Tuesday 27 December 2016

dimensional analysis - Why are the anthropometric units (which are about as big as we are) as large as they are relative to their corresponding Planck units?



This might have some duplicated inquiry that this question or this question had, and while I think I have some of my own opinions about it, I would like to ask the community here for more opinions.


So referring to Duff or Tong, one might still beg the question: Why is the speed of light 299792458 m/s? Don't just say "because it's defined that way by definition of the metre." Before it was defined, it was measured against the then-current definition of the metre. Why is $c$ in the ballpark of $10^8$ m/s and not in the order of $10^4$ or $10^{12}$ m/s?


A similar questions can be asked of $G$ and $\hbar$ and $\epsilon_0$.


To clarify a little regarding $c$: I recognize that the reason that $c\approx10^9$ m/s is that a meter is, by no accident of history, about as big as we are and a second represents a measure of how fast we think (i.e. we don't notice the flashes of black between frames of a movie and we can get pretty bored in a minute).


So light appears pretty fast to us because it moves about $10^9$ lengths about as big as us in the time it takes to think a thought.


So the reason that $c\approx10^9$ m/s is that there are about $10^{35}$ Planck lengths across a being like us ($10^{25}$ Planck lengths across an atom $10^5$ atoms across a biological cell and $10^5$ biological cells across a being like us). Why? And there are about $10^{44}$ Planck times in the time it takes us to think something. Why?


Answer those two questions, and I think we have an answer for why $c\approx 10^9$ in anthropometric units.


The other two questions referred do not address this question. Luboš Motl gets closest to the issue (regarding $c$), but he does not answer it. I think in the previous EDIT and in the comments, I made it (the question) pretty clear. I was not asking so much about the exact values which can be attributed to historical accident. But there's a reason that $c \approx 10^9$ m/s, not $10^4$ or $10^{12}$.


Reworded, I suppose the question could be "Why are the anthropometric units (which are about as big as we are) as large as they are relative to their corresponding Planck units?" (which is asking a question about dimensionless values). If we answer those questions, we have an answer for not just why $c$ is what it is, but also why $\hbar$ or $G$ are what they are.





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