Is H(hbar)/2c^2 a Possible Fundamental Unit of Mass?

[rbj]

Factors of a few $\pi$ are completely arbitrary and up to convention.

okay, so let's just toss in another factor of $4 \pi$ into Gauss's law or take it out.

why not just toss in a factor of 10?

some conventions are cleaner than others.

 

[rbj]

Planck originally used e, c and G and derived a set of units - this was also done by Stoney - there is no logical reason to prefer one set of constants over the other except a prejudice not based upon anything that has been confirmed -

there is a salient difference between using the properties of a prototype object or particle to base units on and not doing so. if it's more important that e is held constant (by the conventional choice of units) than ħ, then use Stoney rather than Planck. which is more "logical" can be disputed.

 

[Chalnoth]

okay, so let's just toss in another factor of $4 \pi$ into Gauss's law or take it out.

why not just toss in a factor of 10?

some conventions are cleaner than others.

The constants take the values they do because historically each provided a simple relationship between to quantities that had certain units. So in certain equations, the constants always end up having no prefactors whatsoever. When we use them in different equations, they naturally end up with some prefactors.

The gravitational constant has no prefactors in Newton's gravitational force equation:

$$F_g = {-G m_1 m_2 \over r^2}\hat{r}$$

The permitivity of free space has no prefactor in Gauss's Law:

$$\nabla \vec{E} = {\rho_f \over \epsilon_0}$$

Which equations you want the constants to have no prefactors in is obviously completely arbitrary, and there's no sense in making up a whole new set of constants that have no prefactors in a different set of equations. All it will do is confuse everybody when you try to show your work to somebody else. So best to just learn the conventions as they are. Anything you make up won't be unequivocally better anyway: it will be better in some areas, worse in others, but generally no different in overall convenience.

 

[rbj]

The gravitational constant has no prefactors in Newton's gravitational force equation:

$$F_g = {-G m_1 m_2 \over r^2}\hat{r}$$

The permittivity of free space has no prefactor in Gauss's Law:

$$\nabla \vec{E} = {\rho_f \over \epsilon_0}$$

so do you wonder why and how we moved from the Coulomb electrostatic force equation (that looks a lot like Newton gravitational force equation) to Gauss's law?

why introduce and use $\epsilon_0$ instead of $k_\mbox{e} = \frac{1}{4 \pi \epsilon_0}$?

Which equations you want the constants to have no prefactors in is obviously completely arbitrary,

obviously.

why not define the unit of force to be whatever force is needed to compress some prototype spring at the BIPM one centimeter? then Newton's 2^nd law is (what he said with words)

$$F = k_\mbox{N} \ \ \frac{dp}{dt}$$

and then every 10 years or so, the BIPM can report to the world what their latest precision measurement for $k_\mbox{N}$ is.

it's obviously arbitrary. what's the matter with doing that?

and there's no sense in making up a whole new set of constants that have no prefactors in a different set of equations. All it will do is confuse everybody when you try to show your work to somebody else.

i thought the problem was not of confusing or showing one's work, but was about fundamental units of nature.

the issue, i thought, was what might be a fundamental unit of Nature. both Newton's law and Coulomb's law are inverse-square and lend themselves directly to the notion of flux and flux density which is what Gauss's law adds up. we see that flux density and field strength are proportional. does the mechanism of Nature herself actually take the flux density (which is naturally associated with the amount or density of "stuff") and she pulls out a little scaler from out of her butt (this would be a true constant or parameter of nature), adjusts that flux density by that scaler to get field strength?

some choice of units require (for humans) such a scaling, but is there evidence that there is an intrinsic difference between flux density and field strength? only a specific choice of units totally loses the differentiation between the two physical quantities.

So best to just learn the conventions as they are. Anything you make up won't be unequivocally better anyway: it will be better in some areas, worse in others, but generally no different in overall convenience.

 

[Chalnoth]

i thought the problem was not of confusing or showing one's work, but was about fundamental units of nature.

