Natural Units

I am building up on this video by Kieran Borovac. I will provide forward (from SI, labeled $\boldsymbol{F}$ ) and backward (to SI, labeled $\boldsymbol{B}$ ) matrices for each system, where the rows represent SI values and columns the corresponding system values. The rows in the backward matrices represent SI units: $s, m, kg, A, K, cd, mol$ , in forward matrices, these are represented by columns. The units are represented by column vectors and are multiplied with the matrix for conversion.

Special Relativity Units

Units: $c, m, kg, A, K, cd, mol$ .

\begin{align*} \boldsymbol{F} &= \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1\end{bmatrix} & \boldsymbol{B} &= \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1\end{bmatrix} \end{align*}

Geometrized Units

Units: $c, G, kg, A, K, cd, mol$ .

\begin{align*} \boldsymbol{F} &= \begin{bmatrix}-3 & -2 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1\end{bmatrix} & \boldsymbol{B} &= \begin{bmatrix}-1 & -2 & 0 & 0 & 0 & 0 & 0\\ 1 & 3 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1\end{bmatrix} \end{align*}

Geometrized Units with Mass

Units: $c, G, M, A, K, cd, mol$ . The matrices are the same as geometrized units.

Unit Converter

To convert values, set the calculator to forward transformation (SI → [system]). To convert from the selected system to SI, multiply by the units and to convert from SI to the selected system divide by the units.