Natural units

See also: Dimensional analysis and Nondimensionalization

In physics, natural units are physical units of measurement based only on universal physical constants. For example, the elementary charge e is a natural unit of electric charge, and the speed of light c is a natural unit of speed.

Fundamental units

A set of fundamental dimensions is a minimal set of units such that every physical quantity can be expressed in terms of this set and where no quantity in the set can be expressed in terms of the others.

Fundamental units:

• Space
• Time (In relativity time has units of imaginary distance)
• Mass
• Charge
• Temperature

Some physicists have not recognized temperature as a fundamental dimension of physical quantity since it simply expresses the energy per particle per degree of freedom which can be expressed in terms of energy.

CGS system of units

From Wikipedia:Centimetre–gram–second system of units:

Quantity	Quantity symbol	CGS unit name	Unit symbol	Unit definition	Equivalent in SI units
length, position	L, x	centimetre	cm	1/100 of metre	= 10−2 m
mass	m	gram	g	1/1000 of kilogram	= 10−3 kg
time	t	second	s	1 second	= 1 s
velocity	v	centimetre per second	cm/s	cm/s	= 10−2 m/s
acceleration	a	gal	Gal	cm/s2	= 10−2 m/s2
force	F	dyne	dyn	g⋅cm/s2	= 10−5 N
energy	E	erg	erg	g⋅cm2/s2	= 10−7 J
power	P	erg per second	erg/s	g⋅cm2/s3	= 10−7 W
pressure	p	barye	Ba	g/(cm⋅s2)	= 10−1 Pa
dynamic viscosity	μ	poise	P	g/(cm⋅s)	= 10−1 Pa⋅s
kinematic viscosity	ν	stokes	St	cm2/s	= 10−4 m2/s
wavenumber	k	kayser	cm−1	cm−1	= 100 m−1
charge	q	Statcoulomb	statC	cm3/2 g1/2 s−1	=

Natural units

From Wikipedia:natural units:

The surface area of a sphere $4\pi r^2$

In Lorentz–Heaviside units (rationalized units), Coulomb's law is:

• $F=\frac{q_1 q_2}{r^2} \frac{1}{4 \pi}$

In Gaussian units (non-rationalized units), Coulomb's law is:

• $F = \frac{q_1 q_2}{r^2}$

Planck units are defined by

c = ħ = G = k_e = k_B = 1,

Stoney units are defined by:

c = G = k_e = e = k_B = 1,

Hartree atomic units are defined by:

e = m_e = ħ = k_e = k_B = 1

c = 1α

Rydberg atomic units are defined by:

e√2 = 2m_e = ħ = k_e = k_B = 1

c = 2α

Quantum chromodynamics (QCD) units are defined by:

c = m_p = ħ = k_B = 1

Natural units generally means:

ħ = c = k_B = 1.

where:

• c is the speed of light,
• ħ is the reduced Planck constant,
• G is the gravitational constant,
• k_e is the Coulomb constant,
• k_B is the Boltzmann constant
• e is the elementary charge,

