Illustrated Physics
physics, drawn
Concept

Planck length — where it appears

The length √(ħG/c³), about 1.6 × 10⁻³⁵ m, at which a particle's quantum wavelength equals the size of the black hole its energy would make. It marks where quantum mechanics and gravity can no longer be treated separately, and no measurement has come near it.

Named by 2 essays across one field — 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 How well a distance can be known, by how long it is. The smallest uncertainty in a distance l, on logarithmic axes, by three rules. Solid: the clock argument, 1.89 (ℓₚ² l)^(1/3), which grows as the cube root of the distance. Dotted: the Planck length, the floor usually quoted, which does not grow at all. Dashed: the guess that Planck-scale jitters add like a random walk, √(ℓₚ l). a metre: 1.2·10⁻²³ m by the clock argument, 4·10⁻¹⁸ m as a random walk; the Earth to the Sun: 6.4·10⁻²⁰ m by the clock argument, 1.6·10⁻¹² m as a random walk; the observable universe: 9.2·10⁻¹⁵ m by the clock argument, 8.4·10⁻⁵ m as a random walk. By the clock argument the size of the observable universe is uncertain by about the size of an atomic nucleus; as a random walk, by about a tenth of a millimetre. All three rules meet at the Planck length.

The clock that cannot be both light and good

The Planck length is usually quoted as the smallest distance that means anything, as though every longer distance could be known to that precision. Time a distance with light against a clock and the floor turns out to rise with the distance. A light clock wanders while the light is away; a heavy clock wanders less but is closer to being a black hole; and the best compromise leaves a metre uncertain by 750 thousand million Planck lengths and the size of the observable universe by the size of an atomic nucleus. The same compromise, counted up, says a region can hold only as many distinguishable places as its surface has Planck areas.

astrophysics · Planck scale

Named alongside it

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

Quantum gravitySchwarzschild radiusBlack hole entropyDimensional analysisEvent horizonHolographic principleMeasurementPlanck unitsUncertaintyUncertainty principleWave packet spreading

All concepts