Illustrated Physics
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Planck scale — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The two lengths every mass has. The Compton wavelength and the Schwarzschild radius of the same mass, against mass, on logarithmic axes. One falls and the other rises, so they cross exactly once — here at 1.539·10⁻⁸ kg and 2.286·10⁻³⁵ m, found by bisecting the difference rather than by writing down √(ħG/c³). The conventional Planck values are 2.176·10⁻⁸ kg and 1.616·10⁻³⁵ m; the crossing sits a factor of 1.414 away from them, which is exactly √2 and is the factor of two in the Schwarzschild radius coming through a square root. That is the whole precision this argument has, and it is worth saying, because a number written to four figures invites a reader to believe the definition is doing more work than it is. Nothing in physics is known at that length.

    Where every model runs out at once

    Every mass carries two lengths — one below which quantum mechanics will not let it be located, one below which gravity will not let anything escape. One falls with mass and the other rises, so they cross exactly once, at a length nothing in physics has ever probed.

    part 1 · astrophysics
  2. Two terms, and the distance neither can beat. The smallest distance a probe of a given momentum can resolve, as the sum of two terms. The falling one is the uncertainty relation: more momentum, shorter wavelength, finer resolution. The rising one is gravity: the probe's own energy curves the region it is probing, and past a point it makes a horizon larger than the thing being looked at. With the gravitational term at 1× the Planck area, the least resolvable distance is 2.286e-35 m — each located by scanning the drawn curve over four hundred thousand momenta rather than by substituting into a formula. There is no momentum at which the resolution is better than that, so the ordinary procedure for measuring a distance has a floor, and the floor is the Planck length up to a factor of order one.

    The length no experiment can resolve

    Measuring a small distance needs a short wavelength, a short wavelength needs a large energy, and a large energy in a small region makes a horizon. Past a point, pushing harder makes the probe bigger — and the distance where that turns round is the Planck length.

    part 2 · astrophysics
  3. The one prediction the Planck scale makes. The energy density the vacuum should have, from summing the zero-point energy of a field's modes up to a cutoff, against where that cutoff is put — thirty decades of cutoff energy and a hundred and thirty of density, both logarithmic. The horizontal line is what is measured: 5.34e-10 joules per cubic metre, the dark energy that accounts for sixty-nine per cent of the universe. Cutting the sum off at the Planck energy — which is where dimensional analysis says every description available runs out — overshoots it by 10^121. That is the largest disagreement between an estimate and a measurement anywhere in physics, and the slope of the line is why it cannot be argued away: the density goes as the fourth power of the cutoff, so cutting off at the electroweak scale still overshoots by 10^54 and cutting off at one electronvolt — below which no physics is in doubt at all — still overshoots by 10^8. The cutoff that would give the right answer is 8.0e-3 electronvolts, which is a wavelength of about a tenth of a millimetre and corresponds to no known physics whatever.

    The estimate that misses by a hundred and twenty

    Every argument about the Planck scale is an argument about consistency rather than about data, with one exception. The zero-point energy of the quantum fields gravitates, dimensional analysis at the Planck cutoff says how much, and what is measured is 10¹²¹ times smaller. It is the largest disagreement between an estimate and a measurement anywhere in physics, and lowering the cutoff does not rescue it.

    part 3 · astrophysics
  4. How large the hidden dimensions would have to be. The size extra dimensions would need, for gravity's true scale to be at a TeV rather than at 10¹⁹ GeV, against how many of them there are — a logarithmic axis of metres, for three choices of the true scale. The relation is the one that makes the arrangement work: the Planck mass observed in four dimensions is M_*^(2+n)Rⁿ, so gravity is weak because its field spreads into a volume nothing else can enter. Solved for R at a true scale of a TeV: 1 dimension needs 2.9e+13 m, 2 dimensions needs 2.4e-3 m, 3 dimensions needs 1.0e-8 m, 4 dimensions needs 2.2e-11 m, 5 dimensions needs 5.4e-13 m, 6 dimensions needs 4.5e-14 m. The two horizontal lines are where experiment has been. One extra dimension would have to be of order a hundred astronomical units, which would have wrecked the orbits of the planets and is excluded absolutely. Two would have to be of order a millimetre — which is what made the arrangement famous, because a millimetre is a distance a laboratory can test, and torsion balances have since verified the inverse-square law down to fifty-two micrometres and excluded it. Three or more sit below a nanometre, where no gravitational measurement reaches, and are untouched.

    The scale that may not be where it looks

    Every Planck number assumes gravity is four-dimensional all the way down. If it is not — if the field spreads into dimensions compact enough to have escaped notice — the true scale where gravity becomes strong could be at a TeV, and the whole remoteness of the Planck scale would be an artefact of where the field lines go. It is the one part of the subject an experiment can address, and the experiments have addressed it.

    part 4 · astrophysics

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