# Natural units

In physics, natural units are physical units of measurement defined in terms of universal physical constants, such that some chosen physical constants each have a numerical value of exactly 1, when expressed in terms of a particular set of natural units.

Natural units are intended to elegantly simplify particular algebraic expressions appearing in physical law or to normalize some chosen physical quantities that are properties of universal elementary particles and that may be reasonably believed to be constant. However, what may be believed and forced to be constant in one system of natural units can very well be allowed or even assumed to vary in another natural unit system. Natural units are natural because the origin of their definition comes only from properties of nature and not from any human construct. Planck units are often, without qualification, called "natural units" but are only one system of natural units among other systems. Planck units might be considered unique in that the set of units are not based on properties of any prototype, object, or particle but are based only on properties of free space.

As with any set of base units or fundamental units the base units of a set of natural units will include definitions and values for length, mass, time, temperature, and electric charge. Some physicists have not recognized temperature as a fundamental dimension of physical quantity since it simply expresses the energy per degree of freedom of a particle which can be expressed in terms of energy (or mass, length, and time). Virtually every system of natural units normalizes the Boltzmann constant to k=1, which can be thought of as simply another expression of the definition of the unit temperature. In addition, some physicists recognize electric charge as a separate fundamental dimension of physical quantity, even if it has been expressed in terms of mass, length, and time in unit systems such as the electrostatic cgs system. Virtually every system of natural units normalizes the permittivity of free space to ε0=(4π)-1, which can be thought of as an expression of the definition of the unit charge.

## Candidate physical constants used in natural unit systems

The candidate physical constants to be normalized are chosen from those in the following table. Note that only a smaller subset of the following can be normalized in any one system of units without contradiction in definition (e.g., me and mp cannot both be defined as the unit mass in a single system).

Constant Symbol Dimension
speed of light in vacuum ${\displaystyle {c}\ }$ L T-1
Gravitational constant ${\displaystyle {G}\ }$ M-1L3T-2
Dirac's constant or "reduced Planck's constant" ${\displaystyle \hbar ={\frac {h}{2\pi }}}$ where ${\displaystyle {h}\ }$ is Planck's constant ML2T-1
Coulomb force constant ${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}}$ where ${\displaystyle {\epsilon _{0}}\ }$ is the permittivity of free space Q-2 M L3 T-2
Elementary charge ${\displaystyle e\ }$ Q
Electron mass ${\displaystyle m_{e}\ }$ M
Proton mass ${\displaystyle m_{p}\ }$ M
Boltzmann constant ${\displaystyle {k}\ }$ ML2T-2Θ-1
${\displaystyle \alpha \ \equiv {\frac {e^{2}}{\hbar c(4\pi \epsilon _{0})}}={\frac {1}{137.03599911}}}$

cannot take on a different numerical value no matter what system of units are used. Judiciously choosing units can only normalize physical constants that have dimension. Since α is a fixed dimensionless number not equal to 1, it is not possible to define a system of natural units that will normalize all of the physical constants that comprise α. Any 3 of the 4 constants: c, ${\displaystyle \hbar }$, e, or 4πε0, can be normalized (leaving the remaining physical constant to take on a value that is a simple function of α, alluding to the fundamental nature of the fine-structure constant) but not all 4.

## Planck units

Quantity Expression Metric value
Length (L) ${\displaystyle l_{P}={\sqrt {\frac {\hbar G}{c^{3}}}}}$ 1.61609735×10-35 m
Mass (M) ${\displaystyle m_{P}={\sqrt {\frac {\hbar c}{G}}}}$ 21.7664598 μg
Time (T) ${\displaystyle t_{P}={\sqrt {\frac {\hbar G}{c^{5}}}}}$ 5.3907205×10-44 s
Electric charge (Q) ${\displaystyle q_{P}={\sqrt {\hbar c(4\pi \epsilon _{0})}}}$ 1.87554573×10-18 C
Temperature (Θ) ${\displaystyle T_{P}={\sqrt {\frac {\hbar c^{5}}{Gk^{2}}}}}$ 1.4169206×1032 K
${\displaystyle c=1\ }$
${\displaystyle G=1\ }$
${\displaystyle \hbar =1\ }$
${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}=1}$
${\displaystyle k=1\ }$
${\displaystyle e={\sqrt {\alpha }}\ }$

