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

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Discussion Overview

The discussion revolves around the relationship between Planck time and the expression \(\sqrt{\frac{\hbar c^5}{G}}\). Participants explore the derivation of this expression and its potential physical significance, particularly in the context of Planck units and their implications for gravity and quantum mechanics.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant derives the expression \(\sqrt{\frac{\hbar c^5}{G}}\) and notes its similarity to Planck time \(\sqrt{\frac{\hbar G}{c^5}}\), questioning any physical significance between the two.
  • Another participant identifies \(\sqrt{\frac{\hbar c^5}{G}}\) as the Planck Energy, describing it as a quantity derived from universal constants and indicating its relevance in contexts where gravity and quantum mechanics cannot be separated.
  • A different participant expresses surprise at discovering the Planck Energy independently and acknowledges the connection to the initial derivation.
  • One participant raises a question regarding the appearance of \(c^5\) in the calculations, seeking to understand if there is a deeper significance beyond dimensional balancing.

Areas of Agreement / Disagreement

Participants generally agree on the identification of the expression as the Planck Energy and its relevance to the intersection of gravity and quantum mechanics. However, the discussion remains unresolved regarding the deeper significance of the \(c^5\) term and its implications.

Contextual Notes

The discussion does not resolve the assumptions or implications surrounding the relationship between the derived expression and Planck time, nor does it clarify the significance of the \(c^5\) term in the context of the calculations.

HelpNowPlease
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take this derivation

\hbar c = GM^2

multiply c^4 on both sides

\hbar c^5 = GM^2c^4

rearrange

\frac{\hbar c^5}{G} = M^2c^4

Take the square root makes

\sqrt{\frac{\hbar c^5}{G}} = E

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

 
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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?
 

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