# Why is Planck mass so big?

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#### Gerinski

If I understand well, Planck units are fundamental in the sense that they don't depend on any arbitrary choice of measurement scale, but they emerge directly from the laws of physics.

For Planck lenght and Planck time, this seems to fit with the widespread belief that they may well also be fundamental in the sense that they are the smallest, indivisible units, below which the terms space or time don't have meaning anymore.
Indeed it seems natural to expect that the fundamental unit emerging from the laws is the smallest possible one, otherwise we would need another smaller unit (or, of course, measure smaller things as fractions of the unit, but that does not feel so "fundamental" anymore)

However, Planck mass is very big by subatomic standards, we know of many things with much smaller mass, so it is not fundamental in the second sense.

Why is it so? Does it have any significance the fact that Planck mass is so big? Shouldn't the fundamental unit of mass be the smallest possible mass?

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#### rbj

Gerinski said:
If I understand well, Planck units are fundamental in the sense that they don't depend on any arbitrary choice of measurement scale, but they emerge directly from the laws of physics.

For Planck lengyh and Planck time, this seems to fit with the widespread belief that they may well also be fundamental in the sense that they are the smallest, indivisible units, below which the terms space or time don't have meaning anymore.

Indeed it seems natural to expect that the fundamental unit emerging from the laws is the smallest possible one, ...
oh? is $c$ the smallest unit of speed? i should think it to be the most fundamental unit of speed.

... otherwise we would need another smaller unit (or, of course, measure smaller things as fractions of the unit, but that does not feel so "fundamental" anymore)

However, Planck mass is very big by subatomic standards, we know of many things with much smaller mass, so it is not fundamental in the second sense.

Why is it so? Does it have any significance the fact that Planck mass is so big? Shouldn't the fundamental unit of mass be the smallest possible mass?
it's a very good, intriguing, and IMO, fundamental question. the way i think about it is, hopefully, the same way Frank Wilczek does (June 2001 Physics Today - http://www.physicstoday.org/pt/vol-54/iss-6/p12.html [Broken] ):

...We see that the question [posed] is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in Natural (Planck) Units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [1/(13 quintillion)]...

a pretty good (IMO) reference is the one at http://en.wikipedia.org/wiki/Planck_units .

my feeling is that the Planck Units (perhaps adjusted by a factor $\sqrt{4 \pi}$ or reciprocal, making "rationalized" Planck units) are the tick marks on Nature's tape measure. they are the units on which Nature operates and any dimensionful quantities that are measured with respect to these rationalized Planck units (resulting in dimensionless numbers) are truly numbers that Nature is dealing with herself. (sorry for anthropomorphizing nature.)

anyway, the Planck Mass isn't really large (it's about the mass of a speck of dust) but, in reality, it's the subatomic particles that have masses so small. perhaps another way to look at it is the Planck Mass seems so large because, from our anthropocentric POV, gravity seems so weak. note that $\sqrt{G}$ is in the denominator for $m_P$ whereas it's in the numerator for $t_P$ and $l_P$.

again, the fundamental question to ask is "why are the particle masses so small? especially when the Elementary Charge is in the ballpark (in fact, it's in the "infield") of the Planck Charge." since the Bohr radius (in terms of the Planck Length) is directly related:

$$a_0 = {{4\pi\epsilon_0\hbar^2}\over{m_e e^2}}= {{m_P}\over{m_e \alpha}} l_P$$

it is akin to asking why the size of atoms are so big (compared to the Planck Length).

now, i don't know why an atom's size is approximately $10^{25} l_P$, but it is, or why biological cells are about $10^{5}$ bigger than an atom, but they are, or why we are about $10^{5}$ bigger than the cells, but we are and if any of those dimensionless ratios changed, life would be different. but if none of those ratios changed, nor any other ratio of like dimensioned physical quantity, we would still be about as big as $10^{35} l_P$, our clocks would tick about once every $10^{44} t_P$, and, by definition, we would always perceive the speed of light to be $$c = \frac{1 l_P}{1 t_P}$$ which is the same as how we do now, no matter how some "god-like" manipulator changes it. so if you get in an argument with someone about theories such as Variable Speed of Light (VSL) or changing the graviational constant, you know where i stand about it. whether or not it's possible, it is, from our POV, meaningless because all of reality is scaled w.r.t. these Planck units.

now if some dimensionless value like $\alpha$ changed, that's different. we would perceive the difference. but to attribute that change to a change in $c$, that case is not defensible. you could argue that the change in $\alpha \$ is due to a change in the speed of light, and i could argue it's a change in Planck's constant or the elementary charge and there is no way to support one over the other.

