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

[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/c²m_\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.

 

[yogi]

How space supports stress is unknown - we do not have an agreed upon model of space - strain in mechanical physics is change in length - but we do not know what this means physically when applied to static space. Normally stress strain relationships are useful to express changes \within some elastic range - but space is not elastic in the conventional sense.

 

[Chalnoth]

How space supports stress is unknown - we do not have an agreed upon model of space - strain in mechanical physics is change in length - but we do not know what this means physically when applied to static space. Normally stress strain relationships are useful to express changes \within some elastic range - but space is not elastic in the conventional sense.

There is no stress in empty space. In fact, the stress-energy tensor is identically zero in empty space.

 

[yogi]

There is no empty space ... the g fields of local matter are negative energy - expansion of energy creates stress.

 

[Chalnoth]

There is no empty space ... the g fields of local matter are negative energy - expansion of energy creates stress.

This is false. This "energy" you speak of doesn't appear in the stress-energy tensor, and there are no stresses in space that contains no matter or light.

 

[yogi]

That is the shortcoming of describing dynamic space in terms of a static stress-energy tensor. Einstein eventually gravitated (excuse the play on words) toward the idea of space a sort of medium ..."being every place conditioned by the presence of matter at a particular location and in neighboring places." In his first 1916 publication of the General Theory, the equation of state was based upon the counter acting factors of static pressure and density. With the introduction of the CC to balance G, the universe gained a dynamic functionality which could not then appreciated.

 

[Chalnoth]

That is the shortcoming of describing dynamic space in terms of a static stress-energy tensor. Einstein eventually gravitated (excuse the play on words) toward the idea of space a sort of medium ..."being every place conditioned by the presence of matter at a particular location and in neighboring places." In his first 1916 publication of the General Theory, the equation of state was based upon the counter acting factors of static pressure and density. With the introduction of the CC to balance G, the universe gained a dynamic functionality which could not then appreciated.

The stress-energy tensor isn't "static". It is defined at every place in space and time. This means, for example, that at any particular spatial point, the value of the stress energy may (and often does) change over time. There are, in some space-times, possible sets of points which have a constant stress-energy tensor over time. But most choices of points typically won't have this situation (unless we're in Minkowski space).

 

[Chalnoth]

Anyway, I think what might help to shine a light on this is to consider the electromagnetic field. The source of the electromagnetic field is electric charge (and quantum spin). There are electromagnetic fields even in locations where there are no charges. The electromagnetic field may take on rather complex and beautiful shapes in areas where there is no charge. Those shapes depend upon the locations where there are charges.

With regard to gravity, the analog of charge is the stress-energy tensor. The analog of the electromagnetic field is the metric. The metric may (and often does) take on very non-trivial behavior in locations where there is no stress-energy at all. Perhaps the most severe example of this is the Schwarzschild metric, which can be thought of as analogous to the electric field of a point charge. Just as the electromagnetic field stretches out to infinity from a point charge, the Schwarzschild metric includes curvature of space-time out to infinity from the black hole (in fact, the long-distance behavior is 1/r^2 in both cases).

 

[Chronos]

Allow me to digress to the OP, the Planck mass is derived from other 'fundamental' physical constants. It is the smallest mass capable of generating a black hole with a Planck length event horizon. It is, obviously, not the smallest mass that exists in nature. The smallest 'masses' in nature are not even well measured. The neutrino mass is incredibly tiny, and even it, at least in theory, cannot definitively claim the title as the smallest natural unit of mass. That should tell us we have missed something along the way.

 

[bobie]

... the Planck mass is derived from other 'fundamental' physical constants. It is the smallest mass capable of generating a black hole with a Planck length event horizon..

Thanks, Chronos, for that explanation. It seems to show that M_p is derived from L_p via the Schwarzschild radius, which probably was unknown to Planck.
- Planck's formula differs by a factor of 2 : L_p = [2] GM/c² * M_p, is that detail irrelevant?

- and what about the claim that M_p is derived equalling the r_s to the Compton wavelength:
(2) GM/c² = h (/4π)/ Mc =>
GM² = \hbarc GM^2 = \hbar c
m_\mbox{P} = \sqrt{\frac{\hbar c}{G}}
this formula differs by a factor of /2π
is it a grounded claim, does it make sense, could it be the real way in which Planck derived all his units?

