# Natural units

As Natural units in physics systems are used by units designated by the values of physical constants are given. The use of such units often simplifies physical formulas. If one also regards the relevant natural constants as " dimensionless ", i.e. as pure numbers, this further simplifies the formulas. If, for example, the speed of light c is set equal to the number 1, the known mass-energy equivalence E = mc 2 is simplified to E = m , and energy, momentum and mass then have the same dimensions .

This is to be distinguished from the definition of units of measurement with the help of natural constants. In the International System of Units ( SI), the speed of light has been used since 1983 and since the reform of 2019 other fundamental constants of nature have been used to define units. These natural constants retain their previous dimension and do not become natural units.

## Basics for natural units

Natural units should be suitable for a particularly simple description of natural processes. So is z. For example, the speed of light in a vacuum is the upper limit for the speed with which physical effects can propagate and is the conversion factor between mass and rest energy of a particle. The elementary charge - and apart from a factor ½ also Planck's constant - are the smallest possible non-zero values ​​for charge or angular momentum. ${\ displaystyle c}$${\ displaystyle c ^ {2}}$ ${\ displaystyle e}$ ${\ displaystyle \ hbar}$

The following can therefore serve as a basis:

as well as properties of important particles such as:

• the electron mass :${\ displaystyle m _ {\ mathrm {e}}}$
• the proton mass :${\ displaystyle m _ {\ mathrm {p}}}$
• the neutron mass :${\ displaystyle m _ {\ mathrm {n}}}$

Since more natural constants are available than the usual system of units has dimensions, different natural systems of units can be formed. Which of these principles is chosen depends on the respective sub-area of ​​physics.

The relevant natural constants, if they are given in the corresponding natural units, all have the numerical value 1. Therefore, the constants do not appear at all when numerical value equations are used in concrete calculations .

Usually the constants are also set as dimensionless, so that all formulas become numerical value equations and look much simpler. This formal advantage is offset by the disadvantage that the results of all calculations are initially obtained as pure numbers. The correct dimensions and units can only be obtained through subsequent conversion into a "normal" system of units. A dimensional consideration of both sides of the equation for error control is not possible with equations in such a natural system of units. Also, the magnitudes of the natural units mostly deviate far from those usual in everyday life and in technology; therefore the general use of a natural system of units instead of e.g. B. the International System of Units (SI) has never been seriously considered.

## Natural systems of units

Overview of natural systems of units and their fundamentals
Planck units Stoney units Particle physics Atomic units theory of relativity Quantum chromodynamics
Speed ​​of light in a vacuum

${\ displaystyle c}$

${\ displaystyle 1}$ ${\ displaystyle 1}$ ${\ displaystyle 1}$ - ${\ displaystyle 1}$ ${\ displaystyle 1}$
Elemental charge

${\ displaystyle e}$

- ${\ displaystyle 1}$ - ${\ displaystyle 1}$ - -
Electric field constant

${\ displaystyle \ varepsilon _ {0}}$

${\ displaystyle {\ frac {1} {4 \ pi}}}$ ${\ displaystyle {\ frac {1} {4 \ pi}}}$ ${\ displaystyle 1}$ ${\ displaystyle {\ frac {1} {4 \ pi}}}$ ${\ displaystyle \ left ({\ frac {1} {4 \ pi}} \ right)}$ -
Coulomb's constant

${\ displaystyle k_ {C} = {\ frac {1} {4 \ pi \ varepsilon _ {0}}}}$

${\ displaystyle 1}$ ${\ displaystyle 1}$ ${\ displaystyle {\ frac {1} {4 \ pi}}}$ ${\ displaystyle 1}$ ${\ displaystyle \ left (1 \ right)}$ -
Gravitational constant

${\ displaystyle G}$

${\ displaystyle 1}$ ${\ displaystyle 1}$ - - ${\ displaystyle 1}$ -
Boltzmann's constant

${\ displaystyle k_ {B}}$

${\ displaystyle 1}$ - ${\ displaystyle 1}$ - - ${\ displaystyle 1}$
Planck's quantum of action

${\ displaystyle h}$

${\ displaystyle 2 \ pi}$ - ${\ displaystyle 2 \ pi}$ ${\ displaystyle 2 \ pi}$ - ${\ displaystyle 2 \ pi}$
Reduced Planck's quantum of action

${\ displaystyle \ hbar = {\ frac {h} {2 \ pi}}}$

${\ displaystyle 1}$ - ${\ displaystyle 1}$ ${\ displaystyle 1}$ - ${\ displaystyle 1}$
Electron mass