Any sensible new set of units you come up with is going to only differ from the ones we have by factors of a few times $\pi$. Such changes will not make any difference in terms of the conclusions we draw from fundamental units, which is generally that you can calculate most things by simply performing the relevant dimensional analysis and get within a factor of a few times $\pi$ of the true result.

And by the way, the "prototype spring" is not a sensible component of fundamental units, because you've added a completely and utterly arbitrary proportionality between force and distance into the equation, and could thus shift the result by any number you want.

 

[rbj]

Of course the prototype spring is unsensible, just as any other prototype object is unsensible for the natural definition of a system of units because you have to decide what the prototype object is and that's where arbitrariness comes in. When physical objects or particles are brought into the picture, they come into it with their properties and the quantitative values of the properties.

My question for you is how are equations of interaction different between the traditional Planck units:

$$t_\mbox{P} = \sqrt{\frac{\hbar G}{c^5}}$$

$$l_\mbox{P} = \sqrt{\frac{\hbar G}{c^3}}$$

$$m_\mbox{P} = \sqrt{\frac{\hbar c}{G}}$$

$$q_\mbox{P} = \sqrt{4 \pi \epsilon_0 \hbar c}$$

and these:

$$t_0 = \sqrt{\frac{\hbar 4 \pi G}{c^5}}$$

$$l_0 = \sqrt{\frac{\hbar 4 \pi G}{c^3}}$$

$$m_0 = \sqrt{\frac{\hbar c}{4 \pi G}}$$

$$q_0 = \sqrt{\epsilon_0 \hbar c}$$

?

what numerical properties of free space (no mention of any particle, yet) are there? the speed of propagation of any of the "instantaneous" interactions or the characteristic impedance of such propagation. do you think that it's really true that this vacuum out there holds intrinsically some special numbers about that? while it may be true that there is some vacuum energy density that is characteristic of the vacuum, that is a parameter that should be measured, just like the cosmological constant or the Hubble constant. i don't think that the vacuum has an intrinsic speed of propagation or characteristic impedance (other than 1) but the vacuum energy density or the mean dark matter density or cosmological constant or the Hubble constant are parameters, no so much of the vacuum, but of this object we call the Universe. our particular universe or pocket universe.

and the difference to the traditional Planck units are a factor of $\sqrt{4 \pi}$ or its reciprocal. not any larger powers of $\pi$. and you're right that it doesn't change any conclusions (except for that factor).

 

[Chalnoth]

and the difference to the traditional Planck units are a factor of $\sqrt{4 \pi}$ or its reciprocal. not any larger powers of $\pi$. and you're right that it doesn't change any conclusions (except for that factor).

Yes. So what's your point? Whether you have those factors in or not is pretty arbitrary.

 

[rbj]

One other thing to point out is, since, in these natural units that remove extraneous scaling factors related to properties of free space make no reference to the elementary charge, then we can express the elementary charge in terms of these natural units and get an important numerical property of nature:

$$e = \sqrt{4 \pi \alpha} \ q_0 = 0.302822 \ q_0$$

one can think of the value of the fine-structure content as a consequence the amount of charge (measured in these natural units) that Nature has bestowed upon the proton and electron and positron. rather than think of α as defining the "strength of the EM interaction", the strength of EM (like gravity) simply is what it is (using Frank Wilczek's language). but the charge on the particles (as well as their mass and other properties inherent to them) is not simply what it is. these particles have specific values of mass and charge and spin that characterize them as objects.

 

[rbj]

Yes. So what's your point? Whether you have those factors in or not is pretty arbitrary.

the point is the same as the point of whether or not we arbitrarily define the unit of force to leave a constant of proportionality in fundamental equations of physical law or if we naturally define the unit of force to eliminate such an extraneous scaling factor. that's the point.

 

[Chalnoth]

Counting one as the cause of another is completely pointless without an actual theory that allows these properties to vary and explains why they take the values they do.

 

[Chalnoth]

the point is the same as the point of whether or not we arbitrarily define the unit of force to leave a constant of proportionality in fundamental equations of physical law or if we naturally define the unit of force to eliminate such an extraneous scaling factor. that's the point.

