Operational Definition of Time and the Ideal Clock

Newton opened the Principia with a sentence that ruled physics for two centuries: "Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external." It is a magnificent sentence and it is unmeasurable. No experiment can be built whose needle deflects when absolute time flows; every experiment ever done has compared one motion with another. Mach said so in 1883, and Einstein, in §1 of the 1905 paper, did the only thing left to do — he threw the sentence away and put a clock in its place. Time at a place is what a clock there reads: not an approximation to something deeper, but the definition itself, with nothing behind it. The move looks like a retreat but is an advance: it converts a metaphysical assertion into a bill of engineering. If time is what a clock reads, you owe an account of what a clock is — and the answer is disarmingly small. Any stable, countable, periodic process — a pendulum, a quartz fork, a caesium atom flipping its outer electron's spin 9 192 631 770 times a second — and time is the count. And inside that innocent account sits an assumption nobody states aloud. Every clock that ever measured anything has been accelerated — carried, spun, dropped, flown. The postulates speak only of inertial frames; they say nothing whatever about a clock being shaken. So what happens to a clock's rate when you accelerate it? Relativity does not answer. It assumes an answer — the clock hypothesis: an accelerated ideal clock ticks at the rate of the inertial clock momentarily at rest beside it. That might have been false. This article states it precisely, shows why it is true — it holds because clocks are small — and how it was tested by spinning muons at $1.1\times10^{18}\,g$ and finding their clocks never noticed.

Key takeaways

An operational definition makes no claim about what time is. It gives a procedure: build a stable periodic process, count its cycles, and declare the count to be the time at that event. The SI does exactly this, fixing $\nu_{\mathrm{Cs}} \equiv 9\,192\,631\,770\ \mathrm{Hz}$ by decree — the second is a counting statement.
Real clocks accelerate; the postulates speak only of inertial ones. Bridging the gap needs the clock hypothesis — the rate is fixed by the instantaneous speed, with no separate dependence on the acceleration. It is an independent assumption, not a theorem of relativity.
Key idea: time is the count on a clock, and a clock is a countable periodic process — nothing more. That definition takes out exactly one loan, the clock hypothesis, which says the rate depends on the speed and not the acceleration. The loan is repaid not by relativity but by scale: a clock feels its acceleration only when $a$ approaches $c^2/L$ , and real clocks are small enough that $10^{18}\,g$ is not even close.

The core insight

Time is what a clock reads: the number of ticks a specified, reproducible periodic process counts between two events on its own worldline. Everything else is built on that count. The one thing the definition cannot supply for itself is what happens when the clock is accelerated — and there relativity does not derive an answer, it assumes one. The clock hypothesis is that assumption, and it survives because acceleration reaches a clock only through the dimensionless ratio $aL/c^2$ built from the clock's own size.

Where this lives today

The operational definition has become a surveying technique. A clock at $10^{-18}$ stability resolves a one-centimetre height difference through $gh/c^2$ — so turn the logic around, carry the clock up a mountain, and read your altitude off its frequency. This is relativistic geodesy: transportable optical clocks have been hauled into tunnels and up mountains, measuring height against distant reference clocks with no line of sight and no levelling rods. The clock does not measure time and then infer height — the clock is the altimeter. The same logic underpins every satellite-navigation fix, where relativistic corrections to orbiting clocks are a precondition, not a refinement (GPS and Relativistic Timing Corrections, S7).

The physical picture

You are handed a sealed box that flashes at regular intervals, and told: this is a clock. You never see time — you see flashes, and you count them. The operational move is to stop treating the count as an approximation to a hidden quantity and declare that the count is the measurement. Physics becomes bookkeeping of coincidences: which flash of your clock coincided with which event.

Figure

The operational chain — any stable countable periodic process is a clock, and its reading at an event is the time there. Below: the clock hypothesis, an accelerated worldline chopped into segments, each borrowing its rate from the momentarily-comoving inertial clock

But here is the crack in the floor. To compute a clock's rate, relativity offers the rate of an inertial clock: the factor $\sqrt{1-\beta^2}$ , derived in The Light Clock and Time Dilation (S1). Nothing in the postulates says anything about a clock being shaken — and every real clock is shaken. A muon in a storage ring is wrenched sideways hard enough to bend its path into a seven-metre circle. Does its internal decay clock care? Relativity is silent. So we assume it does not, and then we check.