The physical constants that Planck units normalize are properties of free space and not properties (such as charge, mass, size or radius) of any object or elementary particle (that would have to be arbitrarily chosen). Being so, the Planck units are defined independently of the elementary charge which, if measured in terms of Planck units, comes out to be the square root of the fine-structure constant, √α. In Planck units a conceivable variation in the value of the dimensionless α would be considered to be due to a variation in the elementary charge.

## Stoney units

Quantity Expression
Length (L) ${\displaystyle l_{S}={\sqrt {\frac {Ge^{2}}{c^{4}(4\pi \epsilon _{0})}}}}$
Mass (M) ${\displaystyle m_{S}={\sqrt {\frac {e^{2}}{G(4\pi \epsilon _{0})}}}}$
Time (T) ${\displaystyle t_{S}={\sqrt {\frac {Ge^{2}}{c^{6}(4\pi \epsilon _{0})}}}}$
Electric charge (Q) ${\displaystyle q_{S}=e\ }$
Temperature (Θ) ${\displaystyle T_{S}={\sqrt {\frac {c^{4}e^{2}}{G(4\pi \epsilon _{0})k^{2}}}}}$
${\displaystyle c=1\ }$
${\displaystyle G=1\ }$
${\displaystyle e=1\ }$
${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}=1}$
${\displaystyle k=1\ }$
${\displaystyle \hbar ={\frac {1}{\alpha }}\ }$

Proposed by George Stoney in 1881. Stoney units fix the elementary charge and allow Planck's constant to float. They can be obtained from Planck units with the substitution:

${\displaystyle \hbar \leftarrow \alpha \hbar ={\frac {e^{2}}{c(4\pi \epsilon _{0})}}}$.

This removes Planck's constant from the definitions and the value it takes on in Stoney units is the reciprocal of the fine-structure constant, 1/α. In Stoney units a conceivable variation in the value of the dimensionless α would be considered to be due to a variation in Planck's constant.

## "Schrödinger" units

Quantity Expression
Length (L) ${\displaystyle l_{\psi }={\sqrt {\frac {\hbar ^{4}G(4\pi \epsilon _{0})^{3}}{e^{6}}}}}$
Mass (M) ${\displaystyle m_{\psi }={\sqrt {\frac {e^{2}}{G(4\pi \epsilon _{0})}}}}$
Time (T) ${\displaystyle t_{\psi }={\sqrt {\frac {\hbar ^{6}G(4\pi \epsilon _{0})^{5}}{e^{10}}}}}$
Electric charge (Q) ${\displaystyle q_{\psi }=e\ }$
Temperature (Θ) ${\displaystyle T_{\psi }={\sqrt {\frac {e^{10}}{\hbar ^{4}(4\pi \epsilon _{0})^{5}Gk^{2}}}}}$
${\displaystyle e=1\ }$
${\displaystyle G=1\ }$
${\displaystyle \hbar =1\ }$
${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}=1}$
${\displaystyle k=1\ }$
${\displaystyle c={\frac {1}{\alpha }}\ }$

The name was coined by Michael Duff[1]. They can be obtained from Planck units with the substitution:

${\displaystyle c\leftarrow \alpha c={\frac {e^{2}}{\hbar (4\pi \epsilon _{0})}}}$.

This removes the speed of light from the definitions and the value it takes on in Schrödinger units is the reciprocal of the fine-structure constant, 1/α. In Schrödinger units a conceivable variation in the value of the dimensionless α would be considered to be due to a variation in the speed of light.

