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# Glossary cards — Chapter ᚁ(B): The actual world

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T3, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch03.md)

*Physical constants and physical laws: seven dimensionless braces, from Λ to the strength of gravity between two electrons, and six laws that hold exactly. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Beith — `ᚁ` {#g-og-beith}

*Ogham letter*

Ogham letter name meaning ‘birch’, one of the few names the medieval glossators did not have to translate.

Beithe means ‘birch-tree’, cognate with Middle Welsh bedw; Latin betula is thought to be a borrowing from the Gaulish cognate. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give féochos foltchain, “withered foot with fine hair”; glaisem cnis, “greyest of skin”; and maise malach, “beauty of the eyebrow.” It is one of only five names the Auraicept itself finds transparent as tree names without glossing, along with fearn, saille, duir, and coll, so here the book’s plant column and the oldest evidence agree. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.

Cites: [2.15](ch02-refs.md#r-2.15), [2.16](ch02-refs.md#r-2.16), [2.17](ch02-refs.md#r-2.17), [2.19](ch02-refs.md#r-2.19)

### cosmological constant — `Λ` {#g-Lambda}

*Physics: measured*

The cosmological constant in Einstein’s field equations, with units m⁻²; the associated vacuum-energy density is derived from Λ, c, and G.

Einstein added Λ to general relativity in 1917 to hold the universe static. It returned to the center of cosmology after the 1998 discovery of accelerated expansion, recognized by the 2011 Nobel Prize. Cosmological observations imply Λ ≈ 1.1 × 10^{−52} m^{−2}; the dimensionless combination ΛGℏ/c³ is about 2.9 × 10^{−122}. Λ itself is not an energy density: the corresponding vacuum-energy density is obtained from Λ, c, and G, with the exact factor depending on whether mass or energy density is meant. The enormous mismatch with naive quantum-field estimates is the cosmological constant problem. Physics: measured through cosmological fits.

Cites: [3.2](ch03-refs.md#r-3.2), [3.21](ch03-refs.md#r-3.21), [3.9](ch03-refs.md#r-3.9)

See also: [Newton’s gravitational constant](ch03-cards.md#g-G), [speed of light](ch03-cards.md#g-c), [the cosmological constant problem](ch03-cards.md#g-ccproblem), [dimensionless numbers](ch03-cards.md#g-dimensionless), [the Einstein field equations](ch03-cards.md#g-efe)

### Newton’s gravitational constant — `G` {#g-G}

*Physics: measured*

One of the least precisely known fundamental constants, measured to about one part in fifty thousand.

Newton’s Principia of 1687 stated the inverse-square law but never wrote G as a number; Cavendish’s torsion-balance experiment of 1798 weighed the Earth and fixed it in effect. CODATA 2018 and 2022 both give G = 6.67430(15) × 10^{−11} m³ kg^{−1} s^{−2}, a relative uncertainty of 22 parts per million, poor by the standards of the CODATA table, because gravity is so weak that laboratory masses barely attract each other. Physics: measured.

Cites: [3.20](ch03-refs.md#r-3.20), [3.7](ch03-refs.md#r-3.7)

### Planck’s constant — `h` {#g-h}

*Physics: fixed by definition*

Fixed exactly at 6.62607015×10⁻³⁴ joule-seconds by the 2019 redefinition of the SI.

Max Planck introduced h in 1900 to explain black-body radiation, calling it a mathematical trick; Einstein took it literally in 1905. Since 20 May 2019 it is fixed by definition at exactly 6.62607015 × 10^{−34} joule-seconds, and the kilogram is defined as whatever mass makes that true, measured by a Kibble balance. Physics: fixed by definition.

Cites: [3.1](ch03-refs.md#r-3.1), [3.20](ch03-refs.md#r-3.20)

See also: [the 2019 redefinition of the SI](ch03-cards.md#g-si2019)

### speed of light — `c` {#g-c}

*Physics: fixed by definition*

Fixed exactly at 299,792,458 meters per second since 1983.

Ole Rømer made the first estimate in 1676 from the moons of Jupiter. Since 1983 the meter has been defined as the distance light travels in 1/299,792,458 of a second, so c = 299,792,458 m/s exactly and can never be measured again, only the meter can. Physics: fixed by definition, a generation before the rest of them.

Cites: [3.1](ch03-refs.md#r-3.1), [3.18](ch03-refs.md#r-3.18)

### Boltzmann constant — `k_{B}` {#g-kB}

*Physics: fixed by definition*

Converts temperature into energy, and was fixed exactly by decree in 2019 to define the kelvin; that changed where we put the ruler, not what the constant does.

Planck introduced the constant in the same 1900 paper as h; it came to be called Boltzmann’s, after the 1877 relation S = k log W it appears in. It converts temperature to energy, one kelvin being about 1.38 × 10^{−23} joules, and since 2019 it has been fixed at exactly 1.380649 × 10^{−23} J/K, so that the kelvin is now defined by it. Physics: fixed by definition, because temperature is energy in disguise.

Cites: [3.1](ch03-refs.md#r-3.1), [3.21](ch03-refs.md#r-3.21)

See also: [Planck’s constant](ch03-cards.md#g-h)

### cosmic microwave background temperature — `T_{CMB}` {#g-Tcmb}

*Physics: measured*

The measured temperature of the oldest light in the universe, 2.7255 kelvin.

Penzias and Wilson found the cosmic microwave background by accident in ~1964, as a hiss their antenna could not shake; the sources give two years for two events: the measurement that clearly showed it was made on 20 May 1964, and the Astrophysical Journal letter that announced it appeared in 1965. It is light released when the universe became transparent, 380,000 years after the beginning, cooled by expansion to 2.7255 ± 0.0006 K (Fixsen, 2009, combining the published measurements). Physics: measured, and the most perfect black-body spectrum ever observed.

Cites: [3.3](ch03-refs.md#r-3.3)

### electron mass — `m_{e}` {#g-me}

*Physics: measured*

The measured rest mass of the electron, whose rest energy m_{e}c² is 511 keV.

J. J. Thomson identified the electron in 1897 and measured its charge-to-mass ratio. CODATA 2022 gives m_{e} = 9.1093837139(28) × 10^{−31} kg, and its rest energy m_{e}c² is 0.51099895 MeV, half the 1.022 MeV released when an electron and a positron annihilate at rest. In the heat bracket it is the yardstick against which the cold of the universe is measured. Physics: measured.

Cites: [3.22](ch03-refs.md#r-3.22), [3.7](ch03-refs.md#r-3.7)

See also: [rest energy](ch03-cards.md#g-emc2)

### Avogadro constant — `N_{A}` {#g-NA}

*Physics: fixed by definition*

Exactly 6.02214076×10²³ per mole since 2019.

