's play with it — running the light clock

The figure below is a running light clock. Between two mirrors (the amber horizontal lines), a point of light travels back and forth. Raise $\beta$ with the slider and the clock moves faster to the right, and the light's path stretches out diagonally (we leave a faint trail). Because the speed of light is constant, the more the diagonal lengthens, the more the up-and-down round trip — the tick — visibly slows. At $\beta\to1$ the light lies almost flat sideways and never reaches the ceiling — time nearly stops. At the lower right, we place the progress of ground time and clock time side by side.

Figure: a running light clock. Light bounces between two mirrors = a tick. Raise β and the light's path stretches diagonally; since c is constant, the tick slows (at β→1 the light lies flat and time stops). Ground time vs clock time = 1 : 1/γ

β = v/c 0.60

light clock (mirrors + bouncing light) the light's diagonal trail

05The face of γ — 1 in everyday life, but it matters for muons and GPS

Let's look at a few values to see how large $\gamma$ gets. At everyday $\beta$, $\gamma$ is all but indistinguishable from 1.

The values of γ, and the everyday approximation

For small β (Taylor expansion)

$$\gamma\approx 1+\tfrac12\beta^2$$

· Shinkansen $\beta\approx3\times10^{-7}$: $\gamma-1\approx5\times10^{-14}$ (negligible)
· $\beta=0.87$: $\gamma\approx2$ (time runs at half speed)
· $\beta=0.99$: $\gamma\approx7.1$ / $\beta=0.999$: $\gamma\approx22$

The slowing kicks in as $\beta^2$, so in everyday life ($\beta\approx0$) it is completely invisible. That is why for thousands of years no one noticed — the same reason as "Newton is β→0" from Episode 1.

Real cases where it matters — muons and GPS Muons, born high in the atmosphere from cosmic rays, are short-lived and by rights should decay before reaching the ground. But they fly at $\beta\approx0.999$ ($\gamma\approx22$), so from the ground their lifetime is stretched by a factor of $\gamma$ and they do reach the surface — passing through the palm of your hand every second. The atomic clocks on GPS satellites, if you don't correct for the special-relativistic slowing due to their motion (plus the general-relativistic shift due to gravity), drift by about 38 μs per day, throwing the position off by as much as 10 km. $\gamma$ is not a textbook formula but engineering in motion.

And the same $\gamma$ also takes on length contraction. A moving object appears contracted by a factor of $1/\gamma$ along its direction of motion (Lorentz contraction). Time stretches by $\gamma$ and length shrinks by $1/\gamma$ — time and space change in opposite directions by the same $\gamma$. This feeling that "time and space change as a set" is the foreshadowing of the next episode's spacetime interval. The unit-bearing time and length change all over the place (stage machinery) for each observer, yet behind it all a single $\gamma$ runs the whole show.

◇ ◇ ◇

The honest line — the trap of "which one slows?"

The derivation of $\gamma=1/\sqrt{1-\beta^2}$ from the light clock, $\Delta t=\gamma\Delta t_0$ (time dilation), the $1/\gamma$ length contraction, and the lifetime extension of muons and the GPS correction (about 38 μs/day) are all established physics, confirmed in experiment and in practice. The one and only assumption the derivation used is "$c$ is the same for all observers" — this is not a matter of units but a principle of relativity (an experimental fact).

But a crucial caution — "runs slow" is relative. If "from my view your clock runs slow," then symmetrically "from your view my clock runs slow" also holds (if both merely move at constant velocity, there is no absolute fact of which is slower). This mutuality, which looks like a contradiction, is untangled by the fact that "simultaneity" differs for each observer (Episode 5). In the famous twin paradox, what really creates a difference is that one of them accelerates and comes back — the acceleration breaks the symmetry. And what is truly invariant, independent of observer, is the proper time τ ticked by a person's own clock, which is the star of next episode's spacetime interval.

Practice problems (solvable with this episode's formulas alone)

