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

[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 = $\hbar$c $$GM^2 = \hbar c$$
$$m_\mbox{P} = \sqrt{\frac{\hbar c}{G}}$$
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 = $\hbar$c $$GM^2 = \hbar c$$ →
$$m_\mbox{P} = \sqrt{\frac{\hbar c}{G}}$$
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 = $\hbar$c $$GM^2 = \hbar c$$ →
$$m_\mbox{P} = \sqrt{\frac{\hbar c}{G}}$$
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.