The clock that cannot be both light and good

The length no experiment can resolve found a floor under distance. To see something small a probe needs a short wavelength, a short wavelength costs energy, and enough energy packed into a small enough region makes a horizon, so beyond a point a harder probe resolves less. The turning point is the Planck length, $1.6 \times 10^{-35}$ metres, and the usual summary is that nothing shorter can be measured.

The summary invites a reading that the argument does not support: that every distance longer than the Planck length can be known to within a Planck length. The argument was about resolving something small. Knowing a long distance precisely is a different measurement, with a different instrument, and when Hans Salecker and Eugene Wigner worked out its quantum limit in 1958 they found that the floor depends on the distance being measured. Adding gravity to their result, as Frigyes Károlyházy did in 1966 and Jack Ng and Hendrik van Dam did more explicitly in 1994, turns it into a floor that grows with the distance, slowly but without end.

Timing a distance #

The cleanest way to measure a distance is to send light to a mirror and time its return. The distance is half the round-trip time multiplied by $c$, and its uncertainty is the uncertainty in the time — which is the uncertainty in the clock.

A clock is a physical object, and like every object its position is uncertain. That matters here because the clock has to be in the same place when the light leaves and when it returns, or the measured interval includes the clock’s own drift. Suppose the clock’s position is known to within $\delta x$ when the light sets out. Then its momentum is uncertain by at least $\hbar/2\delta x$, its velocity by that divided by its mass $m$, and by the time the light comes back, a time $2l/c$ later, the position has spread by about $\hbar l/(mc\,\delta x)$. The two contributions trade against each other: pin the clock down tightly at the start and it spreads fast; leave it loose and it starts uncertain. The best choice balances them, and the smallest total uncertainty is

$$\delta l_{\rm spread} \approx \sqrt{\frac{\hbar l}{mc}}.$$

This is the same competition that keeps a confined particle from sitting still, now played out over the time the light is away. It says that measuring a distance more precisely needs a heavier clock, and that measuring a longer distance to the same precision needs a heavier clock still. Salecker and Wigner stopped there, with a clock that could be made as heavy as wanted.

It is worth seeing how far from this limit real instruments sit, because the comparison explains why nobody had to think about it. A laser-ranging station timing the light’s trip to a reflector on the Moon, 384,000 kilometres away, reaches a few millimetres. The best optical clocks keep time to a part in $10^{18}$, which over the 2.6-second round trip is a few attoseconds, or about a nanometre of light travel. The quantum spread of a one-kilogram clock over the same trip is $\sqrt{\hbar l/mc}$ with $l$ the Earth–Moon distance, which is about $10^{-17}$ metres — eight orders of magnitude below the nanometre. Every real distance measurement is limited by its engineering, the atmosphere and the reflectors long before it meets the clock’s own quantum mechanics. Salecker and Wigner were not describing an obstacle anybody would hit. They were asking what a distance means, in a theory where every ruler and every clock is a quantum object, and their answer was that the meaning is fuzzy by an amount that depends on how much the clock weighs.

The other side of the mass #

A clock cannot be made as heavy as wanted, because a clock heavy enough is inside its own horizon. A clock of mass $m$ has a Schwarzschild radius $2Gm/c^2$, and it must be larger than that or no signal from its face reaches anything. A clock that has to be at least that large cannot locate the instant of the light’s return to better than about its own size, so its mass sets a second floor,

$$\delta l_{\rm horizon} \approx \frac{Gm}{c^2},$$

which rises with the mass. One term falls as $m^{-1/2}$ and the other rises as $m$, and there is a best clock between them.

For a one-metre distance the best clock weighs 5.4 tonnes, and the metre is then uncertain by $1.2 \times 10^{-23}$ metres. That is seventy million times smaller than a proton and comfortably beyond any measurement anyone has made. It is also seven hundred and fifty thousand million Planck lengths. The floor under a metre is not the Planck length; it is the Planck length multiplied by a number of order $10^{12}$.

Written out, the minimum of $\sqrt{\hbar l/mc} + Gm/c^2$ over $m$ is

$$\delta l_{\min} = \frac{3}{2^{2/3}}\left(\ell_P^{\,2}\, l\right)^{1/3} \approx 1.89\,\left(\ell_P^{\,2}\, l\right)^{1/3}, \qquad \ell_P = \sqrt{\hbar G/c^3}.$$

The coefficient 1.89 belongs to the crude way the two floors were added and should not be taken seriously. The exponent is the argument. The uncertainty grows as the cube root of the distance, and at $l = \ell_P$ it is the Planck length itself, so this rule and the probe-energy rule agree exactly where both apply.