Figure

PUNCHLINE — the acceleration ladder against the muon's clock-breaking scale. CERN's ring reached 1.1×10¹⁹ m/s² = 1.1×10¹⁸ g, the most violent acceleration ever applied to a working clock — and it still sits 12.6 decades below c²/λ̄_C = 4.8×10³¹ m/s², where a muon ceases to be a clock

The ring is the most violent acceleration ever applied to something still working as a clock — and it is still twelve and a half orders of magnitude short of where a muon stops being one. That gap is the entire content of the clock hypothesis: it is not magic, it is a ratio, and the ratio is tiny because the clock is tiny.

The core equations

An operational definition is not a formula that predicts; it is a recipe that produces a number. Take a process repeating with period $T_0$ , carry it along a worldline, and between two events count the repetitions $N$ . Count times period, $\tau = N T_0$ , is the elapsed reading: the proper time along that path. Nothing hidden is being approximated — this is a stipulation, so no measurement can show it wrong. What can be wrong is the bet that different clocks agree, and that is checked in a laboratory. The SI turns that stipulation into law by nominating one process and fixing its frequency by decree; since 2019 it is one of seven exactly-fixed defining constants.

\nu_{\mathrm{Cs}} \;\equiv\; 9\,192\,631\,770\ \mathrm{Hz} \qquad\Longrightarrow\qquad 1\ \mathrm{s} \;=\; \frac{9\,192\,631\,770}{\nu_{\mathrm{Cs}}} \qquad (1)

where: $\nu_{\mathrm{Cs}}$ — the frequency radiated when caesium-133 flips the relative orientation of its outer-electron and nuclear spins. Physical reading: the second is a counting statement — build an atomic clock and count 9 192 631 770 cycles. Now the gap. The postulates give the rate of an inertial clock against a frame's coordinate time $t$ : slowed by $\sqrt{1-\beta^2}$ , with $\beta \equiv v/c$ . That factor is not derived here — it belongs to The Light Clock and Time Dilation (S1), and the name conventionally given to its reciprocal to The Lorentz Factor Gamma (S1). We grant its value and pursue what those articles do not address: the clock that is not inertial.

\frac{\mathrm{d}\tau}{\mathrm{d}t}\bigg|_{\text{accelerated, speed } v} \;=\; \frac{\mathrm{d}\tau}{\mathrm{d}t}\bigg|_{\text{inertial, speed } v} \qquad\text{(clock hypothesis)} \qquad (2)

where: the left side is an accelerated clock moving, at this instant, at $v$ ; the right is an inertial clock gliding at the same $v$ . Physical reading: the rate depends on the speed and not on the acceleration. It does not say acceleration is irrelevant to how much a clock reads — acceleration is decisive, being what lets a worldline leave and return. It says acceleration adds no separate term to the rate. This is an assumption, following from neither postulate. Grant it, and the reading on any path follows by addition — chop the worldline into segments, borrow each rate from the inertial clock momentarily at rest beside it, sum the ticks.

\tau \;=\; \sum_i \Delta\tau_i \;\longrightarrow\; \int \sqrt{1 - \frac{v(t)^2}{c^2}}\;\mathrm{d}t \qquad (3)

where: $v(t)$ — the clock's instantaneous speed in the chosen frame. Physical reading: still operational — the clock's own tick-count, assembled segment by segment. Its geometric meaning, as a worldline length measured with the metric, is developed in Proper Time (S1); we borrow nothing from it. Finally, the object that decides whether the assumption survives. A clock is not a point: it has an internal size $L$ , and out of $a$ , $L$ and $c$ there is exactly one dimensionless number to build — $\Pi \equiv aL/c^2 = a/(c^2/L)$ , where $c^2/L$ is the clock's clock-breaking scale. Any correction to the clock hypothesis must be a function of $\Pi$ vanishing when $\Pi$ does, because there is nothing else for it to depend on. The next section turns that into a number.