Avogadro’s 1811 hypothesis that equal volumes of gas hold equal numbers of molecules did not give the number; Jean Perrin, who measured it by several independent methods, named it after Avogadro in 1909. Since 2019 it is exactly 6.02214076 × 10^{23} per mole, and the mole is defined as that many entities. Physics: fixed by definition; times one mole it is a pure count.

Cites: [3.1](ch03-refs.md#r-3.1), [3.22](ch03-refs.md#r-3.22)

See also: [one mole](ch03-cards.md#g-mol)

### one mole — `1 mol` {#g-mol}

*Physics: fixed by definition*

An exact amount of substance, present only to cancel Avogadro’s per-mole.

The SI base unit for amount of substance, adopted in 1971 and redefined in 2019 as exactly 6.02214076 × 10^{23} elementary entities, no longer tied to 12 grams of carbon-12. It appears here only to cancel the per-mole in N_{A}, leaving a pure number. Physics: fixed by definition.

Cites: [3.1](ch03-refs.md#r-3.1), [3.22](ch03-refs.md#r-3.22)

### Stefan–Boltzmann constant — `σ` {#g-sigma}

*Physics: derived*

Derived from Planck’s law as σ = π²k_{B}⁴/(60ℏ³c²), so its bracket is secretly π²/60.

Jožef Stefan found in 1879 that a hot body radiates in proportion to the fourth power of its temperature; his student Boltzmann derived it from thermodynamics in 1884, and Planck’s 1900 law gave the constant its value, σ = π²k_{B}⁴/(60ℏ³c²) ≈ 5.67 × 10^{−8} W m^{−2} K^{−4}. So the bracket σℏ³c²/k_{B}⁴ is exactly π²/60 ≈ 0.1645. Physics: derived; it is not a separate fact about nature.

Cites: [3.23](ch03-refs.md#r-3.23), [3.3](ch03-refs.md#r-3.3)

### reduced Planck constant — `ℏ` {#g-hbar}

*Physics: derived*

Planck’s constant divided by 2π; derived, not independent.

h divided by 2π, written ℏ, a symbol Dirac introduced in his 1930 Principles of Quantum Mechanics, having written plain h for the same quantity since 1926, because the combination appears in almost every quantum formula. Since h is exact, ℏ = 1.054571817… × 10^{−34} J s is exact too, though its decimal expansion never ends. The gravity bracket ΛGℏ/c³ is Λ times the square of the Planck length, ℓ_{P}² = ℏG/c³. Physics: derived.

Cites: [3.1](ch03-refs.md#r-3.1), [3.21](ch03-refs.md#r-3.21)

See also: [cosmological constant](ch03-cards.md#g-Lambda), [Planck’s constant](ch03-cards.md#g-h), [the Planck length](ch03-cards.md#g-plancklen)

### dimensionless numbers — `dimensionless` {#g-dimensionless}

*Measured or derived ratio*

A quantity whose units cancel, so its numerical value does not depend on the chosen units.

Dimensionless numbers can still describe real physical relationships. Ratios such as the fine-structure constant and the proton-to-electron mass ratio compare quantities after their units have canceled.

Cites: [3.17](ch03-refs.md#r-3.17)

See also: [fine-structure constant](ch03-cards.md#g-alpha), [the proton-to-electron mass ratio](ch03-cards.md#g-mpme)

### fine-structure constant — `α` {#g-alpha}

*Physics: measured*

The dimensionless strength of electromagnetism, about 1/137, measured and still unexplained.

Arnold Sommerfeld introduced α in 1916 to explain the fine splitting of spectral lines, as e²/(4πε₀ℏc), the ratio of the electron’s speed in the first Bohr orbit to the speed of light. It is dimensionless, the same in every unit system, and no theory predicts its value; Feynman called it one of the greatest damn mysteries of physics. CODATA 2022 gives 1/α = 137.035999177(21), a relative uncertainty of 1.6 parts in 10¹⁰, from the electron’s magnetic moment and atom-interferometry measurements of h/m; the Rydberg constant and the electron g-factor are both known better. Physics: measured.

Cites: [3.18](ch03-refs.md#r-3.18), [3.4](ch03-refs.md#r-3.4), [3.5](ch03-refs.md#r-3.5)

See also: [the Bohr radius](ch03-cards.md#g-a0), [the electron’s anomalous magnetic moment](ch03-cards.md#g-ae), [speed of light](ch03-cards.md#g-c), [dimensionless numbers](ch03-cards.md#g-dimensionless), [Planck’s constant](ch03-cards.md#g-h), [the Rydberg constant](ch03-cards.md#g-rydberg)

### elementary charge — `e` {#g-echarge}

*Physics: fixed by definition*

The charge of one proton, fixed exactly at 1.602176634×10⁻¹⁹ coulombs in 2019; not Euler’s e, which lives in Chapter ᚂ(L).

The charge of one proton, or minus that of one electron. Millikan and Fletcher’s oil-drop experiment, begun in 1909 and published in final form in 1913, measured it and showed that charge comes in whole multiples of it; since 2019 it is fixed at exactly 1.602176634 × 10^{−19} coulombs, and the ampere is defined from it. Not Euler’s e, which lives in Chapter ᚂ(L). Physics: fixed by definition.

Cites: [3.1](ch03-refs.md#r-3.1), [3.19](ch03-refs.md#r-3.19)

See also: [Euler’s number](ch04-cards.md#g-e)

### vacuum permittivity — `ε_{0}` {#g-eps0}

*Physics: derived*

How strongly empty space resists an electric field; since 2019 it is derived from α, e, ℏ, and c rather than defined.

The permittivity of free space, the constant in Coulomb’s law that says how strongly charges attract across empty space. It was exact before 2019, tied to the old definition of the ampere; since then it is derived, ε₀ = e²/(2αhc) ≈ 8.854 × 10^{−12} F/m, and inherits the tiny uncertainty of α. Physics: derived.

Cites: [3.1](ch03-refs.md#r-3.1)

See also: [fine-structure constant](ch03-cards.md#g-alpha)

### vacuum permeability — `μ_{0}` {#g-mu0}

*Physics: derived*

The magnetic twin of ε₀; μ₀ε₀c² = 1 exactly, so the magnetism bracket is secretly one.

The permeability of free space, the magnetic twin of ε₀. From 1948 to 2019 it was exactly 4π × 10^{−7} H/m by definition of the ampere; since 2019 it is μ₀ = 2αh/(e²c) ≈ 1.25663706 × 10^{−6} H/m, with the uncertainty of α. Maxwell’s 1865 result c² = 1/(μ₀ε₀) makes the bracket μ₀ε₀c² exactly 1. Physics: derived.

