Why is the size of a cell the geometric mean of the size of the observable universe and the Planck length?

Question

The size of the observable universe today measured in elementary units of length (Plank length) is roughly "twice" (order of magnitude) the size of the elementary unit of life, concretely, the typical size of an eukaryote cell (those who have a nucleous).

Observable Universe (today)  \(\sim 10^{27}\, \mbox{m} \sim 10^{62}\, l_P\)

Typical size of a (eukaryote) cell \(\sim 10^{-4}\, \mbox{m}\sim 10^{31}\, l_P \sim \sqrt{\mbox{"longest" distance} \times \mbox{"shortest" distance}}\)

Why is the length of the unit of life similar to the geometric mean of the "shortest" and "longest" distances one can talk about? Any explanation (perhaps an anthropic one)? 

Comments

@ArnoldNeumaier done ...

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@Dilaton @Arnold Neumaier. So, you both are going to move every anthropic-like question to chat, aren't you? I agree that this might very well be a numerical coincidence (as many other anthropic-based questions/arguments), but is this enough to move it to chat? Can you prove that it cannot be answered using science? 

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As a question about a cell it would be a question of biology, not of physics. 

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@drake: http://www.aip.org/publishing/pacs/pacs-reg80#87 shows the range of topics sensibly regarded as part of biophysics (and hence possibly interesting for PO). You topic would fit there nowhere; its certainly not graduate level biophysics.

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@drake: Indeed, all these reasons apply.

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@ArnorldNeumaier That's not right at all. A question about a cell, blood, photosynthesis, etc is a physics question if it asks about physical arguments. For example, the paper below linked by Void (http://www.nr.com/whp/manssize.pdf) tries to relate the size of a human to fundamental constants. Doesn't it belong to physics? I am highly surprised that you, Arnold, say that any question about a cell does belong to biology, but not to physics.

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@ArnoldNeumaier Thanks for the explanation. First reason was that the question "could not really be answered". As you were not able to argue why, your second reason was that it was a "question of biology, not of physics". Now, you change again your reason, saying that it is "certainly not graduate level biophysics". Can you please point an undergraduate-level reference where this question is answered/explained? Thank you.

Answer 1

Note that since physics can be completely geometrized by the $\hbar,c,G$ constants, there is no privileged unit left after doing so. The conventional geometrical unit $l_p \sim t_p \sim E_p...$ obtained by the requirement that the historical physical constants are combined without any numerical prefactor plays mathematically as special a role as picking a unit on the dimensionless real line. You can pick a set of geometrized units based on the combination of SI constants times $10^{35}$ without violating any of their geometric nature. Suddenly, the modified Planck length is $\tilde{l}_p \approx 1.7 m$ and the question falls apart.

Obviously, there is the natural assumption that the Planckian scale as obtained from this combination of historical constants will play an important role in quantum gravity because the conventional form of equations involving these constants do not involve large numerical prefactors.

However, there is an important preliminary assumption in saying that the Planckian scales represent correctly the orders of magnitude of quantum-gravitational interaction and this that the interaction will be geometrically simple. For instance, numerical factors can grow as $n!$ with geometrical dimension $n$, so for the string-motivated $n=10$ that might mean a $10^6$ shift of order in either direction. This is why one should not discuss precise orders of magnitude with respect to the Planckian scale unless the difference is really absurd (such as the cosmological constant $\sim 10^{-122}$ which is $\sim 1/82!$), or unless a specific theory of quantum gravity is at play in the discussion.

Comments

1) Even with such a factor the size of a cell is approximately at the middle.

2) You don't need to speculate about a theory of quantum gravity. You have quantum particles in (classical) Newtonian gravitational fields (see http://www.physicsoverflow.org/10842/schrodinger-gravitational-potential-deterministic-gravitation), which definen a scale of length, don't they?

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1) $10^{54}$ and $10^{25}$ well, that seems somehow contrived to me. In my opinion, the best way to ask the question is why the ratio of the radius of the universe is roughly $10^{31}$ the size of a cell, but that does not have the numerological "wow" feel to it. The answers will be iterative and partial and yes, finally a "Theory of everything" could provide the terminal answer. But we do not have this theory and the answer will very very surely be non-simple.

2) $c$ does not enter these equations hence there is no $l_p$ and full geometrization of units. In fact, the Newtonian (Coulomb) potential is in a certain sense scaleless and only relativity introduces scaleful gravity through quantities such as the Schwarzschild radius.

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First, I agree that there is no c in that case. However, the square of the Planck length (without huge numerical factors) comes out in: i) the first quantum correction to the effective Newtonian potential (factor of order 1). ii) the BH black hole entropy. iii) several thought experiments point out that the Planck length plays a special role in a theory of quantum gravity. It is a fact that the order of magnitude of a cell is roughly the mean of the order of magnitude of the Planck length and the order of magnitude of the size of the observable universe now. 

This may be a pure coincidence, indeed. But is it possible to give a (partial) explanation?

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A start would be this article arguing about the size of humans and references therein: http://www.nr.com/whp/manssize.pdf

From the cell/life size there is the main obstacle of not knowing the reason for the value of the fine structure constant and the masses of protons, neutrons and electrons. From the cosmological side the problem is in not being able to explain the whole set of roughly twenty parameters of the Standard model, the parameters of inflation and the initial conditions of the universe. As already stated, a theory of everything has the ambition to explain all this, but we do not have such a theory.

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@Void Thanks for the link. I upvoted your comment. Please, add that comment to your answer, as long as you consider it convenient, so that I upvote your answer too.

Answer 2

Interesting observation, but it's just a coincidence; the planck length isn't even a special length, it is only obtained through dimensional analysis. 