${\ displaystyle m_ {e}}$

- - - ${\ displaystyle 1}$ - -
Electron volt

${\ displaystyle eV}$

- - ${\ displaystyle \ left (1 \ right)}$ - - -
Proton mass

${\ displaystyle m_ {p}}$

- - - - - ${\ displaystyle 1}$

### Planck units

The most consistent implementation of the natural units can be found in the Planck units proposed by Max Planck in 1899 . In this system of units are set:

• the speed of light: ${\ displaystyle c = 1}$
• the reduced Planck quantum of action : (so is )${\ displaystyle \ hbar = 1}$${\ displaystyle h = 2 \ pi}$
• the Newtonian gravitational constant: ${\ displaystyle G = 1}$
• the Boltzmann constant: ${\ displaystyle k _ {\ mathrm {B}} = 1}$
• the Coulomb constant: (so is ).${\ displaystyle k _ {\ mathrm {C}} = 1}$${\ displaystyle \ varepsilon _ {0} = (4 \ pi k _ {\ mathrm {C}}) ^ {- 1} = (4 \ pi) ^ {- 1}}$

This system of units is considered fundamental because the fundamental constants of nature relate to the most general relationships between space and time and apply to all types of particles and interactions. (The constant is only required here to adapt the temperature scale to the energy scale.) ${\ displaystyle k _ {\ mathrm {B}}}$

With the help of the laws of nature that the above Define constants, the Planck units can also be introduced through the following relationships:

• During the unit of time, light travels a unit of length in a vacuum . (Natural law: )${\ displaystyle r = c \ cdot t}$
• The unit of energy is the quantum energy of an oscillation , the period of which is equal to a unit of time. (Natural law: )${\ displaystyle E = h / t}$
• The unit mass is the mass that is equivalent to one unit of energy. (Natural law: )${\ displaystyle E = m \ cdot c ^ {2}}$
• The unit of length is the distance between two bodies, each with a unit of mass, in which their gravitational energy has the size of a unit of energy. (Natural law: )${\ displaystyle E = G \ cdot m ^ {2} / r}$

### Stoney units

The first natural system of units was proposed by George Johnstone Stoney in 1874 after he had found the last necessary natural constant with the concept of uniform charge carriers in the atoms . In Stony's system of units, 1 is set:

• the elementary charge: ${\ displaystyle e = 1}$
• the speed of light: ${\ displaystyle c = 1}$
• the Newtonian gravitational constant: ${\ displaystyle G = 1}$

To define the charge, Stoney used the electrostatic cgs system , so that is also the Coulomb constant . According to Stoney, the natural units for length, mass and time are therefore smaller by a factor than according to Planck ( is the fine structure constant ). ${\ displaystyle {\ frac {1} {4 \ pi \ varepsilon _ {0}}} = 1}$${\ displaystyle {\ sqrt {\ alpha}} \ approx 0 {,} 085}$${\ displaystyle \ alpha = {\ frac {e ^ {2}} {4 \ pi \ varepsilon _ {0} \ cdot \ hbar c}} \ approx {\ frac {1} {137}}}$

The Stoney units are practically no longer used today, but are of historical interest.

### Particle physics

Some particle physics units in SI units
size unit Value
in SI units
written indeed
energy ${\ displaystyle 1 \, \ mathrm {eV}}$   ${\ displaystyle 1 {,} 60218 \ cdot 10 ^ {- 19}}$ J
length ${\ displaystyle {\ frac {1} {1 \, \ mathrm {eV}}}}$ ${\ displaystyle {\ frac {c \ hbar} {1 \, \ mathrm {eV}}}}$ ${\ displaystyle 1 {,} 97327 \ cdot 10 ^ {- 7}}$ m
time ${\ displaystyle {\ frac {1} {1 \, \ mathrm {eV}}}}$ ${\ displaystyle {\ frac {\ hbar} {1 \, \ mathrm {eV}}}}$ ${\ displaystyle 6 {,} 58212 \ cdot 10 ^ {- 16}}$ s
Dimensions ${\ displaystyle 1 \, \ mathrm {eV}}$ ${\ displaystyle {\ frac {1 \, \ mathrm {eV}} {c ^ {2}}}}$ ${\ displaystyle 1 {,} 78266 \ cdot 10 ^ {- 36}}$ kg
pulse ${\ displaystyle 1 \, \ mathrm {eV}}$ ${\ displaystyle {\ frac {1 \, \ mathrm {eV}} {c}}}$ ${\ displaystyle 5 {,} 34429 \ cdot 10 ^ {- 28}}$ N · s
temperature ${\ displaystyle 1 \, \ mathrm {eV}}$ ${\ displaystyle {\ frac {1 \, \ mathrm {eV}} {k _ {\ mathrm {B}}}}}$ ${\ displaystyle 1 {,} 16044 \ cdot 10 ^ {4}}$ K

