---
title: "Special Relativity: Time Dilation"
book: "High School Physics"
subject: physics
language: en
chapter: 35
exercises: 15
source: https://one-course.com/books/physics/2/en/chapter/35-special-relativity-time-dilation
---

# Chapter 35 — Special Relativity: Time Dilation

Every GPS [satellite](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) carries an atomic clock good to one second in three million years — and before launch, engineers deliberately detune it: left honest, it would gain 38 millionths of a second per day, and by tomorrow evening your phone would place you eleven kilometres from where you stand. The clock is not at fault — time itself ticks differently for the moving and the still: this chapter follows Einstein from one stubborn fact to that conclusion.

## 35.1 One speed refuses to add

Speeds depend on the frame and compose by addition ([Chapter 5](https://one-course.com/books/physics/2/en/chapter/5-relative-motion#ch-g10-relative-motion)): a passenger walking at $4\,\mathrm{km}/\mathrm{h}$ toward the front of a train doing $300\,\mathrm{km}/\mathrm{h}$ passes the trackside observer at $304\,\mathrm{km}/\mathrm{h}$.

**Definition 35.1 (Inertial frame).**

An *inertial frame* is a [reference frame](https://one-course.com/books/physics/2/en/chapter/5-relative-motion#def-g10-relative-motion-frame) ([Chapter 5](https://one-course.com/books/physics/2/en/chapter/5-relative-motion#ch-g10-relative-motion)) in which a force-free body keeps a constant velocity: the ground, near enough, and every frame in uniform straight-line motion relative to it.

Push the rule to the extreme: the train’s headlight beam should pass the ground observer at $c + 300\,\mathrm{km}/\mathrm{h}$. In 1887 Michelson and Morley compared the speed of light along and across the Earth’s $30\,\mathrm{km}/\mathrm{s}$ orbital rush, hunting such differences; to exquisite precision they found none — the same $c$ in every direction, in every season. Einstein, in 1905, believed the experiment and rebuilt time and space around it, on two postulates judged by their consequences — tested for a century now without a single failure.

**Theorem 35.2 (The postulates of special relativity).**

1. *Relativity principle* : the laws of physics take the same form in every [inertial frame](#def-g12-special-relativity-inertial) ; none is “truly at rest”.
2. *Invariance of $c$* : light in vacuum travels at $c = 3.00 \times 10^{8}\,\mathrm{m}/\mathrm{s}$ in every [inertial frame](#def-g12-special-relativity-inertial) , whatever the motion of source or observer.

**Proof.** *Admitted at this level.* ∎

## 35.2 Farewell to “the same instant”

**Proposition 35.3 (Relativity of simultaneity).**

Two [events](#def-g12-special-relativity-proper-time) simultaneous in one [inertial frame](#def-g12-special-relativity-inertial) are, in general, not simultaneous in another: “now, everywhere” is frame-dependent.

**Proof.** Einstein’s train. Two lightning bolts strike the two ends of a moving train, scorching the track. The platform observer $M$, midway between the marks, receives the two flashes together; each crossed the same distance at the same speed $c$, so the strikes were simultaneous. The passenger $M'$, midway along the train, rides *toward* the front flash, which reaches him first; but in his frame too both flashes covered equal distances (half a train) at the same speed $c$ — second postulate! — so the front strike happened first. Each verdict is correct in its own frame; the disagreement, nanoseconds for a train, is why no one had noticed. ∎

![Einstein’s train: the flashes from A and B reach M together; M' rides toward B’s flash, meets it first — and concludes it struck first.](https://one-course.com/images/onecourse/chapters/physics-2/g12-special-relativity/fig-f6d4aa5bf897.svg)

*Einstein’s train: the flashes from $A$ and $B$ reach $M$ together; $M'$ rides toward $B$’s flash, meets it first — and concludes it struck first.*

## 35.3 The light clock and the $\gamma$ factor

**Definition 35.4 (Event, proper time).**

An *event* is a happening at one place and one instant — a tick, a flash, a decay. When a single clock is present at two events, the interval it reads between them is the *proper time* $\Delta t_0$. A frame in which that clock moves measures a generally different interval $\Delta t$.