Um, because that introduces an a number that can take any arbitrary value (whether 1 or 10^100), completely removing the set of units from the underlying physics.

 

[rbj]

Um, because that introduces an a number that can take any arbitrary value (whether 1 or 10^100), completely removing the set of units from the underlying physics.

Bingo.

 

[rbj]

Counting one as the cause of another is completely pointless without an actual theory that allows these properties to vary and explains why they take the values they do.

actually, we are free to select or define any internally consistent system of units we please. but if we measure speed in units of furlongs per fortnight (rather than c), any theory of physics will have extraneous scaling factors tossed in there that will be obvious of having anthropocentric origin and Nature doesn't give a rat's a$s about whatever units we use to describe her.

 

[Chalnoth]

actually, we are free to select or define any internally consistent system of units we please. but if we measure speed in units of furlongs per fortnight (rather than c), any theory of physics will have extraneous scaling factors tossed in there that will be obvious of having anthropocentric origin and Nature doesn't give a rat's a$s about whatever units we use to describe her.

Right. But as I said earlier, if we use "natural" units, a large number of calculations come out within a few factors of $\pi$ of the result you'd estimate from dimensional analysis.

 

[yogi]

$\hbar$ is better understood as being the conversion factor between angular frequency and energy. There is no "fundamental unit" of angular momentum, because angular momentum is a composite unit.

I realize you are a professional cosmologist - and I am only a hobbest... so I don't feel comfortable challenging your statement - but ...

Many years ago as an undergrad I recall a respected and in my memory an insightful professor making the comment that: "in our universe, momentum is a more fundamental entity than mass" I believe it came from some ponderings of Einstein when faced with the decision to treat mass or momentum as conserved in his musings while deriving the transforms of SR.

Perhaps it is not more fundamental, since energy is also a conserved quantity which changes if transformed from mass to other forms - but angular momentum is also a conserved quantity - so perhaps in the holistic context, all conserved quantities are fundamental in one sense. As we all know, subatomic particles have angular momentums in multipiles of hbar/2, except for a few with no angular momentum (which could be justified as counter rotating angular momentums in short lived complex Particles)

Anyway - perhaps some food for thought

 

[Chalnoth]

Perhaps you missed the later discussion, but I conceded that point a while later.

 

[rbj]

Right. But as I said earlier, if we use "natural" units, a large number of calculations come out within a few factors of $\pi$ of the result you'd estimate from dimensional analysis.

but why toss in any unnecessary slop? (i don't see a few factors of $\pi$, i see a few factors of $\sqrt{4 \pi}$ which is more than double. about a half order of magnitude off. i really don't get why Planck knew enough to suggest to normalize ħ instead of h but chose to normalize G instead of 4πG.

i really agree with the notion that "natural units help physicists to reframe questions". with the use of the mostest natural units, i would imagine that this would be helpful in framing or reframing questions the best.

the other issue, is variations of natural units; Planck vs. Stoney vs. Atomic units as well as some others. this is why i like the perspective of Michael Duff regarding fundamental constants (only dimensionless constants are in that set, G and c and ħ and ϵ₀ are not in that set). but, depending on what your units are meant to normalize, then the questions that get framed or reframed are different. i still think that (these rationalized) Planck units are the best and that the elementary charge (measured in these units) becomes a fundamental constant of nature.

L8r...

 

[Chalnoth]

but why toss in any unnecessary slop? (i don't see a few factors of $\pi$, i see a few factors of $\sqrt{4 \pi}$ which is more than double. about a half order of magnitude off. i really don't get why Planck knew enough to suggest to normalize ħ instead of h but chose to normalize G instead of 4πG.

As I said before, it's not about knowing. It's about convention. And shifting things by just one order of magnitude really isn't significant.

i really agree with the notion that "natural units help physicists to reframe questions". with the use of the mostest natural units, i would imagine that this would be helpful in framing or reframing questions the best.

But the problem is that once you get down to a few times $\pi$ as your factors, which set of units is "best" entirely depends upon the context. There is no absolute best.