 

[yogi]

All of which makes consistent sense provided the Planck length can be independently established as a validity. This was Dirac's hang-up and mine - the congruence of the electrical to gravitational force ratio and the ratio of the size of the universe to the size of the electron (10 to the 42 power if we use the force between an electron and positron). Now let's look at the units of G the suspected variable of Dirac's Large Number hypothesis. G has units of volumetric acceleration per unit mass - how is it that volumetric acceleration would be the same for a universe the size of dime and one having a radius of billions of light years - variance in G over the history of cosmic evolution is fatal to the sanctity of Planck numerology.

Now someone will post: - but we know G is constant because the orbits of the planetary moons are stable. To which I will reply "But such measurements only verify the constancy of the MG product" - and since G like inertia is always proportional to mass, inertia and G may simply be interrelated pseudo forces - an idea first suggested by Feynman.

As previously stated, I distrust assertions that elevate Planck units to a preferred status over any other combinations of units that reduce to a single dimension - I would appreciate being taught how Planck units predict any of the known forces. in which case i will change my mind.

Thanking all who replied - Like Weinburg's mass, the value Hh/c^2 probably has no significance - just one more interesting but valueless result of mis-spent doodling with paper and pen

 

[bobie]

.. is fatal to the sanctity of Planck numerology.
... I distrust assertions that elevate Planck units to a preferred status over

I do not know how Plancks units are regarded and if these statements have currency.

I understand that Planck has just established the shortest length possible, (the length under which some phenomena and laws become meaningless), from which he has derived the shortest unit of time (L*c), then the smallest mass that can (in theory) generate a BH (L*c²/G) (which is by no means the smallest mass possible) 16 years before Schwarzschild...
I can't see how L_p has a preferred status, it simply says that 1 cm = 1.6*10³³ L_p, and this might be least meaningful length. Is this correct?

From what I researched it seems that Planck did not explain the genesis of the units, and that nobody has yet been able to deduce it. But it surely must have a rationale and is not numerology, if cosmologist use those units. Is this correct?

I am asking if the source of the units is (or is considered plausible) equating r_s to λ_C because that would be a giant step forward

 

[yogi]

The justification for Planck units that is commonly promoted is that the length is made-up of a G factor - which is then rationalized as being a sort of bridge between the subatomic angular momentum constant h and the macro world governed by G - maybe even a pseudo bridge between quantum mechanics and classical field theory. Much time and effort is made to fit theory to conform with the Planck length - articles have suggested that LQG has been unduly limited thereby. If a dimensional factor is to be useful, it should make a prediction that is unique - Planck units from my biased perspective, seem to be an impediment to progress rather than a tool for revealing something new and wonderful about the world. I could be persuaded otherwise - after all, the fun is in looking for the answers

Yogi

 

[Chalnoth]

The justification for Planck units that is commonly promoted is that the length is made-up of a G factor - which is then rationalized as being a sort of bridge between the subatomic angular momentum constant h and the macro world governed by G - maybe even a pseudo bridge between quantum mechanics and classical field theory. Much time and effort is made to fit theory to conform with the Planck length - articles have suggested that LQG has been unduly limited thereby. If a dimensional factor is to be useful, it should make a prediction that is unique - Planck units from my biased perspective, seem to be an impediment to progress rather than a tool for revealing something new and wonderful about the world. I could be persuaded otherwise - after all, the fun is in looking for the answers

Yogi

Honestly I think you're reading way too much into these units. Units like Planck units really aren't saying anything profound about the universe. At their heart, they're little more than combinations of dimensionful constants that set all of those constants equal to 1 in those units. There are many ways of doing this (see here: https://en.wikipedia.org/wiki/Natural_units), and they all differ somewhat depending upon which constants are set to unity. They are very useful in contexts where one is dealing with relationships between very different units (e.g. length, mass, and time), as they tend to reduce relationships between different dimensionful quantities to rational fractions, perhaps with some multiples of \pi thrown in here and there.

 

[yogi]

Chalnoth - your post 83. I fully concur with the fact that too much is made of Planck units - it is hard to find a book written by any of the popular science writers that doesn't deify Planck length and Planck time - they have a little problem with the significance of the Planck mass.

Setting the constants G and c = to "1' is a quite common practice - but for an old Engineer like myself, its shocking - the operative magnitudes and the dimensionality of the terms frequently reveal surprises as one plows through a derivation in the hope of finding things that will cancel to simplify the result

 

[bobie]

Honestly I think you're reading way too much into these units. Units like Planck units really aren't saying anything profound about the universe. At their heart, they're little more than combinations of dimensionful constants that set all of those constants equal to 1 in those units. There are many ways of doing this .

