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

Question

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.

Answer 1

You are correct in that the core of your question is indeed nontrivial, but you need to be a bit more subtle than that. The fundamental constants don't "take" values - they are what they are, and that's that. It is valid to ask "how did this 3×10⁸ number come about", but the answer lies less with the speed of light than with the units we use to measure it. The real question, then, is

why did we choose units of length and time such that the speed of light has a value of ~3×10⁸ in those units?

Now, the gritty details of why we chose, say, the meter over the yard, are not really that important, and the reasons for those choices are historical - but mostly, we just had to choose something. As you've realized, these gritty details don't really matter; what really matters is the rough ballpark size of those units.

So, why did we choose the meter as the fundamental unit? Quite simply, because we ourselves are roughly meter-sized. This also came to mean that the world we built around us was also roughly meter-sized, so most of the things we measure in everyday life are roughly meter-sized.

On the other hand, we are made of atoms, and the size of atoms is a (fairly) fundamental constant of physics (particularly the Bohr radius $a_0$, which is a function of $ħ,\epsilon_0$ and $m_e$, and which is a good "standard atomic radius"). Let me pose, then, a similar question to yours,

why is the 'standard' size of atoms, $a_0$, roughly 5×10^-11 m?

and rephrase it as before:

why did we choose units of length which are roughly 2×10¹⁰ times the 'standard' atom?

Since our units were chosen because we ourselves are about that size, the real question is therefore

why are humans roughly 2×10¹⁰ 'standard' atoms tall?

The reason I call this the 'real' question is that this is actually (at least in principle) an answerable question. However, it is not a physics question! It is a question about biology and about evolution, and it's not clear what the balance between those two is.

There is some sort of 'allowed' range of lengthscales in which intelligent life is possible (it is hard to imagine intelligence at the level of tens of atoms, and also at the star-sized lengths of 10¹⁹ atoms), but just how broad this range is is a hard question in biology. There is also the historical, evolutionary question of how we came to occupy our particular spot on that range, and that is also a hard question. This is also made much more complicated by the anthropic-principle fact that we only know one instance of intelligent species.

On balance, then, we don't really know the answer to your question. But at least it is possible to ask the right question.

Answer 2

Every dimensionful constant is a comparison to some standard. The oldest standards were based on the sizes of a typical person, because most full-grown people are more or less the same size and you always have your body with you. That's where we got the cubit and the foot. You still measure the height of a horse in hands. There are lots of other units which were historically convenient: for example, an acre is (I believe) the amount of land that a horse can plow in a day. Obviously if you want to be precise, this definition depends on whether you plow on a short winter day or a long summer day, whether your plow is dull or sharp, whether your field is soft or rocky, whether your horse is old or young. But if you have a forty acre plot of land, you know that you'll need to hire a second plow to get it all plowed and planted in under a month.

As engineering got better, it started to matter that not everyone was the same size, and better standards started to appear. This happened haphazardly pretty early — there is a passage in the Old Testament about a surveyor using a measuring rod to tell the size of something or other, so even in the cubit days people knew better than to crawl around putting elbow to fingertip — but the actual sizes of the standards probably still wandered over the years. The meter was an effort to move away from these "variable" standards by establishing a reference to which all people on earth have equal access: the meter was originally one ten-millionth of the distance from the earth's north pole to its equator. But this is an overly optimistic definition, too: that path length depends, slightly, on the longitude, and would be very challenging to actually survey for practical reasons. It's a better standard than the length of the king's arm, but not perfect.

The second has the same sort of history: it was originally intended as a convenient fraction of a 24 hour day. But it's not a convenient fraction of the interval between sunrises, since the sunrise comes earlier and later through the year; nor is it a convenient fraction of the interval between starrises (the "sideral day," shorter than the mean solar day by about four minutes), nor moonrises, nor any other obvious brief interval. It takes a year of averaging before you can properly talk about a "mean solar day." Then you run into the fact that there are not an integer number of days in a year, requiring an extra day in leap years, nor are there an integer number of days in four years, requiring leap years to behave differently every four centuries. A calendar error of one day in four centuries is comparable to a clock error of one second per day. It takes care to define the second relative to a "natural" reference interval. And once you have defined the second relative to the length of a day, you discover that the length of the day is not really constant. Big earthquakes change the earth's moment of inertia; the 2011 earthquake in Japan lengthened the day by a few milliseconds.

If we lived on a different planet with a different circumference and a different day, these same sensible but arbitrary choices for natural reference lengths and times would have given us a different numerical value for the speed of light, and for the other constants.

Answer 3

I can always choose units where $c = 1$. This is almost always done in context of General Relativity, Special Relativity and Quantum Field Theory. As ACuriousMind said in the comments above, how we measure quantities tells us nothing useful about the physics of it, since units can be rescaled.

As of why the speed of light is in the order of $10^8$? In my eyes, it is simply because compared to ordinary speeds where humans are used to, the speed of light is simply almost $10^8$ greater!

Edit: You have asked in the comments why human speeds are so small compared to $c$. Well recall that the speed of light is the speed a massless particle travels through the universe. For massive particles you need energy to accelerate them close to c. Humans are a collection of massive particles and hence you require a lot of energy to accelerate them at those high speeds. That energy is way too high than what we extract out of food/fuel etc.