**Theorem 35.5 (Time dilation).**

A clock moving at constant speed $v$ through an [inertial frame](#def-g12-special-relativity-inertial) is measured there to run slow: between two [events](#def-g12-special-relativity-proper-time) at the clock,

$$
\Delta t = \gamma\, \Delta t_0, \qquad
\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \geq 1 .
$$

**Proof.** Build the simplest clock imaginable: two facing mirrors a distance $L$ apart, a [photon](https://one-course.com/books/physics/2/en/chapter/34-the-quantum-world-photons-and-energy-levels#def-g12-quantum-world-photon) bouncing between them; one round trip is one tick, and at rest a tick lasts $\Delta t_0 = 2L/c$. Now bolt the clock to a ship crossing at speed $v$, mirrors perpendicular to the motion, and time one tick from the ground: while the [photon](https://one-course.com/books/physics/2/en/chapter/34-the-quantum-world-photons-and-energy-levels#def-g12-quantum-world-photon) climbs, the mirrors slide sideways, so the [photon](https://one-course.com/books/physics/2/en/chapter/34-the-quantum-world-photons-and-energy-levels#def-g12-quantum-world-photon) flies a slanted path — two motions at once, like the [projectile](https://one-course.com/books/physics/2/en/chapter/26-free-fall-and-projectile-motion#def-g12-projectile-motion-projectile) of [Chapter 24](https://one-course.com/books/physics/2/en/chapter/24-kinematics-in-two-dimensions#ch-g12-kinematics-2d). If the tick lasts $\Delta t$, each half-tick advances $v\,\Delta t/2$ horizontally and $L$ [vertically](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight); and — the crucial step — the second postulate fixes the [photon](https://one-course.com/books/physics/2/en/chapter/34-the-quantum-world-photons-and-energy-levels#def-g12-quantum-world-photon)’s ground speed at $c$, so the slanted half-path measures $c\,\Delta t/2$. Pythagoras:

$$
\Bigl(\frac{c\,\Delta t}{2}\Bigr)^{\!2} = L^2 + \Bigl(\frac{v\,\Delta t}{2}\Bigr)^{\!2}
\;\Longrightarrow\;
\Delta t = \frac{2L/c}{\sqrt{1 - v^2/c^2}} = \gamma\,\Delta t_0 .
$$

And any clock riding beside it — wristwatch, heartbeat, decaying [nucleus](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-ladder) — must stay in step, or the comparison would betray the ship’s motion and violate the first postulate. ∎

![The light clock. At rest the photon climbs L; from the ground it flies the hypotenuse at the same c — so the tick must take longer.](https://one-course.com/images/onecourse/chapters/physics-2/g12-special-relativity/fig-14a9c053d047.svg)

![The light clock. At rest the photon climbs L; from the ground it flies the hypotenuse at the same c — so the tick must take longer.](https://one-course.com/images/onecourse/chapters/physics-2/g12-special-relativity/fig-a18bb72868de.svg)

*The light clock. At rest the [photon](https://one-course.com/books/physics/2/en/chapter/34-the-quantum-world-photons-and-energy-levels#def-g12-quantum-world-photon) climbs $L$; from the ground it flies the hypotenuse at the same $c$ — so the tick must take longer.*

**Example 35.6 (Gamma in numbers).**

| $v/c$ | 0.01 | 0.1 | 0.5 | 0.8 | 0.9 | 0.99 | 0.999 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| $\gamma$ | 1.00005 | 1.005 | 1.155 | 1.667 | 2.294 | 7.09 | 22.4 |

Nothing happens for pages, then everything: $\gamma$ hugs $1$ up to a third of light speed, passes $2$ near $0.9c$, and blows up as $v \to c$.