When $\beta=0.6$, what is $\gamma$? At what fraction of the ground clock's rate does the moving clock run?
Show answer
$\gamma=1/\sqrt{1-0.36}=1/\sqrt{0.64}=1/0.8=1.25$. The moving clock runs at $1/\gamma=0.8$ the rate (it advances only 0.8 s while 1 s passes on the ground).
To make $\gamma=2$, what should $\beta$ be?
Show answer
$2=1/\sqrt{1-\beta^2}\Rightarrow\sqrt{1-\beta^2}=1/2\Rightarrow1-\beta^2=1/4\Rightarrow\beta=\sqrt{3}/2\approx0.87$. About 87% of the speed of light.
For the Shinkansen ($\beta\approx3\times10^{-7}$), estimate $\gamma-1\approx\tfrac12\beta^2$. Can the time dilation be felt?
Show answer
$\tfrac12(3\times10^{-7})^2\approx4.5\times10^{-14}$. Even over a day (about 90,000 s) that's $4\times10^{-9}$ s = 4 nanoseconds. Utterly imperceptible. Because everyday life is β→0.
What is the "one and only assumption" used in the light-clock derivation? If the speed of light differed for each observer, would γ come out?
Show answer
The one assumption is "the speed of light c is the same for all observers." If light sped up for the running clock, it could cover the longer diagonal path faster and no slowing would result. The invariance of c is the source of time dilation.

Episode 2 summarytime dilation is a function of β alone, γ

A fast-moving clock runs slow by a factor of $1/\gamma$. That $\gamma$ comes out from just one principle, "$c$ is the same for everyone," and the Pythagorean theorem. Run a light clock sideways and the light takes a longer diagonal path, but at the same speed $c$ — so one ground tick $\Delta t$ is longer than the intrinsic $\Delta t_0$, and solving $(c\Delta t)^2=(v\Delta t)^2+(c\Delta t_0)^2$ gives $\Delta t=\gamma\Delta t_0,\ \gamma=1/\sqrt{1-\beta^2}$.

$\gamma$ is a pure function of the unitless $\beta$ alone: $\gamma\approx1$ in everyday life ($\beta\approx0$), infinite near the speed of light. Time stretches by $\gamma$ and length shrinks by $1/\gamma$ — time and space, as a set, are governed by a single $\gamma$. It is engineering in motion, proven by muons and GPS. But "runs slow" is relative, and what is truly invariant is each observer's own proper time τ — which leads to the next episode's spacetime interval. Unit-bearing time and length are each observer's stage machinery; what binds them is the β-function γ.

Next time — Episode 3 "The spacetime interval — the one invariant that no observer can split apart." Time stretches by $\gamma$ and length shrinks by $1/\gamma$ — all over the place for each observer. So is there nothing at all that is the same no matter who measures it? There is. Combine time and space with that subtraction, $s^2=(c t)^2-x^2$ — the spacetime interval alone agrees for every observer. This quantity, joined not by Pythagoras's "+" but by "−", is the true geometry of spacetime. The unit-bearing $t$ and $x$ are stage machinery; only $s$ is invariant — the backbone of relativity comes into view.

This document is Episode 2 of the "Relativity That Clicks" series, a reading piece for physics-loving high-schoolers and undergraduates. The time dilation $\Delta t=\gamma\Delta t_0$ derived from the principle of the constancy of the speed of light and the light clock, the Lorentz factor $\gamma=1/\sqrt{1-\beta^2}$, the $1/\gamma$ length contraction along the direction of motion, the small-$\beta$ expansion $\gamma\approx1+\beta^2/2$, the lifetime extension of atmospheric muons, and the relativistic correction of GPS (net about 38 μs/day from special plus general relativity) are all established, standard physics. That time dilation is mutual (symmetric) between inertial frames and there is no "absolute fact of which is slower," that the asymmetry of the twin paradox arises from acceleration, and that the observer-invariant quantity is the proper time $\tau$ (Episode 3's spacetime interval) are stated in the "honest line" in the main text. The figure is a schematic of a running light clock's path and time dilation; the apparent speed of the light path is a device for on-screen display (the actual speed of light is invariant). — To print, use your browser's "Print" and "Save as PDF" (in the print version the animation and answers are static/hidden). Adjacent episodes: Episode 1, β / Contents.

Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, run the light clock with the β slider and the light path stretches diagonally as the tick slows. "Show answer" opens each solution.

γ — a moving clock runs slow, set only by β A fast-moving clock ticks slowly. How much it slows is a single function set only by β, γ = 1/√(1−β²).There is one seed — "the speed of light c is the same as seen by anyone." After that, it comes out from the Pythagorean theorem alone.

01A moving clock runs slow — but why?

02The light clock — starting from "c is the same for everyone"

03Pythagoras gives us γ

04Let's play with it — running the light clock

05The face of γ — 1 in everyday life, but it matters for muons and GPS

Episode 2 summarytime dilation is a function of β alone, γ

γ — a moving clock runs slow, set only by β A fast-moving clock ticks slowly. How much it slows is a single function set only by β, γ = 1/√(1−β²).
There is one seed — "the speed of light c is the same as seen by anyone." After that, it comes out from the Pythagorean theorem alone.