In particle physics ( high energy physics ) , gravity only plays a subordinate role. Therefore the gravitational constant is left in the SI system here . Only:

• the speed of light: ${\ displaystyle c = 1}$
• Planck's quantum of action: ${\ displaystyle \ hbar = 1}$
• the Boltzmann constant: ${\ displaystyle k _ {\ mathrm {B}} = 1}$
• the electric field constant: ${\ displaystyle \ varepsilon _ {0} = 1}$

The unit of energy is not determined thereby; The electron volt unit is usually used for this . All other units can then be expressed in terms of powers of this energy unit (see table). So the electron-volt is at the same time the unit of mass ; this makes the equivalence of mass and energy particularly clear. According to the concept of spacetime, time and space get the same dimension and the unit 1 / eV.

### Atomic units

In atomic physics , the system of atomic units is used. Here are set to 1:

• the electron mass :${\ displaystyle m _ {\ mathrm {e}} = 1}$
• the elementary charge: ${\ displaystyle e = 1}$
• Planck's quantum of action: ${\ displaystyle \ hbar = 1}$
• the Coulomb constant: ${\ displaystyle 1 / (4 \ pi \ varepsilon _ {0}) = 1}$

### theory of relativity

In the general theory of relativity , 1 is set:

• the speed of light: ${\ displaystyle c = 1}$
• the gravitational constant: ${\ displaystyle G = 1}$

and often in situations with a dominant crowd

• the central mass: ${\ displaystyle M = 1}$
• the Coulomb constant: ${\ displaystyle k _ {\ mathrm {C}} = 1}$

### Quantum chromodynamics

In quantum chromodynamics , the proton of central interest. Here are set to 1:

• the speed of light: ${\ displaystyle c = 1}$
• the proton mass :${\ displaystyle m _ {\ mathrm {p}} = 1}$
• Planck's quantum of action: ${\ displaystyle \ hbar = 1}$
• the Boltzmann constant: ${\ displaystyle k _ {\ mathrm {B}} = 1}$

## Other suggestions

With CODATA -2014 were proposed

• a list with seven natural units (nu) , some of which use unusual natural quantities: for length, for time, for mass (charge and temperature are not listed),${\ displaystyle \ hbar / (m _ {\ mathrm {e}} c)}$${\ displaystyle \ hbar / (m _ {\ mathrm {e}} c ^ {2})}$${\ displaystyle m _ {\ mathrm {e}}}$
• and a list with 23 atomic units (au) , also with some unusual natural quantities: Bohr's radius for the length, for the time ( is the Hartree energy ), for the mass, for the charge (the temperature is not listed).${\ displaystyle a_ {0}}$${\ displaystyle \ hbar / E _ {\ mathrm {h}}}$${\ displaystyle E _ {\ mathrm {h}}}$${\ displaystyle m _ {\ mathrm {e}}}$${\ displaystyle e}$

## literature

• Helmut Hilscher: Elementary Particle Physics. Vieweg, 1996, ISBN 3-322-85004-8 , pp. 6-7.
• Peter Pohling: Through the universe with natural constants. Verlag Books on Demand, Norderstedt 2013, ISBN 978-3-7322-6236-6 .
• Michael Ruhrländer: Ascent to the Einstein equations. Introduction to quantitative general relativity. Pro Business, Berlin 2014, ISBN 978-3-86386-779-9 , p. 578.

## Individual evidence

1. Andreas Müller: Lexicon of Astronomy
2. Steven Fuerst, Kinwah Wu: Radiation Transfer of Emission Lines in Curved Space-Time . P. 4.
3. ^ Ezra Newman, Tim Adamo (Scholarpedia, 2014): Kerr-Newman metric . doi: 10.4249 / scholarpedia.31791
4. Peter J. Mohr, David B. Newell, Barry N. Taylor: CODATA Recommended Values ​​of the Fundamental Physical Constants: 2014 . In: Zenodo . 2015, doi : 10.5281 / zenodo.22826 , arxiv : 1507.07956 .