 

[rbj]

As I said before, it's not about knowing. It's about convention.

then we're round the maypole again. some conventions are better than others. this:

$$F = \frac{dp}{dt}$$

is better than

$$F = k_\mbox{N} \ \ \frac{dp}{dt}$$

And shifting things by just one order of magnitude really isn't significant.

for cosmology, maybe. but once you really want to know how big the black hole is, or how much mass was needed to collapse it, i don't think you want to be off by 10.

But the problem is that once you get down to a few times $\pi$ as your factors, which set of units is "best" entirely depends upon the context. There is no absolute best.

i disagree that normalizing $4 \pi \epsilon_0$ is ever better than normalizing $\epsilon_0$.

c'mon, admit it. some conventions were prematurely adopted and it's just inertia that keeps them going in their premature form.

 

[Chalnoth]

for cosmology, maybe. but once you really want to know how big the black hole is, or how much mass was needed to collapse it, i don't think you want to be off by 10.

If you want to know the precise answer, you're not going to be using dimensional analysis in natural units to try to find the answer, are you?

 

[rbj]

If you want to know the precise answer, you're not going to be using dimensional analysis in natural units to try to find the answer, are you?

no, you won't. i wouldn't be using dimensional analysis for the purpose of getting quantitative values in a physical problem in the first place.

i presume what we use are either established physical law (that is normally good only for the circumstances that such physics was developed in the first place) or something new (to sort of test it out on a problem that is difficult or impossible to describe with the old physics). these laws relate physical quantities that we measure usually with anthropocentric units (like SI or cgs). because of that certain physical "constants", that have been determined (in terms of these anthropocentric units) over the years, are needed in these physical laws to transform quantities that, except for this physical law, are independent.

e.g. Newton's second law. all it really says is that the rate of change of momentum is proportional to this other concept we call "force". we don't have to equate change of momentum to force, but, since we didn't yet define a unit of force, we could do that and we do do that. so, by the choice of unit definition, that constant of proportionality is exactly 1 and doesn't crap up the equations. now, does that mean that the time rate of change of momentum is exactly the same as net force? i dunno, but it's an interesting concept. i tend to not believe so, because force exists as a concept in contexts of stress and pressure and has some effect on the atomic level, even when the momentum of bodies are not changing.

another e.g.: electrostatic interaction. this physical constant we call ϵ₀ relates two, otherwise unrelated, quantities: "flux density" (which is just defined because you have a pile of charge somewhere and you're at some distance where the "effect", something we call "flux", of that charge distributed over little pieces of area can be directly determined) to "electrostatic field". then you notice that, proportional to the amount of charge of a test charge, this test charge accelerates as if a force acts on it. now these two quantities (which are dimensionally not the same at all: QL^-2 vs. MLT^-2Q^-1) don't have to be related, but Coulomb's law says they are and 1/ϵ₀ is the thing that converts one species of animal to the other. but are they really different? is it possible that flux density is field strength? the same thing? not two different things that just happen to be related by this anthropocentric scaler that we measured very carefully because of the unit definitions we pulled out of our human butt?

what Planck units (or these rationalized Planck units that I've been advocating) do is make it clear that these constants are not intrinsic properties of free space, just a manifestation of the units we came up with to measure things. they are not fundamental physical constants.

i'm not advocating using dimensional analysis to solve physical problems (perhaps to check one's work, to make sure they are getting the correct dimension of stuff in their answer), I'm only advocating using either established or proposed physical law. you can leave the constants in if you wish, but there might be some insight in knowing that space-time curvature is the same as stress-energy not just proportional to it.

 

[yogi]

Perhaps you missed the later discussion, but I conceded that point a while later.

O yes - your post 29 - I recall now

 

[yogi]

like[/i] Planck units because they are not based on any prototype object or particle. it's like Planck units are based on nothing, leaving little room for arbitrarily choosing some prototype object or particle. i think that normalizing $4 \pi G$ would be better than normalizing $G$ and normalizing $\epsilon_0$ would be better than normalizing $4 \pi \epsilon_0$ as Planck units do.