## Atomic units (Hartree)

Quantity Expression
Length (L) ${\displaystyle l_{A}={\frac {\hbar ^{2}(4\pi \epsilon _{0})}{m_{e}e^{2}}}}$
Mass (M) ${\displaystyle m_{A}=m_{e}\ }$
Time (T) ${\displaystyle t_{A}={\frac {\hbar ^{3}(4\pi \epsilon _{0})^{2}}{m_{e}e^{4}}}}$
Electric charge (Q) ${\displaystyle q_{A}=e\ }$
Temperature (Θ) ${\displaystyle T_{A}={\frac {m_{e}e^{4}}{\hbar ^{2}(4\pi \epsilon _{0})^{2}k}}}$
${\displaystyle e=1\ }$
${\displaystyle m_{e}=1\ }$
${\displaystyle \hbar =1\ }$
${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}=1}$
${\displaystyle k=1\ }$
${\displaystyle c={\frac {1}{\alpha }}\ }$

First proposed by Douglas Hartree to simplify the physics of the Hydrogen atom. Michael Duff[2] calls these "Bohr units". The unit energy in this system is the total energy of the electron in the first circular orbit of the Bohr atom and called the Hartree energy, Eh. The unit velocity is the velocity of that electron, the unit mass is the electron mass, me, and the unit length is the Bohr radius, ${\displaystyle a_{0}=4\pi \epsilon _{0}\hbar ^{2}/m_{e}e^{2}\ }$. They can be obtained from "Schrödinger" units with the substitution:

${\displaystyle G\leftarrow \alpha G\left({\frac {m_{P}}{m_{e}}}\right)^{2}={\frac {e^{2}}{4\pi \epsilon _{0}m_{e}^{2}}}\ }$.

This removes the speed of light (as well as the gravitational constant) from the definitions and the value it takes on in atomic units is the reciprocal of the fine-structure constant, 1/α. In atomic units a conceivable variation in the value of the dimensionless α would be considered to be due to a variation in the speed of light.

## Electronic system of units

Quantity Expression
Length (L) ${\displaystyle l_{e}={\frac {e^{2}}{c^{2}m_{e}(4\pi \epsilon _{0})}}}$
Mass (M) ${\displaystyle m_{e}=m_{e}\ }$
Time (T) ${\displaystyle t_{e}={\frac {e^{2}}{c^{3}m_{e}(4\pi \epsilon _{0})}}}$
Electric charge (Q) ${\displaystyle q_{e}=e\ }$
Temperature (Θ) ${\displaystyle T_{e}={\frac {m_{e}c^{2}}{k}}}$
${\displaystyle c=1\ }$
${\displaystyle e=1\ }$
${\displaystyle m_{e}=1\ }$
${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}=1}$
${\displaystyle k=1\ }$
${\displaystyle \hbar ={\frac {1}{\alpha }}\ }$

Michael Duff[3] calls these "Dirac units". They can be obtained from Stoney units with the substitution:

${\displaystyle G\leftarrow \alpha G\left({\frac {m_{P}}{m_{e}}}\right)^{2}={\frac {e^{2}}{4\pi \epsilon _{0}m_{e}^{2}}}\ }$.

They can be also obtained from Atomic units with the substitution:

${\displaystyle \hbar \leftarrow \alpha \hbar ={\frac {e^{2}}{c(4\pi \epsilon _{0})}}}$.

Similarly to Stoney units, a conceivable variation in the value of α would be considered to be due to a variation in Planck's constant.

## Quantum electrodynamical system of units (Stille)

Quantity Expression
Length (L) ${\displaystyle l_{\mathrm {QED} }={\frac {e^{2}}{c^{2}m_{p}(4\pi \epsilon _{0})}}}$
Mass (M) ${\displaystyle m_{\mathrm {QED} }=m_{p}\ }$
Time (T) ${\displaystyle t_{\mathrm {QED} }={\frac {e^{2}}{c^{3}m_{p}(4\pi \epsilon _{0})}}}$
Electric charge (Q) ${\displaystyle q_{\mathrm {QED} }=e\ }$
Temperature (Θ) ${\displaystyle T_{\mathrm {QED} }={\frac {m_{p}c^{2}}{k}}}$
${\displaystyle c=1\ }$
${\displaystyle e=1\ }$
${\displaystyle m_{p}=1\ }$
${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}=1}$
${\displaystyle k=1\ }$
${\displaystyle \hbar ={\frac {1}{\alpha }}\ }$

Similar to the electronic system of units except that the proton mass is normalized rather that the electron mass. Also a conceivable variation in the value of α would be considered to be due to a variation in Planck's constant.

