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
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