Clarification

What you are saying is that \(\frac{\ell_{\mathrm{cell}}^2}{\ell_P}=\ell_{\mathrm{universe}} \). This clearly gives some importance to the planck length, and therefore doesn't make sense. The planck length isn't the shortest distance. I'm sure you know this. Saying the size of the observable universe is somehow the "longest distance" is extreme hand-waving. 

Also, I don't understand any special role of the length of the universe, or the cell having to be eukaryotic (what's wrong with having a nucleoid?), or whatever. Cell sizes also vary a lot, with the human egg cell having a width of around 0.1mm, while the red blood cell being much smaller.

This really sounds no more sensible than astrology, but it is still an interesting observation of course. 

Comments

I don't know what you exactly mean by "no special". What I am saying is that in natural units ($l_P=1$) the (length of the) unit of life is the square root of (length of) the universe, which is a meaningful physical statement. 

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@drake I have updated my answer. By the way, I am not the downvoter. 

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The claim $l_{cell}^2=l_P\times l_{universe}$ is independent of the systems of units. Of course, it depends on the Planck length, regardless of the system of units one is using. According to your reasoning the statements $\rho_{obs}\sim 10^{-12} eV^4 \ll M_P^4$,         $S\gg \hbar$, or   $v\leq c$ lack of physical meaning, because they "give some importance" to the Planck mass, Planck constant, and speed of light, right?

The Planck length is supposed to be the threshold for length measurements or length-time intervals measurements (\(\Delta x \cdot \Delta t\geq l_P^2 \)). In this sense, $l_P$ gives the order of magnitude of the space(-time) resolution. 

Cell sizes indeed vary a lot. Sizes of cells with a nucleus, which are the elementary units of complex life, also vary, but not that much, around a couple of order of magnitudes. And regardless of the specific order of magnitude, this can be approximated as the geometric mean of the Planck length and the size of the observable universe today.

I agree that the question may sound as astrology, but not more than other questions one can read in the literature about the values of fundamental constants such as the fine structure constant or the cosmological constant (why now problem). One can usually find anthropic arguments (perhaps arguably, one can usually make up ad hoc arguments) for these values. I am asking about the existence of such arguments for the numerical coincidence of my question.

Answer 3

One can talk about arbitrarily small and arbitrarily large distances; in particular, wikipedia  talks (reasonably meaningfully) towards the end of the page about distances so large that even the geometric mean with the Planck length is astronomically large, very far from the size of a cell. 

Moreover, the size of the visible universe is highly variable in time, and the size of a cell is time-independent but depends strongly on which kind of cell you are looking at - ranging over more than 6 magnitudes, from 0.6-0.8$\mu$m for a red blood cell (according to the above source) to 3m for a nerve cell of giraffe's neck.

Comments

Your first paragraph: you know the question says "Planck length" and "observable universe today". How accurate/rigorous to call them shortest and largest, respectively, is secondary, although it is not meaningless at all.

"variable in time": In the question you can read: observable universe TODAY.

"depends strongly on": In the question you can read in regard to the order of magnitude: "roughly twice", "similar to", "typical size".

"which kind of cell": In the question you can read:  "TYPICAL size of an eukaryote cell". The sourceYOU LINKED says that "MOST eukaryote cells" are in the range 10-100 $\mu$m, thus it agrees (or is compatible) with the typical size of a cell written in the question (100 $\mu$m)

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@drake: You'd need to explain why the universe is TODAY in a special state where your relation holds true, and why MOST eukaryote cells are special among all cells, or all living objects, so that your special relations holds for them but not for others.

If there were a physical explanation it should hold without having to pick a special date and a special class of objects.

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1. If you are not going to allow any anthropic-principle question, then I do understand you don't accept my question. Then I'm okay with your decision. But you need to state somewhere that anthropic based questions are not permitted on this website. Some (many?) physicists think anthropic arguments are not physical, but as far as I know they are not forbidden in technical journals, university seminars, etc.  

2.  "need to explain why the universe is TODAY in a special state where your relation holds true". Same thing is to be said about the "cosmic coincidence problem" or "why now problem", for example.

3. Today (in this context) = period of time when complex and intelligent life exist.

3. Eukaryote cells = elementary building block of complex (or intelligent) life.

4. You claimed that eukaryote cell size ranges 6 orders of magnitude (although you seem to accept that most of them are within only one order of magnitude). That's right. I agree. But, because of that, the question says "roughly", "similar to" in order of magnitude. In your worst case scenario (1 meter), arguably somewhat extreme for the size of the unit of complex life (and NOT TYPICAL at all), the Planck length is 35 orders of magnitude away, whereas the size of the observable universe is only 27 orders of magnitude, that is, 1 meter is in the middle of the range within about a 10% of error.  Not too bad for such an extreme case.

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@drake Well, nobody was suggesting it should be off-topic, but if you're putting it forth as an anthropic question, it isn't physics any longer - you're basically talking biology.

Anyway, I'm not sure why you think that there should be anything favourable in having a cell whose length happens to be the geometric mean of the diameter of the observable universe and the planck length. The planck length is an arbitrary length in this regard, it's not the string length or something like that.

There are non-eukaryotic multicellular organisms. Some weird protists and stuff - don't know much of the details. Eukaryotic cells vary greatly in size. Neurons are pretty long, and ova are some 300 times wider than erythrocytes. The "average eukaryotic cell" is sort of arbitrary.

Also this.

Answer 4

If this counts as anthropic - able as we are to at once contemplate and theorize planck scales and the observable universe, we must be in the general ballpark...