Derivation AThe clock hypothesis is an assumption — and this is exactly how big an assumption it is

The postulates constrain inertial frames and nothing else, so they cannot fix the rate of an accelerated clock — yet every clock ever used has been accelerated. Something must bridge that gap, and it is a separate physical hypothesis. The goal is to state it precisely, show it is not derivable from what we have, and bound its failure by the only route open: dimensional analysis on the clock's internal size. The payoff is a number — the acceleration at which a clock stops being one.

Step 1. For an inertial clock gliding at constant $v$ the postulates give $\mathrm{d}\tau/\mathrm{d}t = \sqrt{1-\beta^2}$ — two mirrors, a bouncing pulse — and that is The Light Clock and Time Dilation's (S1), granted, not repeated. Notice what is missing: the light clock glides; the muon in a ring does not. The hypothesis fills the gap by asserting that, with proper acceleration $a$ , the instantaneous rate is a function of $v$ alone — no dependence on $a$ , $\dot a$ , or any higher derivative — so the reading is additive: chop finely, hand each piece to the momentarily-comoving inertial clock, sum. Why: the light-clock derivation assumes the clock's parts hold fixed relative positions while the pulse crosses; an accelerated clock's do not. Additivity is the practical content — it lets us compute a muon's decay from a speed history alone.

Step 2. Show it is not a theorem. Nothing in P1 or P2 forbids an extra acceleration term, so write the general possibility, with $\zeta$ and $a_0$ unknown:

\frac{\mathrm{d}\tau}{\mathrm{d}t} \;=\; \sqrt{1-\beta^2}\,\Big[\,1 \;+\; \zeta\,\frac{a}{a_0} \;+\; \mathcal{O}(a^2)\,\Big]

Relativity has no opinion whatever about $\zeta$ ; it must be measured or bounded. Why: this is the crux, routinely skipped — textbooks present the clock hypothesis as though it fell out of the postulates. It did not.

Step 3. Find $a_0$ by dimensional analysis, then compute it. A clock is an extended object of internal size $L$ ; from $a$ , $L$ and $c$ there is exactly one dimensionless group, $\Pi = aL/c^2$ , since $[\mathrm{m/s^2}][\mathrm{m}]/[\mathrm{m^2/s^2}] = 1$ . Any correction must be a function of $\Pi$ vanishing at $\Pi = 0$ , pinning the natural scale at $a_0 \sim c^2/L$ . A muon's internal size is its reduced Compton wavelength, $\bar\lambda_{\mathrm C} = \hbar/m_\mu c = 1.87\times10^{-15}\ \mathrm{m}$ , so $c^2/\bar\lambda_{\mathrm C} = 4.8\times10^{31}\ \mathrm{m/s^2}$ ; for a caesium atom ( $L \approx 10^{-10}\ \mathrm{m}$ ) it is $9\times10^{26}$ , and for a one-metre pendulum a mere $9\times10^{16}\ \mathrm{m/s^2}$ . Why: there is no other candidate — introduce a second length and you have introduced a second clock. Big clocks are fragile; the toughest clock we own is a subatomic particle.

Step 4. Turn the knob as hard as anyone ever has. In the CERN muon storage ring the muons ran on $R = 7.0\ \mathrm{m}$ at $\beta = 0.99942$ , giving a dilation factor $1/\sqrt{1-\beta^2} = 29.33$ and a proper acceleration $a = v^2/[R(1-\beta^2)] = 1.10\times10^{19}\ \mathrm{m/s^2} = 1.1\times10^{18}\,g$ . Yet the only knob acceleration has reads

\Pi \;=\; \frac{a\,\bar\lambda_{\mathrm C}}{c^{2}} \;=\; \frac{(1.10\times10^{19})(1.87\times10^{-15})}{8.988\times10^{16}} \;=\; 2.3\times10^{-13}

