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

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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, out 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 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
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: – dmckee Jul 9 '15 at 14:59

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