Physical Constants and SI Units

Contents

Mathematical constants
The seven defining constants of the SI
CODATA 2018 derived and measured constants
Planck units
SI base units

Since the revision that entered into force on 20 May 2019, the International System of Units (SI) is defined by assigning exact numerical values to seven defining constants of nature; every SI unit is then realised as a consequence of these fixed values [BIPM:2019]. All other physical constants are determined experimentally, and this treatise uses throughout the recommended values of the CODATA 2018 adjustment [Tiesinga:2021]. The present appendix collects, first, the seven defining constants with their exact values (The seven defining constants of the SI); second, the derived and measured constants that recur in the main text, quoted with their standard uncertainties (CODATA 2018 derived and measured constants); and third, the seven SI base units (SI base units).

Mathematical constants

Three dimensionless numbers precede every physical constant in this appendix, since they are not measured but proven:

\begin{equation}\tag{B.1} \ee = 2.718281828\ldots\ec\qquad \pi = 3.141592653\ldots\ec\qquad \varphi = 1.6180339887\ldots \end{equation}

All three are derived in full in Section 7.9.1 of Real Analysis — $\ee$ as the sum $\sum 1/n!$ and the limit $\lim(1+1/n)^n$ (Definition 7.53 and Proposition 7.57); $\pi$ as twice the first positive zero of the cosine series and independently as the circle constant (Definition 7.74 and Proposition 7.76); and $\varphi$, the golden ratio, as the positive root of $x^{2}=x+1$ (Definition 7.90 and Proposition 7.91). They are listed here only for numerical reference; their proofs are not repeated.

The three are not on a level, and Remark 7.94 draws the line. $\varphi$ is algebraic of degree $2$ — one square root writes it exactly — while $\ee$ and $\pi$ are transcendental, satisfying no polynomial equation with integer coefficients [Lindemann:1882] [Hilbert:1893]. $\varphi$ is also far the rarest of the three in this treatise, and it is listed here for its irrationality rather than its size: $2\cos(2\pi/5)=1/\varphi$ is irrational, which is what forbids a crystal a five-fold axis (Proposition 125.2) and hence what put the quasicrystal of Phenomenon 125.1 in contradiction with periodicity; and the same number is the worst-approximable winding ratio of Proposition 32.55.

The digits are a separate claim from the existence of the constants, and worth separating explicitly. A definition that converges establishes that a number exists; it does not by itself justify writing down a single decimal. For $\ee$ both steps are carried out: the truncation bound $\frac{1}{(n+1)!}\le\ee-\sum_{k\le n}1/k!<\frac{1}{n\cdot n!}$ (Lemma 7.54) traps $\ee$ between two explicit rationals, and twelve terms already force the nine decimals quoted above (Corollary 7.55). The trailing $\ldots$ is a theorem too, not a typographical hedge: $\ee$ is irrational (Proposition 7.56), so no finite decimal in this appendix equals it. Every entry below is a truncation; the only question is whether its error is stated.

For $\varphi$ the same two steps are cheaper than for either of the others, because one integer square root settles them: the greatest integer $a$ with $a^{2}\le5\cdot10^{20}$ gives $a/10^{10}\le\sqrt{5}<(a+1)/10^{10}$ and hence the ten decimals quoted in Equation (B.1) (Proposition 7.91); and $\varphi$ is irrational because $5$ is not a perfect square (Proposition 7.89), so its expansion does not terminate either.

The same two claims hold for $\pi$, and are established the same way. The digits come from Archimedes' construction: the inscribed and circumscribed regular polygons bracket $\pi$ between two computable numbers whose separation falls below $12/4^{n}$ after $n$ doublings (Proposition 7.79), and carrying the recursion to $n=17$ forces the nine decimals quoted above (Corollary 7.81), by a finite computation in exact integer arithmetic with every rounding taken outward. The trailing $\ldots$ is again a theorem: $\pi$ is irrational (Proposition 7.85, after [Niven:1947]), so no finite decimal in this appendix equals it either. Both constants are therefore quoted here with the same warrant — a proof of existence, a certified error bound behind the digits, and an irrationality theorem behind the dots.

The seven defining constants of the SI

The 2019 revision fixes the numerical values listed in Table B.1 exactly, by definition and without uncertainty [BIPM:2019]. Each constant anchors one base unit, although the definitions are interlocking: the kilogram, for instance, is realised through the fixed value of the Planck constant together with the definitions of the second and the metre.

