The length no experiment can resolve

Where every model runs out at once assembles the Planck quantities from the three constants and shows that several independent estimates of “where the theories fail” land on the same numbers. That is a statement about theories.

There is a sharper statement available, and it is about instruments: at that scale the standard procedure for measuring a distance stops working, for a reason internal to the procedure.

Two terms with opposite slopes #

The procedure for measuring a small distance has been the same since Rutherford: throw something at it and see what comes back.

The resolution is set by the probe’s wavelength, so sharpness has to be paid for gives the first term — resolving a distance $\Delta x$ needs a momentum of at least $\hbar/2\Delta x$. More momentum, finer resolution, without limit as far as quantum mechanics is concerned.

Gravity supplies the second term. The probe carries energy, energy gravitates, and a lump of energy $E$ confined to a small region has a Schwarzschild radius $2GE/c^4$ around it. That radius grows in proportion to the energy, so the region the probe itself makes inaccessible grows as the probe is made harder.

The uncertainty in the position of whatever is being probed is then the sum of the two, and a sum of a falling term and a rising one has a minimum. The figure finds it by searching rather than by differentiating, and it comes out at $\sqrt2$ times the Planck length.

There is no momentum at which the resolution is better. That is a different kind of statement from “the theories disagree there”: it says the question has no operational meaning below that scale, whatever theory is used to ask it.

The crossing, in real units #

Drawn against energy rather than momentum, the same statement takes a form that can be read against real machines.

An optical photon resolves half a micrometre. An X-ray resolves an ångström. The LHC resolves about $10^{-19}$ metres, and the highest-energy cosmic ray ever recorded would resolve $10^{-26}$.

Extending the line, the wavelength reaches the Planck length at about $10^{19}$ GeV — and the probe’s own Schwarzschild radius reaches the same value at the same energy. That is not a coincidence; it is the definition of the Planck energy, and the crossing of the two curves is what the definition means.

Above the crossing the useful length is the larger of the two, because a probe surrounded by a horizon reveals nothing inside it. So the achievable resolution is the upper envelope of the two curves, and the envelope has a minimum at the crossing.

That has a striking corollary. Beyond the Planck energy, a scattering experiment produces black holes rather than information about structure — and the black holes get bigger with energy, so the apparent size of whatever is being hit increases. Past a point, harder collisions look at larger things.

The same trade at other strengths #

The coefficient in front of the gravitational term is unknown, and it is worth seeing how much that matters.

It matters very little. The minimum sits at $\ell_P\sqrt{2\beta}$, so a coefficient wrong by a factor of sixteen moves the floor by four. Against the twenty decades separating a laboratory from the scale, a factor of four is nothing, and every statement in this essay survives any plausible value.

That insensitivity is characteristic of dimensional arguments and is their main virtue. What the construction establishes is that a floor exists and that its order of magnitude is fixed by the three constants; the coefficient is a matter for whatever theory turns out to be right, and no experiment will distinguish the candidates.

It also explains why the several routes to the Planck scale agree so well. A quantity fixed by dimensional analysis is fixed up to a factor of order one, and when four unrelated arguments all give the same combination of $\hbar$, $c$ and $G$, their disagreement about the numerical factor is not evidence that one of them is wrong.

How far away it still is #

The practical distance is worth stating in a form that does not soften it.

The LHC resolves about $6\times10^{15}$ Planck lengths. For comparison, the observable universe is about $10^{26}$ metres and a person is about one, so the ratio a laboratory falls short by is not far off the ratio between a person and everything.

The historical rate does not help. From the first cyclotron to the LHC is about seven decades of energy over ninety years, and the cost of a machine rises roughly in proportion to its energy — so covering the remaining thirteen decades by the same method is not a matter of patience.

Nature does better and not nearly well enough. The highest-energy cosmic ray ever recorded carried about $10^{20}$ electronvolts, which is a macroscopic amount of energy in one particle and is still eight decades short.

Every claim about physics at the Planck scale is therefore a claim about consistency rather than about data, and that is the honest description of the subject rather than a complaint about it. The arguments that have any traction are arguments that some combination of quantum mechanics and general relativity is internally inconsistent, and those can be made on paper.