i know that was more answer than your question, but it is like asking "why is the speed of light what it is?" or "why is the graviational constant what it is?".

r b-j

...

a little postscript: a point i forgot to make when i quoted Wilczek was that if the proton mass was closer to (or, heaven forbid, larger than) the Planck mass, then the gravitational attraction of two protons (alone in free space) would rival the electrostatic repulsion between these two protons. they're both inverse-square actions and if $$m_p = \sqrt{\alpha} m_P \approx \frac{m_P}{11}$$, the opposing gravitational and electrostatic forces would be exactly equal. life, sure as hell, would be different.

r b-j

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#### Gerinski

Too bad there's not an illuminating answer, but that was great help anyway, thanks a lot !!

#### rbj

Gerinski said:
Too bad there's not an illuminating answer, but that was great help anyway, thanks a lot !!
sorry if it wasn't illuminating enough. i tried to cast some dim light on it saying that the factor that the Planck mass is bigger than the particle masses is about the same factor that atoms are bigger than the Planck length (wouldn't it be weird if the Bohr radius were about as small as the Planck length?). and also that if the Planck mass wasn't a helluva lot bigger, then interparticle gravitation would be an issue in the physics of the atom (wouldn't that be weird?).

so, i s'pose one can liberally toss "anthropic principle" at this for a hand-waving answer. there are about 26 fundamental dimensionless constants in the universe (many are particle masses that, i s'pose are normalized against the Planck mass), and if they be significantly different, the universe would be different, perhaps enough that atoms do not form or galaxies/stars/planets do not form (or survive long enough) or something else that would prevent the development of (quasi-)intelligent beings like us to invent the internet and sit on our behinds and gaze into the heavens (or at our navels) and ask this very question: why?

i incorrectly compared this question to asking "why is the speed of light what it is?" or "why is the gravitational constant what it is?". it is not the same kind of question. you are asking why this particular dimensionless number (the ratio of some particle mass to the Planck mass) is what it is. that is closer to asking "why is the Fine-structure constant, $\alpha$, equal to the value it is?" asking "why is the speed of light what it is?" or "why is the graviational constant what it is?" is tantamount to asking why we chose the meter, kilogram, and second to be what they are.

r b-j

#### JesseM

Science Advisor
Gerinski said:
If I understand well, Planck units are fundamental in the sense that they don't depend on any arbitrary choice of measurement scale, but they emerge directly from the laws of physics.

For Planck lenght and Planck time, this seems to fit with the widespread belief that they may well also be fundamental in the sense that they are the smallest, indivisible units, below which the terms space or time don't have meaning anymore.
Indeed it seems natural to expect that the fundamental unit emerging from the laws is the smallest possible one, otherwise we would need another smaller unit (or, of course, measure smaller things as fractions of the unit, but that does not feel so "fundamental" anymore)

However, Planck mass is very big by subatomic standards, we know of many things with much smaller mass, so it is not fundamental in the second sense.

Why is it so? Does it have any significance the fact that Planck mass is so big? Shouldn't the fundamental unit of mass be the smallest possible mass?
One speculation I have heard about the Planck mass is that it it is the largest possible mass that can fit in the smallest meaningful volume of space, which would be about equal to the Planck length cubed (maybe there'd be a small constant like 2*pi in front or something). If this were true, I think the Planck mass would also be about equal to the mass of the smallest possible black hole--after that it just evaporates thanks to Hawking radiation. Another way of thinking about it is that the Planck mass divided by the Planck length would be the greatest possible meaningful density of mass/energy, which might be the same as the density of the universe one Planck time after the Big Bang.

This page talks more about the significance of the Planck mass:

http://www.physlink.com/Education/AskExperts/ae635.cfm

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#### mccrone

This is a good question. It may help to consider the possibility that the Planck scale is not just about a limit on smallness, but largeness as well. To put this in terms of an ancient dichotomy, the Planck scale is the smallest scale for "form" - it is the smallest coherent unit of spacetime. And also the largest scale for "substance" - the density of energy/mass/temperature that can fit into a unit of spacetime.

So first step is to think of the Planck scale as a complex package that gives you both upper and lower limits. Then as the Universe expands, both the form and the substance "fall" towards their other extreme. So expansion increases spacetime - the form is heading towards maximum largeness. Yet at the same time the substance, the energy density of the Universe, is falling towards its lowest possible level, its lowest possible temperature. The "unpacking" of the Planck scale is thus a move from a hot point to a cold void. What was small at the beginning grows large, and what was large grows small - so a conservation of scale going on.

