How Are Planck Time and the Expression \(\sqrt{\frac{\hbar c^5}{G}}\) Related?

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HelpNowPlease
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take this derivation

[tex]\hbar c = GM^2[/tex]

multiply [tex]c^4[/tex] on both sides

[tex]\hbar c^5 = GM^2c^4[/tex]

rearrange

[tex]\frac{\hbar c^5}{G} = M^2c^4[/tex]

Take the square root makes

[tex]\sqrt{\frac{\hbar c^5}{G}} = E[/tex]

Now this [tex]\sqrt{\frac{\hbar c^5}{G}}[/tex] expression inside the radical expression is fascinatingly similar to the Planck time [tex]\sqrt{\frac{\hbar G}{c^5}}[/tex]. I began to wonder if there was any physical significance between the two? Is there any?

[tex][/tex]
 
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What you wrote down is call the Planck Energy. It is a quantity with dimensions of energy that you can get, as you saw, by simply rearranging the universal constants. The Planck time, length, mass, etc are scales where people expect that you can't talk about gravity and quantum mechanics separately.
 
So it is... I never thought to check the plank energy. I found this independantly lol
 
I have a new question, why does c^5 show up in the calculations, is there a significant reason for this, other than balancing the dimensions?