Planck force

Planck force is the force associated with any mass self-gravitating around its own gravitational radius:

$F_{P}={\frac {Gm^{2}}{r_{G}^{2}}}={\frac {c^{4}}{G}}=1.244\times 10^{44}{\mbox{N}}$

where G is the Gravitational constant, c is the speed of light, m is any mass and r_G is the gravitational radius of the given mass:

$r_{G}={\frac {Gm}{c^{2}}}$

Since the dimensions of force, Newtons, are equivalent to Joules/metres, the Planck force can also be understood as the distribution of energy over a gravitational radius :

$F_{P}={\frac {mc^{2}}{\frac {Gm}{c^{2}}}}={\frac {c^{4}}{G}}$

The force is named 'Planck force' because it has a unique association with the Planck mass or, more specifically, Planck energy and Planck length. In this case, force is expressed as a distribution of energy over the Compton wavelength of that energy :

$F={\frac {mc^{2}}{\frac {\hbar }{mc}}}={\frac {m^{2}c^{3}}{\hbar }}$

Here the force is different for every mass - it is Planck force only for the Planck mass (approximately $2.18\times 10^{-8}$ Kilograms). This follows from the fact that the Compton wavelength of the Planck mass is also the gravitational radius of the Planck mass:

${\frac {\hbar }{m_{P}c}}={\frac {Gm_{P}}{c^{2}}}$

which in turn follows from another relation of fundamental significance:

$c\hbar =Gm_{P}^{2}$

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