The Planck scale is of course set by various other constants - c (speed of light) and g (strength of gravity). The Planck time is how long it takes light to cross the Planck space.

It also seems intuitive to me that you get the Planck energy out of lightspeed considerations. The shorter the wavelength, the higher the energy. So if you have a world as small as the Planck scale, there is only room for a single crisp oscillation of about the Planck length. This would be the reason for a crisp upper bound on energy/mass/temperature. If anything "substantial" is happening at the Planck-scale, it would have to start at the shortest possible wavelength, and thus the highest possible energy level.

Someone better informed may be able to contradict this simplistic interpretation. I would certainly be interested in a better answer.

Cheers - John McCrone.

#### Chronos

Science Advisor
Gold Member
JesseM has it right. The Planck mass is the mass required to form a black hole with an event horizon of a Planck length - which basically means nothing smaller than this can collapse into a black hole. This is a good thing if you consider the incredible density of an atomic nucleus. And to elaborate on what mccrone was saying, the Planck temperature [highest possible temperature] is radiation with a wavelength of a Planck length. Not coincidentally, this happens to be the same as the temperature of the universe one Planck time after the big bang.

#### pi-r8

Chronos- you're saying that the Planck mass is the MINIMUM amount necessary to form a black hole of the Planck length, right? I thought what JesseM said was that it's the MAXIMUM aount that can fit into that space.

#### mccrone

Chronos said:
The Planck mass is the mass required to form a black hole with an event horizon of a Planck length - which basically means nothing smaller than this can collapse into a black hole.
Yes, but does this "explain" why there is a mass limit in some natural way? If you are imagining gravity as the curvature of spacetime, then the Planck mass is where spacetime gets so warped it gets closed off. But then QM uncertainty/compton wavelength has to be invoked as to why there is not then a complete collapse of spacetime to a singularity.

So my feeling here is that gravity = spacetime = form. Mass/energy/temperature as such do not feature in this explanatory chain. So that is why I am interested in the alternative formulation that yields "substance" - mass = wavelength = substance.

Cheers - John McCrone.

#### JesseM

Science Advisor
pi-r8 said:
Chronos- you're saying that the Planck mass is the MINIMUM amount necessary to form a black hole of the Planck length, right? I thought what JesseM said was that it's the MAXIMUM aount that can fit into that space.
For any given length, there is only one possible mass that will give a black whole whose radius is exactly that length (see the formula for Schwarzschild radius as a function of mass here). The Planck mass represents the minimum for the size of a black hole, since anything smaller would be a black hole smaller than a Planck length, which probably wouldn't even make sense according to quantum gravity. But what I was also saying is that the Planck mass is probably the maximum mass that can be packed into a physically meaningful unit of space--if you try to add mass, you'll just get a black hole with a radius larger than the Planck length, whose density will be lower (which means the mass per unit volume is lower).

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#### Chronos

Science Advisor
Gold Member
A black hole with an event horizon smaller than a Planck length is not definable by any known mathematical model. In other words, it has no observational consequences. That renders it a meaningless solution. You can't change that without rewriting the books on quantum mechanics, and that is a tall order. In other words I again agree with JesseM [aside from some trivial issues].

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#### memarf1

Oh My Gosh

This website elludes me. I can't believe the questions some people ask. First, The Planck Mass is VERY VERY VERY SMALL!!! Second, c, the speed of light is a VERY VERY VERY LARGE SPEED, NOT SMALL!!!

#### JesseM

Science Advisor
memarf1 said:
This website elludes me. I can't believe the questions some people ask. First, The Planck Mass is VERY VERY VERY SMALL!!!
Not really, it's about the same mass as a flea.

#### memarf1

Planck Mass = 2.18E-8 kg
Planck Length = 1.6161E-35 m
Planck Time = 5.39E-44 s

I say these are VERY VERY VERY SMALL!!!

#### JesseM

Science Advisor
memarf1 said:
Planck Mass = 2.18E-8 kg
Planck Length = 1.6161E-35 m
Planck Time = 5.39E-44 s

I say these are VERY VERY VERY SMALL!!!
You think the Planck mass is very small in the context of physics rather than everyday life? Again, the planck mass is about the same as that of a flea, and it's many orders of magnitude larger than any fundamental particle, or any atom or molecule. For comparison, the mass of a proton (which is itself composed of three quarks) is about 1.67E-27 kg, so the Planck mass is about 13,000,000,000,000,000,000 times larger than that. And a proton is a pretty massive object in the context of particle physics. Check out this page to see the masses of various fundamental particles and atoms and molecules in kilograms, all far smaller than the Planck mass.

