Natural Units
Concept
Natural units are popular in high-energy physics. They help to discard
constants from physics formulas, simplify expressions and make physical
relations between seemingly disparate quantities like mass, linear momentum
amd emergy glaringly obvious.
The introdction of the SI system of units, defining units by fixing the
values of certain universal constants, makes introducing natural units
more intuitive. Instead of fixing the values of the defining constants one
goes a step further: set their values to unity and drop their units.
That approach ultimately leads to a single unit, which is often energy
measured in Joule or electron-volt. Energy is a good choice since it exists
in various manifestations of energy, but can neither increase nor decrease
in its total amount of energy.
Background
The approach described here was inspired by a paper published by
Prof. Alan L. Myers of the University of Pennsylvania. That paper
can be found here:
Prof. Myers' approach was extended to handle both natural and geoemtric
units. Python scripts allow to convert between SI and natural units
with ease and to perform calculations using natural units.
Download a local zip file with the code here:
Basics
SI units are defined, starting with the hyperfine transition frequency
of cesium-133, by the values of the constants c, $\hbar$, e, $k_B$,
$N_A$ and a constant for luminous efficiency.
To create natural units, all the constants are set to unity, except for,
to remain consistent with Sommerfeld's fine-structure constant, electric
charge. For electromagnetism $k_Q = \sqrt{\hbar\,c\,\varepsilon_0} = 1$.
For the remainder one sets $c = \hbar = k_B = N_A = 1$ and drops their units.
This implies that $\varepsilon_0 = \mu_0 = 1$,
since $\varepsilon_0\,\mu_0\,c^2 = 1$.
Conversion
In order to derive a matrix that transforms between SI and natural units,
one first chooses a base unit X. A natural choice for natural units is
energy measure in Joule or elecztonvolt. The conversion condition is
$$
\large{kg^{i_1}\ m^{i_2}\ s^{i_3} A^{i_4} K^{i_5} mol^{i_6} cd^{i_7} =
X^{v_1}\ c^{v_2}\ \hbar^{v_3} k_Q^{v_4} k_B^{v_5} N_A^{v_6} k_{cd}^{v_7}}
$$
One writes the $i_n$ as a column vector and each of the $U^{v_n}$ as a
column in a matrix. For instance, if [X] = Joule, then
$[X]^{v_1} = v_1\,(1,\,2,\,-2,\,0\,...)^T$, since $[J] = kg^1\,m^2\,s^{-2}$.
Hence the equation above leads to
$$
\Large{
\text{K} =
\begin{pmatrix}
u_{11} & u_{21} & \dots & u_{71}\\
u_{12} & u_{22} & \dots & u_{72} \\
\vdots & \vdots & \ddots & \vdots \\
u_{17} & u_{27} & \dots & u_{77} \\
\end{pmatrix}}
$$
and the equation becomes
$$\Large{\vec{i}^T = K\cdot\vec{v}^T}$$
Physical Units in Natural Units
Once a base unit has been chosen, using the above conversion matrix,
physical quantities can be expressed in this base unit. If one wishes
to convert natural back to SI units, one also needs to keep track of
the constants involved in the conversion. otherwise they can be ignored.
The table below shows some physical quantities exoressed in natural
units. The SI cilumn refers to |X| = 1, e.g. |mass| = 1 kg.
$$
\large{
\begin{array}{|c|c|c|c|}
\hline
\text{SI} & \text{NU} & \text{X} & \text{Units} \\
\hline\hline
mass & 8.9876e+16 & J & c^{-2} \\
length & 3.1630e+25 & J^{-1} & c\ \hbar \\
time & 9.4825e+33 & J^{-1} & \hbar \\
velocity & 3.3356e-09 & 1 & c \\
acceleration & 3.5177e-43 & J & c\ \hbar^{-1} \\
momentum & 2.9979e+08 & J & c^{-1} \\
force & 3.1615e-26 & J^2 & c^{-1}\ \hbar^{-1} \\
energy & 1.0000e+00 & J & \\
density & 2.8401e-60 & J^4 & c^{-5}\ \hbar^{-3} \\
pressure & 3.1600e-77 & J^4 & c^{-3}\ s\hbar^{-3} \\
charge & 1.8901e+18 & 1 & k_Q \\
temperature & 1.3806e-23 & J & k_B^{-1} \\
\hline
\end{array}}
$$
One can then easily see, why potentials like the gravitational potential
need to be $r^{-1}$ potentials.
$$
\large{
\begin{align}
U &= G\frac{M_1\,M_2}{r}\\
[U] &= [G]\,[M]^2\,[r^{-1}]\\
\text{J} &= \text{J}^{-2}\times\text{J}^2\times\text{J}
\end{align}}
$$
Were the r-dependence of gravitation any different, the constant G
would need to have different units. While that also follows from
the physical units representation, it is much easier to spot when
using natural units. To convert natural to SI units one would use.
with $UNITS = |Units|\times [Units]$
$$
\large{
\text{SI} = \text{NU}\times UNITS}
$$