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:

## Natural units

From natural units:

The surface area of a sphere

4
π

r
2

{\displaystyle 4\pi r^2}

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

# * F

q
1

q
2

r
2

1

4
π

{\displaystyle F=\frac{q\_1 q\_2}{r^2} \frac{1}{4 \pi}}

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

# * F

q
1

q
2

r
2

{\displaystyle 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* = 1*α*
**Rydberg atomic units** are defined by:

*e*√2 = 2*m*e = *ħ* = *k*e = *k*B = 1
*c* = 2*α*
**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

* *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 Fine-structure constant:

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

# α

1

4
π

ε
0

e
2

ℏ
c

=

μ
0

4
π

e
2

c

ℏ

=

k
e

e
2

ℏ
c

=

c

μ
0

2

R
K

=

e
2

4
π

Z
0

ℏ

=

1
137.036

{\displaystyle
\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} = \frac{1}{137.036}
}

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
* *ε*0 is the electric constant or permittivity of free space
* *µ*0 is the magnetic constant or permeability of free space
* *k*e is the Coulomb constant
* *R*K is the von Klitzing constant
* *Z*0 is the vacuum impedance or impedance of free space

#### Gravitational coupling constant

From Gravitational coupling constant:

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

α

G

=

G

m

e

2

ℏ
c

=

(

m

e

m

P

)

2

≈
1.751751596
×

10

−
45

{\displaystyle \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

### Gravitoelectromagnetism

See also: Einstein\_field\_equations
From 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:

∇
⋅

E

g

=
−
4
π
G

ρ
g

{\displaystyle \nabla \cdot \mathbf{E}\_\text{g} = -4 \pi G \rho\_\text{g} \ }

∇
⋅

E

=

ρ

ϵ
0

{\displaystyle \nabla \cdot \mathbf{E} = \frac {\rho} {\epsilon\_0}}

∇
⋅

B

g

=
0

{\displaystyle \nabla \cdot \mathbf{B}\_\text{g} = 0 \ }

∇
⋅

B

=
0

{\displaystyle \nabla \cdot \mathbf{B} = 0 \ }

∇
×

E

g

=
−

∂

B

g

∂
t

{\displaystyle \nabla \times \mathbf{E}\_\text{g} = -\frac{\partial \mathbf{B}\_\text{g} } {\partial t} \ }

∇
×

E

=
−

∂

B

∂
t

{\displaystyle \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B} } {\partial t} \ }

∇
×

B

g

=
−

4
π
G

c
2

J

g

1

c
2

∂

E

g

∂
t

{\displaystyle \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} }

∇
×

B

=

1

ϵ
0

c
2

J

1

c
2

∂

E

∂
t

{\displaystyle \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*g**v**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;
* *ε*0 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

1
2

q
2

r

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

Therefore:

2
E
⋅

r

q
2

=
# k
c
o
n
s
t
a
n
t

{\displaystyle {\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

q
1

q
2

d
2

{\displaystyle F=k\_e \frac{q\_1 q\_2}{d^2}}

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

# F

F
⋅
d

d

q
2

q
1

q
2

d
2

{\displaystyle F = {\color{red}F \cdot d \frac{d}{q^2}} \frac{q\_1 q\_2}{d^2}}

### Gravity

Newton's law of universal gravitation states that:

# F
G

m
1

m
2

d
2

{\displaystyle F = G \frac{m\_1 m\_2}{d^2} }

where:

# F
F
⋅
d

d

m
2

m
1

m
2

d
2

{\displaystyle F = F \cdot d \frac{d}{m^2} \frac{m\_1 m\_2}{d^2}}

But its probably better to say that:

# a

d

t
2

=
G

m

d
2

{\displaystyle 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

m
1

m
2

d
2

=

k
e

q
1

q
2

d
2

{\displaystyle 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 = α

{\displaystyle \sqrt{\alpha}}

 planck masses = 1 Stoney mass

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

### Boltzmann constant

For monatomic gases:

P

V

5
3

=
C
o
n
s
t
a
n
t

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

From 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

V
n

{\displaystyle k\_B T = P \frac{V}{n}}

where:

* kB 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

v
x

v
y

v
z

{\displaystyle v = \sqrt{v\_x + v\_y + v\_z}}

 is:

p
r
e
s
s
u
r
# e

f
o
r
c
e

a
r
e
a

=

m
o
m
e
n
t
u
m

t
i
m
e

d
2

=

2
m

v
x

2
d

/

v
x

d
2

=

m

v
x
2

d
3

=

2

E
x

V
0

=

2

E
3

V
0

{\displaystyle 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.
* vx 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.
* Ex is the kinetic energy per particle
	+ E = Ex +Ey + Ez

Therefore:

V
0

=

V
n

{\displaystyle V\_0 = \frac{V}{n}}

Therefore:

# T
p

V
n

=
p

V
0

=
m

v
2

=
2
E

{\displaystyle 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

{\displaystyle T = 2 E}

#### Blackbody radiation

From Black-body radiation:

Planck's law states that

B
ν

(
ν
,
T
# )

2
h

ν
3

c
2

1

e

h
ν

/

k
T

−
1

,

{\displaystyle 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 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
3

q
2

a
2

4
π

ε
0

c
3

=

q
2

a
2

6
π

ε
0

c
3

 (SI units)

{\displaystyle 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

## References

1. ↑ Wikipedia:Base unit (measurement)
2. ↑ at -181 C
3. ↑ Engineering ToolBox, (2003). *Specific Heat and Individual Gas Constant of Gases.* [online]
Available at: https://www.engineeringtoolbox.com/specific-heat-capacity-gases-d\_159.html
[Accessed 20-4-2019].

## External links

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