![= 1/√1 - v2/c2: indistinguishable from 1 at everyday speeds, diverging at the asymptote v = c — the speed no massive body reaches.](https://one-course.com/images/onecourse/chapters/physics-2/g12-special-relativity/fig-0b4659d89b31.svg)

*$\gamma = 1/\sqrt{1 - v^2/c^2}$: indistinguishable from $1$ at everyday speeds, [diverging](https://one-course.com/books/physics/2/en/chapter/10-lenses-images-and-the-eye#def-g11-lenses-and-eye-lens) at the asymptote $v = c$ — the speed no massive body reaches.*

**Remark 35.7 (Why daily life never noticed).**

For $v \ll c$, write $x = v^2/c^2$: since $(1-x)(1+x) \approx 1$ and $(1 + x/2)^2 \approx 1 + x$, we get $\gamma \approx 1 + v^2/(2c^2)$. A car at $130\,\mathrm{km}/\mathrm{h}$ has $\gamma - 1 \approx 7 \times 10^{-15}$ (one second lost per four million years of driving); a jet manages $3.5 \times 10^{-13}$, one second per ninety thousand years aloft. Our clocks were simply too coarse.

## 35.4 The verdict of experiment

**Example 35.8 (Muons reach the ground).**

Cosmic rays striking the upper atmosphere create *muons* — short-lived particles of proper lifetime $\Delta t_0 = 2.2\,\text{µ}\mathrm{s}$ — about $15\,\mathrm{km}$ up, moving at nearly $c$. Without relativity a muon covers $c\,\Delta t_0 \approx 660\,\mathrm{m}$ per lifetime: the descent takes some $23$ lifetimes, and the surviving fraction $\mathrm{e}^{-23} \approx 10^{-10}$ should make sea-level muons vanishingly rare. Yet detectors count about one per square centimetre per minute. [Time dilation](#thm-g12-special-relativity-dilation) resolves it: arriving with $\gamma \approx 30$, the muons live $30 \times 2.2\,\text{µ}\mathrm{s} = 66\,\text{µ}\mathrm{s}$ on our clocks — [range](https://one-course.com/books/physics/2/en/chapter/26-free-fall-and-projectile-motion#def-g12-projectile-motion-range) $\approx 20\,\mathrm{km}$, the whole atmosphere.

![The muon’s journey: one classical lifetime dies out 660\, m below the birth altitude; the dilated lifetime spans the atmosphere.](https://one-course.com/images/onecourse/chapters/physics-2/g12-special-relativity/fig-978859812ebb.svg)

*The muon’s journey: one classical lifetime dies out $660\,\mathrm{m}$ below the birth altitude; the dilated lifetime spans the atmosphere.*

Confirmations pile up. In 1971 Hafele and Keating flew caesium atomic clocks ([Chapter 28](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#ch-g12-oscillators-and-time)) around the world on airliners; on landing they differed from ground clocks by the predicted fractions of a microsecond. Accelerators bank on dilation daily, their particles boosted to $\gamma$ in the thousands surviving laps they could never classically complete. And GPS is a running experiment with a price tag.

**Remark 35.9 (The relativity bill of GPS).**

A GPS [satellite](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) [orbits](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) at about $3.9\,\mathrm{km}/\mathrm{s}$, so [special relativity](#thm-g12-special-relativity-postulates) slows its clock by some $7\,\text{µ}\mathrm{s}$ per day. Gravity, weaker at altitude, [works](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-work) the other way and wins: general relativity (Einstein’s theory of [gravitation](https://one-course.com/books/physics/2/en/chapter/13-the-fundamental-interactions#def-g11-fundamental-interactions-four), told in the university volumes) speeds it by about $45\,\text{µ}\mathrm{s}$ per day; net, $38\,\text{µ}\mathrm{s}$ per day fast. Positioning converts clock readings to distances at speed $c$: an uncorrected day would smear positions by $c \times 38\,\text{µ}\mathrm{s} \approx 11\,\mathrm{km}$ — hence the detuned clocks of the opening paragraph.

