Where every model runs out at once

Every mass has two lengths attached to it, and they come from theories that have nothing to do with one another.

The first is its Compton wavelength, $\hbar/mc$, which is roughly the scale below which quantum mechanics stops allowing the object to be treated as being in one place: try to localise it more sharply and the momentum uncertainty is large enough to create more particles of the same kind. It falls as the mass rises.

The second is its Schwarzschild radius, $2Gm/c^2$, the size to which the mass would have to be squeezed to make a horizon. It rises as the mass rises.

One falls and the other rises, so they cross exactly once.

What the crossing is an argument about #

The two curves are not merely a graph. The crossing has an operational meaning, and it is the reason the number is taken seriously.

Suppose a measurement of position to a precision $\Delta x$. By the uncertainty relation that requires a momentum uncertainty of at least $\hbar/\Delta x$, so an energy of at least $\hbar c/\Delta x$ has to be involved — a shorter probe wavelength means a more energetic probe, which is why particle physics builds larger machines to see smaller things.

The relation that fixes the first curve is that a packet localised in space is built from a spread of wavelengths, and narrowing one widens the other. Nothing about gravity is in it, and nothing about it changes as the mass rises — which is why the Compton length falls smoothly across forty decades of the mass axis with no feature anywhere on it.

Now bring in gravity. That energy has a Schwarzschild radius of its own, $2GE/c^4$. As $\Delta x$ is made smaller, the energy needed grows and its Schwarzschild radius grows with it — and at some point the radius exceeds the precision being sought. Beyond that, concentrating more energy into the region makes a bigger horizon rather than a sharper measurement, and the attempt defeats itself.

The other curve is the radius to which each mass would have to be squeezed to make a horizon, and tabulating it is a useful corrective. The proton’s is $2.5\times10^{-54}$ m — nineteen orders of magnitude below the Planck length, and utterly meaningless as a physical statement. The formula is being applied far outside anywhere anything is known, and the crossing of the two curves is precisely the point at which it stops being applied outside and starts being applied at the boundary.

Setting the two equal is the crossing in the figure. It says: below this scale, the theory used to describe the probe and the theory used to describe the geometry cannot both be applied without one wrecking the other’s assumptions. That is not the same as saying anything about what is there.

The argument has a pleasing symmetry to it that is worth pointing out. Both curves are statements about the failure of a description, and they fail in opposite directions. Quantum mechanics stops a thing from being small because making it small requires energy; gravity stops a thing from being small because energy makes a horizon. Each theory alone permits an arbitrarily precise measurement — quantum mechanics by supplying more energy, general relativity by supplying a smaller probe — and it is only the combination that closes off both routes at once. That is why the scale belongs to neither theory and to the pair of them.

The numbers, and the one that is strange #

The three basic combinations are

$$\ell_P = \sqrt{\frac{\hbar G}{c^3}} = 1.62\times10^{-35}\ \text{m}, \quad m_P = \sqrt{\frac{\hbar c}{G}} = 2.18\times10^{-8}\ \text{kg}, \quad t_P = \frac{\ell_P}{c} = 5.4\times10^{-44}\ \text{s}.$$

Two of them are far outside anything: the length is twenty orders of magnitude below a proton’s radius, and the time is correspondingly absurd. The mass is not. Twenty-two micrograms is a visible speck — the mass of a flea’s egg, or of a fifth of a millimetre cube of water — and it sits in the middle of ordinary experience.

That oddity is worth dwelling on because it is the whole of why gravity is negligible in particle physics. Take two protons and compare the gravitational and electrostatic forces between them: the ratio is $Gm_p^2/(ke^2) \approx 8\times10^{-37}$, independent of separation because both forces fall as the inverse square. That $10^{-36}$ is not a coincidence — it is essentially $(m_p/m_P)^2$, the square of how far a proton falls short of the Planck mass. Gravity is weak between particles precisely because particles are so light compared with the mass at which gravity would matter.