A floor that rises #

Across sixty decades of distance the cube root covers twenty decades of uncertainty. The distance from the Earth to the Sun is uncertain by $6 \times 10^{-20}$ metres, and the size of the observable universe, $4.4 \times 10^{26}$ metres, by about $10^{-14}$ metres — the size of an atomic nucleus. No measurement anywhere comes close to either. The point is not that these uncertainties are large but that they exist at all, and that they grow.

The figure also carries a second rule, which the cube root should be compared with. If spacetime were made of Planck-sized cells whose sizes fluctuated independently, a distance of $l$ would cross $l/\ell_P$ of them and the fluctuations would add like steps of a random walk, giving an uncertainty of $\sqrt{\ell_P l}$. That grows much faster: it would leave a metre uncertain by $4 \times 10^{-18}$ metres and the observable universe by a tenth of a millimetre. The clock argument does not produce a random walk. Its fluctuations are correlated across the whole distance, because the limit comes from one clock and one mirror rather than from a sum of independent cells.

The clock each distance asks for #

The best clock grows with the distance it measures, as the cube root.

At the Planck length the optimal clock weighs 0.63 Planck masses, some fourteen micrograms, and its quantum spread and its horizon are the same size: the measurement is at the place where every model runs out at once, and the two halves of this argument are the two lengths that cross there. At a metre the clock is the mass of a truck. At the size of the observable universe it is four million million kilograms — a small asteroid — and the clock is still, by construction, outside its own horizon.

None of these clocks is a proposal. The argument does not describe a device; it asks what the laws of physics would allow the best possible device to achieve, and the answer is a limit no real clock is anywhere near. That is the sense in which the result is about spacetime rather than about instruments. If no possible clock can locate the end of a metre more precisely than $10^{-23}$ metres, then the statement that the metre has a length more precise than that has no operational content.

Counting places #

The cube root has a consequence that the scaling figure hides and that was not part of anybody’s intention when the argument was first made.

Take a cube of side $l$. If distances inside it can be fixed only to within $\delta l \approx (\ell_P^2 l)^{1/3}$, then the cube can be divided into at most $(l/\delta l)^3$ cells that a measurement could tell apart. Substituting,

$$\left(\frac{l}{\delta l}\right)^3 \sim \frac{l^3}{\ell_P^{\,2}\, l} = \left(\frac{l}{\ell_P}\right)^2.$$

The count is an area, not a volume. A region can hold as many distinguishable places as its surface has Planck areas, and no more. For the observable universe that is about $10^{122}$, where counting Planck volumes would give $10^{184}$.

The same root from counting ticks #

There is a third route to the cube root, and it uses a clock as a counter rather than as a ruler. A clock measures time by passing through a sequence of distinguishable states — the ticks — and a state cannot stop being itself faster than its energy allows. The Margolus–Levitin theorem puts the limit at $\pi\hbar/2E$ per tick, so a clock of mass $m$ can tick at most $2mc^2/\pi\hbar$ times a second. To tick faster it needs more mass.

Now ask a region of size $l$ to act as a clock for a time $l/c$, the time light takes to cross it, and to resolve that time as finely as possible. The more mass it holds the more ticks it can make, but it cannot hold more than the mass that would make it a black hole of its own size, about $lc^2/2G$. Put that mass in and the number of ticks in the crossing time comes out of order $(l/\ell_P)^2$, and the shortest interval it can resolve, the crossing time divided by the number of ticks, corresponds to a distance $\ell_P^2/l$. That is a different statement — it is the finest time resolution available to the whole region acting together — and Seth Lloyd and Jack Ng used it to argue that the total number of operations the region can perform, and so the number of cells it can tell apart, again scales with its area.

Three different questions — how far a clock wanders, how many places a region can distinguish, how often a region can tick — keep returning the same power. Each assumes the uncertainty principle and the existence of horizons and nothing else, and each finds that gravity caps what the uncertainty principle would otherwise allow at a rate set by area rather than by volume. The black-hole result came first and by a different road altogether.

That is exactly the scaling Jacob Bekenstein and Stephen Hawking found for the entropy of a black hole, which lives on its horizon’s area at a quarter of the area in Planck units rather than in its volume, and which is why a horizon’s area is not allowed to shrink in any classical process. It is also the content of the holographic principle, the conjecture that the number of states any region can hold is bounded by its surface area. Nobody put area into the clock argument. It came out of balancing the uncertainty principle against the Schwarzschild radius for a single measurement, and arrived at the same counting that black-hole thermodynamics reached by an entirely different route. That two arguments with nothing in common give the same power of the size is the strongest reason to take either seriously.

Can it be seen? #

A growing uncertainty in distance ought to show itself in light that has travelled a long way. If the length of a photon’s path from a quasar to a telescope is uncertain by $\delta l$, its phase is uncertain by $2\pi\,\delta l/\lambda$, and if that reaches a radian the wavefronts arriving at different parts of the telescope are no longer in step, the interference that forms an image fails, and a point source blurs.