**Proposition 35.10 (Length contraction).**

A body of length $L_0$ in its rest frame is measured shorter along its direction of motion in a frame where it moves at speed $v$: $L = L_0/\gamma$.

**Proof.** *Admitted at this level.* ∎

**Remark 35.11 (The muon’s own story).**

[Length contraction](#prop-g12-special-relativity-contraction) is [time dilation](#thm-g12-special-relativity-dilation)’s flip side (derived honestly in the university volumes): the muon’s clock ticks normally in its own frame, but the atmosphere rushing up is contracted to $15\,\mathrm{km}/30 =
500\,\mathrm{m}$, crossed well within $2.2\,\text{µ}\mathrm{s}$ — both frames agree the muon reaches the ground.

## 35.5 Mass, energy, and the classical world regained

**Theorem 35.12 (Mass–energy equivalence).**

A body of mass $m$ possesses, by its mass alone, the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy)

$$
E = m c^2 .
$$

Mass is a concentrated form of [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy); [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy), conversely, has mass.

**Proof.** *Admitted at this level.* ∎

**Remark 35.13 (Where you have met it).**

This is the ledger behind [nuclear energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-forms) ([Chapter 33](https://one-course.com/books/physics/2/en/chapter/33-nuclear-energy-fission-fusion-e-mc2#ch-g12-nuclear-energy)): the [mass defect](https://one-course.com/books/physics/2/en/chapter/33-nuclear-energy-fission-fusion-e-mc2#def-g12-nuclear-energy-defect) of a reaction, times $c^2$, is the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) released. One gram of mass is $9.0 \times 10^{13}\,\mathrm{J}$ — a full day’s output of a large [power](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-power) station. The honest derivation belongs to the university volumes; the reactors and the stars run on it regardless.

**Remark 35.14 (Newton was not wrong).**

Set $v \ll c$: $\gamma \to 1$, times agree, lengths uncontract, simultaneity turns absolute — the mechanics of the last three years reappears, intact, as the low-speed limit. Relativity did not demolish Newton; it drew the boundary of his empire. That is how physics grows: by refining theories, each new one handing back the old as a limiting case.

## 35.6 Exercises

**Exercise 35.1 ★.**

Compute $\gamma$, to three [significant figures](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-sigfig), for $v = 0.1c$, $0.6c$, $0.8c$, $0.995c$.

**Solution of Exercise 35.1.**

$\gamma = 1/\sqrt{1 - (v/c)^2}$: $1.01$ ($0.1c$); $1.25$ ($0.6c$); $1.67$ ($0.8c$); $10.0$ ($0.995c$).

**Exercise 35.2 ★.**

A ship cruises at $0.6c$; its metronome beats every $10.0\,\mathrm{s}$ by the ship’s clock. What interval does the ground measure? Which is the [proper time](#def-g12-special-relativity-proper-time), and why?

**Solution of Exercise 35.2.**

$\gamma = 1.25$, so the ground measures $1.25 \times 10.0 =
12.5\,\mathrm{s}$. The ship’s $10.0\,\mathrm{s}$ is the [proper time](#def-g12-special-relativity-proper-time): the metronome (one clock) is present at both beats.

**Exercise 35.3 ★.**

At what speed (as a fraction of $c$) is $\gamma = 2$? And $\gamma = 10$?

**Solution of Exercise 35.3.**

$v = c\sqrt{1 - 1/\gamma^2}$: $\gamma = 2$ gives $v = 0.866c$; $\gamma = 10$ gives $v = 0.995c$.

**Exercise 35.4 ★.**

A car drives at $130\,\mathrm{km}/\mathrm{h}$. Compute $v/c$, then $\gamma - 1$ ([Remark 35.7](#rem-g12-special-relativity-everyday)); how long must it drive for its clock to fall $1\,\mathrm{s}$ behind?

**Solution of Exercise 35.4.**

$v = 36.1\,\mathrm{m}/\mathrm{s}$, $v/c = 1.2 \times 10^{-7}$; $\gamma - 1 \approx (1.2 \times 10^{-7})^2/2 \approx 7.2 \times 10^{-15}$. One second lost after $1/7.2 \times 10^{-15} \approx 1.4 \times 10^{14}\,\mathrm{s}$ — about four million years of driving.

