Illustrated Physics
physics, drawn
Concept

Planck units — where it appears

The unique combinations of G, ħ and c that have the dimensions of a length, a mass and a time, and belong to no laboratory's conventions. The length is 1.6 × 10⁻³⁵ metres, twenty orders of magnitude below anything measured, and it marks where gravity, quanta and relativity would all matter at once.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

astrophysics · Planck scale One mass, two entropies. The entropy of a solar mass as ordinary gas, generously counted at ten Boltzmann constants per proton, against the entropy of a solar-mass horizon. The first is 1.19·10⁵⁸ k and the second 1.05·10⁷⁷ k — a factor of 8.83·10¹⁸. This is why a horizon had to be given an entropy: without one, dropping anything at all through it destroys entropy and the second law fails.

The entropy that lives on a surface

Throw a cup of tea through a horizon and the entropy of the outside world falls. Either the second law is wrong or the horizon has an entropy of its own — and the only quantity available for it turns out to be its area, in units of a length made from gravity, quantum mechanics and the speed of light together.

thermodynamics · Entropy

Named alongside it

The objects these essays reach for when they reach for this one.

Event horizonBlack hole entropyDimensional analysisEntropyHawking temperatureHolographic boundMicrostatesPlanck lengthQuantum gravitySchwarzschild radiusThe second lawUncertainty

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