Comparing Planck's Mass, Length, Time and Energy

[George Keeling]

TL;DR Summary

Planck's mass, length, time and energy. I wanted to compare them with actual things. Some don't seem extreme.

Sean Carroll gives the Planck's set of four dimensioned quantities: Planck's mass, length, time and energy. I wanted to compare them with actual things.\begin{align}

m_p=\sqrt{\frac{\hbar c}{G}}&=2.18\times{10}^{-8}\rm{kg}&\rm{{10}^{7}\ E. coli}\phantom {100000000000000000000}&\phantom {10000}(1)\nonumber\\

l_p=\sqrt{\frac{\hbar G}{c^3}}&=1.63\times{10}^{-35}\rm{m}&\rm{Radius\ of\ proton\ ={10}^{-15}\ m}\phantom {10000}&\phantom {10000}(2)\nonumber\\

t_p=\sqrt{\frac{\hbar G}{c^5}}&=5.39\times{10}^{-44}\rm{s}&\rm{Cosmic\ inflation\ ends\ at\ {10}^{-32}s}\phantom {10000}&\phantom {10000}(3)\nonumber\\

E_p=\sqrt{\frac{\hbar c^5}{G}}&=1.95\times{10}^9\rm{J}&\rm{Sun\ emits{\ 10}^{26}\ Js^{-1}. \text{ A-bomb}\rm={10}^{12}}\ J&\phantom {10000}(4)\nonumber\\

&=1.22\times{10}^{19}\rm{GeV}&

&\phantom {10000}\nonumber

\end{align}He then says "Most likely, quantum gravity does not become important until we consider particle masses greater than $m_p$, or times shorter than $t_p$, or lengths smaller than $l_p$, or energies greater than $E_p$; at lower scales classical GR should suffice. Since these are all far removed from observable phenomena, constructing a consistent theory of quantum gravity is more an issue of principle than of practice."

Whilst it is unimaginable that we will see things shorter than $t_p$ or smaller than $l_p$, particle masses ('point masses') greater than $m_p$ are commonplace in GR and energies greater than $E_p$ are happening all the time. Can anybody help me make sense of the $m_p,E_p$ parts? And why are those 'greater than' and the others 'less than'?

 

[mitochan]

From your post I calculate "Planck momentum" of M_p c= 6.54 kg m/s which amounts a pitched baseball with 100 miles/hour speed. We can see it in major league game.

 

[Ibix]

He means energies involved in a single particle-particle collision. The LHC can accelerate protons to about a millionth of the Planck energy (if memory serves) (edit:) of a Joule, so we're going to need a bigger accelerator.

 

[George Keeling]

From your post I calculate "Planck momentum" of M_p c= 6.54 kg m/s which amounts a pitched baseball with 100 miles/hour speed. We can see it in major league game.

Much better than collecting 10 million bacteria!

 

[George Keeling]

The LHC can accelerate protons to about a millionth of the Planck energy (if memory serves), so we're going to need a bigger accelerator

https://en.wikipedia.org/wiki/Large_Hadron_Collider say "After upgrades it reached 6.5 TeV per beam 13 TeV (= $1.3\times 10^4$ GeV) total collision energy," so that's $10^{15}$ times bigger!

 

[Ibix]

Correct - my memory was faulty. It's one millionth of a Joule they can reach (source). So we're going to need an even bigger accelerator.