Probably the units are not deified, but Yogi is not completely wrong:
from what you gather from cosmology, string theory etc. Plancks units appears to be a lot more than an arbitrary combination of constants, and not one of the many ways in which you can do it:

Planck time, the smallest observable unit of time...before which science is unable do describe the universe

it would become impossible to determine the difference between two locations less than one Planck length apart.
...In string theory, Planck length is the order of magnitude of the oscillating strings that form elementary particles, and shorter lengths do not make physical sense.

They are something special, the do have a particular importance.

Of course if you put the unit of length 3*10¹⁰ shorter than the unit of time, c becomes 1:
c_p = 10⁴³ L / 10⁴³ T = 1 L/T, but ...
could you show how G becomes 1 if we decide that 1 cm = 1.6*10³³ cm and the mass of the Earth is expressed in M_p?, wouldn't it become
G_p = 5*10⁶⁷ (6.67*10^-11*1.6*10³⁵²*5⁷) and g = G_pM/ L_p² = 9.8 10³⁵ L_p/ 10⁴³ T_p?

On the contrary, the unit of mass M_p seems no big deal as it doesn't say anything new or strange: even if Planck units did not exist, the Schwarzschild mass at 1.6*10^-33 cm could be calculated as 2*10^-5 g.

Besides the importance in cosmology etc, they have a huge theoretical importance and might be saying something profound if , (as it seems) they suggest that the universe is de facto discrete, although this be not generally acknowledged. Is that wrong?
Thanks, Chalnoth, for your help

 

[Chalnoth]

They are something special, the do have a particular importance.

These are generally little more than speculation. We do not know the smallest unit of time that it is possible to measure, because we haven't probed physics anywhere near the Planck scale.

There may be reasons to suspect that the true limit is close to these values, but there's little to no reason to believe it is exactly at those values.

 

[bobie]

...speculations...There may be reasons to suspect that the true limit is close to these values, but there's little to no reason to believe it is exactly at those values.

I only reported what you read in cosmology.
- What do you think of the derivation of M_p from r_s = λ_C
- Do you agree about the absolute irrelevance of M_p, and that G is not = 1?

 

[Chalnoth]

I only reported what you read in cosmology.
- What do you think of the derivation of M_p from r_s = λ_C

I don't understand what you're trying to say here.

- Do you agree about the absolute irrelevance of M_p, and that G is not = 1?

No. The value of G as a dimensionful parameter is meaningless: it's just a unit conversion factor. What are meaningful are dimensionless ratios of fundamental constants. So the use of dimensionless constants is that it gets the cruft of meaningless units out of the way, and exposes more meaningful relationships between various things.

So it's not irrelevant. It's very useful. And it does, potentially, tell us something about physics at very high energies. But we can't really say for sure: as I said, it's speculation. It's informed speculation, but speculation nonetheless.

 

[bobie]

I only reported what you read in cosmology.
- What do you think of the derivation of M_p from r_s = λ_C

I don't understand what you're trying to say here.

I mentioned in post #79 that here at PF I read somewhere that the unit M_p was not derived finding the mass that would generate a BH at L_p, but

equalling the formulas of the r_s and the Compton wavelength (divided by a factor o 2π):
Schwarzschild radius ((2) GM/c² )= (h (/4π)/ Mc) Compton wavelength →
GM² = \hbarc GM^2 = \hbar c →
m_\mbox{P} = \sqrt{\frac{\hbar c}{G}}
it seems suggestive,
What sense does it make to you, when the Schwarzschild radius of a mass is equal to its minimal wavelength? or, does it make sense at all?

 

[Chalnoth]

I mentioned in post #79 that here at PF I read somewhere that the unit M_p was not derived finding the mass that would generate a BH at L_p, but

equalling the formulas of the r_s and the Compton wavelength (divided by a factor o 2π):
Schwarzschild radius ((2) GM/c² )= (h (/4π)/ Mc) Compton wavelength →
GM² = \hbarc GM^2 = \hbar c →
m_\mbox{P} = \sqrt{\frac{\hbar c}{G}}
it seems suggestive,
What sense does it make to you, when the Schwarzschild radius of a mass is equal to its minimal wavelength? or, does it make sense at all?

As Planck first derived these units decades before the Schwarzschild metric, this isn't a possible motivation. I think it was just an exercise in deriving a series of units where G=\hbar=c=k_C=k_B=1.

The fact that a black hole with Planck mass has an r_s that is very close to the Planck length is, in large part, due to the bit that I mentioned before about how rendering these things in such units reduces the results of most calculations to rational fractions not too far from one times some power of \pi. You'd be surprised how many calculations done in natural units come up this way.