Your like is the thing that bothers me most about Planck units - looking at the complexity of the expressions that were put together to sift out a single dimension of either length, time or mass, the whole process appears to be nothing but an exercise in manipulation, totally devoid of physics - the dimensions did not evolve from a derivation that has any physical reality
In contrast take hbar/2 - it is a physical constant that pervades the quantum world - it is a consequence of the intrinsic uncertainty of angular position - and is therefore foundational to physics. Someone has already raised the question, since we already have a Planck time - why do we need another one? My answer, one of them is of no physical significance, and maybe neither one is for the purpose of finding fundamentals. But, if there is something deep to be revealed, the very fact we have a short time constant derived from Planck manipulations and a long time constant 1/Ho that measures the Hubble time, which one is likely to turn out to be numerology

 

[ryan albery]

From much admitted ignorance, I wonder if time is the unit in question with this post, rather than mass? My thinking is that a photon, gluon, or other massless particle like perhaps a graviton... they have no mass, so they have no time? Can there be time without mass, or mass without time?

 

[bobie]

It's a curious fact that gravity is 'orthogonal' to electromagnetism.

The source of gravity is the stress-energy tensor. The source of E&M is electromagnetic charge.

Hi, can anyone explain the meaning of these two sentences in simple concepts?
Thanks

 

[Chalnoth]

Hi, can anyone explain the meaning of these two sentences in simple concepts?
Thanks

Basically, 'orthogonal' in this context means that you can think of the electromagnetic force and gravity as being two completely different things. You don't have to know how the electromagnetic force is behaving to understand how gravity is behaving (for the most part).

As for the stress-energy tensor, this is a mathematical object that contains energy, momentum, pressure (compressing/stretching forces), and strain (twisting forces).

For most matter most of the time, the mass-energy is so much larger than the other components of the stress-energy tensor that we can just ignore them and only consider the mass-energy. This is why Newtonian gravity, which only looks at mass, works so well.

But this breaks down for light and for extremely compact objects like neutron stars. With regard to light, for example, if you were to take a simple Newtonian estimate of how much masses tend to bend light as it passes by them, you'd get half the measured value (General Relativity gives the correct prediction). This is because the momentum of photons is the same as their energy, and the Newtonian estimate only looks at the energy.

 

[bobie]

As for the stress-energy tensor, this is a mathematical object that contains energy, momentum, pressure (compressing/stretching forces), and strain (twisting forces).

For most matter most of the time, the mass-energy is so much larger than the other components of the stress-energy tensor that we can just ignore them and only consider the mass-energy. .

Thanks, Chalnoth, that is amazingly clear, but just to understant it fully,
- what is the tensor of the earth? we know mass-energy (10²⁴, the momentum is related to speed 10⁴, is there stress and strain? how do we measure it and what is its value? and
- what is the tensor of a photon 3*10¹⁴ Hz, energy is 1.2 eV, momentum 10⁴ (the same as its temperature), what is stress and strain? and what is its G-field?

 

[Chalnoth]

Thanks, Chalnoth, that is amazingly clear, but just to understant it fully,
- what is the tensor of the earth? we know mass-energy (10²⁴, the momentum is related to speed 10⁴, is there stress and strain? how do we measure it and what is its value? and

There is no single stress-energy tensor for the Earth. The stress-energy tensor is defined at every point in space and time.

However, as I pointed out before, the pressure is pretty much negligible compared to the mass-energy, so we can approximate the stress-energy tensor of the Earth to simply contain the mass-energy density at a particular point, and that's it. This will change, of course, depending upon whether you're measuring near the surface of the Earth, or under water, or within rock, or near the core.

- what is the tensor of a photon 3*10¹⁴ Hz, energy is 1.2 eV, momentum 10⁴ (the same as its temperature), what is stress and strain? and what is its G-field?

I don't recall offhand, unfortunately. It will, at the very least, have energy and momentum components of the stress-energy tensor. I don't remember offhand whether a single photon has pressure components, but for sure a photon gas does.

But by the way, if the energy is 1.2eV, then the momentum has to be 1.2eV/c.