**Exercise 35.5 ★.**

In Einstein’s train experiment, explain — citing the postulate used — why the passenger concludes the *front* strike came first, and why neither observer is mistaken.

**Solution of Exercise 35.5.**

The passenger rides toward the front flash, so it reaches him first; by the second postulate both flashes travel at the same $c$ over the equal half-train distances *in his frame*, so an earlier arrival means an earlier strike. Simultaneity is frame-dependent: each verdict is correct in its own frame.

**Exercise 35.6 ★★.**

Charged pions live $\Delta t_0 = 2.6 \times 10^{-8}\,\mathrm{s}$. A beam leaves an accelerator at $0.99c$. Compute $\gamma$, the lifetime in the lab, the classical [range](https://one-course.com/books/physics/2/en/chapter/26-free-fall-and-projectile-motion#def-g12-projectile-motion-range) $v\,\Delta t_0$, and the actual mean [range](https://one-course.com/books/physics/2/en/chapter/26-free-fall-and-projectile-motion#def-g12-projectile-motion-range).

**Solution of Exercise 35.6.**

$\gamma = 1/\sqrt{1 - 0.99^2} = 7.09$; lab lifetime $7.09 \times 2.6 \times 10^{-8} = 1.8 \times 10^{-7}\,\mathrm{s}$. Classical [range](https://one-course.com/books/physics/2/en/chapter/26-free-fall-and-projectile-motion#def-g12-projectile-motion-range) $0.99 \times 3.00 \times 10^{8} \times 2.6 \times 10^{-8} \approx 7.7\,\mathrm{m}$; actual $7.09 \times 7.7 \approx 55\,\mathrm{m}$.

**Exercise 35.7 ★★.**

From the $\gamma$ curve, read the speeds giving $\gamma = 1.5$ and $\gamma = 3$ (check by computation). What does the vertical asymptote at $v = c$ say about accelerating a massive body?

**Solution of Exercise 35.7.**

$\gamma = 1.5$ at $v = c\sqrt{1 - 1/2.25} \approx 0.75c$; $\gamma = 3$ at $\approx 0.94c$. The asymptote: $\gamma$ (and the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) cost of further speed) diverges as $v \to c$ — no finite effort brings a massive body to light speed.

**Exercise 35.8 ★★.**

A light clock has mirrors $L = 1.5\,\mathrm{m}$ apart. Compute its tick at rest, then its tick as measured from the ground when it cruises at $0.8c$.

**Solution of Exercise 35.8.**

$\Delta t_0 = 2L/c = 3.0/3.00 \times 10^{8} = 1.0 \times 10^{-8}\,\mathrm{s}$; at $0.8c$, $\gamma = 5/3$, so $\Delta t = 1.7 \times 10^{-8}\,\mathrm{s}$.

**Exercise 35.9 ★★.**

Who measures the [proper time](#def-g12-special-relativity-proper-time) between: (a) two ticks of a wristwatch — the wearer, or a passer-by? (b) birth and decay of a muon — the muon, or the lab? (c) leaving Earth and reaching Mars — pilot, or mission control?

**Solution of Exercise 35.9.**

[Proper time](#def-g12-special-relativity-proper-time) belongs to the clock present at both [events](#def-g12-special-relativity-proper-time): (a) the wearer; (b) the muon; (c) the pilot — only the ship is at both the departure and the arrival.

**Exercise 35.10 ★★.**

The space station [orbits](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) at $7.7\,\mathrm{km}/\mathrm{s}$. Compute $\gamma - 1 \approx v^2/2c^2$, then by how much an astronaut’s clock trails a ground clock after six months ($1.58 \times 10^{7}\,\mathrm{s}$).