Base units

Base units

Dimension	Planck (L-H)	Planck (Gauss)	Stoney	Hartree	Rydberg	Natural (L-H)	Natural (Gauss)	QCD (Original)	QCD (L-H)	QCD (Gauss)
Length (L)	$\sqrt{4 \pi \hbar G \over c^3}$	$\sqrt{\frac{\hbar G}{c^3}}$	$\sqrt{\frac{G k_\text{e} e^2}{c^4}}$	$\frac{\hbar^2 (4 \pi \epsilon_0)}{m_\text{e} e^2}$	$\frac{\hbar^2 (4 \pi \epsilon_0)}{m_\text{e} e^2}$	$\frac{\hbar c}{1\,\text{eV}}$	$\frac{\hbar c}{1\,\text{eV}}$	$\frac{\hbar}{m_\text{p} c}$	$\frac{\hbar}{m_\text{p} c}$	$\frac{\hbar}{m_\text{p} c}$
Time (T)	$\sqrt{4 \pi \hbar G \over c^5}$	$\frac{\hbar}{m_\text{P}c^2} = \sqrt{\frac{\hbar G}{c^5}}$	$\sqrt{\frac{G k_\text{e} e^2}{c^6}}$	$\frac{\hbar^3 (4 \pi \epsilon_0)^2}{m_\text{e} e^4}$	$\frac{2 \hbar^3 (4 \pi \epsilon_0)^2}{m_\text{e} e^4}$	$\frac{\hbar}{1\,\text{eV}}$	$\frac{\hbar}{1\,\text{eV}}$	$\frac{\hbar}{m_\text{p} c^2}$	$\frac{\hbar}{m_\text{p} c^2}$	$\frac{\hbar}{m_\text{p} c^2}$
Mass (M)	$\sqrt{\hbar c \over 4 \pi G}$	$\sqrt{\frac{\hbar c}{G}}$	$\sqrt{\frac{k_\text{e} e^2}{G}}$	$m_\text{e} \$	$2 m_\text{e} \$	$\frac{1\,\text{eV}}{c^2}$	$\frac{1\,\text{eV}}{c^2}$	$m_\text{p} \$	$m_\text{p} \$	$m_\text{p} \$
Electric charge (Q)	$\sqrt{\hbar c \epsilon_0}$	$\frac{e}{\sqrt{\alpha}}$	$e \$	$e \$	$\frac{e}{\sqrt{2}} \$	$\frac{e}{\sqrt{4\pi\alpha}}$	$\frac{e}{\sqrt{\alpha}}$	$e$	$\frac{e}{\sqrt{4\pi\alpha}}$	$\frac{e}{\sqrt{\alpha}}$
Temperature (Θ)  with $f=2$	$\sqrt{\frac{\hbar c^5}{4 \pi G {k_\text{B}}^2}}$	$\frac{m_\text{P} c^2}{k_\text{B}} = \sqrt{\frac{\hbar c^5}{G k_\text{B}^2}}$	$\sqrt{\frac{c^4 k_\text{e} e^2}{G {k_\text{B}}^2}}$	$\frac{m_\text{e} e^4}{\hbar^2 (4 \pi \epsilon_0)^2 k_\text{B}}$	$\frac{m_\text{e} e^4}{2 \hbar^2 (4 \pi \epsilon_0)^2 k_\text{B}}$	$\frac{1\,\text{eV}}{k_\text{B}}\cdot\frac{2}{f}$	$\frac{1\,\text{eV}}{k_\text{B}}\cdot\frac{2}{f}$	$\frac{m_\text{p} c^2}{k_\text{B}}$	$\frac{m_\text{p} c^2}{k_\text{B}}$	$\frac{m_\text{p} c^2}{k_\text{B}}$

Summary table

From Wikipedia:natural units:

Quantity / Symbol	Planck (L-H)	Planck (Gauss)	Stoney	Hartree	Rydberg	"Natural" (L-H)	"Natural" (Gauss)	QCD (original)	QCD (L-H)	QCD (Gauss)
Speed of light $c \,$	$1\,$	$1\,$	$1\,$	$\frac{1}{\alpha} \$	$\frac{2}{\alpha} \$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$
Reduced Planck constant $\hbar = \frac{h}{2 \pi}$	$1\,$	$1\,$	$\frac{1}{\alpha} \$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$
Elementary charge $e \,$	$\sqrt{4\pi\alpha} \,$	$\sqrt{\alpha} \,$	$1\,$	$1\,$	$\sqrt{2} \,$	$\sqrt{4\pi\alpha}$	$\sqrt{\alpha}$	$1\,$	$\sqrt{4\pi\alpha} \,$	$\sqrt{\alpha} \,$
Vacuum permittivity $\varepsilon_0 \,$	$1\,$	$\frac{1}{4\pi}$	$\frac{1}{4\pi}$	$\frac{1}{4\pi}$	$\frac{1}{4\pi}$	$1\,$	$\frac{1}{4\pi}$	$\frac{1}{4 \pi \alpha}$	$1\,$	$\frac{1}{4\pi}$
Vacuum permeability $\mu_0 = \frac{1}{\epsilon_0 c^2} \,$	$1\,$	$4\pi$	$4\pi$	$4 \pi \alpha^2$	$\pi \alpha^2$	$1\,$	$4\pi$	$4 \pi \alpha$	$1\,$	$4\pi$
Impedance of free space $Z_0 = \frac{1}{\epsilon_0 c} = \mu_0 c \,$	$1\,$	$4\pi$	$4\pi$	$4 \pi \alpha$	$2\pi\alpha$	$1\,$	$4\pi$	$4 \pi \alpha$	$1\,$	$4\pi$
Josephson constant $K_\text{J} =\frac{e}{\pi \hbar} \,$	$\sqrt{\frac{4\alpha}{\pi}} \,$	$\frac{\sqrt{\alpha}}{\pi} \,$	$\frac{\alpha}{\pi} \,$	$\frac{1}{\pi} \,$	$\frac{\sqrt{2}}{\pi} \,$	$\sqrt{\frac{4\alpha}{\pi}} \,$	$\frac{\sqrt{\alpha}}{\pi} \,$	$\frac{1}{\pi} \,$	$\sqrt{\frac{4\alpha}{\pi}} \,$	$\frac{\sqrt{\alpha}}{\pi} \,$
von Klitzing constant $R_\text{K} =\frac{2 \pi \hbar}{e^2} \,$	$\frac{1}{2\alpha}$	$\frac{2\pi}{\alpha} \,$	$\frac{2\pi}{\alpha} \,$	$2 \pi \,$	$\pi\,$	$\frac{1}{2\alpha}$	$\frac{2 \pi}{\alpha}$	$2 \pi \,$	$\frac{1}{2\alpha}$	$\frac{2\pi}{\alpha} \,$
Coulomb constant $k_e=\frac{1}{4 \pi \epsilon_0}$	$\frac{1}{4\pi}$	$1\,$	$1\,$	$1\,$	$1\,$	$\frac{1}{4\pi}$	$1\,$	$\alpha$	$\frac{1}{4\pi}$	$1\,$
Gravitational constant $G \,$	$\frac{1}{4\pi}$	$1\,$	$1\,$	$\frac{\alpha_\text{G}}{\alpha} \,$	$\frac{8 \alpha_\text{G}}{\alpha} \,$	$\frac{\alpha_\text{G}}{{m_\text{e}}^2} \,$	$\frac{\alpha_\text{G}}{{m_\text{e}}^2} \,$	$\mu^2 \alpha_\text{G}$	$\mu^2 \alpha_\text{G}$	$\mu^2 \alpha_\text{G}$
Boltzmann constant $k_\text{B} \,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$
Proton rest mass $m_\text{p} \,$	$\mu \sqrt{4 \pi \alpha_\text{G}} \,$	$\mu \sqrt{\alpha_\text{G}} \,$	$\mu \sqrt{\frac{\alpha_\text{G}}{\alpha}} \,$	$\mu \,$	$\frac{\mu}{2} \,$	$938 \text{ MeV}$	$938 \text{ MeV}$	$1\,$	$1\,$	$1\,$
Electron rest mass $m_\text{e} \,$	$\sqrt{4 \pi \alpha_\text{G}} \,$	$\sqrt{\alpha_\text{G}} \,$	$\sqrt{\frac{\alpha_\text{G}}{\alpha}} \,$	$1\,$	$\frac{1}{2} \,$	$511 \text{ keV}$	$511 \text{ keV}$	$\frac{1}{\mu}$	$\frac{1}{\mu}$	$\frac{1}{\mu}$

where:

• α is the dimensionless fine-structure constant
• α_G is the dimensionless gravitational coupling constant
• µ is dimensionless proton-to-electron mass ratio

Fine-structure constant

From Wikipedia:Fine-structure constant:

The Fine-structure constant, α, in terms of other fundamental physical constants:

$\alpha = \frac{1}{4 \pi \varepsilon_0} \frac{e^2}{\hbar c} = \frac{\mu_0}{4 \pi} \frac{e^2 c}{\hbar} = \frac{k_\text{e} e^2}{\hbar c} = \frac{c \mu_0}{2 R_\text{K}} = \frac{e^2}{4 \pi}\frac{Z_0}{\hbar}$

where:

• e is the elementary charge
• π is the mathematical constant pi
• ħ is the reduced Planck constant
• c is the speed of light in vacuum
• ε₀ is the electric constant or permittivity of free space
• µ₀ is the magnetic constant or permeability of free space
• k_e is the Coulomb constant
• R_K is the von Klitzing constant
• Z₀ is the vacuum impedance or impedance of free space

Gravitational coupling constant

From Wikipedia:Gravitational coupling constant:

The Gravitational coupling constant, α_G, is typically defined in terms of the gravitational attraction between two electrons. More precisely,

$\alpha_\mathrm{G} = \frac{G m_\mathrm{e}^2}{\hbar c} = \left( \frac{m_\mathrm{e}}{m_\mathrm{P}} \right)^2 \approx 1.751751596 \times 10^{-45}$

where:

• G is the gravitational constant
• m_e is the electron rest mass
• c is the speed of light in vacuum
• ħ is the reduced Planck constant
• m_P is the Planck mass

Maxwell's equations

From Wikipedia:Lorentz–Heaviside units:

Name	SI units	Gaussian units	Lorentz–Heaviside units
Gauss's law (macroscopic)	$\nabla \cdot \mathbf{D} = \rho_\text{f}$	$\nabla \cdot \mathbf{D} = 4\pi\rho_\text{f}$	$\nabla \cdot \mathbf{D} = \rho_\text{f}$
Gauss's law (microscopic)	$\nabla \cdot \mathbf{E} = \rho/\epsilon_0$	$\nabla \cdot \mathbf{E} = 4 \pi \rho$	$\nabla \cdot \mathbf{E} = \rho$
Gauss's law for magnetism:	$\nabla \cdot \mathbf{B} = 0$	$\nabla \cdot \mathbf{B} = 0$	$\nabla \cdot \mathbf{B} = 0$
Maxwell–Faraday equation:	$\nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}$	$\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}$	$\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}$
Ampère–Maxwell equation (macroscopic):	$\nabla \times \mathbf{H} = \mathbf{J}_{\text{f}} + \frac{\partial \mathbf{D}} {\partial t}$	$\nabla \times \mathbf{H} = \frac{4\pi}{c}\mathbf{J}_{\text{f}} + \frac{1}{c}\frac{\partial \mathbf{D}} {\partial t}$	$\nabla \times \mathbf{H} = \frac{1}{c}\mathbf{J}_{\text{f}} + \frac{1}{c}\frac{\partial \mathbf{D}} {\partial t}$
Ampère–Maxwell equation (microscopic):	$\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \frac{1}{c^2}\frac{\partial \mathbf{E}} {\partial t}$	$\nabla \times \mathbf{B} = \frac{4\pi}{c}\mathbf{J} + \frac{1}{c}\frac{\partial \mathbf{E}} {\partial t}$	$\nabla \times \mathbf{B} = \frac{1}{c}\mathbf{J} + \frac{1}{c}\frac{\partial \mathbf{E}} {\partial t}$

Gravitoelectromagnetism

See also: Einstein_field_equations

From Wikipedia:Gravitoelectromagnetism:

According to general relativity, the gravitational field produced by a rotating object (or any rotating mass–energy) can, in a particular limiting case, be described by equations that have the same form as in classical electromagnetism. Starting from the basic equation of general relativity, the Einstein field equation, and assuming a weak gravitational field or reasonably flat spacetime, the gravitational analogs to Maxwell's equations for electromagnetism, called the "GEM equations", can be derived. GEM equations compared to Maxwell's equations in SI units are:

GEM equations	Maxwell's equations
$\nabla \cdot \mathbf{E}_\text{g} = -4 \pi G \rho_\text{g} \$	$\nabla \cdot \mathbf{E} = \frac {\rho} {\epsilon_0}$
$\nabla \cdot \mathbf{B}_\text{g} = 0 \$	$\nabla \cdot \mathbf{B} = 0 \$
$\nabla \times \mathbf{E}_\text{g} = -\frac{\partial \mathbf{B}_\text{g} } {\partial t} \$	$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B} } {\partial t} \$
$\nabla \times \mathbf{B}_\text{g} = -\frac{4 \pi G}{c^2} \mathbf{J}_\text{g} + \frac{1}{c^2} \frac{\partial \mathbf{E}_\text{g}} {\partial t}$	$\nabla \times \mathbf{B} = \frac{1}{\epsilon_0 c^2} \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{E}} {\partial t}$

where:

• E_g is the static gravitational field (conventional gravity, also called gravitoelectric in analogous usage) in m⋅s⁻²;
• E is the electric field;
• B_g is the gravitomagnetic field in s⁻¹;
• B is the magnetic field;
• ρ_g is mass density in kg⋅m⁻³;
• ρ is charge density:
• J_g is mass current density or mass flux (J_g = ρ_gv_ρ, where v_ρ is the velocity of the mass flow generating the gravitomagnetic field) in kg⋅m⁻²⋅s⁻¹;
• J is electric current density;
• G is the gravitational constant in m³⋅kg⁻¹⋅s⁻²;
• ε₀ is the vacuum permittivity;
• c is the speed of propagation of gravity (which is equal to the speed of light according to general relativity) in m⋅s⁻¹.

Electromagnetism

The total energy in the electric field surrounding a hollow spherical shell of radius r and charge q is:

$E = k \frac{1}{2} \frac{q^2}{r}$

Therefore:

${\color{red}2E \cdot \frac{r}{q^2} } = k =constant$

The constant k is a property of space. It is the "stiffness" of space. (If space were stiffer then c would be faster.)

