# Natural units

By Marcus Wilson 01/08/2011

Physics is all about describing physical quantities. Whether it’s length, velocity, force, electric current or heat flux, it takes physics to describe what it is and what it does. Central to this is our system of units. The three really common base units (in the S.I. system) are the metre (unit of length), the second (unit of time) and kilogram (unit of mass). With these three, we can construct a whole range of units for other quantities – e.g. the newton for force, pascal for pressure, joule for energy and so on.

The ‘metre’, ‘second’, and ‘kilogram’, have a long history behind them. And so they are not necessarily the ‘easiest’ or ‘best’ units for describing certain processes. When we get into particle physics, for example, and describe things of tiny mass (e.g. an electron is about ten to the power of minus 30 kilograms) we end up using rather large multipliers on our units. That makes them a bit awkward.

However, we are not confined to using a ‘metre’, ‘second’ and ‘kilogram’. Any length, time and mass unit could be chosen. One choice that’s popular with particle physicists is to make as many fundamental constants as possible equal to 1 something – e.g. choose a natural length unit and a natural time unit such that the speed of light is equal to 1 natural length per natural time.

With three units to play with, we can choose three things to ‘set to 1’. A common choice is the speed of light, Planck’s constant divided by 2 pi (called h-bar), and the rest mass of the proton. Particles people will claim this makes things easier, because you don’t have to bother writing down the symbols ‘c’, ‘h-bar’, and ‘m_0’ for the speed of light, h-bar and the mass of the proton respectively whenever you have an equation.

However, I’m not so sure it simplifies things – at least not in understanding what’s going on. When a constant ‘disappears’ from an expression, it’s hard to see just what that expression entails. For example, the rest mass energy of an electron in natural units becomes m_e;   (m_e being mass of an electron); it comes from the famous E equals m c squared (mass times speed of light squared), but since the speed of light in these units is one, and the mass we have is the mass of the electron, then it’s just E equals  m_e.

Now, do you agree that this is a touch confusing?  If we have E = m_e, it looks like we have an energy on the left hand side of the equals sign, and a mass on the right hand side. That can’t work in physics. An energy equals an energy; a mass equals a mass.  It’s because our particle physicists take a few liberties with the units when they say c = 1, h-bar = 1, m_0 = 1. (Or worse still, c = h-bar = m_0 = 1).  And I see plenty of textbooks that are written like this. Writing this is WRONG; what should be written is c = 1 natural length unit per natural time unit; h-bar = 1 natural mass unit natural length unit squared per natural time unit, and m_0 = 1 natural mass unit.  The units are not dispensable. Take them out and you start losing the physics. Saying c = h-bar = m_0 = 1 is claiming that the speed of light equals the mass of the proton. The two are utterly different entities.

Science papers using natural units therefore bug me somewhat, and students won’t catch me using them in lectures.