**Solution of Exercise 35.10.**

$\gamma - 1 \approx 7700^2/(2 \times 9.0 \times 10^{16}) \approx
3.3 \times 10^{-10}$; over $1.58 \times 10^{7}\,\mathrm{s}$ the astronaut trails by $3.3 \times 10^{-10} \times 1.58 \times 10^{7} \approx 5.2 \times 10^{-3}\,\mathrm{s}$ — about five milliseconds younger.

**Exercise 35.11 ★★.**

Compute the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) content $E = mc^2$ of $1.0\,\mathrm{g}$ of matter. Compare it with the daily output of a $1.0\,\mathrm{GW}$ [power](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-power) station, and with the $8.5 \times 10^{5}\,\mathrm{J}$ [kinetic energy](https://one-course.com/books/physics/2/en/chapter/18-mechanical-energy-and-its-conservation#def-g11-mechanical-energy-kinetic) of a car at $130\,\mathrm{km}/\mathrm{h}$.

**Solution of Exercise 35.11.**

$E = 1.0 \times 10^{-3} \times (3.00 \times 10^{8})^2 = 9.0 \times 10^{13}\,\mathrm{J}$. A $1.0\,\mathrm{GW}$ station delivers $1.0 \times 10^{9} \times 86400 \approx
8.6 \times 10^{13}\,\mathrm{J}$ per day — one gram is one day of its output, and $9.0 \times 10^{13}/8.5 \times 10^{5} \approx 10^{8}$: a hundred million speeding cars.

**Exercise 35.12 ★★★.**

Show from $\gamma = 1/\sqrt{1 - v^2/c^2}$ that $v = c\sqrt{1 - 1/\gamma^2}$; deduce the speed of the $\gamma = 30$ muons, to five [significant figures](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-sigfig) in [units](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of $c$.

**Solution of Exercise 35.12.**

Square and invert: $\gamma^2(1 - v^2/c^2) = 1$, so $v^2/c^2 = 1 - 1/\gamma^2$, hence $v = c\sqrt{1 - 1/\gamma^2}$. For $\gamma = 30$: $v = c\sqrt{1 - 1/900} = 0.99944c$.

**Exercise 35.13 ★★★.**

Retell the muon’s journey from its own frame: how thick is the atmosphere at $\gamma = 30$, how long does it sweep past at $0.99944c$, and does that fit within one proper lifetime?

**Solution of Exercise 35.13.**

Contracted thickness $L = 15\,\mathrm{km}/30 = 500\,\mathrm{m}$; sweep time $500/(0.99944 \times 3.00 \times 10^{8}) \approx 1.7 \times 10^{-6}\,\mathrm{s} <
2.2\,\text{µ}\mathrm{s}$. Both frames agree the muon reaches the ground — one blames a stretched lifetime, the other a shrunken atmosphere.

**Exercise 35.14 ★★★.**

A linear accelerator pushes electrons to $\gamma = 1.0 \times 10^{5}$ along a $3.2\,\mathrm{km}$ tube. How long does the trip take in the lab? How long in the electron’s frame, and how “long” is the tube there?

**Solution of Exercise 35.14.**

Lab: $3200/3.00 \times 10^{8} \approx 1.1 \times 10^{-5}\,\mathrm{s}$ (speed $\approx c$). Electron frame: tube contracted to $3200/1.0 \times 10^{5} = 3.2\,\mathrm{cm}$, crossed in $1.1 \times 10^{-5}/1.0 \times 10^{5} \approx 1.1 \times 10^{-10}\,\mathrm{s}$.

**Exercise 35.15 ★★★.**

Proxima Centauri lies $4.2$ [light-years](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-lightyear) away (a [light-year](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-lightyear) is the distance light covers in a year). A probe cruises there at $0.9c$: how long does the trip take on Earth’s calendar, and on the probe’s clock?

**Solution of Exercise 35.15.**

Earth: $4.2/0.9 \approx 4.7\,\mathrm{years}$. With $\gamma = 1/\sqrt{1 - 0.81} = 2.29$, the probe’s clock logs $4.7/2.29 \approx 2.0\,\mathrm{years}$.

