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Unlike most Planck units named after "Planck" such as Planck length, Planck temperature, etc, the Planck mass seems more closed to daily life. It is about $10^{-5}$g, same order of magnitude of one eyebrow hair or a flea egg.

I am just wondering is there any interesting explanation about the relation between Planck mass and the mass of small lives such as flea.

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You may be interested in the "Planck impedance" - it's an amount of resistance very close to everyday life, not at all extreme in electronics and radio. en.wikipedia.org/wiki/Planck_units –  DarenW Nov 6 '12 at 21:52
    
I looked at @DarenW's link, and I found that the Planck momentum is almost exactly the momentum of a baseball thrown at 100 mph, which is also almost exactly the world record pitch. This is downright uncanny. –  Alan Rominger Nov 7 '12 at 20:37
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Wow, it seems interesting that we have lots of daily Planck things! –  Yingfei Gu Nov 8 '12 at 6:11

5 Answers 5

The reason the Planck mass is big is the same reason that the Planck length is small--- we are living on a scale which is enormous in Planck units. So everything around us is made from enormous atoms which have tiny, tiny masses, and you need a large number of atoms to make 1 Planck mass, just as you need a large number of Planck lengths to make 1 meter. The inverse relationship is because of the uncertainty principle, short distances are large energies.

The number of atoms you need is roughly the size of a few million cells, so what. It's not very significant, except the cells are about half way from the Planck length to the radius of the universe. The radius of the universe is from the cosmological constant scale, and the Higgs scale is about half-way between the Planck scale and the cosmological scale in log-energy. There's no explanation for this.

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There is a popular physics book (similar to The Elegant Universe, but different) (EDIT: a comment suggested this is The Black Hole War, and that sounds right, although I can't reference the exact figure) that I remember addressing the significance of the Planck Mass relative to the idea of elementary particles versus black holes. For now, Wikipedia will have to suffice, which I will reference here:

http://en.wikipedia.org/wiki/Planck_mass

The Planck mass can be derived approximately by setting it as the mass whose Compton wavelength and Schwarzschild radius are equal.

So why does this matter? The argument as I remember in the book goes like this:

Both elementary particles and black holes are "singular" objects. An atom, by comparison, is a collection of elementary particles, with structure to boot. A black hole can have almost any given mass. If a particle falls into a black hole, its mass increases by that amount. If you include massless particles, the permitted mass of black holes is almost a continuum. Not so for elementary particles. They have a certain rest mass (if any), and this rest mass comes from the fundamental properties of the universe.

The Planck Mass, the argument goes, is like the boundary between these two regions of elementary particles and black holes. This book had a very good image illustrating this. Unfortunately I can't find it anywhere online, so I will reproduce it here:

Planck mass significance

A notable observation is that there are much fewer particles with low rest masses, like the electron. This is consistent with what we know. As particle physics advances, we also produce more high mass particles, like the Higgs. By this line of thinking (which I'm not 100% confident is true), there will be a much higher density of particles at higher masses as they approach the Planck mass. Once you get higher mass than that, you're talking about a valid black hole.

That region, however, is relatively unimportant from a practical perspective because both high mass elementary particles (see again, the Higgs) and low mass black holes are incredibly unstable. Thus, on either side on that divide the particles are particularly short lived. You have to go far right or far left to get something stable.

Allow me to make the obvious argument that the absence of stable particles and black holes at our physical scale is important. Why? Because that means that for the mass ranges from quarks to almost stellar-mass black holes, the universe has no choice but to make complex things, made of many elementary particles, but not collapsed into a black hole. I hope the Anthropic argument is then obvious. We should be thankful that our cells are not commonly intruded by frequently interacting 100s of GeV or TeV particles, as this would not be good for cell chemistry. We can also be thankful that small black holes are not stable... I hope the reason for that is obvious.

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"There is a popular physics book (similar to The Elegant Universe, but different) that I remember addressing the significance of the Planck Mass relative to the idea of elementary particles versus black holes." I think it is The Black Hole War by Leonard Susskind –  Leos Ondra Nov 8 '12 at 17:02
    
@LeosOndra That sounds right! I was sure other people have seen the figure as well, but I can't find it on the internet. –  Alan Rominger Nov 8 '12 at 17:34

No, not really.

If you managed to compress a flea egg so it became a black hole then the conflicting descriptions of General Relativity and Quantum Mechanics could come into play.

