[qnmxrrox] Stoney units

Let the matrix A=

2  -1  -3   4
0  -1   3  -2
1   0   0   1
0   0   1  -1

The columns correspond to ampere, kilogram, meter, and second respectively. The rows correspond to the exponents of the units of the Coulomb constant 1/(4πε₀), gravitational constant G, fundamental charge e, and speed of light c. Then 2*inv(A)=

-1  -1   0   6
 1  -1   2   0
 1   1   2  -4
 1   1   2  -6

from which can read off Stoney units expressed in SI units:

You have: sqrt(coulombconst^-1 G^-1 e^0 c^6)
You want: 
 Definition: 3.4788728e+25 A
You have: sqrt(coulombconst^1 G^-1 e^2 c^0)
You want: 
 Definition: 1.8592089e-09 kg
You have: sqrt(coulombconst^1 G^1 e^2 c^-4)
You want: 
 Definition: 1.3806783e-36 m
You have: sqrt(coulombconst^1 G^1 e^2 c^-6)
You want: 
 Definition: 4.6054472e-45 s

We can also get conversion to planck units. e/sqrt(alpha)/plancktime is planck current, not built into the "units" program

You have: sqrt(coulombconst^-1 G^-1 e^0 c^6)
You want: e/sqrt(alpha)/plancktime
 * 1
 / 1
You have: sqrt(coulombconst^1 G^-1 e^2 c^0)
You want: planckmass
 * 0.085424543
 / 11.706238
You have: sqrt(coulombconst^1 G^1 e^2 c^-4)
You want: plancklength
 * 0.085424543
 / 11.706238
You have: sqrt(coulombconst^1 G^1 e^2 c^-6)
You want: plancktime
 * 0.085424543
 / 11.706238

The recurring constant 0.0854 is sqrt(alpha), where alpha is the fine structure constant.

I've patched my units.dat to get planckmass more correct:

--- /usr/share/misc/units.dat 2009-12-12 19:38:32.000000000 -0500
+++ units.dat 2012-08-27 03:25:37.458785728 -0400
@@ -902,7 +902,7 @@
                                          #   binding energy of the deuteron
 # Planck constants
 
-planckmass              2.1767e-8 kg     # sqrt(hbar c / G)
+planckmass              sqrt(hbar c / G)
 m_P                     planckmass
 plancktime              hbar / planckmass c^2
 t_P                     plancktime

because otherwise you get this:

You have: sqrt(hbar c G^-1)
You want: planckmass
 * 0.99987785
 / 1.0001222