In physics, natural unit systems are measurement systems for which certain physical constants have been set to 1 through nondimensionalization of physical units. For example, the speed of light c may be set to 1, and it may then be omitted, equating mass and energy directly E=m rather than using c as a conversion factor in the typical mass–energy equivalence equation E=mc^{2}. A purely natural system of units has all of its dimensions collapsed, such that the physical constants completely define the system of units and the relevant physical laws contain no conversion constants.
While natural unit systems simplify the form of each equation, it is still necessary to keep track of the non-collapsed dimensions of each quantity or expression in order to reinsert physical constants (such dimensions uniquely determine the full formula). Dimensional analysis in the collapsed system is uninformative as most quantities have the same dimensions.
Quantity | Planck | Stoney | Atomic | Particle and atomic physics | Strong | Schrödinger |
---|---|---|---|---|---|---|
Defining constants | , , , | , , , | , , , | , , , | , , | , , , |
Speed of light | ||||||
Reduced Planck constant | ||||||
Elementary charge | — | — | ||||
Vacuum permittivity | — | — | ||||
Gravitational constant |
where:
Quantity | Expression | Approx. metric value |
---|---|---|
Length | 1.380×10^{−36} m^{[1]} | |
Mass | 1.859×10^{−9} kg^{[1]} | |
Time | 4.605×10^{−45} s^{[1]} | |
Electric charge | 1.602×10^{−19} C |
The Stoney unit system uses the following defining constants:
where c is the speed of light, G is the gravitational constant, k_{e} is the Coulomb constant, and e is the elementary charge.
George Johnstone Stoney's unit system preceded that of Planck by 30 years. He presented the idea in a lecture entitled "On the Physical Units of Nature" delivered to the British Association in 1874.^{[2]} Stoney units did not consider the Planck constant, which was discovered only after Stoney's proposal.
Quantity | Expression | Approx. metric value |
---|---|---|
Length | 1.616×10^{−35} m^{[3]} | |
Mass | 2.176×10^{−8} kg^{[4]} | |
Time | 5.391×10^{−44} s^{[5]} | |
Temperature | 1.417×10^{32} K^{[6]} |
The Planck unit system uses the following defining constants:
where c is the speed of light, ħ is the reduced Planck constant, G is the gravitational constant, and k_{B} is the Boltzmann constant.
Planck units form a system of natural units that is not defined in terms of properties of any prototype, physical object, or even elementary particle. They only refer to the basic structure of the laws of physics: c and G are part of the structure of spacetime in general relativity, and ħ is at the foundation of quantum mechanics. This makes Planck units particularly convenient and common in theories of quantum gravity, including string theory.^{[citation needed]}
Planck considered only the units based on the universal constants G, h, c, and k_{B} to arrive at natural units for length, time, mass, and temperature, but no electromagnetic units.^{[7]} The Planck system of units is now understood to use the reduced Planck constant, ħ, in place of the Planck constant, h.^{[8]}
Quantity | Expression | Approx. metric value |
---|---|---|
Length | 2.593×10^{−32} m | |
Mass | 1.859×10^{−9} kg | |
Time | 1.185×10^{−38} s | |
Electric charge | 1.602×10^{−19} C^{[9]} |
The Schrödinger system of units (named after Austrian physicist Erwin Schrödinger) is seldom mentioned in literature. Its defining constants are:^{[10]}^{[11]}
Defining constants:
The geometrized unit system,^{[12]}^{: 36 } used in general relativity, the base physical units are chosen so that the speed of light, c, and the gravitational constant, G, are set to one.
Quantity | Expression | Metric value |
---|---|---|
Length | 5.292×10^{−11} m^{[13]} | |
Mass | 9.109×10^{−31} kg^{[14]} | |
Time | 2.419×10^{−17} s^{[15]} | |
Electric charge | 1.602×10^{−19} C^{[16]} |
The atomic unit system^{[17]} uses the following defining constants:^{[18]}^{: 349 }^{[19]}
The atomic units were first proposed by Douglas Hartree and are designed to simplify atomic and molecular physics and chemistry, especially the hydrogen atom.^{[18]}^{: 349 } For example, in atomic units, in the Bohr model of the hydrogen atom an electron in the ground state has orbital radius, orbital velocity and so on with particularly simple numeric values.
Quantity | Expression | Metric value |
---|---|---|
Length | 3.862×10^{−13} m^{[20]} | |
Mass | 9.109×10^{−31} kg^{[21]} | |
Time | 1.288×10^{−21} s^{[22]} | |
Electric charge | 5.291×10^{−19} C |
This natural unit system, used only in the fields of particle and atomic physics, uses the following defining constants:^{[23]}^{: 509 }
where c is the speed of light, m_{e} is the electron mass, ħ is the reduced Planck constant, and ε_{0} is the vacuum permittivity.
The vacuum permittivity ε_{0} is implicitly used as a nondimensionalization constant, as is evident from the physicists' expression for the fine-structure constant, written α = e^{2}/(4π),^{[24]}^{[25]} which may be compared to the correspoding expression in SI: α = e^{2}/(4πε_{0}ħc).^{[26]}^{: 128 }
Quantity | Expression | Metric value |
---|---|---|
Length | 2.103×10^{−16} m | |
Mass | 1.673×10^{−27} kg | |
Time | 7.015×10^{−25} s |
Defining constants:
Here, m_{p} is the proton rest mass. Strong units are "convenient for work in QCD and nuclear physics, where quantum mechanics and relativity are omnipresent and the proton is an object of central interest".^{[27]}
In this system of units the speed of light changes in inverse proportion to the fine-structure constant, therefore it has gained some interest recent years in the niche hypothesis of time-variation of fundamental constants.^{[28]}
In this set of units the speed of light changes in inverse proportion to the fine structure constant. From this we can conclude that if c changes but e and ℏ remain constant then the speed of light in Schrödinger units, c_{ψ} changes in proportion to c but the speed of light in Planck units, c_{P} stays the same. Whether or not the "speed of light" changes depends on our measuring system (three possible definitions of the "speed of light" are c, c_{P} and c_{ψ}). Whether or not c changes is unambiguous because the measuring system has been defined.