Planck units - introduction

In this post we are going to take a look at one of my current favourites in Physics, Planck units. These are one of those things that often don't get mentioned in physics documentaries, but should. These things are amazing and hopefully in this post I will explain why.

Firstly, as with all things like this, Wiki has a really good page on the subject. I would advise you to take a read of this its brilliant. To sum up Wiki, Planck units are based on 5 constants of nature;

For those who like the dramatic these are sometimes called "God's Units" because they are linked at a fundamental physical level, something that is beyond current understanding. They are listed here, this is taken straight from Wikipedia.

Table 1: Fundamental physical constants

Constant	Symbol	Dimension	Value in SI units with uncertainties[6]
Speed of light in vacuum	c	L T −1	2.99792458×108 m s−1 (exact by definition of meter)
Gravitational constant	G	L3 M−1 T −2	6.67384(80)×10−11 m3 kg−1 s−2[7]
Reduced Planck constant	ħ = h/2π where h is Planck constant	L2 M T −1	1.054571726(47)×10−34 J s[8]
Coulomb constant	(4πε0)−1 where ε0 is the permittivity of free space	L3 M T −2 Q−2	8.9875517873681764×109 kg m3 s−2 C−2 (exact by definitions of ampere and meter)
Boltzmann constant	kB	L2 M T −2 Θ−1	1.3806488(13)×10−23 J/K[9]

Key: L = length, M = mass, T = time, Q = electric charge, Θ = temperature.

Each of these is associated with a fundamental physical theory as follows;

the gravitational constant - general relativity and Newtonian gravity
the reduced Planck constant - quantum mechanics
the speed of light in a vacuum - electromagnetism and special relativity
the Coulomb constant - electrostatics
the Boltzmann constant - statistical mechanics and thermodynamics

The key above mentions Length, Mass, Time, Charge & Temperature, below are the Planck equivalents. These can be derived a number of different ways, in a previous post the idea of units was introduced. The post explored how you can look at units to get a better understanding of the underlying values. Instead of using Kg, m, s it is possible to use arbitrary units, dimensions, M, L, T (mass, length, time). This idea can be used to derive the Planck units.

Consider Planck Length, the unit of length is the meter, m , the "dimension" listed above is Length. So how do we get L? One way is to take a look at the dimensions of c, G and ħ

c - L T ⁻¹
G - L³ M⁻¹ T ⁻²
ħ - L² M T ⁻¹

so multiple G and ħ . This is the same as adding the dimensions, so you have

$ G \hbar = L^3 M^{-1} T^{-2} + L^2 M^{+1} T^{-1} = L^5 T^{-3}$

Then take 

$ c^3 = L^1 T^{-1} + L^1 T^{-1} + L^1 T^{-1} = L^3 T^{-3}$

So dividing one by the other now gives

$ \frac{G \hbar}{c^3} = L^5 T^{-3} - L^3 T^{-3} = L^2  = l_p^2$

Finally giving the Planck length,

$l_p = \sqrt{\frac{\hbar G}{c^3}}$

We can do a similar thing to get Planck mass,

$ \frac{\hbar}{G} = L^2 M^{+1} T^{-1} - L^3 M^{-1} T^{-2} = L^{-1} M^2 T^1$

Then multiply by c to give 

$\frac{\hbar c}{G} = L^{-1} M^2 T^1 + L^1 T^{-1} = M^2 = m_p^2$

So Planck mass is given by

$m_p = \sqrt{\frac{\hbar c}{G}}$

This can then be repeated to find values for time, charge, temperature, area, volume, energy, force, pressure, momentum, density, current and so on. To see a more detailed list take a look at the Wiki page.

This is not the only approach to deriving the Planck units, but for me it makes it clear that we are deriving a unit, even if the final equation looks very odd.

So, OK, this is interesting, sort off, but what is the big deal? Well, the big deal is that we still don't really know what these numbers represent, if they represent anything at all. The Planck length is mind numbingly small

l_p = 1.616199 x 10^-35 m

They do raise some interesting questions though. Is space quantized in the same way as energy? is the smallest length possible actually the Planck length? Why is the Planck charge not the same as the charge on an electron? Why is the Planck mass so large compared to the mass of an electron? Who know's, probably not that many at the moment.

What is amazing is that there are a number of things you can derive when starting out with a Planck unit. In several of my coming posts I will show some of the results I have obtained from an analysis of Planck units.