Why: two parts in ten trillion. Bailey and co-workers found the lifetimes obeying the dilation factor to $2\times10^{-3}$ with no acceleration term — and could not have found anything else, since the ring turns the knob far below its own sensitivity. What the ring does buy is a direct floor: any linear correction needs a scale $a_0 \gtrsim a/(2\times10^{-3}) = 5.5\times10^{21}\ \mathrm{m/s^2}$ , consistent with the $4.8\times10^{31}$ dimensional analysis predicted. *Verified in verify_aq-sr-operational-definition-of-time.py: $\Pi$ is the unique dimensionless group in $(a, L, c)$ by Buckingham counting; $\bar\lambda_{\mathrm C} = 1.868\times10^{-15}\ \mathrm{m}$ ; $c^2/\bar\lambda_{\mathrm C} = 4.81\times10^{31}$ ; $a = 1.10\times10^{19}\ \mathrm{m/s^2} = 1.12\times10^{18}\,g$ ; $\Pi = 2.3\times10^{-13}$ ; $a_0 > 5.5\times10^{21}\ \mathrm{m/s^2}$ .*

Go deeper · Derivation B · 1 minHow good an ideal clock can be built — and the centimetre at which "the time here" stops meaning anything

Derivation BHow good an ideal clock can be built — and the centimetre at which "the time here" stops meaning anything

The operational definition is only as good as the hardware, so ask what the hardware now does. The answer doubles back on the definition itself: the best clocks are so stable that two of them, on one bench at slightly different heights, no longer agree — not because either is faulty, but because there is genuinely no such thing as "the time in this room".

Step 1. A clock with fractional stability $\sigma$ accumulates $\sigma$ seconds of error per second, so the time to be wrong by a full second is $1/\sigma$ . Cash out $\sigma = 10^{-18}$ , the level optical lattice and single-ion clocks now reach: $10^{18}\ \mathrm{s}$ , and dividing by the Julian year $3.156\times10^{7}\ \mathrm{s}$ gives $3.17\times10^{10}$ years — 2.3 times the age of the universe ( $1.38\times10^{10}\ \mathrm{y}$ ). Why: caesium, on which the SI still runs, sits near $10^{-16}$ — optical clocks are already a hundredfold better than the definition of the unit they calibrate against.

Step 2. Now watch the definition bite back. Raise such a clock by a height $h$ and its rate shifts fractionally by $gh/c^2$ . (That effect is gravitational, not kinematic; it is derived in Gravitational Time Dilation and the Equivalence Principle (S6) and merely quoted here.) At $h = 1\ \mathrm{cm}$ ,

\frac{gh}{c^{2}} \;=\; \frac{(9.81)(0.01)}{8.988\times10^{16}} \;=\; 1.09\times10^{-18}

Why: that is the clock's own resolution. Lift it by the thickness of a finger and you have moved it a full unit of its precision.

Step 3. Check it on the bench, then draw the moral. Chou, Hume, Rosenband and Wineland raised one aluminium-ion optical clock $33\ \mathrm{cm}$ above another in 2010: predicted $gh/c^2 = 3.6\times10^{-17}$ , measured $(4.1 \pm 1.6)\times10^{-17}$ . So ask "what time is it here?" and the honest reply is: here where, to the centimetre? There is no universal time to be had, not even inside one room. Why: Newton's absolute time is no longer merely unmeasurable — it is refuted by two atoms on a bench a hand's breadth apart. *Verified in verify_aq-sr-operational-definition-of-time.py: $1/10^{-18} = 10^{18}\ \mathrm{s} = 3.17\times10^{10}\ \mathrm{y} = 2.3\times$ the age of the universe; $gh/c^2 = 1.09\times10^{-18}$ at $h = 1\ \mathrm{cm}$ and $3.60\times10^{-17}$ at $h = 33\ \mathrm{cm}$ , within $1\sigma$ of the measured $(4.1 \pm 1.6)\times10^{-17}$ .*

Limiting cases

Case	Result	Meaning
$c \to \infty$ (Galilean limit)	$\sqrt{1-\beta^2} \to 1$ , $c^2/L \to \infty$ , so $\mathrm{d}\tau = \mathrm{d}t$ on every path	Newton's absolute time is exactly what the operational definition returns when light is infinitely fast — why "time flows equably" served for two centuries
$\Pi = aL/c^2 \ll 1$	hypothesis holds; rate set by $v$ alone	every clock in every experiment ever done. At CERN, $\Pi = 2.3\times10^{-13}$
$a \to c^2/L$	hypothesis fails	and not by "running wrong" — the clock is torn apart and stops being a clock
Larger clock, $L\uparrow$	breaking scale $c^2/L$ falls	pendulum $9\times10^{16}$ · caesium $9\times10^{26}$ · muon $4.8\times10^{31}$ . Small is robust