Cites: [3.22](ch03-refs.md#r-3.22), [3.4](ch03-refs.md#r-3.4)

See also: [fine-structure constant](ch03-cards.md#g-alpha), [vacuum permittivity](ch03-cards.md#g-eps0)

### von Klitzing constant — `R_{K}` {#g-RK}

*Physics: derived*

The quantum of electrical resistance, h/e², so e²R_K/h = 1 exactly; another famous R, and the collision with ᚏ is deliberate.

Klaus von Klitzing discovered in 1980 that the Hall resistance of a two-dimensional electron gas in a strong magnetic field comes in exact steps of h/e², the quantum Hall effect, which won the 1985 Nobel Prize. R_{K} = h/e² = 25,812.80745… ohms, exact since 2019 because h and e are, so e²R_{K}/h is 1 identically. Another famous R, and the collision with ᚏ is deliberate. Physics: derived.

Cites: [3.23](ch03-refs.md#r-3.23), [3.4](ch03-refs.md#r-3.4)

See also: [elementary charge](ch03-cards.md#g-echarge), [Planck’s constant](ch03-cards.md#g-h), [the letter Ruis](ch01-cards.md#g-og-ruis)

### impedance of free space — `Z_{0}` {#g-Z0}

*Physics: derived*

About 376.73 ohms, equal to μ₀c, so Z₀/(μ₀c) = 1 exactly.

The impedance of free space, the ratio of electric to magnetic field in a light wave traveling through vacuum, Z₀ = μ₀c ≈ 376.730313 ohms. It was exact before 2019 and is now derived from α; either way Z₀/(μ₀c) is 1 by definition. Physics: derived.

Cites: [3.24](ch03-refs.md#r-3.24), [3.4](ch03-refs.md#r-3.4)

See also: [fine-structure constant](ch03-cards.md#g-alpha)

### Wien’s displacement constant — `b` {#g-wien}

*Physics: derived*

The wavelength at which a hot body glows brightest, times its temperature: b = 2.897771955… mm·K, exact since 2019 because it is hc/(x·k_B).

Wilhelm Wien found in 1893 that the peak wavelength of black-body radiation times the temperature is a constant, b = λ_{max}T. Planck’s law fixes it as b = hc/(x_{W}k_{B}), where x_{W} = 4.965114… is the root of x = 5(1 − e^{−x}), so since 2019 b = 2.897771955… × 10^{−3} m K is exact; the “…” marks only that its digits, like the root’s, run on past those printed. For the 2.7255 K background it gives a peak near 1.06 mm, which is why the oldest light is microwave. The bracket b·x_{W}·k_{B}/(hc) is exactly 1. Physics: derived.

Cites: [3.21](ch03-refs.md#r-3.21), [3.24](ch03-refs.md#r-3.24), [3.3](ch03-refs.md#r-3.3)

### Wien’s root — `x_{W}` {#g-wienx}

*Proven transcendental*

The number 4.965114231…, the root of x = 5(1 − e^(−x)), which sits inside Wien’s constant; it is 5 + W(−5e^(−5)) and transcendental.

The peak of Planck’s black-body curve in wavelength is where x = hc/(λk_{B}T) satisfies x = 5(1 − e^{−x}); the nonzero root is 4.965114231…, and in closed form x_{W} = 5 + W_{0}(−5e^{−5}) with W the Lambert function of Chapter ᚐ(A). It is transcendental by Lindemann–Weierstrass: if it were algebraic, e^{−x} = 1 − x/5 would be algebraic too, which is impossible for a nonzero algebraic exponent. Type: proven transcendental.

Cites: [3.3](ch03-refs.md#r-3.3)

See also: [the Lindemann–Weierstrass theorem](ch04-cards.md#g-lindweier), [Wien’s displacement constant](ch03-cards.md#g-wien)

### the molar gas constant — `R` {#g-gasR}

*Physics: derived*

The R of PV = nRT, exactly N_A·k_B = 8.314462618… J/(mol·K) since 2019; one more famous R.

The gas constant of Clapeyron’s ideal-gas law PV = nRT (1834), which gathers Boyle, Charles, and Avogadro into one line. It is Boltzmann’s constant per molecule scaled up to a mole, R = N_{A}k_{B}, so since 2019 it is exact, 8.31446261815324 J mol^{−1} K^{−1}, and the bracket R/(N_{A}k_{B}) is exactly 1. The letter is one more famous R in the piece, after ᚏ, R_{K}, and R_{∞}. Physics: derived.

Cites: [3.1](ch03-refs.md#r-3.1), [3.20](ch03-refs.md#r-3.20)

See also: [Boltzmann constant](ch03-cards.md#g-kB), [the letter Ruis](ch01-cards.md#g-og-ruis)

### the Faraday constant — `F` {#g-faraday}

*Physics: derived*

The charge of one mole of electrons, exactly N_A·e = 96485.33212… C/mol since 2019.

Faraday’s laws of electrolysis of 1834 say that the mass deposited at an electrode is proportional to the charge passed; the constant of proportionality per mole of electrons is F = N_{A}e, later named in his honor. We could not confirm when: no source we found dates the naming. Since 2019 it is exact, 96485.33212331… C/mol, and the bracket F/(N_{A}e) is exactly 1. Physics: derived.

Cites: [3.1](ch03-refs.md#r-3.1), [3.19](ch03-refs.md#r-3.19)

See also: [elementary charge](ch03-cards.md#g-echarge), [one mole](ch03-cards.md#g-mol)

### Coulomb’s constant — `k_{e}` {#g-coulombk}

*Physics: derived*

The constant in Coulomb’s law of 1785, k_e = 1/(4πε₀) ≈ 8.98755 × 10⁹ N·m²/C², so k_e·4πε₀ = 1 exactly.

Charles-Augustin de Coulomb measured the inverse-square law of electric force with a torsion balance in 1785. The constant in F = k_{e}q₁q₂/r² is k_{e} = 1/(4πε₀) = 8.9875517862(14) × 10⁹ N m² C^{−2} (computed from CODATA 2022’s ε₀), and the bracket k_{e}·4πε₀ is 1 by definition. Physics: derived.

Cites: [3.19](ch03-refs.md#r-3.19), [3.4](ch03-refs.md#r-3.4)

### the Rydberg constant — `R_{∞}` {#g-rydberg}

*Physics: measured*

The constant of atomic spectra, 10973731.568157(12) per meter, among the most precisely measured constants in physics, to about one part in 10¹²; equal to α²m_ec/(2h).