#### mccrone

memarf1 said:
First, The Planck Mass is VERY VERY VERY SMALL!!! Second, c, the speed of light is a VERY VERY VERY LARGE SPEED, NOT SMALL!!!
Talk of the Planck mass is misleading here. It is the Planck mass density that is large and maximal. The Planck mass density would equate to 10^96 grams per cubic centimetre.

#### memarf1

ok, planck mass density is another story, but the original question was of planck mass, and I did say that it was very small, but I was thinking about the planck length when I said that. In reference to particle physics yes, the planck mass is large, but in the grand scheme of things its very small. Actually the argument should be that its really relitive to what you are comparing it to.

#### rbj

memarf1 said:
First, The Planck Mass is VERY VERY VERY SMALL!!! Second, c, the speed of light is a VERY VERY VERY LARGE SPEED, NOT SMALL!!!
no. that is only what you think from an anthropocentric POV. the Planck mass is sorta small compared to our mass (or the mass of things we usually hold) and the speed of light is very, very, very fast compared to how fast i drive, but both are simply what they are, neither fast nor slow nor heavy nor light.

memarf1 said:
ok, planck mass density is another story, but the original question was of planck mass, and I did say that it was very small, but I was thinking about the planck length when I said that.
which is very small compared to the dimensions (poor word when i mean "size") of any known physical particle.

In reference to particle physics yes, the planck mass is large, but in the grand scheme of things its very small.
actually not. only if you compare it to some object or "thing" of anthropocentric interest.

compared to subatomic particles is it enormously big, but, those are other objects or "things". imagine a world where there were no particular particles or "things" to base your concept of what is big and what is little. no electron of other particular particles. there would still be $G$, $\epsilon_0$, $c$, and $\hbar$ as something to reference to. then the Planck mass would be the only mass that you could make any reference to without tossing in some possibly contrived numbers.

Actually the argument should be that its really relative to what you are comparing it to.
again, the real question to ask is not "why is the Planck mass so large" but is "why are the particle masses so small." like $G$, $\epsilon_0$, $c$, and $\hbar$, the Planck mass simply is what it is. the Natural unit of mass. (again, a more natural unit of mass might differ from the Planck mass by a factor of $\sqrt{4 \pi}$.)

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G

#### Gerinski

memarf1 said:
This website elludes me. I can't believe the questions some people ask. First, The Planck Mass is VERY VERY VERY SMALL!!! Second, c, the speed of light is a VERY VERY VERY LARGE SPEED, NOT SMALL!!!
First, the original question stated clearly enough that the Planck mass is very very very large by subatomic standards, and in the context of other Planck units which seem to define the smallest meaningfull value, as opposed to the case with mass. As already remarked by many here, this is unquestionable.

Second, who says the speed of light is small??? I re-read all the posts and I don't find such statement anywhere.

Third: no offence, but while everybody else here just aimed to give valuable input for my ignorance (which they indeed did, for which I'm very thankful to all of them), you just not only make 2 useless remarks, but on top of it, do not give any valuable information or opinion on the issue.

#### memarf1

I believe rbj just wanted to use some very large words. He quoted me correctly but not completely. I stated that the size was relative to what it is compared to.

To the other guy, yes, the original question was about particle physics and as I also stated in some of my above responses, I was thinking about the planck length, not mass.

So in closing, before responding to what I say, please read my entire response before critisizing what I write.

Thank you.

#### rbj

memarf1 said:
I believe rbj just wanted to use some very large words.
maybe just one very large word. however, the word was fewer keystrokes than typing in the meaning by use of other words. it's the right word.

Planck units mean a few different things to different people. Planck units mean to some people (like Chronos, perhaps) a scale of physical quantity where some tremendously extreme things may have been happening. i take an earlier pedagogical POV (oh, there i go, another large word) of Planck units. i don't think Planck was thinking about Compton wavelengths and Schwarzschild radii.

but he was thinking about a definition of units that might also be arrived at by the aliens on the planet Zog (how might we communicate concepts like how big we are or how big Earth is, etc. to the Zoggians). these Zoggians, being pretty smart to receive radio signals from our part of the heavens, are thinking that their units might have meaning on Zog, but won't mean diddley to folks like us. so how do they get away from their Zog-centric units so they can understand numerical values of dimensional physical quantities referred to by us?

pedagogically, in a conceptual world where no particular particle, object, or "thing" is used as a prototype as the definition of some fundamental unit of measure, only Planck units (or some small adjustment to them, i think they should be scaled by $\sqrt{4 \pi}$) survive as something that you can use as a basis for defining units. they are the "Natural units" and they are what you compare any other length or mass or time or property of any other physical thing, if you want to communicate that quantity to someone who has absolutely no reference to the human experience.