Common trapThe clock hypothesis is not a theorem of relativity — and it does not say that acceleration slows clocks

Two errors, pointing in opposite directions, and most readers commit one of them. The first is to think the clock hypothesis follows from the postulates. It does not. The postulates concern inertial frames, and the light-clock derivation assumes the clock's parts hold still while the pulse crosses; nothing in P1 or P2 forbids an extra term in $a$ . The hypothesis is an independent empirical commitment, and the CERN ring is how we pay for it. The second is to over-correct — to hear "the rate does not depend on acceleration" and conclude acceleration is irrelevant. It is not. Acceleration is decisive: it is the only thing that lets a worldline leave, turn round, and come back to be compared. The travelling twin ages less because she accelerated — but not as the phrase suggests: turning round changed her speed history, and it is the integral of $\sqrt{1-\beta^2}$ over that history that comes out short. The hypothesis says the integrand has no $a$ in it — not that the answer does.

Worked examples

1 · The most violently shaken clock ever built

In the CERN muon storage ring (1977) muons circulated on radius $R = 7.0\ \mathrm{m}$ with dilation factor $1/\sqrt{1-\beta^2} = 29.33$ ; the muon's rest mean lifetime is $\tau_0 = 2.197\ \mu\mathrm{s}$ . What lifetime should the laboratory see, what proper acceleration were the muons enduring, and how close did that come to breaking their internal clock?

Step 1. Granting the inertial rate from The Light Clock and Time Dilation (S1), the laboratory should measure $\tau_0/\sqrt{1-\beta^2} = 29.33 \times 2.197\ \mu\mathrm{s} = \mathbf{64.44\ \mu s}$ . Bailey and co-workers measured $\tau^{+} = 64.419 \pm 0.058$ and $\tau^{-} = 64.368 \pm 0.029\ \mu\mathrm{s}$ — agreement to $2\times10^{-3}$ at 95 % confidence, with no acceleration term anywhere.

Step 2. From $1/\sqrt{1-\beta^2} = 29.33$ : $1-\beta^2 = 1/29.33^2 = 1.162\times10^{-3}$ , so $\beta = 0.99942$ and $v = 2.996\times10^{8}\ \mathrm{m/s}$ . For circular motion,

a \;=\; \frac{v^{2}}{R\,(1-\beta^{2})} \;=\; \frac{(2.996\times10^{8})^{2}}{(7.0)(1.162\times10^{-3})} \;=\; \mathbf{1.10\times10^{19}\ m/s^{2}}

Dividing by $g = 9.81\ \mathrm{m/s^2}$ : that is $\mathbf{1.1\times10^{18}\,g}$ — a billion billion times the pull you are sitting in.

Step 3. Turn it into the only number that matters. With $\bar\lambda_{\mathrm{C}} = 1.87\times10^{-15}\ \mathrm{m}$ the muon's clock-breaking scale is $c^2/\bar\lambda_{\mathrm{C}} = 4.8\times10^{31}\ \mathrm{m/s^2}$ , so the knob reads $\Pi = (1.10\times10^{19})/(4.8\times10^{31}) = \mathbf{2.3\times10^{-13}}$ . The harshest treatment ever handed to a working clock dialled the hypothesis's one free parameter to two parts in ten trillion, while the experiment could only see departures above $2\times10^{-3}$ : the null was guaranteed in advance by the smallness of the muon. What it does buy is a direct floor on any new acceleration scale, $a_0 \gtrsim \mathbf{5.5\times10^{21}\ m/s^{2}}$ .

2 · A clock so good it can feel a staircase

A modern optical clock reaches fractional stability $10^{-18}$ . How long before it is wrong by one second — and how far must you lift it before that stability stops meaning anything?