Defining constant	Symbol	Exact value	Anchors
hyperfine transition frequency of $^{133}$Cs	$\Delta\nu_{\mathrm{Cs}}$	$9192631770\,\mathrm{Hz}$	$\mathrm{s}$
speed of light in vacuum	$c$	$299792458\,\mathrm{m}/\mathrm{s}$	$\mathrm{m}$
Planck constant	$h$	$6.62607015\times 10^{-34}\,\mathrm{J}\,\mathrm{s}$	$\mathrm{kg}$
elementary charge	$e$	$1.602176634\times 10^{-19}\,\mathrm{C}$	$\mathrm{A}$
Boltzmann constant	$k$	$1.380649\times 10^{-23}\,\mathrm{J}/\mathrm{K}$	$\mathrm{K}$
Avogadro constant	$N_{\mathrm{A}}$	$6.02214076\times 10^{23}\,/\mathrm{mol}$	$\mathrm{mol}$
luminous efficacy of $540\times 10^{12}\,\mathrm{Hz}$ radiation	$K_{\mathrm{cd}}$	$683\,\mathrm{lm}/\mathrm{W}$	$\mathrm{cd}$

The seven defining constants of the SI (2019 revision). All values are exact by definition [BIPM:2019].

CODATA 2018 derived and measured constants

Table B.2 lists the constants used most frequently in this treatise, with the CODATA 2018 recommended values and standard uncertainties [Tiesinga:2021]. Two entries deserve comment. The reduced Planck constant $\hbar = h/2\pi$ is exact because $h$ is exact; its decimal representation is merely truncated in the table. The standard acceleration of free fall $g_0$ is not a constant of nature at all but a conventional exact value, adopted by the third Conférence Générale des Poids et Mesures in 1901 and retained in the SI Brochure [BIPM:2019]; the local acceleration of free fall differs from it at the level of a few parts in $10^{3}$ and is what absolute gravimeters actually measure [Niebauer:1995].

Quantity	Symbol	Value	$u_{\mathrm{r}}$
Newtonian constant of gravitation	$G$	$6.67430(15)\times 10^{-11}\,\mathrm{m}^{3}/\mathrm{kg}/\mathrm{s}^{2}$	$2.2\times 10^{-5}$
reduced Planck constant	$\hbar$	$1.054571817\times 10^{-34}\,\mathrm{J}\,\mathrm{s}$	exact^a
electron mass	$m_{\mathrm{e}}$	$9.1093837015(28)\times 10^{-31}\,\mathrm{kg}$	$3.0\times 10^{-10}$
proton mass	$m_{\mathrm{p}}$	$1.67262192369(51)\times 10^{-27}\,\mathrm{kg}$	$3.1\times 10^{-10}$
vacuum magnetic permeability	$\mu_0$	$1.25663706212(19)\times 10^{-6}\,\mathrm{N}/\mathrm{A}^{2}$	$1.5\times 10^{-10}$
vacuum electric permittivity	$\epsilon_0$	$8.8541878128(13)\times 10^{-12}\,\mathrm{F}/\mathrm{m}$	$1.5\times 10^{-10}$
fine-structure constant	$\alpha$	$7.2973525693(11)\times 10^{-3}$	$1.5\times 10^{-10}$
Bohr magneton	$\mu_{\mathrm{B}}$	$9.2740100783(28)\times 10^{-24}\,\mathrm{J}/\mathrm{T}$	$3.0\times 10^{-10}$
standard acceleration of free fall	$g_0$	$9.80665\,\mathrm{m}/\mathrm{s}^{2}$	exact^b

Derived and measured constants, CODATA 2018 recommended values [Tiesinga:2021]. The column $u_{\mathrm{r}}$ gives the relative standard uncertainty.

^a\,Exact because $\hbar = h/2\pi$ with $h$ fixed; the decimal digits shown are truncated. ^b\,Conventional exact value, not a constant of nature [BIPM:2019].

Planck units

The constants $\hbar$, $c$, and $G$ together fix a length, mass, and time with no free numerical factor left undetermined by dimensional analysis alone.

Proposition B.1 (Planck units).

The unique (up to a dimensionless constant) combinations of $\hbar$, $c$, $G$ with dimensions of length, mass, and time are

\begin{equation}\tag{B.2} \ell_{\mathrm P} = \sqrt{\frac{\hbar G}{c^{3}}}\ec\qquad m_{\mathrm P} = \sqrt{\frac{\hbar c}{G}}\ec\qquad t_{\mathrm P} = \sqrt{\frac{\hbar G}{c^{5}}} = \frac{\ell_{\mathrm P}}{c}\ep \end{equation}

Rests on Theorem 2.3 and Axiom 2.2.

Proof.

Derives Proposition B.1. Write $\hbar^{a}c^{b}G^{d}$ and demand dimension $\mathsf{L}$ (length). In SI, $[\hbar] = \mathsf{M}\mathsf{L}^{2}\mathsf{T}^{-1}$, $[c] = \mathsf{L}\mathsf{T}^{-1}$, $[G] = \mathsf{M}^{-1}\mathsf{L}^{3}\mathsf{T}^{-2}$. Matching powers of $\mathsf{M}, \mathsf{L}, \mathsf{T}$ gives the linear system $a - d = 0$ ($\mathsf{M}$), $2a + b + 3d = 1$ ($\mathsf{L}$), $-a - b - 2d = 0$ ($\mathsf{T}$); solving, $a = d = 1/2$, $b = -3/2$, i.e. $\hbar^{1/2}c^{-3/2}G^{1/2} = \sqrt{\hbar G/c^{3}}$. By the Buckingham $\pi$ theorem (Theorem 2.3), three quantities built from three dimensionally independent constants leave zero dimensionless combinations, so this exponent solution is unique and no undetermined dimensionless prefactor can be fixed by dimensional analysis alone — exactly the freedom absorbed into the convention that defines $\ell_{\mathrm P}$ with prefactor $1$. The mass and time forms follow by the same linear system with target dimension $\mathsf{M}$ and $\mathsf{T}$ respectively (or, more quickly, from $m_{\mathrm P} = \hbar/(\ell_{\mathrm P} c)$ and $t_{\mathrm P} = \ell_{\mathrm P}/c$, using $[\hbar] = \mathsf{M}\mathsf{L}^{2}\mathsf{T}^{-1}$ to fix the mass combination and dimensional consistency to fix the time combination).

