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Uh.... I think your units are off to say the least.
I'm assuming e is the electron charge. The constants hbar, c and G don't have units of charge to cancel that out.
Yes, e stands for the electron charge. However, hbar times c divided by e squared is the fine structure constant to the minus 1. (hbar * c/e^2)= approx. 137 (hbar * c/e^2)raised to the third power equals: 2.57338006 E6 [(hbar * c/e^2)]^3 * [(hbar * c/G^3)]^3 = (pi^3 * 10^20) moving (pi^3 * 10^20) to the left denominator and G^9 to the right numerator, we have: [(hbar * c/e^2)]^3 times [(hbar * c * pi-1)^3]/10^20] = G^9 [2.57338006 E+6] X [1.019159191E-71] = 2.6226839407 E-65 taking 2.6226839407E-65 to the ninth root we get: 6.6727538776 E-8 in cgs unitsBecause (hbar * c/e^2) is dimensionless and,because (hbar * c/G^3) is dimensionless,this equation works just as well when done in SI units.6.6727538776 E-8 in cgs units and 6.6727538776 E-11 in SI units agrees very closely with NIST figures.
Remembering that (hbar * c/e^2) is dimensionless, the first part of the equation will remain the same whether figured in cgs or SI units.
Now plug in the values in the second part of the equation with SI values and you'll find the answer for G will also be registered in SI values. This should prove that (hbar * c/G^3) is also dimensionless.
Quote from: Ethos on 11/03/2009 08:20:49Remembering that (hbar * c/e^2) is dimensionless, the first part of the equation will remain the same whether figured in cgs or SI units.No, hc/e2 is dimensionless in c.g.s. only. In M.K.S. the dimensionless expression is 4πε0hc/e2.QuoteNow plug in the values in the second part of the equation with SI values and you'll find the answer for G will also be registered in SI values. This should prove that (hbar * c/G^3) is also dimensionless. No, that expression is not dimensionless neither in c.g.s. nor in M.K.S.
Dimensionless is a term equal to a ratio, and when changing from cgs to SI, if the ratio does not change
Quote from: Ethos on 11/03/2009 14:26:10Dimensionless is a term equal to a ratio, and when changing from cgs to SI, if the ratio does not changeWho told you that it doesn't change? hc/e2 ≈ 137 in c.g.s. while it's ≈ 1.24*1012 J*m*C-2 = 1.24*1012 kg*m3*A-2*s-4 in M.K.S..
Very interesting Ethos; I'm assuming your maths correct; what is the implication? Have you shown a relationship that was not previously known? Are you showing a new relationship between gravity and the electromagnetic field?
Now for those that don't like me using (hbar c/e^2) in this equation, let's use this equation instead: (hbar c/(re me c^2))