## Geometrized units

${\displaystyle c=1\ }$
${\displaystyle G=1\ }$

The geometrized unit system is not a completely defined or unique system. In this system, the base physical units are chosen so that the speed of light and the gravitational constant are set equal to unity leaving latitude to also set some other constant such as the Boltzmann constant and Coulomb force constant equal to unity:

${\displaystyle k=1\ }$
${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}=1}$

If Dirac's constant (also called the "reduced Planck's constant") is also set equal to unity,

${\displaystyle \hbar =1\ }$

then geometrized units are identical to Planck units.

## N-body units

Quantity Expression
Length (R) ${\displaystyle {\frac {1}{R}}={\frac {1}{N(N-1)}}\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {1}{r_{j}-r_{i}}}}$
Mass (M) ${\displaystyle M=\sum _{i=1}^{N}m_{i}}$
${\displaystyle M=1\ }$
${\displaystyle G=1\ }$
${\displaystyle R=1\ }$

N-body units are a completely self-contained system of units used for N-body simulations of self gravitating systems in astrophysics. In this system, the base physical units are chosen so that the total mass (M), the gravitational constant (G) and the virial radius (R) are set equal to unity. The underlying assumption is that the system of N objects (stars) satisfies the virial theorem. The consequence of standard N-body units is that the velocity dispersion of the system is ${\displaystyle v=1/{\sqrt {2}}}$ and that the dynamical -crossing- time scales as ${\displaystyle t=2{\sqrt {2}}}$. The first mention of standard N-body units was by Michel Hénon (1971) [4]. They were taken up by Haldan Cohn (1979) [5] and later widely advertised and generalized by Douglas Heggie and Robert Mathieu (1986) [6].

## SI units

The metric system, or International System of Units (SI) as it is currently known, is not a natural system of units. Historically, metric units were not defined in terms of universal physical constants, nor were they defined in such a manner that some chosen set of physical constants would each have a numerical value of exactly 1.

There has been a trend in the last few decades, however, to redefine the units of the SI in terms of universal physical constants. In 1983, the seventeenth CGPM redefined the metre in terms of time and the speed of light, thus fixing the speed of light at exactly 299,792,458 m/s. And in 1990, the eighteenth CGPM adopted conventional values for the Josephson constant and the von Klitzing constant, fixing the conventional Josephson constant at exactly 483,597.9 ×109 Hz/V, and the conventional von Klitzing constant at exactly 25 812.807 Ω.

When the conventional values of the Josephson and von Klitzing constants are taken in conjunction with the definition of the meter, one obtains a metric system with units which are not natural, but which are derived from natural units through multiplicative factors. The relationship is illustrated in the following table:

Quantity / Symbol Planck Stoney Schrödinger Atomic Electronic SI
speed of light in vacuum
${\displaystyle c\,}$
${\displaystyle 1\,}$ ${\displaystyle 1\,}$ ${\displaystyle {\frac {1}{\alpha }}\ }$ ${\displaystyle {\frac {1}{\alpha }}\ }$ ${\displaystyle 1\,}$ ${\displaystyle 299792458\ }$
Planck's constant
${\displaystyle h\,}$
${\displaystyle 2\pi \,}$ ${\displaystyle {\frac {2\pi }{\alpha }}\ }$ ${\displaystyle 2\pi \,}$ ${\displaystyle 2\pi \,}$ ${\displaystyle {\frac {2\pi }{\alpha }}\ }$ ${\displaystyle {\frac {4\times 10^{-18}}{(25812.807)(483597.9)^{2}}}\ }$
Dirac's constant
${\displaystyle \hbar ={\frac {h}{2\pi }}}$
${\displaystyle 1\,}$ ${\displaystyle {\frac {1}{\alpha }}\ }$ ${\displaystyle 1\,}$ ${\displaystyle 1\,}$ ${\displaystyle {\frac {1}{\alpha }}\ }$ ${\displaystyle {\frac {2\times 10^{-18}}{\pi (25812.807)(483597.9)^{2}}}\ }$
elementary charge
${\displaystyle e\,}$
${\displaystyle {\sqrt {\alpha }}\,}$ ${\displaystyle 1\,}$ ${\displaystyle 1\,}$ ${\displaystyle 1\,}$ ${\displaystyle 1\,}$ ${\displaystyle {\frac {2\times 10^{-9}}{(25812.807)(483597.9)}}\ }$
Josephson constant
${\displaystyle K_{J}={\frac {2e}{h}}\,}$
${\displaystyle {\frac {\sqrt {\alpha }}{\pi }}\,}$ ${\displaystyle {\frac {\alpha }{\pi }}\,}$ ${\displaystyle {\frac {1}{\pi }}\,}$ ${\displaystyle {\frac {1}{\pi }}\,}$ ${\displaystyle {\frac {\alpha }{\pi }}\,}$ ${\displaystyle 483597.9\times 10^{9}\,}$
von Klitzing constant
${\displaystyle R_{K}={\frac {h}{e^{2}}}\,}$
${\displaystyle {\frac {2\pi }{\alpha }}\,}$ ${\displaystyle {\frac {2\pi }{\alpha }}\,}$ ${\displaystyle 2\pi \,}$ ${\displaystyle 2\pi \,}$ ${\displaystyle {\frac {2\pi }{\alpha }}\,}$ ${\displaystyle 25812.807\,}$
characteristic impedance of vacuum
${\displaystyle Z_{0}=2\alpha R_{K}\,}$
${\displaystyle 4\pi \,}$ ${\displaystyle 4\pi \,}$ ${\displaystyle 4\pi \alpha \,}$ ${\displaystyle 4\pi \alpha \,}$ ${\displaystyle 4\pi \,}$ ${\displaystyle 2\alpha (25812.807)\,}$
electric constant (vacuum permittivity)
${\displaystyle \epsilon _{0}={\frac {1}{Z_{0}c}}\,}$
${\displaystyle {\frac {1}{4\pi }}\,}$ ${\displaystyle {\frac {1}{4\pi }}\,}$ ${\displaystyle {\frac {1}{4\pi }}\,}$ ${\displaystyle {\frac {1}{4\pi }}\,}$ ${\displaystyle {\frac {1}{4\pi }}\,}$ ${\displaystyle {\frac {1}{2\alpha (25812.807)(299792458)}}\ }$
magnetic constant (vacuum permeability)
${\displaystyle \mu _{0}={\frac {Z_{0}}{c}}\,}$
${\displaystyle 4\pi \,}$ ${\displaystyle 4\pi \,}$ ${\displaystyle 4\pi \alpha ^{2}\,}$ ${\displaystyle 4\pi \alpha ^{2}\,}$ ${\displaystyle 4\pi \,}$ ${\displaystyle {\frac {2\alpha (25812.807)}{299792458}}\ }$
Newtonian constant of gravitation
${\displaystyle G\,}$
${\displaystyle 1\,}$ ${\displaystyle 1\,}$ ${\displaystyle 1\,}$ ${\displaystyle -\,}$ ${\displaystyle -\,}$ ${\displaystyle -\,}$
electron mass
${\displaystyle m_{e}\,}$
${\displaystyle -\,}$ ${\displaystyle -\,}$ ${\displaystyle -\,}$ ${\displaystyle 1\,}$ ${\displaystyle 1\,}$ ${\displaystyle -\,}$
caesium ground state hyperfine
transition frequency
${\displaystyle -\,}$ ${\displaystyle -\,}$ ${\displaystyle -\,}$ ${\displaystyle -\,}$ ${\displaystyle -\,}$ ${\displaystyle 9\ 192\ 631\ 770\,}$