# Planck mass is about the mass of one eyebrow hair

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## Solution 1

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:

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.

## Solution 2

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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### Yingfei Gu

Updated on August 01, 2022

• Yingfei Gu 5 months

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.

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
• Alan Rominger about 10 years
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.
• Yingfei Gu about 10 years
Wow, it seems interesting that we have lots of daily Planck things!
• sammy gerbil over 4 years
-1 Just because A is about the same mass/length/duration etc as B, why should that suggest there might a connection? You (or somebody else) could have chosen any number of naturally occurring objects to use for the comparison.
• Yingfei Gu about 10 years
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?
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.
Huh? This doesn't make any sense.
@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.
• Yingfei Gu about 10 years
Oh, I see. Thank you very much for reminding me of this.
• Yingfei Gu about 10 years
So, one point is "Does Newton constant G appear in biology theory?"
• Yingfei Gu about 10 years
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.
• anna v about 10 years
No, it need not as gravitational forces on earth are very very small with respect to all other forces effecting matter.
• anna v about 10 years
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 about 10 years
@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.
• Ron Maimon about 10 years
-1: this is a terrible answer.
• Eduardo Guerras Valera about 10 years
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.
• anna v about 10 years
@Eduardo I am handwaving about a lower limit due to the unsurmountable limitations that quantum mechanical dimensions would impose on what we call "living organisms". Not upper limits
• Leos Ondra about 10 years
"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
• Alan Rominger about 10 years
@LeosOndra That sounds right! I was sure other people have seen the figure as well, but I can't find it on the internet.