Automated Natural Unit Conversions in Mathematica

Introduction and Review

Natural units are a way to simplify repetitive fundamental constants in theoretical physics equations. In my research, I used three particular unit systems: geometric ($c=G=1$), natural ($c=\hbar=1$), and Planck units ($c=\hbar=G=1$). However, for me it is time consuming to convert to and from natural units. I decided that I would write an extension to Mathematica’s units package to remedy this.

A great summary of natural units goes in more detail in an excellent document here. Given a SI quantity $Q_{\text{SI}}$, the quantity in natural units is given by $Q_{\text{geometric}} = Q_{\text{SI}}/A$ for some factor $A$. For the rest of this post, I will use geometric units. Suppose that the quantity has SI units of the form

$[Q_{\text{SI}}] = M^\alpha \times L^\beta \times T^\gamma \, ,$

then, one can find $Q_{\text{geometric}}$ with the following factor

$A = G^{-\alpha} c^{2\alpha-\gamma} \, .$

So all we need to do to convert between the units is to find $A$.

Mathmatica Implementation

The first thing that we need to do is write a helper function that will return the $\alpha$, $\beta$, and $\gamma$ from our previous equation. All I need to do was write a wrapper class for the UnitDimensions[] fuction that Mathematica has.

Next, I wrote a function to calculate the constant $A$.

Finally, I used a function that will find $Q_{\text{geometric}}$ by dividing $Q_{\text{SI}}$ by $A$. Afterwards, it converts the result into meters to the $n$ power as is used in geometric units.

The full version of these functions which includes geometric, natural, and Planck units is publicly available on my Github. Feel free to use it and I hope that it will save you time in your research!

Automated Natural Unit Conversions in Mathematica

Introduction and Review

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