# Natural units

## Limitations of the SI system

The SI system of measurements which is based on the metric system is more rigid than the older traditional systems used in different parts of the world due to the precise definitions of what the magnitudes of the units are and the the coherence within the system. Seven base units are chosen and the rest are derived from these, some have special names, others not. Why are these particular seven units chosen? There is no irrefutable rational behind the choices, they could have been different and the same coherence can be applied.

Moreover, the definitions for the length of a meter, the mass of a kilogram and so on are arbitrary. Six of the base units are defined on certain principals of physics. An example is the second being equal to the duration of a particular amount of periods of radiation of the caesium atom. This means that you can recreate this in a laboratory and measure exactly one second. The seventh unit, the kilogram, is as previously discussed defined by a particular object, the International Prototype Kilogram, making it impossible to measure exactly one kilogram in a laboratory since you can only have a nearly exact replica of the prototype in your laboratory.

A more mathematically and scientifically beautiful system would not rely on so many arbitrary choices for definitions.

## What is a natural system of units?

A system of natural units would be based on universal physical constants. The reason for this is the elegant simplicity this gives in algebraic physical equations and the rationale of defining units based on properties of nature and not human constructs. For example, the unit of electric charge could equal the elementary charge e. The unit of speed could be the speed of ligth c and so on.

By normalizing the units in this way equations become easier, think of Einstein's famous equality of energy and mass E = mc2. Exchange c for the unit 1 and it becomes even more apparent that energy and mass is the same thing E = m.

The choices for normalization can be done in different ways depending on what types of problem the system of units is intended to handle. Two examples of natural unit system used in the field of electromagnetism are the Lorentz-Heaviside units and Gaussian units.

A common choice is to define the natural constants kB, the Boltzmann constant, ħ, the reduced Planck constant and c, the speed of light as one.

kB = ħ = c = 1

## Planck units

The most commonly used system of natural units are the Planck units proposed by the German physisict Max Planck in 1899. He takes the concept of natural units one step further and defines the base units based on physical constants but also avoids arbitrary choices of particle masses and charges in definitions. All definitions are grounded in properties of free space.

The physical constants normalized to 1 in Planck units are

Physical constant Symbol Value in SI system
Gravitational constant G 6.67384·10-11 m3/kg s2
Speed of light c 2.99792458·108 m/s
Coulomb constant 1/4πε0 8.9875517873681764·109 kg m3/s2 C2
Reduced Planck constant ħ = h/ 1.054571726·10-34 J s
Boltzmann constant kB 1.3806488·10-23 J/K

This means that the Planck units are defined by normalizing these constants equaling 1. These constants are taken from the laws of physics as we understand them today, c and G are part of the structure of spacetime in general relativity. ħ is the key to the relationship between energy and frequency in quantum mechanics.

kB = ħ = c = G = 1

If these constants are normalized to one, we can solve our physics equations for the quantities of length, mass, time, temperature and electric charge using these constants. In this way we can come up with numbers for what a unit of Planck length is in terms of meters, what a Planck mass is in terms of kilograms and so on. The units are often denoted with a subscript p to show that they are Planck units.

Planck unit Derivation Value in SI system
Planck length lp = √(ħG/c3) 1.616·10-35 m
Planck mass mp = √(ħc/G) 2.176·10-8 kg
Planck time tp = √(ħG/c5) 5.3912·10-44 s
Planck temperature Tp = √(ħc5/GkB2) 1.417·1032 K
Electric charge (Lorentz-Heaviside) qp = e/√(4πα) 5.291·10-19 C
Electric charge (Gaussian) qp = e/√α 1.876·10-18 C

Planck's units are sometimes called God's unit since they are not anthropocentric in their definitions and are likely the best candidate for a system of units to use when discussing with an extra terrestrial engineer.