Johannes Rydberg’s 1888 formula for the lines of hydrogen-like spectra has one constant, which Bohr’s 1913 atom explained as R_{∞} = α²m_{e}c/(2h) = m_{e}e⁴/(8ε₀²h³c). CODATA 2022 gives 10 973 731.568 157(12) m^{−1}, a relative uncertainty of 1.1 parts in 10¹², from hydrogen and deuterium spectroscopy; among the most precisely measured constants in physics, though CODATA 2022 knows the electron g-factor better still, to 1.8 parts in 10^{13}. The bracket 2hR_{∞}/(α²m_{e}c) is exactly 1 and is the identity that ties the spectra to the fine-structure constant. Physics: measured.

Cites: [3.23](ch03-refs.md#r-3.23), [3.4](ch03-refs.md#r-3.4)

See also: [fine-structure constant](ch03-cards.md#g-alpha), [speed of light](ch03-cards.md#g-c)

### the magnetic flux quantum — `Φ_{0}` {#g-fluxq}

*Physics: derived*

The smallest amount of magnetic flux a superconducting ring can hold, h/(2e) = 2.067833848… × 10⁻¹⁵ Wb, exact since 2019.

In 1961 Deaver and Fairbank at Stanford and Doll and Näbauer in Herrsching independently found that the magnetic flux through a superconducting ring comes in steps of h/2e, the 2 because electrons pair up. Φ_{0} = h/(2e) = 2.067833848… × 10^{−15} Wb is exact since 2019, and its product with the Josephson constant 2e/h is exactly 1. Physics: derived.

Cites: [3.19](ch03-refs.md#r-3.19), [3.20](ch03-refs.md#r-3.20), [3.4](ch03-refs.md#r-3.4)

See also: [Planck’s constant](ch03-cards.md#g-h), [the Josephson constant](ch03-cards.md#g-josephson)

### the Josephson constant — `K_{J}` {#g-josephson}

*Physics: derived*

The frequency-to-voltage ratio of a Josephson junction, 2e/h = 483597.8484… GHz per volt, exact since 2019; the reciprocal of the flux quantum.

Brian Josephson predicted in 1962 that a supercurrent tunnels between two superconductors and that a voltage V across the junction produces an oscillation at frequency 2eV/h. The constant K_{J} = 2e/h = 483 597.8484… × 10⁹ Hz V^{−1} is exact since 2019 and underlies the Josephson voltage standard, which gives the standard representation of the volt. Its product with Φ_{0} = h/2e is 1 identically. Physics: derived.

Cites: [3.21](ch03-refs.md#r-3.21), [3.4](ch03-refs.md#r-3.4)

### the gravitational coupling constant — `α_{G}` {#g-alphaG}

*Physics: derived*

Gravity’s answer to α: Gm_e²/(ℏc) ≈ 1.75 × 10⁻⁴⁵, the dimensionless strength of gravity between two electrons, 10⁴² times weaker than electromagnetism.

The gravitational coupling constant α_{G} = Gm_{e}²/(ℏc) = (m_{e}/m_{P})² ≈ 1.7518 × 10^{−45} is the dimensionless strength of gravity between two electrons, the exact analogue of the fine-structure constant α ≈ 1/137 for electromagnetism. Their ratio, about 4 × 10⁴², is one statement of the hierarchy problem, and one of Dirac’s large numbers. It closes the formula’s particle-physics brace, and, with the gravity bracket, it is one of the two numbers that drag the whole factor down to 10^{−154}. Physics: derived.

Cites: [3.12](ch03-refs.md#r-3.12), [3.7](ch03-refs.md#r-3.7)

See also: [fine-structure constant](ch03-cards.md#g-alpha), [dimensionless numbers](ch03-cards.md#g-dimensionless), [the hierarchy problem](ch03-cards.md#g-hierarchy)

### the proton-to-electron mass ratio — `m_{p}/m_{e}` {#g-mpme}

*Physics: measured*

1836.152673426(32), the dimensionless ratio of the proton’s mass to the electron’s.

The proton is 1836.152673426(32) times heavier than the electron (CODATA 2022), a fundamental dimensionless constant, like α: no theory predicts it, and it must be measured. It opens the formula’s particle-physics brace. Physics: measured.

Cites: [3.7](ch03-refs.md#r-3.7)

See also: [fine-structure constant](ch03-cards.md#g-alpha), [dimensionless numbers](ch03-cards.md#g-dimensionless), [proton mass](ch28-cards.md#g-mp)

### the electron’s anomalous magnetic moment — `a_{e}` {#g-ae}

*Physics: measured*

The electron’s magnetic moment exceeds Dirac’s prediction by 0.00115965218…, which quantum electrodynamics computes and experiment confirms to more than ten significant figures.

Dirac’s 1928 equation gives the electron a g-factor of exactly 2; quantum electrodynamics corrects it by a_{e} = (g − 2)/2 = 0.00115965218…, first computed by Schwinger in 1948 as α/2π. The 2022 Northwestern measurement gives 1.15965218059(13) × 10^{−3}, and the quantum-electrodynamic prediction agrees with it to more than ten significant figures. One of the five ratios in the formula’s particle-physics brace. Physics: measured.

Cites: [3.18](ch03-refs.md#r-3.18), [3.5](ch03-refs.md#r-3.5)

See also: [fine-structure constant](ch03-cards.md#g-alpha), [the Bohr magneton](ch03-cards.md#g-bohrmag), [quantum electrodynamics](ch03-cards.md#g-qed)

### the strong coupling constant — `α_{s}` {#g-alphas}

*Physics: measured*

The strength of the strong nuclear force at the Z boson’s energy, 0.1180(9): the coupling of the Yang–Mills theory of the epilogue once quarks are added.

The coupling of quantum chromodynamics, the SU(3) Yang–Mills theory of the epilogue with quarks added; the epilogue’s mass-gap problem is about the pure theory, without quarks. It runs with energy, and the standard reference value is at the Z boson mass: α_{s}(M_{Z}) = 0.1180(9) (Particle Data Group, Review of Particle Physics, 2026). It grows at low energies, the regime in which quarks are confined, though confinement has not been proved from the theory. One of the five ratios in the formula’s particle-physics brace. Physics: measured.

Cites: [3.5](ch03-refs.md#r-3.5)

### the dark-energy density parameter — `Ω_{Λ}` {#g-OmegaL}

*Physics: measured*

The dark-energy density divided by critical density; about 0.685 in Planck’s base flat ΛCDM fit.

Ω_{Λ} is the dark-energy density divided by the critical density. Planck’s base flat ΛCDM fit gives a value near 0.685, with spatial curvature fixed to zero and the radiation share, tiny today, normally left out when the rounded dark-energy and matter values are quoted as adding to 1. The exact symbolic Friedmann identity Ω_{Λ} + Ω_{m} + Ω_{r} + Ω_{k} = 1 is retained, but numerical values from different model fits must not be mixed. Physics: measured within a specified model.