## 35.7 Problem: The Muon and the Twin

**Problem 35.1.**

Weekend problem — the muon’s journey and the astronaut’s twin: a particle outliving its clock, a light clock rebuilt, a traveler younger than her twin, and the bookkeeping that keeps GPS honest

Data: muon proper lifetime $\Delta t_0 = 2.2\,\text{µ}\mathrm{s}$; muons born about $15\,\mathrm{km}$ up at nearly $c$; sea-level flux about one muon per $\mathrm{cm}^{2}$ per minute; $c = 3.00 \times 10^{8}\,\mathrm{m}/\mathrm{s}$.

**Part I — The muon’s impossible arrival.**

1. Without relativity, how far does a muon travel in one lifetime?
2. How many lifetimes does the $15\,\mathrm{km}$ descent then take?
3. After $n$ lifetimes a fraction $\mathrm{e}^{-n}$ survives: estimate it, confront the measured flux, and pass verdict on classical physics in one sentence.
4. These muons in fact arrive with $\gamma \approx 30$ . Compute their lifetime as measured from the ground and the resulting [range](https://one-course.com/books/physics/2/en/chapter/26-free-fall-and-projectile-motion#def-g12-projectile-motion-range) .
5. Compute their speed, using $v = c\sqrt{1 - 1/\gamma^2}$ , to five [significant figures](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-sigfig) in [units](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of $c$ .

**Part II — The light clock, rebuilt.** A light clock ($L = 1.5\,\mathrm{m}$) rides a ship at $v = 0.6c$.

6. Compute the tick $\Delta t_0$ read on board.
7. Seen from the ground, a tick lasts $\Delta t$ : give the horizontal advance per half-tick, the [photon](https://one-course.com/books/physics/2/en/chapter/34-the-quantum-world-photons-and-energy-levels#def-g12-quantum-world-photon) ’s vertical climb, and — naming the postulate that fixes it — the slanted half-path’s length.
8. Apply Pythagoras to the half-tick triangle and solve for $\Delta t$ , showing that $\Delta t = \gamma\,\Delta t_0$ .
9. Compute $\gamma$ and $\Delta t$ numerically.
10. Which observer measures the [proper time](#def-g12-special-relativity-proper-time) between two ticks? Justify from the definition.

**Part III — The traveler and her twin.** An astronaut leaves her twin on Earth for a round trip at $0.8c$; Earth clocks time the whole trip at $10.0\,\mathrm{years}$.

11. Compute $\gamma$ at $0.8c$ (exact fraction, then decimal).
12. How much time elapses on the astronaut’s clock?
13. How far out (Earth frame, in [light-years](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-lightyear) ) did the turnaround lie?
14. What is the age difference on reunion, and which twin is younger?
15. “But motion is relative — the astronaut could claim Earth did the traveling!” Resolve the objection in one sentence.

**Part IV — The GPS bill.** A GPS [satellite](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) [orbits](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) at radius $r = 2.66 \times 10^{7}\,\mathrm{m}$, [period](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) $T = 12\,\mathrm{h}$.

16. Compute its orbital speed $v = 2\pi r/T$ .
17. Compute $\gamma - 1 \approx v^2/2c^2$ .
18. Deduce, in microseconds, how far the [satellite](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite) clock falls behind ground clocks per day ( $86\,400\,\mathrm{s}$ ).
19. If this drift alone went uncorrected, positions err by $c \times$ (drift): compute the error after one day.
20. Gravity, weaker at altitude, runs the clock *fast* by $45\,\text{µ}\mathrm{s}$ per day (general relativity): compute the net drift and the daily position error in kilometres — the number that obliges engineers to detune every GPS clock.

**Solution of Problem 35.1.**

**1.** $c\,\Delta t_0 = 3.00 \times 10^{8} \times 2.2 \times 10^{-6} \approx
660\,\mathrm{m}$.

**2.** $15\,000/660 \approx 23$ lifetimes.

**3.** $\mathrm{e}^{-23} \approx 10^{-10}$: fewer than one muon in ten billion should arrive, yet the flux is one per $\mathrm{cm}^{2}$ per minute — classical physics is flatly contradicted by the sky.