Step 1. Time to be wrong by a second: $1/\sigma = 10^{18}\ \mathrm{s}$ . In years (Julian year $3.156\times10^{7}\ \mathrm{s}$ ) that is $\mathbf{3.17\times10^{10}\ years}$ — about 2.3 times the age of the universe ( $1.38\times10^{10}\ \mathrm{y}$ ).

Step 2. Now lift it. The gravitational fractional rate shift is $gh/c^2$ (quoted, not derived — see Gravitational Time Dilation and the Equivalence Principle, S6). At $h = 1\ \mathrm{cm}$ : $(9.81)(0.01)/(8.988\times10^{16}) = \mathbf{1.09\times10^{-18}}$ .

Step 3. Compare the two numbers. They are the same number. A one-centimetre lift moves the clock by one full unit of its own precision — the instrument that could keep faith with the universe for thirty billion years cannot tell you the time in a room to better than a hand's breadth of height. Confirming on the bench: Chou et al. (2010) raised one aluminium-ion clock by $33\ \mathrm{cm}$ — predicted $3.6\times10^{-17}$ , measured $(4.1\pm1.6)\times10^{-17}$ . The operational definition was never a scruple; it is the only definition the hardware permits.

Physics of the portrait

The portrait here is not a person but a method — one that outlived every argument made for it. Bridgman named it; Einstein used it, and changed physics.

Origins · 2 minOperationalism and how this was discovered

Operationalism

1883–1927 · Mach's critique · Einstein's 1905 §1 · Bridgman gives it a name

The idea is older than its name. Ernst Mach, in Die Mechanik in ihrer Entwicklung (1883), attacked Newton's absolute time as a fiction: what we ever actually do, he insisted, is compare one motion with another — the Earth's rotation against a pendulum's swing — never with time itself. Einstein read Mach as a student and applied the razor with lethal precision. Section 1 of "Zur Elektrodynamik bewegter Körper" (1905) does not argue that absolute time is false; it observes that nobody has said what it would mean, then supplies a meaning of its own — the time of an event is the reading of a clock there, and distant simultaneity is settled by definition, not discovered. Percy Williams Bridgman (1882–1961), a Harvard experimentalist who squeezed matter to pressures nobody had reached and won the 1946 Nobel Prize for it, watched relativity dissolve simultaneity and drew the general lesson. In The Logic of Modern Physics (1927) he coined the term operational definition and stated the doctrine at full strength: a concept is synonymous with the corresponding set of operations. Length is what a ruler does; time is what a clock does; a question irreducible to a set of operations is not a hard question — it is not a question. Taken literally that is too strong — length-by-ruler and length-by-radar would become two concepts that merely happen to agree — and the strict doctrine died as a theory of meaning. But as a rule of hygiene (before asserting something exists, say what you would do to find it) it may be the most productive habit twentieth-century physics acquired.

History

1687

— Newton opens the Principia: "Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external."

1905

— Einstein, §1 of "Zur Elektrodynamik bewegter Körper", defines the time at a place as the reading of a clock there, and distant simultaneity as a definition rather than a discovery.

1955

— Louis Essen and Jack Parry build the first caesium-beam atomic clock at the National Physical Laboratory, Teddington.

1967

— the 13th CGPM redefines the second as 9 192 631 770 periods of the caesium-133 hyperfine radiation, the integer chosen to match the astronomical second then in use.

1977

— Bailey and co-workers, at the CERN muon storage ring, measure the lifetimes of muons circulating with dilation factor 29.33 under a proper acceleration near 10¹⁸ g: the clock hypothesis holds to 2×10⁻³.

2010

— Chou, Hume, Rosenband and Wineland raise one aluminium-ion optical clock 33 cm above another and watch it gain, by (4.1 ± 1.6)×10⁻¹⁷.

Going further

Texts. Taylor & Wheeler, Spacetime Physics, 2nd ed., Ch. 1 · Rindler, Introduction to Special Relativity, §§1–3 · P. W. Bridgman, The Logic of Modern Physics (1927) · J. Bailey et al., Nature 268 (1977) 301 · C. W. Chou et al., Science 329 (2010) 1630.