Why a horizon is the obstacle #

It is worth being explicit about why a horizon is fatal to the measurement rather than merely inconvenient, because the argument uses a specific property of one.

A dense object that is not a black hole can be probed: send something in, wait, and it comes out carrying information about what it met. Density alone does not hide anything, and neutron stars are studied this way with neutrinos and with gravitational waves.

A horizon is different because the surface that only lets things in is exactly what it is. Nothing that crosses it returns, so a probe that has surrounded its target with a horizon has not merely disturbed the target — it has removed it from the region any signal can reach. The measurement does not become noisy; it becomes impossible.

What comes out instead is the thermal spray the hole that outlives everything and then does not computes, and it is thermal — it depends on the hole’s mass and on nothing else, so it carries no information about what fell in. That is the same fact as the information paradox, arriving here as a limit on measurement rather than as a puzzle about unitarity.

The obstacle is not that the probe is too violent but that it has changed the causal structure, and that is why no cleverness in the design of the experiment gets round it. A gentler probe of the same energy has the same Schwarzschild radius.

The quantity that is not extreme #

The Planck mass is the odd one in the family and its oddness is informative.

Twenty-two micrograms is a speck of dust: small, and entirely within the range of things that can be weighed on an ordinary balance. Every other Planck quantity is twenty or thirty decades away from anything anybody handles.

The reason is that the Planck mass is the mass at which a particle’s Compton wavelength equals its own Schwarzschild radius — the mass at which quantum mechanics and gravity become comparable for a single object. Gravity is extraordinarily weak, so that happens at a mass enormously larger than any elementary particle’s: an electron is $10^{22}$ times lighter, and a proton $10^{19}$. The same weakness is why the mass no cold matter can hold up is a couple of solar masses rather than a gram.

The extremity of the Planck length is the weakness of gravity, seen in a different unit. The two are the same fact, and the reason the scale is unreachable is not that it is small but that reaching it requires concentrating a dust-speck’s worth of energy into one particle.

That also explains why the Planck energy sounds unimpressive. Two gigajoules is a tankful of petrol, and it is the energy of a mosquito in flight multiplied by a few million. What is impossible is not the amount but the concentration: putting it into a single quantum, where at present the record is $10^{-9}$ of it.

What would be seen instead #

If a scattering experiment above the crossing makes black holes rather than resolving structure, it is fair to ask what it would look like, because the answer is testable in the extra-dimensional scenarios where the scale is lowered.

The collision would produce a small black hole, which by the hole that outlives everything and then does not would evaporate essentially instantly — a hole of a few Planck masses lives for a few Planck times. What comes out is thermal: a spray of every kind of particle in roughly equal numbers, with no memory of what went in, and a multiplicity far higher than an ordinary hard collision produces.

That is a distinctive signature and it has been searched for. It has not been seen, and the non-detection is the constraint mentioned above: if there are large extra dimensions, the effective Planck scale is above several TeV.

The interesting feature is the direction of the effect. An ordinary collision at higher energy probes shorter distances and produces jets with more structure; a trans-Planckian one produces something with less structure the harder it is hit. A signature that gets simpler as the energy rises is the opposite of everything particle physics has seen so far, and it would be unmistakable.

That the search is possible at all rests on the scale possibly being much lower than the four-dimensional value. In the ordinary case the energy is $10^{15}$ times the LHC’s, and nothing about the signature helps.

The four routes, and why they agree #

The Planck scale is arrived at by several arguments that share nothing but their constants, and setting them beside one another is the reason to believe the answer.

Dimensional analysis. As where every model runs out at once sets out, there is exactly one combination of $\hbar$, $c$ and $G$ with the dimensions of a length, and it is $\ell_P$. That establishes a scale and says nothing about what happens there.

The Compton–Schwarzschild crossing. A particle’s quantum size and its gravitational size are equal at the Planck mass, which is the statement that quantum mechanics and gravity are equally important for one object at that mass.