One difficulty here is that the precise modeling of the photon in the stress-energy tensor is kind arbitrary. Do you simply use an infinite-extent plane wave? Or do you use a Gaussian wave packet? Or something else?

I find it a bit easier to deal with a photon gas than an individual photon. When you're measuring at rest with respect to the photon gas, then the energy density component is simply the energy density of the photon gas, and the pressure components of the stress-energy tensor are all one third of that. All other components are zero (no twisting forces are possible with a photon gas, and when you're at rest with respect to the gas, it has no net movement in any direction).

 

[yogi]

I have not posted on these forums for some time, even temporarily forgot my pw - tonight in looking for something stimulating to read, I opened the forums and to my surprise, the first thing I saw was a post I had started some time back. Nice to find a new interest in the subject.

Anyone interest in some thoughts on the subject of natural constants can email me and I will be glad to discuss my own speculations --- since they are not peer reviewed, they cannot be posted here.

Yogi

 

[bobie]

But by the way, if the energy is 1.2eV, then the momentum has to be 1.2eV/c.

Can we express the momentum of light/ a photon in eV? what are its units? not m*v?

 

[bobie]

In another thread it is stated that the unit of mass was derived equalling the Schwarzschikd radius to the Compton wavelength (2)Gm/c² = h/mc, and then the L_p was derived multiplying it by the constant (2)G/c²$$m_\mbox{P} = \sqrt{\frac{\hbar c}{G}}$$
$$l_\mbox{P} = \frac{G}{c^2} * \sqrt{\frac{\hbar c}{G}} = \sqrt{\frac{\hbar G}{c^3}}$$
Does it make any sense to you? can it be true? It is anyway incorrect by a factor of 2!
It seems more logical that M_p was obtained from L_p, as it is just a hypothetical, theoretical quantity of mass that in reality you could never stuff into that space.
Moreover I suppose that gravity make no sense at at distance less than 10^-15m, how could it work at 10^-35,
do you agree?
They say also that at 1,6*10^-35m the laws of physics break down, what laws?
When Planck presented his units did not show what unit he found in the first place and how he derived it?

 

[Chalnoth]

Can we express the momentum of light/ a photon in eV? what are its units? not m*v?

This is the relativistic momentum, which can be defined using:

$$E^2 = m^2 c^4 + p^2 c^2$$

For light, which has no mass:

$$E = pc$$

Now, the energy of light is the Planck constant times frequency:

$$E = hf = pc$$

So the momentum is:

$$p = {hf \over c}$$

But as frequency and wavelength are related to one another by the speed of light, this is simply:

$$p = {h \over \lambda}$$

 

[bobie]

Thanks
Do you know where I can find how Planck derived his units?

 

[bobie]

The stress-energy tensor is defined at every point in space and time.
... the stress-energy tensor of the Earth to simply contain the mass-energy density at a particular point, and that's it.

If I got it right the stress-energy tensor is in space and is determined by mass-energy, mass has (almost ) no stress tensor and so light.
- If it is so what is and what tetermines the stress-bit of the tensor?
- light reacts (passively) to stress tensor in space, but what aspect of it reacts to it? what determines red shift, the loss of energy-frequency? the mass-stress tensor acts on what?

 

[Chalnoth]

If I got it right the stress-energy tensor is in space and is determined by mass-energy, mass has (almost ) no stress tensor and so light.
- If it is so what is and what tetermines the stress-bit of the tensor?
- light reacts (passively) to stress tensor in space, but what aspect of it reacts to it? what determines red shift, the loss of energy-frequency? the mass-stress tensor acts on what?

Your questions don't really make sense to me. Pressure is a compressing or pulling force. Strains are twisting forces. The other components of the stress-energy tensor are energy and momentum density.

As far as gravity is concerned, for most matter, the momentum, pressure and strain are irrelevant, because the mass-energy is so large. This isn't to say that matter doesn't have momentum, pressure, or strain, just that the mass-energy completely overwhelms them as far as gravity is concerned.

Light reacts to the stress-energy tensor in the exact same way all matter does: it follows the space-time curvature produced by the stress-energy tensor.