Provenance. Every number is recomputed in verify_aq-sr-operational-definition-of-time.py from $c = 2.998\times10^{8}\ \mathrm{m/s}$ (exact SI $299\,792\,458$ ) and $g = 9.81\ \mathrm{m/s^2}$ ; $\nu_{\mathrm{Cs}} = 9\,192\,631\,770\ \mathrm{Hz}$ is the BIPM definition (13th CGPM, 1967). Storage-ring parameters (dilation factor $29.33$ , $R = 7.0\ \mathrm{m}$ , $\tau_0 = 2.197\ \mu\mathrm{s}$ , measured $64.419(58)$ and $64.368(29)\ \mu\mathrm{s}$ , holding to $2\times10^{-3}$ at 95 % confidence) are Bailey et al. (1977); the $33\ \mathrm{cm}$ shift $(4.1\pm1.6)\times10^{-17}$ is Chou et al. (2010). Two corrections to the dry source are in the traps below.

Builds on. Absolute Space and Absolute Time · Inertial Frames of Reference. Leads to. The Constancy of the Speed of Light · The Principle of Relativity · Galilean Transformations. Granted by forward reference, not derived here. The Light Clock and Time Dilation (S1) · The Lorentz Factor Gamma (S1) · Proper Time (S1) · Gravitational Time Dilation and the Equivalence Principle (S6).

If time is just "what a clock reads", isn't that circular? You need time to define a clock.

No. The definition never mentions time: it says build a process that repeats, and count the repetitions. "Repeats" is not a temporal notion smuggled in — it says the system has returned to a state indistinguishable from its starting state, and you can check that without a clock. What the definition does quietly bet is that all such counters keep step once calibrated. That bet is empirical, not circular — and for pendulums against caesium it does fail, which is how caesium got promoted.

The muons at CERN were accelerated sideways. Surely a transverse push can't reach an internal decay clock?

That intuition is a guess about the clock's internal structure, and the point of the clock hypothesis is that relativity supplies no such guess. No theorem says a transverse push is gentler on an internal process than a longitudinal one, and a muon's decay has no obvious direction to protect — which is why it had to be measured rather than asserted. What rescues the intuition is not the direction of the push but its size relative to $c^2/\bar\lambda_{\mathrm C}$ : by that measure, $2.3\times10^{-13}$ , it is a whisper.

If a clock's rate doesn't depend on acceleration, why does the travelling twin come back younger?

Both halves are true and do not conflict. Acceleration is what makes the comparison possible: without it she never turns round, and there is no reunion event at which to compare the clocks. But the ageing is not caused by the turn-around as a local event — it is the integral $\int\sqrt{1-v^2/c^2}\,\mathrm{d}t$ along her whole path that comes out short, and the clock hypothesis is the claim that no extra term in $a$ appears in that integrand. Acceleration is the reason there are two paths, not an extra term in the rate along either.

What actually happens if you accelerate a clock past c²/L?

It stops being a clock. This matters, because people imagine it quietly "running wrong" in a way one could correct for. It does not. At $a \sim c^2/L$ the acceleration across the clock's own body becomes comparable to the internal forces holding it together: a pendulum at $10^{17}\ \mathrm{m/s^2}$ is not a slow pendulum, it is a smear of metal. The failure is not a correction term in a formula — it is the disintegration of the thing the formula was about. Which is why the hypothesis is so robust: where it fails there is nothing left to measure.

Doesn't the caesium frequency 9 192 631 770 look suspiciously arbitrary?

It is entirely arbitrary. The integer was chosen in 1967 to make the new caesium second match, as closely as measurements then allowed, the astronomical second already in use. Nobody wanted the world's timetables to jump — so the SI second carries a fossil of the Earth's orbit inside it, encoded as a nine-digit integer. That is what an operational definition is: a nomination chosen for stability and continuity, not a discovery about the fabric of things. And it can be renominated.

If two clocks a centimetre apart in height genuinely disagree, hasn't the operational definition destroyed itself?

Quite the opposite — it has been vindicated in the most uncomfortable way available. What has been destroyed is the assumption that there is one time in the room which all clocks approximate. That is Newton's assumption, and it is now false on a laboratory bench. The operational definition never claimed it: it said name the clock, name the event, report the reading. That survives intact.