The scattering limit above, which says the measurement procedure fails.

And the failure of perturbation theory. Treating gravity as a quantum field with a coupling gives a dimensionful constant, so the effective coupling grows with energy and reaches one at the Planck energy. Beyond it every loop correction is as large as the term before, and the expansion has nothing left to say.

The four are not independent — they all rest on the same three constants — but they are independent in what they assume. The last one knows nothing about black holes; the second knows nothing about scattering; the first knows nothing about anything. Four arguments with different assumptions and one answer is what makes the scale a real feature of the theories rather than an artefact of one way of writing them.

Where the model stops #

The gravitational term is heuristic. The extra term in the uncertainty relation is not derived from a theory of quantum gravity — there is not one — but from the plausible statement that a probe’s energy makes a horizon. Its coefficient is unknown, which is why the figure draws several, and different candidate theories give different values including, in some string-theoretic settings, a different functional form entirely.

The minimum is a minimum of an estimate. What the argument establishes is that the naive scattering procedure fails, not that spacetime has a granularity. Those are different claims, and several approaches to quantum gravity have a minimum length while others do not.

Everything here is a single probe hitting a single target. Interferometric arguments — measuring a distance by counting fringes rather than by scattering — reach different limits, some of them weaker, and there is a genuine literature on whether a length can be measured more finely by an extended apparatus than by a point-like one.

And the black-hole production claim assumes gravity is four-dimensional all the way down. If there are extra dimensions of a large enough size, gravity strengthens at short distance, the effective Planck scale drops, and the crossing could in principle be within reach of an accelerator. Searches for exactly that at the LHC have found nothing, which is a genuine experimental constraint on the only version of this subject that is testable.

The target is treated as passive. Both the probe and whatever it hits carry energy, and in a symmetric collision the horizon that matters is the one the pair makes together. That changes the numbers by a factor of order one and the argument not at all, but it means the “probe energy” on the axes is a centre-of-mass energy rather than a beam energy — a distinction that costs a factor of a hundred at a fixed-target machine and nothing at a collider.

And the whole account assumes semiclassical gravity holds up to the crossing. It is applied at exactly the scale where it is expected to fail, which is a familiar and uncomfortable position: the argument uses general relativity to establish that general relativity cannot be tested there. What saves it is that every candidate replacement produces a minimum length of the same order, so the conclusion is more robust than the derivation.

What the pictures cannot show #

The tradeoff figure adds two uncertainties as though they were independent contributions to one quantity, which is a heuristic rather than a theorem. A proper treatment would need a theory in which position is an operator with a gravitational correction, and the whole point is that no such theory is agreed.

The reach figure compares resolutions across instruments as though resolution were a single number, and it is not: a machine’s reach depends on what it is looking for, and an experiment sensitive to a rare process can constrain physics at energies far above its beam energy through virtual effects. The bars measure one thing about each machine and not the most useful thing.

A third omission is what “resolution” means for something that is not a particle. All the lengths here are the sizes of things a scattering experiment could distinguish, and the more interesting question at the Planck scale is whether spacetime itself has a structure — whether there is a smallest volume, a discreteness, a granularity. Nothing in these figures bears on that. What they establish is that no experiment of the kind physics currently knows how to do could see it, which leaves the question open in a particularly unsatisfying way: a property of the world that may exist and is by construction unobservable.

Where the ladder goes next #

The planck-scale ladder began with where every model runs out at once, which assembles the units from the constants and shows that several unrelated estimates agree. This rung asks what happens to a measurement there, and finds that the procedure defeats itself before the theories have a chance to disagree. The rungs after it: the holographic bound, which limits how much can be inside a region by the area of its boundary rather than its volume; the black-hole entropy count, where the Planck area appears as a unit of information; and the possibility that the scale is lower than it looks, which is the one part of the subject an experiment can address.

The habit worth carrying away is that a limit on measurement is a stronger statement than a limit on a theory. A theory can be replaced and a procedure that defeats itself cannot, and the argument here needs no commitment about what happens at the Planck scale in order to say that nothing can be seen there.