Natural units

Fundamental units:

• Time
• Space
• Mass
• Charge

Coulomb's law states that:

$F=k_{e}{\frac {q_{1}q_{2}}{d^{2}}},$

where:

• $k_e$ is the Coulomb constant which is the "stiffness" of space.

If the Coulomb constant is "negative pressure" then it has units of Force/area which gives:

$F={\frac {F}{d^{2}}}{\frac {q_{1}q_{2}}{d^{2}}},$

Therefore:

$q=d^{2},$

Charge is therefore no longer a fundamental unit but instead has units of distance squared.

Natural units

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.

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,

Summary table

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 $c \,$	$1\,$	$1\,$	$1\,$	$1\,$	$\frac{1}{\alpha} \$	$\frac{2}{\alpha} \$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$
Reduced Planck constant $\hbar = \frac{h}{2 \pi}$	$\frac{1}{4 \pi \alpha}$	$1\,$	$1\,$	$\frac{1}{\alpha} \$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$
Elementary charge $e \,$	$1\,$	$\sqrt{4\pi\alpha} \,$	$\sqrt{\alpha} \,$	$1\,$	$1\,$	$\sqrt{2} \,$	$\sqrt{4\pi\alpha}$	$\sqrt{\alpha}$	$1\,$	$\sqrt{4\pi\alpha} \,$	$\sqrt{\alpha} \,$
Vacuum permittivity $\varepsilon_0 \,$	$1\,$	$1\,$	$\frac{1}{4\pi}$	$\frac{1}{4\pi}$	$\frac{1}{4\pi}$	$\frac{1}{4\pi}$	$1\,$	$\frac{1}{4\pi}$	$\frac{1}{4 \pi \alpha}$	$1\,$	$\frac{1}{4\pi}$
Vacuum permeability $\mu_0 = \frac{1}{\epsilon_0 c^2} \,$	$1\,$	$1\,$	$4\pi$	$4\pi$	$4 \pi \alpha^2$	$\pi \alpha^2$	$1\,$	$4\pi$	$4 \pi \alpha$	$1\,$	$4\pi$
Impedance of free space $Z_0 = \frac{1}{\epsilon_0 c} = \mu_0 c \,$	$1\,$	$1\,$	$4\pi$	$4\pi$	$4 \pi \alpha$	$2\pi\alpha$	$1\,$	$4\pi$	$4 \pi \alpha$	$1\,$	$4\pi$
Coulomb constant $k_e=\frac{1}{4 \pi \epsilon_0}$	$\frac{1}{4\pi}$	$\frac{1}{4\pi}$	$1\,$	$1\,$	$1\,$	$1\,$	$\frac{1}{4\pi}$	$1\,$	$\alpha$	$\frac{1}{4\pi}$	$1\,$
Gravitational constant $G \,$	$\frac{1}{4\pi}$	$\frac{1}{4\pi}$	$1\,$	$1\,$	$\frac{\alpha_\text{G}}{\alpha} \,$	$\frac{8 \alpha_\text{G}}{\alpha} \,$	$\frac{\alpha_\text{G}}{{m_\text{e}}^2} \,$	$\frac{\alpha_\text{G}}{{m_\text{e}}^2} \,$	$\mu^2 \alpha_\text{G}$	$\mu^2 \alpha_\text{G}$	$\mu^2 \alpha_\text{G}$
Boltzmann constant $k_\text{B} \,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$	$1\,$
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 $m_\text{e} \,$		$\sqrt{4 \pi \alpha_\text{G}} \,$	$\sqrt{\alpha_\text{G}} \,$	$\sqrt{\frac{\alpha_\text{G}}{\alpha}} \,$	$1\,$	$\frac{1}{2} \,$	$511 \text{ keV}$	$511 \text{ keV}$	$\frac{1}{\mu}$	$\frac{1}{\mu}$	$\frac{1}{\mu}$
Planck mass $m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}},$	${\sqrt {\frac {1}{\alpha }}},$

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

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

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

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:

$pV=Nk_{B}T=Nk_{B}mv^{2}=Nk_{B}2E,$

where:

• p is the pressure
• V is the volume
• N is the number of molecules of gas.
• k_B is the Boltzmann constant
• T is the temperature
• v is the velocity
  • Twice the velocity means twice as much momentum transferred per collision and twice as many collisions per unit time.
• E is the kinetic energy per particle

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

• $T = 2 E$

Electromagnetism

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 (Faraday's law of induction):	$\nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}$	$\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}$	$\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}$
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

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.

CGS

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	=

References