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Gravitational Force and the Planck Scale
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From Newton, FG = Gm2/r2
Frpm Special Relativity, m = E/c2
From Quantum Mechanics,E = pc
Therefore,
m = p/c
From the Uncertainty Principle, ΔxΔp = h => p = h/r
Therefore,
m = h/rc
The gravitational force becomes:
FG = Gh2/r4c2
The Coulomb force between 2 identical charges, F = ke2/r2
We can write this in terms of the dimensionless Fine
structure constant, α = e2/hc as follows:
FC = hcα/r2
Equating FG = Equating FC gives:
r = √(Gh/c3α)
= √(Gh/c3)√(1/α)
~ 10lP
The gravitational force is very small compared to the other
forces. However, at distances approaching the Planck length
the gravitational and Coulomb forces become comparable.
At these scales, there is a 'new physics' where quantum
gravity becomes important (indicated by the Planck mass
(energy) ~ 1.22 x 1019 GeV).