Turned round: an object at the Planck mass would have gravity and quantum mechanics equally important, and it would be a speck a person could see. What makes it unreachable is not the mass but the requirement that the mass be confined within its own Compton wavelength, which for twenty-two micrograms is $10^{-35}$ m.

Units chosen by nature rather than by a committee #

There is a second way into the same numbers, and it is the one Planck himself took in 1899 — before relativity, before quantum mechanics, and before anything the scale is now associated with.

Planck’s observation was that a system of units could be built from constants that belong to the world rather than to human bodies or to the size of the Earth. A metre is the length of a particular bar, then a fraction of a meridian, then a number of wavelengths, then a fixed fraction of a light-second: each redefinition is an improvement in reproducibility and none of them stops the unit from being arbitrary. But $c$, $G$ and $h$ are not arbitrary, and there is exactly one way to combine them into a length, one into a mass and one into a time.

He wrote that such units “necessarily retain their meaning for all times and for all cultures, including extraterrestrial and non-human ones”, which is a remarkable thing to have said about a set of numbers whose physical significance would not be understood for another quarter of a century. His motivation was aesthetic. The interpretation as a scale where physics changes came later, from the localisation argument above.

The modern system of units has moved a long way in Planck’s direction without going the whole way: the kilogram is now defined by fixing $h$, the metre by fixing $c$, and the second by an atomic transition. Three of the seven base units are fixed constants of nature. What stops the programme from completing is $G$, which is known to about one part in $10^5$ — by far the worst-measured of the fundamental constants, and the reason the Planck length’s third significant figure is not settled by measurement either.

What the crossing does not say #

Three claims regularly attach themselves to this number and none of them follows from the argument.

It is not a pixel size. Nothing in the derivation produces a lattice, a discreteness or a minimum distance. Some theories of quantum gravity do predict a minimum length and others do not; the dimensional argument is agnostic and cannot distinguish them. A statement that “space is quantised at the Planck length” is a statement about a theory, not about this calculation.

It is not measured. The shortest length scale probed by experiment is around $10^{-19}$ m. The gap between that and $10^{-35}$ is sixteen orders of magnitude — the same ratio as between a millimetre and the distance to the nearest star. No result at any accelerator constrains physics at this scale directly.

It is not exact. The combination is fixed only up to dimensionless factors, and different routes give different ones. The bisection in the hero figure produces $2.29\times10^{-35}$ m, which is $\sqrt2$ times the conventional value — the factor of two in the Schwarzschild radius, surviving a square root. Other conventions put $8\pi$ into $G$ and shift the answer again. What the argument fixes is the exponent.

There is a fourth misreading worth adding, because it is the most common one in careful company: that the Planck scale is where quantum gravity effects become large. Strictly, it is where the standard way of treating gravity as a small quantum correction stops converging — which is a statement about a calculational method rather than about nature. Effects could be visible far earlier if some symmetry conspires to amplify them, and there are proposals of that kind; or the scale could be exactly where the naive argument puts it. The argument on this page does not distinguish those cases, and neither does any measurement.

Where the scale shows up anyway #

Although nothing has been measured there, the combination appears in every calculation that has tried to put the two theories together, which is some evidence that it is the right variable.

A horizon’s temperature combines $\hbar$, $c$, $G$ and $k$ in one expression, so it is already a quantum-gravitational quantity — an answer computed in a regime where no theory has been checked. Extrapolate it to a Planck mass and it gives $5.6\times10^{30}$ K and a lifetime of about $10^{-39}$ seconds, some ten thousand Planck times. At that point the object is one Planck unit of nearly everything, and the formula that produced the number is certainly no longer trustworthy.

A horizon’s entropy is a quarter of its area in units of the Planck length squared, which is the cleanest appearance of the scale anywhere: a formula combining thermodynamics, gravity and quantum mechanics, in which the Planck area is the natural unit of information. Nobody put it there.