**4.** Ground lifetime $30 \times 2.2\,\text{µ}\mathrm{s} =
66\,\text{µ}\mathrm{s}$; [range](https://one-course.com/books/physics/2/en/chapter/26-free-fall-and-projectile-motion#def-g12-projectile-motion-range) $\approx 3.00 \times 10^{8} \times 6.6 \times 10^{-5}
\approx 20\,\mathrm{km} > 15\,\mathrm{km}$: the arrival is explained.

**5.** $v = c\sqrt{1 - 1/900} = 0.99944c$.

**6.** $\Delta t_0 = 2L/c = 3.0/3.00 \times 10^{8} = 1.0 \times 10^{-8}\,\mathrm{s}$.

**7.** Horizontal $v\,\Delta t/2$; vertical $L$; the slanted half-path measures $c\,\Delta t/2$ because the second postulate fixes the [photon](https://one-course.com/books/physics/2/en/chapter/34-the-quantum-world-photons-and-energy-levels#def-g12-quantum-world-photon)’s speed at $c$ in the [ground frame](https://one-course.com/books/physics/2/en/chapter/5-relative-motion#def-g10-relative-motion-frames) too.

**8.** $(c\,\Delta t/2)^2 = L^2 + (v\,\Delta t/2)^2$ gives $\Delta t^2(c^2 - v^2) = 4L^2$, so $\Delta t =
(2L/c)/\sqrt{1 - v^2/c^2} = \gamma\,\Delta t_0$.

**9.** $\gamma = 1/\sqrt{1 - 0.36} = 1.25$; $\Delta t = 1.25 \times 10^{-8}\,\mathrm{s}$.

**10.** The ship’s: its one clock is present at both ticks, so it reads the [proper time](#def-g12-special-relativity-proper-time); the ground needs two synchronized clocks.

**11.** $\gamma = 1/\sqrt{1 - 0.64} = 1/0.6 = 5/3 \approx 1.67$.

**12.** $10.0/(5/3) = 6.0\,\mathrm{years}$.

**13.** Outbound $5.0\,\mathrm{years}$ at $0.8c$: $4.0$ [light-years](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-lightyear).

**14.** $10.0 - 6.0 = 4.0\,\mathrm{years}$: the astronaut is four years younger than her twin.

**15.** The situation is not symmetric: only the astronaut turns around — she accelerates and changes [inertial frame](#def-g12-special-relativity-inertial), while the Earth twin stays in one — so only her clock logs the shorter time.

**16.** $v = 2\pi \times 2.66 \times 10^{7}/43\,200 \approx
3.9 \times 10^{3}\,\mathrm{m}/\mathrm{s}$.

**17.** $\gamma - 1 \approx (3.87 \times 10^{3})^2/(2 \times
9.0 \times 10^{16}) \approx 8.3 \times 10^{-11}$.

**18.** $8.3 \times 10^{-11} \times 86\,400 \approx 7.2 \times 10^{-6}\,\mathrm{s}
= 7.2\,\text{µ}\mathrm{s}$ per day slow.

**19.** $c \times 7.2 \times 10^{-6} \approx 2.2\,\mathrm{km}$ of position error per day.

**20.** Net drift $45 - 7.2 \approx 38\,\text{µ}\mathrm{s}$ per day fast; error $c \times 38 \times 10^{-6} \approx 11\,\mathrm{km}$ per day — the relativity bill that every GPS clock pays in advance, detuned on the ground so it ticks true in [orbit](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-satellite).

Here ends the physics of school — not the physics. In the university volumes, mechanics returns armed with calculus; electricity and magnetism reveal themselves as one structure in which light, and relativity itself, was hiding all along; the quantum world, glimpsed last chapter, becomes a working theory of matter. Each theory will be obliged, like relativity, to hand back what you now know as a limiting case. Carry it forward: it is the part that will not change.