Cites: [3.2](ch03-refs.md#r-3.2)

See also: [the radiation density parameter](ch03-cards.md#g-Omegar), [the ΛCDM model](ch03-cards.md#g-lcdm)

### the matter density parameter — `Ω_{m}` {#g-Omegam}

*Physics: measured*

Total matter divided by critical density; about 0.315 in Planck’s base flat ΛCDM fit.

Ω_{m} is total matter density—dark matter plus baryonic matter and the model’s neutrino contribution—over critical density. Planck’s base flat ΛCDM fit gives a value near 0.315. That result belongs to the same model assumptions as its quoted dark-energy complement; it must not be combined numerically with a curvature value from an extended model. Physics: measured within a specified model.

Cites: [3.2](ch03-refs.md#r-3.2)

See also: [the ΛCDM model](ch03-cards.md#g-lcdm)

### the radiation density parameter — `Ω_{r}` {#g-Omegar}

*Physics: derived*

The share of the universe that is light and neutrinos today: about 9 × 10⁻⁵, negligible now, dominant in the first fifty thousand years.

Ω_{r} is the density of photons and relativistic neutrinos over critical density. Its present value is of order 9 × 10^{−5}, derived from the background temperature, neutrino assumptions, and the chosen Hubble parameter. It belongs in the exact Friedmann balance, but a rounded value should be quoted only with the same cosmological model as the other Ω terms. Physics: derived from measured inputs.

Cites: [3.2](ch03-refs.md#r-3.2)

See also: [the Hubble constant](ch03-cards.md#g-H0)

### the curvature density parameter — `Ω_{k}` {#g-Omegak}

*Physics: derived*

The curvature term in the Friedmann balance; fixed to zero in flat ΛCDM and fitted only in curved extensions.

Ω_{k} is the spatial-curvature term in the Friedmann equation. It is fixed to zero in the base flat ΛCDM model and allowed to vary in curved extensions, which produce different correlated estimates for the other parameters. Algebraically Ω_{k} := 1 − Ω_{Λ} − Ω_{m} − Ω_{r}, so the four-symbol bracket is exactly one. The book keeps that identity and does not attach a curved-fit number to flat-fit values. Physics: derived within the displayed identity.

Cites: [3.2](ch03-refs.md#r-3.2)

See also: [the ΛCDM model](ch03-cards.md#g-lcdm)

### the weak mixing angle — `sin²θ_{W}` {#g-weakmix}

*Physics: measured*

How the two neutral electroweak fields, W³ and B, are mixed to make the photon and the Z boson: 0.22305(23) on-shell (CODATA 2022), a number the Standard Model does not predict.

The Weinberg angle θ_{W} of the 1967 electroweak theory sets the ratio of the W and Z boson masses, cos θ_{W} = m_{W}/m_{Z}, and is the angle through which the theory’s two neutral fields, W^{3} and B, are rotated into the photon and the Z. CODATA 2022 gives sin²θ_{W} = 0.22305(23) in the on-shell scheme; the value depends on the scheme and the energy, so the formula quotes the on-shell value. Physics: measured.

Cites: [3.23](ch03-refs.md#r-3.23), [3.5](ch03-refs.md#r-3.5)

See also: [the Standard Model](ch03-cards.md#g-smodel)

### the cosmological constant problem {#g-ccproblem}

*Status open*

The gap between the measured Λ and the value quantum field theory naively predicts: about 120 orders of magnitude, and unexplained.

In Planck units the measured cosmological constant is about 3 × 10⁻¹²², while a naive quantum-field estimate of the vacuum energy gives something of order 1. The mismatch of roughly 120 orders of magnitude has been called the worst theoretical prediction in the history of physics, a phrase attributed to Hobson, Efstathiou, and Lasenby’s General Relativity: An Introduction for Physicists (2006), p. 187. No accepted resolution exists: proposals range from modified gravity to anthropic selection in a landscape of vacua. Status: open.

Cites: [3.2](ch03-refs.md#r-3.2)

See also: [cosmological constant](ch03-cards.md#g-Lambda)

### the hierarchy problem {#g-hierarchy}

*Status open*

Why gravity is so much weaker than the other forces: between two electrons, electromagnetism beats gravity by about 10⁴², and no accepted theory says why.

Dividing the fine-structure constant by the gravitational coupling constant gives α/α_G ≈ 4 × 10⁴², the factor by which electromagnetism exceeds gravity between two electrons. The same disparity appears in the Standard Model as the gap between the electroweak scale and the Planck scale, where it takes the form of the Higgs-mass fine-tuning problem: quantum corrections should drive the Higgs mass up to the Planck scale unless something cancels them to many decimal places. Proposed answers include supersymmetry, large extra dimensions, and compositeness; none is established. Not to be confused with the fast-growing hierarchy of Chapter ᚂ(L). Status: open.

Cites: [3.12](ch03-refs.md#r-3.12)

See also: [fine-structure constant](ch03-cards.md#g-alpha), [the gravitational coupling constant](ch03-cards.md#g-alphaG), [the fast-growing hierarchy](ch04-cards.md#g-fgh), [the Planck length](ch03-cards.md#g-plancklen), [the Standard Model](ch03-cards.md#g-smodel)

### the 2019 redefinition of the SI — `h, e, kʙ, Nₐ` {#g-si2019}

*Function or operator*

The day the kilogram stopped being a lump of metal: four base units redefined by fixing exact values for four constants, effective 20 May 2019.

The 26th General Conference on Weights and Measures voted unanimously on 16 November 2018 to redefine four of the seven SI base units, and the change took effect on 20 May 2019, the 144th anniversary of the Metre Convention. The kilogram, ampere, kelvin, and mole are now defined by fixing exact numerical values for the Planck constant, the elementary charge, the Boltzmann constant, and the Avogadro constant; the second, meter, and candela were already defined by constants. This is what the blue entries in this chapter’s key mean: a constant marked as fixed by definition has no uncertainty because a committee gave it a number, and every quantity built only from such constants inherits that exactness. It is also why several entries here say “since 2019”—the same number can have been measured before that date and derived after it. An act of definition, uncolored.

Cites: [3.16](ch03-refs.md#r-3.16)

See also: [Avogadro constant](ch03-cards.md#g-NA), [elementary charge](ch03-cards.md#g-echarge), [Planck’s constant](ch03-cards.md#g-h), [Boltzmann constant](ch03-cards.md#g-kB)

### the interval along a light ray — `ds²_{light}` {#g-dslight}

*Integer or rational*

In relativity the spacetime interval between two events on a light ray is exactly 0. In the formula it is added in as the zero it is.