The same combination appears as the energy at which the naive quantum theory of gravity stops making sense, as the coupling scale in string theory, and as the natural cutoff in every effective field theory that includes gravitons. The consistency is the argument for taking the number seriously, in the absence of any measurement.

How far away it is, stated properly #

Every discussion of this scale should carry an honest statement of the size of the extrapolation, because the number is quoted so often that it acquires an unearned familiarity.

The energy corresponding to the Planck mass is $1.2\times10^{19}$ GeV. The largest accelerator reaches about $1.4\times10^4$ GeV in the centre of mass. The ratio is $10^{15}$.

To feel that, take the ratio the other way. If the energies reached so far were represented by the width of a human hair, the Planck energy would be a hundred million kilometres away. Or: the improvement in energy from the first cyclotron in 1931 to the largest machine now is about six orders of magnitude in ninety years, so at the historical rate — which has already flattened — the remaining fifteen would take some two centuries and a machine the size of a planet.

That is why the arguments in this area are made with consistency rather than with data, and why physicists are careful to describe them as such. The Planck scale is not a frontier being approached; it is a place where two theories are known to be mutually inconsistent, identified by dimensional analysis and by a heuristic, and everything said about what happens there is theory constrained by other theory.

There is one loophole worth naming, because it is the reason the subject is not purely formal. The argument assumes gravity keeps its familiar strength all the way down. If space has extra dimensions that are curled up at a small but not absurdly small size, gravity’s field spreads into them and is stronger at short distances than the extrapolation suggests — and the scale at which quantum gravity becomes relevant could then be far lower, conceivably within reach. Experiments have looked for exactly that, by testing whether the inverse-square law of gravity holds below a millimetre and by searching for missing energy at colliders. It does hold, and nothing has been found, which constrains the size of any such dimensions rather than ruling out the idea.

The Planck units that have no quantum in them #

Not every combination of these constants involves $\hbar$, and the ones that do not behave quite differently from the length and the time.

Divide the Planck mass times the Planck length by the Planck time squared and the quantum constant cancels, leaving $c^4/G$ — about $1.2\times10^{44}$ newtons. That is a force built out of relativity and gravity alone, and it appears in general relativity as the inverse of the coefficient in the field equations: the equations say that curvature equals the stress-energy divided by $c^4/G$, so this force is the stiffness of spacetime. Some formulations of the theory take it, or a fixed fraction of it, to be a maximum force that no physical situation can exceed.

The corresponding power, $c^5/G$, is $3.6\times10^{52}$ watts, and it is the one Planck quantity that observation has come within a factor of a thousand of.

Nothing electromagnetic goes near it — the whole observable universe’s starlight is some $10^{49}$ watts. But a pair of black holes in their final orbit radiates gravitationally, and the calculation of how much gives a peak of order $10^{-3}$ times $c^5/G$ regardless of the masses, because the masses cancel out of the ratio. The first merger detected peaked at about $3.6\times10^{49}$ watts, for a few thousandths of a second, which briefly exceeded the combined light of everything else in the visible universe.

So an essay about a scale nothing has probed can close with one Planck unit that has been measured to within three decades — and it is measured, characteristically, in an event where no quantum mechanics is involved at all.

The worst prediction anybody has made #

The Planck density, $10^{96}$ kilograms per cubic metre, is where the same scale produces the largest discrepancy in physics.

Quantum field theory assigns a zero-point energy to every mode of every field, and summing over modes gives an energy density that diverges — so the sum is cut off, and the only defensible place to cut it off is the scale at which the theory is expected to fail. Using the Planck scale gives a vacuum energy density of around $10^{113}$ joules per cubic metre.

That energy gravitates, and the amount of vacuum energy in the universe is measured: it is what makes the expansion accelerate, and it comes to about $10^{-9}$ joules per cubic metre.