The random walk fails at once. Over 3.2 thousand million light-years it would put 277 radians of phase noise on visible light, and quasars at that distance are imaged sharply by large telescopes and by interferometers. Richard Lieu and Lloyd Hillman made that argument in 2003, and nothing since has rescued the random walk.

The cube-root rule survives easily in visible light, at $5 \times 10^{-8}$ radians, and in X-rays, at $2 \times 10^{-5}$. Only for γ-rays does it approach a tenth of a radian, and analyses published between 2011 and 2015 argued that sharp γ-ray images of distant sources rule it out. Others have answered that the calculation assumes the whole uncertainty in the distance lands on every photon’s phase independently, when the fluctuations the clock argument describes are correlated across the wavefront and across photons, and would largely cancel. The dispute is not about the data. It is about what the cube-root rule actually predicts for a wave, which the clock argument — a statement about one measurement of one distance — does not say.

A different kind of test was built at Fermilab between 2014 and 2017. The Holometer was a pair of interferometers forty metres long, placed close together, looking for correlated jitter in their arm lengths of the kind a particular holographic model predicted, at a level set by the Planck length. It found none, and excluded that model with high confidence. Other models, including the one here, predict noise that the Holometer’s design was not built to see.

Where the argument stops #

The argument is a heuristic built from two inequalities, each true to within a factor of order one, added as though they were independent. It assumes the clock is an ordinary quantum object obeying the uncertainty principle and that its gravity behaves classically, as a Schwarzschild radius. Near the Planck scale neither assumption is secure, and that is where the result matters most. A full quantum theory of gravity might change the exponent, or show that the clock and mirror cannot be treated separately, or that the measurement can be arranged in a way the argument did not consider — several clocks, an entangled clock, a clock whose mass is spread out.

Two features of the argument are easy to miss. The first is that it treats the mirror as perfect and fixed, putting all the uncertainty in the clock; a real mirror is another quantum object with its own spread, and including it changes the coefficient but not the exponent. The second is that it treats the measured distance as flat. Over cosmological distances the geometry itself is curved and expanding, the distance between two galaxies is not even defined until a way of slicing spacetime into moments is chosen, and what counts as now across such a distance is itself a convention. The cube root for the observable universe is therefore a statement about scale, not about any particular cosmological distance.

It also says nothing about what spacetime is. A limit on what any clock can measure is compatible with a smooth spacetime that simply cannot be probed more finely, with a spacetime that is genuinely uncertain at that level, and with a spacetime made of something else entirely whose description at large scales happens to obey the limit. What it rules out is the position that a long distance has a meaning at the precision of the Planck length.

And the step from the uncertainty in one distance to the phase noise on a wave — the step every astronomical test needs — is not part of the argument at all. That is why the observational record is clear only for the random walk.

What the pictures cannot show #

The curves in the scaling figure are straight lines on logarithmic axes, and they look like laws. They are orders of magnitude, with coefficients that could be several times larger or smaller. The 5.4 tonnes, the $1.2 \times 10^{-23}$ metres and the $10^{122}$ cells are the coefficients of a crude model and should be read as “about”. The figures cannot show the correlation structure of the uncertainty, which is the thing that decides whether astronomy can see it, and the phase figure assumes the opposite of what the clock argument suggests, deliberately, to show the most any test could find.

The domain of the argument is a measurement of a single distance by a single light signal against a single clock, with gravity entering only as a minimum size for the clock. Inside it, the floor grows as the cube root of the distance. Outside it — many clocks, distributed masses, quantum gravity proper — the argument is silent.

Still open: whether the foam has the shape the clock suggests #

The cube root has appeared from several directions: from Károlyházy’s analysis of how gravity limits the sharpness of a mass’s position, from the clock argument, from counting the operations a region could perform before collapsing, and indirectly from the holographic bound. Whether those are one fact seen several ways or a coincidence of dimensional analysis is not settled, and neither is what the cube-root uncertainty does to a travelling wave — whether its correlations make it invisible to every astronomical test, or leave a signature in the arrival of the highest-energy photons that next-generation γ-ray telescopes could find. Proposals to look for it in tabletop interferometers, in the coherence of atom interferometers, and in correlated noise between gravitational-wave detectors exist; none has reached the predicted level.

The argument that produced the floor is two lines long. A clock light enough to stay small wanders while the light is away, and a clock heavy enough not to wander is too close to being a black hole, so the best possible measurement of a distance l is uncertain by about $1.89\,(\ell_P^2 l)^{1/3}$ — 750 thousand million Planck lengths for a metre, a nucleus for the observable universe — and a region can hold only as many distinguishable places as its surface has Planck areas. The Planck length is where that floor starts, not where it stays.