Minkowski’s 1908 reformulation of special relativity gives spacetime a distance, ds² = c²dt² − dx² − dy² − dz², that all observers agree on. For two events joined by a light ray it is exactly 0, a null interval: in spacetime terms light goes nowhere and takes no time. Every observer agrees, which is another way of saying the speed of light is the same for everyone. Type: exactly 0, written as a law rather than a measurement.

Cites: [3.11](ch03-refs.md#r-3.11), [3.6](ch03-refs.md#r-3.6)

See also: [speed of light](ch03-cards.md#g-c)

### no magnetic monopoles — `∇·𝐁` {#g-divB}

*Integer or rational*

Maxwell’s second equation: in classical electromagnetism, with no magnetic monopoles, the divergence of the magnetic field is 0 everywhere. In the formula it is added in as the zero it is.

Gauss’s law for magnetism, ∇·𝐁 = 0, one of the four Maxwell equations assembled in 1861–65 and put into vector form by Heaviside in the 1880s. It says the magnetic field has no sources or sinks: field lines never begin or end, and there are no magnetic monopoles. Searches for monopoles continue, and every one so far has found nothing, which is the point. Type: exactly 0.

Cites: [3.19](ch03-refs.md#r-3.19), [3.6](ch03-refs.md#r-3.6)

See also: [Maxwell’s equations](ch11-cards.md#g-maxwell)

### Noether’s conservation — `dQ/dt` {#g-noether}

*Integer or rational*

Under the theorem’s hypotheses, every continuous symmetry of an action gives a quantity whose rate of change is 0; energy, momentum, and charge are conserved for this reason. In the formula it is added in as the zero it is.

Emmy Noether’s work on the theorem began in 1915, when she was helping Felix Klein and David Hilbert with general relativity; the invariant-variation theorem was presented for her by Klein, presumably because she was not a member of the Göttingen society, and published in 1918. Under the theorem’s hypotheses, every differentiable symmetry of an action yields a conservation law: time translation gives energy, space translation momentum, and rotational symmetry angular momentum. The conserved quantity Q satisfies dQ/dt = 0. Type: exactly 0 within its stated scope.

Cites: [3.22](ch03-refs.md#r-3.22), [3.6](ch03-refs.md#r-3.6)

### rest energy — `E − mc²` {#g-emc2}

*Integer or rational*

Einstein’s 1905 equation written as a zero: a body at rest has energy E = mc², so E − mc² = 0. In the formula it is added in as the zero it is.

In a paper received in September 1905 and published that November, Einstein showed that a body’s inertia depends on its energy content, the result now written E = mc²; for a body at rest the total energy is exactly mc², so E − mc² is exactly 0, and in motion E = γmc² with γ the Lorentz factor. It has been called the world’s most famous equation, and it explains the source of the Sun’s energy and the energy released by fission in nuclear weapons and nuclear power. Type: exactly 0, written as a law.

Cites: [3.19](ch03-refs.md#r-3.19), [3.6](ch03-refs.md#r-3.6)

See also: [the Lorentz factor](ch03-cards.md#g-lorentz)

### the contracted Bianchi identity — `∇_{μ}G^{μν}` {#g-bianchi}

*Integer or rational*

The Einstein tensor has zero divergence, identically, which underlies local energy-momentum conservation in general relativity. In the formula it is added in as the zero it is.

The Bianchi identity, which Luigi Bianchi published in 1902 and which Levi-Civita says Ricci had already found in about 1889, says the covariant derivatives of the Riemann curvature tensor cancel in a cyclic sum; contracted twice it gives ∇_{μ}G^{μν} = 0 for the Einstein tensor G^{μν} = R^{μν} − ½Rg^{μν} — and that contracted form, the one this entry is named for, had been published by Aurel Voss in 1880, twenty-two years before the identity got its name. Einstein’s field equations set G^{μν} + Λg^{μν} proportional to the stress–energy tensor, so the identity forces ∇_{μ}T^{μν} = 0: energy and momentum are conserved because geometry has no choice. Type: exactly 0, a law.

Cites: [3.6](ch03-refs.md#r-3.6)

See also: [the Einstein field equations](ch03-cards.md#g-efe)

### conservation of charge — `∂_{μ}j^{μ}` {#g-continuity}

*Integer or rational*

The continuity equation: the four-divergence of the electric current is 0, so charge is never created or lost. In the formula it is added in as the zero it is.

∂_{μ}j^{μ} = ∂ρ/∂t + ∇·𝐉 = 0 says that charge leaving a region must flow out through its boundary. It follows from Maxwell’s equations of 1865 (take the divergence of the Ampère–Maxwell law) and, by Noether’s theorem, from the phase symmetry of electromagnetism. Type: exactly 0, a law.

Cites: [3.19](ch03-refs.md#r-3.19), [3.6](ch03-refs.md#r-3.6)

See also: [Maxwell’s equations](ch11-cards.md#g-maxwell), [Noether’s conservation](ch03-cards.md#g-noether)

### the Planck length — `ℓ_{P}` {#g-plancklen}

*Physics: derived*

√(ℏG/c³) = 1.616255(18) × 10⁻³⁵ m, a scale at which gravity and quantum mechanics are expected to meet; the gravity bracket is Λ times its square.

Max Planck noticed in 1899 that G, c, and his own new constant combine into one natural length; in the modern form, which uses ℏ where he used h, it is ℓ_{P} = √(ℏG/c³) = 1.616255(18) × 10^{−35} m (CODATA 2022), a scale at which quantum effects of gravity are expected to dominate. The gravity bracket ΛGℏ/c³ is exactly Λℓ_{P}², which is what “the cosmological constant in Planck units” means. Not in the formula; a key entry. Physics: derived, with the uncertainty of G.

Cites: [3.23](ch03-refs.md#r-3.23), [3.7](ch03-refs.md#r-3.7)

See also: [Newton’s gravitational constant](ch03-cards.md#g-G), [cosmological constant](ch03-cards.md#g-Lambda), [speed of light](ch03-cards.md#g-c), [Planck’s constant](ch03-cards.md#g-h), [reduced Planck constant](ch03-cards.md#g-hbar)

### the Planck time — `t_{P}` {#g-plancktime}

*Physics: derived*

ℓ_P/c = 5.391247(60) × 10⁻⁴⁴ s, a scale at which quantum-gravity effects are expected to become essential.

The time light takes to cross a Planck length, t_{P} = √(ℏG/c⁵) = 5.391247(60) × 10^{−44} s. It is a natural scale built from G, ℏ, and c, not a proved smallest possible tick. Present theories indicate that quantum-gravity effects should become essential near this scale, but do not establish a literal boundary at which known physics stops. Physics: derived.

