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I have been curious to find what could be the significance of Planck force? It is calculated by the formula $c^4/G = 1.21031359\times 10^{44} \, \mathrm{N}$, where $c$ is the speed of light and $G$ is the gravitational constant.

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There is no particular physical significance; it's just a unit. Of course, in any system where such a large force is exerted, our current theories should not be accurate, and a quantum theory of gravity or some as-yet-unknown theory would be needed to accurately describe its behavior.

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I wish people made it clearer there is no physical significance to any Planck unit. – OrangeDog Jun 1 at 8:44

When describing the formation of a black hole (or the merger of two equally-sized black holes) in a Newtonian gravity framework, Planck-scale gravitational forces of the order $c^4/G$ enter into the description. This is independent of the mass of the black hole. Such can easily be seen by modeling an infalling spherical shell of dust with mass $M$ under the influence of a Newtonian gravitational force.

Of course, we know that black hole formation and black hole mergers require a general relativistic description, and studying these phenomena in a Newtonian framework means stretching Newtonian gravity beyond its range of applicability. So treat the statement "Planck-scale forces are the forces that occur in the formation of black holes" as nothing more than a rough intuitive scaling argument.

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Planck's constant is a force associated with each cycle of a photon. For instance a photon with the 500 nm wavelength actually oscillates at a frequency more than 600 trillion times per second. Each time it oscillates, Planck's Constant is applied.

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What do you mean when you say "Planck's constant is applied?" How do you "apply" a constant? – Asher Apr 7 at 21:33

I think, the physical signifcance of the Planck force comes fromthe formula: looks like it's the force needed to accelerate the Planck mass to the speed of light in the Planck time right?

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That is the definition of plank force, not the significance thereof. – gonenc Jul 9 '15 at 11:36
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There does not exist a force capable of accelerating the Planck mass to the speed of light in any finite amount of time – Jim Jul 9 '15 at 13:14
    
You might be interested in the help center page on merging multiple accounts: physics.stackexchange.com/help/merging-accounts. – dmckee Jul 9 '15 at 14:59

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