Coulomb's law states that:

$F=k_e \frac{q_1 q_2}{d^2}$

The Coulomb constant has units of Energy * distance/charge² which gives:

$F = {\color{red}F \cdot d \frac{d}{q^2}} \frac{q_1 q_2}{d^2}$

The factor of 1/2 in the first equation above comes from the fact that the field diminishes to zero as it penetrates the shell.

Gravity

Newton's law of universal gravitation states that:

$F = G \frac{m_1 m_2}{d^2}$

where:

$F = F \cdot d \frac{d}{m^2} \frac{m_1 m_2}{d^2}$

But its probably better to say that:

$a = \frac{d}{t^2} = G \frac{m}{d^2}\$

The obvious unit of charge is one electron but there is no obvious unit of mass. We can, however, create one by setting the electric force between two electrons equal to the gravitational force between two equal masses:

$G \frac{m_1 m_2}{d^2} = k_e\frac{q_1 q_2}{d^2}$

Solving we get m = 1.859 × 10^-6 g = $\sqrt{\alpha}$ planck masses = 1 Stoney mass

The Schwarzschild radius of a Stoney mass is 2 Stoney lengths.

Boltzmann constant

Gas Specific heat ratio Degrees of freedom Helium 1.667 3 Neon 1.667 3 Argon 1.667 3 Hydrogen 1.597 3.35 Hydrogen 1.41 4.88 Nitrogen 1.4 5 Oxygen 1.395 5.06 Chlorine 1.34 5.88 Carbon dioxide 1.289 6.92 Methane 1.304 6.58 Ethane 1.187 10.7

Engineering ToolBox (2003)

For monatomic gases:

$P V^{\frac{5}{3}} = Constant$

From Wikipedia:Boltzmann constant:

The Boltzmann constant, k, is a scaling factor between macroscopic (thermodynamic temperature) and microscopic (thermal energy) physics.
Macroscopically, the ideal gas law states:

$k_B T = P \frac{V}{n}$

where:

• k_B is the Boltzmann constant
• T is the temperature
• P is the pressure
• V is the volume
• n is the number of molecules of gas.

Single particle

The pressure exerted on one face of a cube of length d by a single particle bouncing back and forth perpendicular to the face with mass m and velocity $v = \sqrt{v_x + v_y + v_z}$ is:

$pressure = \frac{force}{area} = \frac{\frac{momentum}{time}}{d^2} = \frac{\frac{2 m v_x}{2 d / v_x}}{d^2} = \frac{m v_x^2}{d^3} = \frac{2 E_x}{V_0} = \frac{2 \frac{E}{3}}{V_0}$

where:

• V₀ = d³ is the volume occupied by a single particle.
• v_x is the velocity perpendicular to the face
  • Twice the velocity means twice as much momentum transferred per collision and twice as many collisions per unit time.
• E_x is the kinetic energy per particle
  • E = E_x +E_y + E_z

Therefore:

$V_0 = \frac{V}{n}$

Therefore:

$T = p \frac{V}{n} = p V_0 = m v^2 = 2 E$

Therefore temperature is twice the energy per degree of freedom per particle

• $T = 2 E$

Blackbody radiation

From Wikipedia:Black-body radiation:

Planck's law states that

$B_\nu(\nu, T) = \frac{2h\nu^3}{c^2}\frac{1}{e^{h\nu/kT} - 1},$

where

B_ν(T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency ν radiation per unit frequency at thermal equilibrium at temperature T.

h is the Planck constant;

c is the speed of light in a vacuum;

k is the Boltzmann constant;

ν is the frequency of the electromagnetic radiation;

T is the absolute temperature of the body.

Most of the electromagnetic radiation is emitted (and absorbed) during the brief but intense acceleration's during the atomic collisions.

From Wikipedia:Larmor formula:

For velocities that are small relative to the speed of light, the total power radiated is given by the Larmor formula:

$P = {2 \over 3} \frac{q^2 a^2}{ 4 \pi \varepsilon_0 c^3}= \frac{q^2 a^2}{6 \pi \varepsilon_0 c^3} \mbox{ (SI units)}$

Uncertainty principle

• https://www.ias.ac.in/article/fulltext/reso/004/02/0020-0023
• Uncertainty principle
• Gaussian function

External links

• The Spectrum of Riemannium
• wisc.edu
• Stoney Scale and Large Number Coincidences