Cites: [3.7](ch03-refs.md#r-3.7)

See also: [Newton’s gravitational constant](ch03-cards.md#g-G), [speed of light](ch03-cards.md#g-c), [reduced Planck constant](ch03-cards.md#g-hbar), [the Planck length](ch03-cards.md#g-plancklen)

### the Planck mass — `m_{P}` {#g-planckmass}

*Physics: derived*

√(ℏc/G) = 2.176434(24) × 10⁻⁸ kg, about a flea’s egg; roughly the mass at which a particle’s Compton wavelength equals its Schwarzschild radius.

The Planck mass m_{P} = √(ℏc/G) = 2.176434(24) × 10^{−8} kg, about 22 micrograms, is the one Planck base unit on a human scale. It is the mass at which a particle’s quantum size and its black-hole size roughly coincide, and the ratio (m_{e}/m_{P})² is the gravitational coupling constant. Not in the formula; a key entry. Physics: derived.

Cites: [3.21](ch03-refs.md#r-3.21), [3.7](ch03-refs.md#r-3.7)

See also: [Newton’s gravitational constant](ch03-cards.md#g-G), [the gravitational coupling constant](ch03-cards.md#g-alphaG), [the Compton wavelength](ch03-cards.md#g-compton), [the Schwarzschild radius](ch03-cards.md#g-schwarzschild)

### the Hubble constant — `H_{0}` {#g-H0}

*Physics: measured*

The expansion rate of the universe today, 67.4 ± 0.5 km/s per megaparsec from the microwave background and 73.0 ± 1.0 from nearby supernovae; the gap is the Hubble tension.

Hubble’s 1929 law says galaxies recede at a speed proportional to their distance, v = H_{0}d. The Planck satellite’s 2018 fit to the microwave background gives H_{0} = 67.4 ± 0.5 km s^{−1} Mpc^{−1}; the SH0ES team’s 2022 distance ladder of Cepheids and supernovae gives 73.04 ± 1.04. The disagreement, about five standard deviations, is the Hubble tension, the most statistically significant tension in the standard cosmological model. Its reciprocal is about 14 billion years, the age scale of the universe. Not in the formula; a key entry. Physics: measured.

Cites: [3.20](ch03-refs.md#r-3.20), [3.7](ch03-refs.md#r-3.7)

See also: [the age of the universe](ch03-cards.md#g-age)

### the Bohr radius — `a_{0}` {#g-a0}

*Physics: derived*

The radius of the hydrogen atom in Bohr’s 1913 model, 5.29177210544(82) × 10⁻¹¹ m, equal to ℏ/(α m_e c).

Bohr’s 1913 model of hydrogen puts the electron’s innermost orbit at a_{0} = 4πε₀ℏ²/(m_{e}e²) = ℏ/(αm_{e}c) = 5.29177210544(82) × 10^{−11} m (CODATA 2022), about half an ångström, and quantum mechanics keeps it as the most probable distance of the electron from the proton. It is the unit of length of atomic physics. Not in the formula; a key entry. Physics: derived.

Cites: [3.18](ch03-refs.md#r-3.18), [3.7](ch03-refs.md#r-3.7)

See also: [speed of light](ch03-cards.md#g-c), [reduced Planck constant](ch03-cards.md#g-hbar)

### the Compton wavelength — `λ_{C}` {#g-compton}

*Physics: derived*

h/(m_{e}c) = 2.42631023538(76) × 10⁻¹² m, the length scale at which an electron stops being a particle you can locate.

Arthur Compton explained the scattering of X-rays by electrons in 1923 with a shift proportional to h/(m_{e}c); the constant, λ_{C} = 2.42631023538(76) × 10^{−12} m (CODATA 2022), is the wavelength of a photon with the electron’s rest energy, and the scale below which quantum field theory becomes essential. Not in the formula; a key entry. Physics: derived.

Cites: [3.18](ch03-refs.md#r-3.18), [3.7](ch03-refs.md#r-3.7)

See also: [speed of light](ch03-cards.md#g-c), [rest energy](ch03-cards.md#g-emc2), [Planck’s constant](ch03-cards.md#g-h)

### the classical electron radius — `r_{e}` {#g-re}

*Physics: derived*

e²/(4πε₀m_{e}c²) = 2.8179403205(13) × 10⁻¹⁵ m, the size an electron would have if its mass were all electrostatic energy; it is α² times the Bohr radius.

The classical electron radius, 2.8179403205(13) × 10^{−15} m (CODATA 2022), comes from setting the electrostatic energy of a charged sphere equal to m_{e}c²; the real electron is pointlike as far as experiment can tell, but r_{e} still sets the Thomson scattering cross-section. The three lengths of the electron nest by powers of α: r_{e} = αλ_{C}/2π = α²a₀. Not in the formula; a key entry. Physics: derived.

Cites: [3.7](ch03-refs.md#r-3.7)

See also: [the Bohr radius](ch03-cards.md#g-a0), [fine-structure constant](ch03-cards.md#g-alpha)

### the Bohr magneton — `μ_{B}` {#g-bohrmag}

*Physics: derived*

eℏ/(2m_{e}) = 9.2740100657(29) × 10⁻²⁴ J/T, the natural unit of magnetic moment; the electron’s is very slightly more than one of them.

The Bohr magneton μ_{B} = eℏ/(2m_{e}), obtained by Procopiu and by Bohr in 1913 and named by Pauli in 1920, is the magnetic moment of an electron in the first Bohr orbit; CODATA 2022 gives 9.2740100657(29) × 10^{−24} J T^{−1}. The electron’s actual moment is (1 + a_{e}) of it, which is where the anomalous moment in this key comes from. Not in the formula; a key entry. Physics: derived.

Cites: [3.18](ch03-refs.md#r-3.18), [3.7](ch03-refs.md#r-3.7)

See also: [the Bohr radius](ch03-cards.md#g-a0)

### the Fermi coupling constant — `G_{F}` {#g-fermiG}

*Physics: measured*

The strength of the weak nuclear force, G_{F}/(ℏc)³ = 1.1663787(6) × 10⁻⁵ GeV⁻²; the weak force’s own coupling, which the formula leaves out in favor of the mixing angle.

Enrico Fermi’s 1934 theory of beta decay had one constant, the strength of the four-fermion contact interaction; in modern units G_{F}/(ℏc)³ = 1.1663787(6) × 10^{−5} GeV^{−2} (CODATA 2022), measured from the muon lifetime. It is the weak force’s answer to α and α_{s}, and it has units because the weak interaction is carried by heavy bosons. Not in the formula; a key entry. Physics: measured.

Cites: [3.19](ch03-refs.md#r-3.19), [3.7](ch03-refs.md#r-3.7)

See also: [fine-structure constant](ch03-cards.md#g-alpha)

### the Planck temperature — `T_{P}` {#g-planckT}

*Physics: derived*

√(ℏc⁵/Gk_{B}²) = 1.416784(16) × 10³² K, the temperature at which the wavelength of thermal radiation reaches the Planck length.

