Fun with dimensional analysis 5 – Planck units

In is often the case in equations of physics that a constant will be included to make the units work out, and to match the variable quantities with the results the universe actual gives. For example, Newton’s law of gravitation is

\displaystyle \vec{F} = G \frac{ m_1 m_2 }{r^2} \hat{r}

The G out front has some value in newtons, meters, kilograms, and seconds that is not strictly important for this discussion. What is important is why it is has that particular value? Obviously because of the way we humans chose the values for newtons, meters, etc. If we had chosen them some other way, it would have a different value in those units. The question naturally arises: What system of units could we choose such that \displaystyle G = 1? This would allow us to be lazy and omit the constant from our equations.

There are infinitely many possible such systems of units of course, but if we also set some other constants equal to zero we’d have something interesting. There are many parameters in the physical universe that we could normalize; for example, the mass of the electron. Why the electron and not say the muon? The choice is totally arbitrary, but there are certain very interesting, dare I say even cosmic, parameters: those of free space. The constants to be normalized would be:

  • The speed of light (from special relativity), c;
  • The gravitational constant (from general relativity), G;
  • Planck’s reduced constant (from quantum mechanics), ħ;
  • The vacuum permittivity (from electrodynamics), ε0;
  • Boltzmann’s constant (from statistical mechanics), kB;

Combined, these contain all the dimensions of length, time, mass, electric charge, and temperature. Specifically:

\displaystyle \lbrack c \rbrack = LT^{-1} \\ \lbrack G \rbrack = L^3M^{-1}T^{-2} \\ \lbrack \hbar \rbrack = L^2MT^{-1} \\ \lbrack \varepsilon _0 \rbrack = L^{-3}M^{-1}T^2 Q^2 \\ \lbrack k_B \rbrack = L^2MT^{-2} \Theta ^{-1}

We should like to see how we can recapture the values for length, time, etc. It should also be noted that such units are called Planck units and often have physical significance, for example the Planck length is the smallest length that makes sense to talk about with our present physical understanding (since it is the distance that light, the fastest speed possible, travels in one Planck time, the shortest interval of time allowed). This isn’t always the case, the Planck values being the greatest or smallest physically sensible values, though. For example, 1 Planck impedance is about 30 ohms.

Now, let’s figure out what a Planck length is. The electric constant has a charge dimensions and Boltzmann’s constant has temperature, so they’ll contribute nothing. We have:

\displaystyle \lbrack \ell _P \rbrack = L = \lbrack c^{ \alpha } G^{ \beta } \hbar ^{ \gamma }\rbrack = L^\alpha T^{ -\alpha }L^{3 \beta }M^{- \beta }T^{-2 \beta} L^{2 \gamma} M^\gamma T^{-\gamma} \Rightarrow \\ 1 = \alpha + 3\beta + 2\gamma \\ 0 = 0 -3\beta + \gamma \\ 0 = \alpha +2\beta +\gamma

Solving this system of equations is left as an exercise to the reader. When done, you should (hopefully) arrive at the solutions

\displaystyle \alpha = -3/2 \\ \beta = 1/2 \\ \gamma = 1/2 \\ \Rightarrow \ell _P = \sqrt{ \frac{ G \hbar }{c^3} }

Similar exercises can be done for the other units like Planck time, Planck mass, etc. I think it would make for some decent dimensional analysis practice. A full table of results can be found on Wikipedia.

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