∎

Numerically, from the CODATA values of Table B.2,

\begin{equation}\tag{B.3} \ell_{\mathrm P} \approx 1.616\times 10^{-35}\,\mathrm{m}\ec\qquad m_{\mathrm P} \approx 2.176\times 10^{-8}\,\mathrm{kg}\ec\qquad t_{\mathrm P} \approx 5.391\times 10^{-44}\,\mathrm{s}\ec \end{equation}

the length and time scales at which quantum and gravitational effects are expected to become comparable in magnitude, and the mass scale against which the hierarchy problem of The Free Parameters of Physics is stated quantitatively.

SI base units

The seven base units of the SI, from which all coherent derived units are formed by products of powers, are collected in Table B.3 [BIPM:2019]. Every quantity in this treatise is expressed in these units or in coherent derived units built from them, in accordance with the axiom stated in the front matter.

Base quantity	Base unit	Symbol
time	second	$\mathrm{s}$
length	metre	$\mathrm{m}$
mass	kilogram	$\mathrm{kg}$
electric current	ampere	$\mathrm{A}$
thermodynamic temperature	kelvin	$\mathrm{K}$
amount of substance	mole	$\mathrm{mol}$
luminous intensity	candela	$\mathrm{cd}$

The seven SI base units [BIPM:2019].

Defining constant	Symbol	Exact value	Anchors
hyperfine transition frequency of $^{133}$Cs	$\Delta\nu_{\mathrm{Cs}}$	\(9192631770\,\mathrm{Hz}\)	\(\mathrm{s}\)
speed of light in vacuum	$c$	\(299792458\,\mathrm{m}/\mathrm{s}\)	\(\mathrm{m}\)
Planck constant	$h$	\(6.62607015\times 10^{-34}\,\mathrm{J}\,\mathrm{s}\)	\(\mathrm{kg}\)
elementary charge	$e$	\(1.602176634\times 10^{-19}\,\mathrm{C}\)	\(\mathrm{A}\)
Boltzmann constant	$k$	\(1.380649\times 10^{-23}\,\mathrm{J}/\mathrm{K}\)	\(\mathrm{K}\)
Avogadro constant	$N_{\mathrm{A}}$	\(6.02214076\times 10^{23}\,/\mathrm{mol}\)	\(\mathrm{mol}\)
luminous efficacy of \(540\times 10^{12}\,\mathrm{Hz}\) radiation	$K_{\mathrm{cd}}$	\(683\,\mathrm{lm}/\mathrm{W}\)	\(\mathrm{cd}\)

Quantity	Symbol	Value	$u_{\mathrm{r}}$
Newtonian constant of gravitation	$G$	\(6.67430(15)\times 10^{-11}\,\mathrm{m}^{3}/\mathrm{kg}/\mathrm{s}^{2}\)	\(2.2\times 10^{-5}\)
reduced Planck constant	$\hbar$	\(1.054571817\times 10^{-34}\,\mathrm{J}\,\mathrm{s}\)	exact^a
electron mass	$m_{\mathrm{e}}$	\(9.1093837015(28)\times 10^{-31}\,\mathrm{kg}\)	\(3.0\times 10^{-10}\)
proton mass	$m_{\mathrm{p}}$	\(1.67262192369(51)\times 10^{-27}\,\mathrm{kg}\)	\(3.1\times 10^{-10}\)
vacuum magnetic permeability	$\mu_0$	\(1.25663706212(19)\times 10^{-6}\,\mathrm{N}/\mathrm{A}^{2}\)	\(1.5\times 10^{-10}\)
vacuum electric permittivity	$\epsilon_0$	\(8.8541878128(13)\times 10^{-12}\,\mathrm{F}/\mathrm{m}\)	\(1.5\times 10^{-10}\)
fine-structure constant	$\alpha$	\(7.2973525693(11)\times 10^{-3}\)	\(1.5\times 10^{-10}\)
Bohr magneton	$\mu_{\mathrm{B}}$	\(9.2740100783(28)\times 10^{-24}\,\mathrm{J}/\mathrm{T}\)	\(3.0\times 10^{-10}\)
standard acceleration of free fall	$g_0$	\(9.80665\,\mathrm{m}/\mathrm{s}^{2}\)	exact^b