The two numbers differ by some hundred and twenty orders of magnitude. It is not a discrepancy of the kind a better calculation repairs; it is the largest mismatch between a theoretical estimate and a measurement in the history of the subject, and no accepted explanation exists.

Two readings are available and both are instructive. One is that the estimate is simply illegitimate — that using a cutoff this way is not a prediction of anything, since the answer depends entirely on a scale nobody can justify. The other is that it is a genuine prediction of a genuine theory and is therefore a genuine failure, pointing at a cancellation of a hundred and twenty digits that some symmetry ought to be enforcing and none is known to.

Either way it belongs on this page, because the number that goes wrong is the one the crossing produced. An estimate built on the Planck scale is either meaningless or catastrophically wrong, and distinguishing those is one of the open problems this essay’s honest caveats are about.

What it costs, and where the model stops #

Dimensional analysis proves nothing about physics. Combining constants to make a length always succeeds; whether the length means anything depends on whether the physics actually changes there. The same procedure applied to $e$, $\hbar$ and $c$ gives the fine-structure constant, which is dimensionless and tells nothing about a scale at all. What makes the Planck length more than an arithmetic exercise is the operational argument about localisation, and that argument is a heuristic rather than a derivation.

Both input theories are being used outside their tested range. The Schwarzschild radius of a proton is computed from a solution of general relativity applied to a quantum object, and the Compton wavelength of a microgram mass is computed from a relation established for particles. Neither is a legitimate application; the crossing is where they would meet if both continued to hold, which is precisely the assumption in question.

A hierarchy is not an explanation. That the Planck mass exceeds a proton’s by $10^{19}$ is an observation, not a mechanism, and why the two differ so enormously is one of the standing open questions. Any theory that makes gravity strong at a much lower energy moves this whole picture, and such theories have been proposed and constrained.

The Compton wavelength is being used loosely. The strict statement is that a particle cannot be localised more sharply than its Compton wavelength without the energy involved being enough to create another particle of the same species — which is a statement about a field, not about a single object, and does not apply to a microgram mass in the way the smooth curve suggests. The curve is drawn across forty decades because the formula is defined there, not because the physics behind it is.

Nothing here is about the early universe. The Planck time is often quoted as a moment in cosmic history; that application belongs to cosmology and is a separate argument with separate assumptions. What is claimed on this page is only what the two lengths do as functions of mass.

What the figure is actually claiming #

It is worth closing on what the hero figure does and does not assert, since a picture of two lines crossing is a strong-looking thing.

It asserts three facts, each of which is arithmetic: that the Compton wavelength falls as $1/m$, that the Schwarzschild radius rises as $m$, and that two monotonic curves of opposite slope cross once. None of that is in doubt.

It does not assert that either curve means anything at the crossing. Both are extrapolations from regimes where they are established — one from particle physics, one from the weak-field solutions of general relativity — into a region where nothing has been measured, and the honest description of the crossing is that it marks where the extrapolations become mutually inconsistent rather than where either individually fails.

That is a weaker claim than the number’s fame suggests, and it is the strongest claim the argument supports. What makes it worth drawing anyway is that the same combination of constants keeps arriving from unrelated directions — in horizon entropy, in the breakdown of perturbative quantum gravity, in the coupling scales of candidate theories. A number that several independent routes agree on is worth taking seriously even when no experiment has been near it.

The ladder from here #

Later rungs on this anchor: the derivation of the localisation limit properly, with the energy needed to probe a scale set against the horizon that energy makes; Planck units as a system, and what it means that every quantity becomes one in them; the appearance of the Planck area in horizon entropy, which is the one place the scale carries a factor nobody chose; the hierarchy problem, and models in which the true scale of gravity is far lower; and the experimental frontier, which is the honest statement of how far away all of this is.

The neighbouring ladders are the uncertainty relation, which supplies one of the two curves, and horizons, which supply the other.