The Planck temperature, m_{P}c²/k_{B} = 1.416784(16) × 10^{32} K (CODATA 2022), is the temperature at which the wavelength of thermal radiation reaches the Planck length. Present theories are not expected to hold above it, though nothing proves a hard boundary there, and it is the temperature usually attached to the universe one Planck time after the beginning. It completes the set of Planck units in this key with the length, time, and mass. Not in the formula; a key entry. Physics: derived.

Cites: [3.7](ch03-refs.md#r-3.7)

See also: [the Planck time](ch03-cards.md#g-plancktime)

### the age of the universe — `t_{0}` {#g-age}

*Physics: measured*

13.787 ± 0.020 billion years since the beginning, from the Planck 2018 fit with baryon acoustic oscillations.

The age of the universe follows from H₀ and the density parameters through the Friedmann equation; the Planck 2018 fit gives t₀ = 13.787 ± 0.020 billion years once the microwave background is combined with baryon acoustic oscillations, and the oldest stars agree. It is the one number in this key with a date attached to everything. Not in the formula; a key entry. Physics: measured.

Cites: [3.18](ch03-refs.md#r-3.18), [3.7](ch03-refs.md#r-3.7)

See also: [the Hubble constant](ch03-cards.md#g-H0)

### the Lorentz factor — `γ` {#g-lorentz}

*Function or operator*

1/√(1 − v²/c²): how much time slows and mass-energy grows at speed v; equal to 1 at rest, which is why E − mc² is 0 there.

The Lorentz factor γ = 1/√(1 − v²/c²) appeared in Lorentz’s transformations of 1904 and Einstein’s special relativity of 1905; it multiplies time dilation, length contraction, and energy, E = γmc². At the LHC it is about 7000. Type: a function of the framework.

Cites: [3.21](ch03-refs.md#r-3.21), [3.6](ch03-refs.md#r-3.6)

See also: [rest energy](ch03-cards.md#g-emc2)

### the Schwarzschild radius — `r_{s}` {#g-schwarzschild}

*Physics: derived*

2GM/c², the radius a mass must be squeezed inside to become a black hole: 3 km for the Sun, 9 mm for the Earth.

Karl Schwarzschild solved Einstein’s field equations for a point mass in 1916, months after they were published and months before his death; the solution has a horizon at r_{s} = 2GM/c², about 3 km for the Sun and 9 mm for the Earth. The Planck mass is the mass whose Schwarzschild radius equals its Compton wavelength, up to a factor. Not in the formula; a key entry. Physics: derived.

Cites: [3.21](ch03-refs.md#r-3.21)

See also: [the Compton wavelength](ch03-cards.md#g-compton), [the Einstein field equations](ch03-cards.md#g-efe), [the Planck mass](ch03-cards.md#g-planckmass)

### the Einstein field equations — `G_{μν} + Λg_{μν} = 8πG T_{μν}/c⁴` {#g-efe}

*Function or operator*

General relativity in one line, 1915: curvature on the left, matter and energy on the right, and Λ, the cosmological constant of the gravity bracket, in between.

Einstein’s field equations of November 1915 equate the Einstein tensor, which measures curvature, to the stress–energy tensor, which measures matter and energy, with 8πG/c⁴ as the conversion; Λ was added in 1917. They are ten coupled nonlinear equations, they contain Newton’s law as a limit, and their contracted Bianchi identity is one of the zeros in the laws brace. Type: a law of the framework.

Cites: [3.19](ch03-refs.md#r-3.19), [3.2](ch03-refs.md#r-3.2), [3.6](ch03-refs.md#r-3.6)

See also: [cosmological constant](ch03-cards.md#g-Lambda), [the contracted Bianchi identity](ch03-cards.md#g-bianchi)

### the Standard Model — `SU(3)×SU(2)×U(1)` {#g-smodel}

*Function or operator*

The theory that owns three of the four forces, defined by its gauge group, and carrying 19 numbers it cannot predict—among them the gauge couplings behind this chapter’s strong coupling and weak mixing angle.

The Standard Model is the quantum field theory of the strong, weak, and electromagnetic interactions, and what defines it is a symmetry rather than a list of particles: the local gauge group SU(3) × SU(2) × U(1), whose three factors give the three forces. Writing down the most general Lagrangian with that symmetry leaves 19 numbers the theory does not fix—quark and lepton masses, mixing angles, the three couplings, the Higgs mass—and every one of them has to be measured. That is the sense in which the strong coupling and the weak mixing angle in the chapter’s particle-physics brace are measurements and not derivations. Gravity is not in it; neutrino mass is believed to need another seven or eight constants. A theory, uncolored.

Cites: [3.14](ch03-refs.md#r-3.14)

### the ΛCDM model — `ΛCDM` {#g-lcdm}

*Function or operator*

Six fitted numbers, a cosmological constant, and cold dark matter: the model that the chapter’s four density parameters are read out of, and the reason they may be added.

ΛCDM is the standard cosmological model—a cosmological constant Λ for dark energy, cold dark matter, and a flat, expanding universe. The Planck collaboration’s version is fitted with six parameters: the baryon density, the dark-matter density, the angular size of the sound horizon, the amplitude and the spectral index of the primordial fluctuations, and the reionization optical depth. Six is the smallest number that fits the data; everything else is either fixed at a natural value (total density 1, dark-energy equation of state −1) or calculated, including the Hubble constant and the dark-energy density themselves. This matters for the Ω bracket: the fit fixes the total density at 1, so the four density parameters add to 1 within that one fit. A model, uncolored.

Cites: [3.15](ch03-refs.md#r-3.15)

See also: [the Hubble constant](ch03-cards.md#g-H0), [cosmological constant](ch03-cards.md#g-Lambda)

### quantum electrodynamics — `QED` {#g-qed}

*Function or operator*

The quantum field theory of light and charged matter, whose prediction for the electron’s anomalous magnetic moment agrees with experiment to more than ten significant figures.

Quantum electrodynamics unites quantum mechanics with special relativity to describe how electrons, positrons, and photons interact; Feynman, Schwinger, and Tomonaga each developed it independently in the late 1940s and shared the 1965 Nobel Prize for it. Its central success is the electron’s anomalous magnetic moment, first computed by Schwinger in 1948 and now matched by measurement to more than ten significant figures, one of the most accurately verified predictions in the history of physics. A theory, uncolored.

Cites: [3.26](ch03-refs.md#r-3.26)

See also: [the electron’s anomalous magnetic moment](ch03-cards.md#g-ae)