He quoted me correctly but not completely. I stated that the size was relative to what it is compared to.
what did i leave out? i quoted you completely and then responded that, fundamentally, the Planck units are neither large nor small. they are simply what they are and it's the other quantities, the masses of particles or of people that are small or large. rather than compare the Planck mass to the proton's mass and ask "why is it so big?", i'm suggesting that we ask why the proton's mass is so small. the Planck mass is the unit. it is neither large nor small.

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#### Chronos

Science Advisor
Gold Member
We are in full agreement, rbj. Planck units are basically dimensionless numbers. Most any other sufficiently intelligent beings would immediately recognize the ratios they represent.

#### rbj

i didn't mean to imply that there was any disagreement between us, it's just that i have heard a lot of folks ascribe the genesis of Planck Units beginning with the Planck Mass as such a mass that the Compton Wavelength is equal to the Swartzchild Radius. it's true, but i consider that to be the consequence of "solving a physics problem". pedagogically they are the units so that when $G, \hbar, c$ are expressed in terms of these units, their numerical value goes to 1.

dunno if i would say that Planck units are dimensionless. physical quantities expressed in terms of Planck units are dimensionless. at least that's how i think it.

r b-j

#### mccrone

These discussions seem rather off the point. The original question was -

"Why is it so? Does it have any significance the fact that Planck mass is so big? Shouldn't the fundamental unit of mass be the smallest possible mass?"

So Gerinski was asking, if the Planck scale is supposed to be the "smallest" scale, then why is the Planck mass larger than the regular masses of particles. He was probably thinking that mass comes in discrete quanta of about the Planck size and would get glued together to make particles with various masses.

I pointed out that the Planck mass (as are the Planck energy and temperature) are density measurements and they define an upper limit on how much can be packed into how little.

Furthermore, there is a natural explanation in terms of "what happens" if you imagine reducing a whole world to a Planck scale point. Several people mentioned the standard answer - gravity curls up spacetime into a blackhole at that density and so no more mass can "get in". I said you can also look at it from the point of view of the speed of light and frequency of oscillation. The Planck length would define the shortest possible wavelength. And a Planck scale world would indeed seem to be only able to contain this one "massive" wavelength.

None of this is "dimensionless" talk as it is comparing the mass of particles to the mass as it would be at the Planck scale upper limit. And Gerinski wanted some answer as to why Planck mass was larger than a typical particle, not fundamentally smaller.

The fact that Gerinski even framed the question this way is what I think is significant. It is normal in physics to attempt to reduce everything to "smallness". And to thus be puzzled by largeness. Yet most things make more sense when you consider them as dichotomisations towards opposing limits of scale.

So you have the speed of light and the apparently weird fact that nothing travels faster. You can reframe your thinking by saying that the speed of things in the Universe is actually bound by two limits - the upper bound of c and the lower bound of rest, or temperature-wise, absolute zero. A mass cannot go slower than "rest". And QM uncertainty says it can't even achieve rest, just approach such a limit asymptotically.

One further interesting point is that of course - according to current theory - all particles were massless (and so flew along only at c, no "slower") until the symmetry breaking of the Higgs field at 10^-10 seconds into the Big Bang when the average temperature has dropped to 10^15 degrees. So it took about 30 orders of magnitude in time (ie: a cooling expansion) and 15 orders in terms of a drop in temperature for massiveness to become a concrete fact.

So it would seem from this that the Planck mass should be considered a bit of a fiction as there would only be a Planck energy density?

Cheers - John McCrone.

#### rbj

... And Gerinski wanted some answer as to why Planck mass was larger than a typical particle, not fundamentally smaller. ...
i dunno if that can be answered to Gerinski's or any of our satifaction. i only tried to respond:

1. the Planck mass isn't large, it's just the unit, neither large nor small (i guess side-stepping the contention some might have that units should be smaller than whatever they measure). it's the particle masses that are small.

2. if the particle masses were closer to the Planck mass, gravitation could not be ignored in the physic of the atom.

3. if the electrom mass was closer to the Planck mass, the size of atoms would be closer to the Planck length. that might make for a weird reality that would be difficult for matter and life as we know it to evolve and ask "why is the Planck mass so large?" (weak anthropic principle)

although the Planck mass and Planck length (or "Planck scale") have that property of equating a majore GR metric to the salient QM metric, that's well and good but i don't see how that addresses Gerinski's question.

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