But fleas are not able to achieve the extreme density required for this, so the similar magnitude of the two masses is largely irrelevant.

Similarly, the Planck energy unit being the same order of magnitude as the chemical energy in an automobile's gasoline tank is a curiosity but is not significant, except perhaps to suggest that Planck units may not always be fundamental.

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Yeah, I agree there might be no connection between these two scale. However I am still curious about "What determines the scale of life?" More specifically, let's say, the scale of the smallest species, for example the small bug. I think this must be related to how many "information" a life need at least, but I've not figured out the next step. Do you have any idea about the issue? –  Yingfei Gu Nov 6 '12 at 8:37
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If you look at small bacteria such as Pelagibacter ubique then you are many orders of magnitude smaller than a Planck mass. Most viruses are even smaller, with the Rous sarcoma virus being particularly small. So the minimum size is more about having enough biological information to be reproduced than anything to do with physical constants. –  Henry Nov 6 '12 at 8:44
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@YingfeiGu The scale of cells is determined more by surface area-to-volume ratios and rates of diffusion across membranes. Looking for it in terms of $\hbar$, $c$, or whatever is just numerology. –  Chris White Nov 6 '12 at 8:52
    
Oh, I see. Thank you very much for reminding me of this. –  Yingfei Gu Nov 6 '12 at 8:53
    
So, one point is "Does Newton constant G appear in biology theory?" –  Yingfei Gu Nov 6 '12 at 8:55

Well, this is a question that asks for opinions. In my case I think there exists a tentative relationship between very small live organisms and the Planck mass. From wikipedia

enter image description here = 2.17651(13)×10−8 kg, (or 21.7651 µg)

The name honors Max Planck because the unit measures the approximate scale at which quantum effects, here in the case of gravity, become important. Quantum effects are typified by the magnitude of Planck's constant, enter image description here .

My hand waving argument goes: Live organisms have to exist in a classical environment, where there is evident causality for energy input and surroundings. Therefore the Planck mass must be the lower limit in mass value, in order for a primary seed not to be driven this way and that by quantum mechanical potentials. This would certainly be true if the seeds of life came during the evolution of the Big Bang.

In our earth environment quantum mechanical dimensions are gauged by h, and one would have to check small organisms against that. I see that they are going to atto ( 10^-3 nano) units to measure viruses (viz the comment on the other answer). h is a very small number 6.6*10^-37 ergs sec and evolution could develop smaller sizes than the planck mass without hitting quantum mechanical uncertainties.

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Huh? This doesn't make any sense. –  Chris White Nov 6 '12 at 8:51
    
Wow! But I think the Planck mass here is not the lower limit in mass value. It seems quite different from the Planck length or Planck time. –  Yingfei Gu Nov 6 '12 at 8:56
    
I only set it as a limit if the seeds of life started during the evolution from the Big Bang, or in strong gravitational fields. I edited the answer –  anna v Nov 6 '12 at 9:01
    
@ChrisWhite The dimensions when quantum mechanical phenomena become strong are a lower limit for the possibility of life to exist, by life I mean roughly :energy input, growth, death and some voluntary behavior. Depending on the problem then the size and/or mass of a living tissue has a lower limit, imo. –  anna v Nov 6 '12 at 9:03
    
The brain cells of whales are enormous. I don't know if cells size information is preserved in fossils, but many animals in past geological times where giants because of a larger fraction of oxygen in the atmosphere... A virus size has to do with its functions and related to the size of the host bacteria... I don't thing Planck mass has anything to do with the size scales of life, but rather much more complex and unrelated macroscopic reasons. –  Eduardo Guerras Valera Nov 6 '12 at 18:54

Planck mass being so large and "classical", as well as the fact that Planck length and time are extremely small and "quantum mechanical", could point to the possibility that it is not a fundamental unit of mass. It could also point to the possibility that Newton's constant "G" is not a fundamental universal constant but composed of other universal constants like h and c.

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-1: this is a terrible answer. –  Ron Maimon Nov 6 '12 at 14:37
    
Ron, to make the statement that Planck mass is large due to the uncertainty principle because Planck length and time are so tiny, is a plausible argument. Max Planck based his scale on the existing universal constants h, c and G that he was familiar with at the time. But why should G have the same footing as a universal constant as h and c? –  Farhâd Nov 9 '12 at 14:11

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