[PaulTalbot (OP)]

If the equations don't correspond to something physical then it is just math and not physics.

I agree with this last formulation "something physical" but not necessarily "a phenomenon".
When we write E=mc², we assume that mass and energy are related that way, whatever the phenomenon.
The "physical things" are the physical quantities "mass" and "energy". If this relation is right, then all observed physical processes should comply with this relation.

[Bored chemist]

The assumed relations must comply with dimensional analysis, that is both sides of the equations must have the same dimension. Many questions have been posted on the subject, and it is OK. If you believe that some of the assumed relations do not comply with this rule, please let me know.

"Could the Hubble constant be related to the proton radius?"
One has units of 1/time and the otehr has units of distance.
Also, one is observably constant and the other changes.

[PaulTalbot (OP)]

Could the Hubble constant be related to the proton radius?"
One has units of 1/time and the otehr has units of distance.

Good question.
The assumed relation between H (the Hubble constant) and r_p (the proton radius) is
r_p = 2 αc / βH, or
H = 2 αc / β r_p, which is the same.
α is the fine structure constant (e² / 2 ε₀ h c), and
β, the ratio of the electrostatic to gravitational force previously described and calculated.

Since you are a chemist, you may already know the physical meaning of the value αc. It is the speed of the electron in the ground state of the hydrogen atom. From https://medium.com/starts-with-a-bang/ask-ethan-what-is-the-fine-structure-constant-and-why-does-it-matter-178c1a68471b [nofollow]
"If you modeled a hydrogen atom as Bohr did, but took the ratio of a ground-state electron’s velocity and compared it to the speed of light, you’d get a very specific value, which Sommerfeld called α: the fine structure constant."

As for the value of βH, it is the assumed frequency associated with the electron-proton couple (f_ep) as described in the text of the preview.

So the physical meaning of this assumed relation can be stated:
The proton radius equals twice the speed of the electron in the gound state of the hydrogen atom, divided by the frequncy associated with the electron-proton couple.

As there was a typographic glitch in the previously attached chart of the assumed relations, I attach the corrected version to this post. As you can see "Dimensional Numerology" (this is a term I just invented after the comments in this thread) allows linking physical quantities in new ways.

Only observation can tell us if those relations are right, or if they are the result of coincidences.

[Origin]

The assumed relation between H (the Hubble constant) and rp (the proton radius) is
rp = 2 αc / βH

I tried using your equation above and I did not get the radius of a proton.  Could you work out the above equation so I can see where we disagree?

[PaulTalbot (OP)]

I tried using your equation above and I did not get the radius of a proton.  Could you work out the above equation so I can see where we disagree?

The value of α is 7.29735 × 10^-3
That of β is 2.26866 × 10³⁹
and that of H is 2.3338 × 10^-18 s^-1, which is the SI standard notation for the calculated value.
The result of 2 αc / βH is thus 8.264 × 10^-16 m or 8.264 fm, which matches (within 1σ) the proton radius according to recent experiments and calculations [14] [15] [16] [17] (in the preview).

[Origin]

The result of 2 αc / βH is thus 8.264 × 10-16 m or 8.264 fm, which matches

There's the issue, the sources I saw all said the radius is about 0.85 fm or an order of magnitude lower than your number.  Another question, why are the constants in the numerator multiplied by 2 in the equation?

[PaulTalbot (OP)]

There's the issue, the sources I saw all said the radius is about 0.85 fm or an order of magnitude lower than your number.  Another question, why are the constants in the numerator multiplied by 2 in the equation?

For the precision of the recent observed values of r_p, please refer to the references provided int the text (attached). As for the factor 2 in the equation, this results from the way α is calculated. It is the result of incorporating α in
e² / ε₀ E_ep ≈ 0.826 4 fm, (14), which does not include this factor 2.
According to (2) and (5), E_ep can be replaced by β h H. Using (9), and the fine-structure constant (α = e² / 2 ε₀ h c), equation (14) of the proton radius then becomes
r_p = 2 αc / βH = 2 αc / f_ep, (15)

[PaulTalbot (OP)]

Sorry for the second to last post,
8.264 × 10^-16 m is  0.8264 fm.

[Bored chemist]

the speed of the electron in the ground state of the hydrogen atom

LOL.

[Origin]

β, the ratio of the electrostatic to gravitational force previously described and calculated.

Just to be clear β the ratio of the electrostatic to gravitational force between an electron and a proton.  To get a more general relationship you would need to have the masses equal, as has been mentioned before.
I am still not clear on what the relationship is between the fine structure constant, the speed of light,  the Hubble constant and β.  Why would these constants have anything to do with the diameter of the proton?

As for the factor 2 in the equation, this results from the way α is calculated.

Could you explain that a bit more, I still do not understand why the constants in the numerator were multiplied by 2.

[PaulTalbot (OP)]

Could you explain that a bit more, I still do not understand why the constants in the numerator were multiplied by 2.

I would be cautious on that question, since I am way out of my comfort zone.
We can get rid of the factor of two by writing:
r_p = αc / ½ f_ep.
This is not unusual as we can see in the calculation of the kinetic energy : E_k = ½ mv².
As for the physical meaning of that factor ½ preceding the frequency, I think it may be related to the spin of the charged particles (½).
It is as if it took two cycles for the electron-proton couple to fully return to its initial state.

[PaulTalbot (OP)]

Just to be clear β the ratio of the electrostatic to gravitational force between an electron and a proton.

Yes, there seems to be something special about the electron-proton couple, as if they were two sides of the same coin.

[PaulTalbot (OP)]

To get a more general relationship you would need to have the masses equal, as has been mentioned before.

And why so?

[PaulTalbot (OP)]

I am still not clear on what the relationship is between the fine structure constant, the speed of light,  the Hubble constant and β.  Why would these constants have anything to do with the diameter of the proton?

The proton radius I refer to is the rms charge radius of the proton, where rms stands for "root mean square".
https://en.wikipedia.org/wiki/Charge_radius [nofollow]
"The problem of defining a radius for the atomic nucleus is similar to that of defining a radius for the entire atom; neither atoms nor their nuclei have definite boundaries. However, the nucleus can be modeled as a sphere of positive charge for the interpretation of electron scattering experiments: because there is no definite boundary to the nucleus, the electrons "see" a range of cross-sections, for which a mean can be taken. The qualification of "rms" (for "root mean square") arises because it is the nuclear cross-section, proportional to the square of the radius, which is determining for electron scattering."

The value r_p thus refers to the radius of the cross-section of the electron-proton interaction, which has the dimension of an area.
Now, this is precisely what the value f_ep refers to: the frequency of the electron-proton interaction, that is the exchange of virtual photons between those particles. Of course, this reasoning is somewhat speculative, yet it sounds logical to me.

[Bored chemist]

Quote from: Bored chemist on 19/03/2022 20:47:41
The fact that it fails on dimensional analysis is beside the point unless he can explain why something constant is the same as something that's changing.

[PaulTalbot (OP)]

The fact that it fails on dimensional analysis is beside the point unless he can explain why something constant is the same as something that's changing.

OK, I will try to answer your question in a different way.
First, have a look at the attached modified chart on the assumed relations between physical constants.
There is no more reference to the Hubble constant in this chart. All values are considered constant, so your question becomes irrelevant.
The proton radius is still calculated in the same way and with the same accuracy.
The Wesson's quantum of mass is still calculated in the same way and with the same accuracy.
According to known physics, this should be the result of two unlikely and unrelated coincidences.
Those relations assume the existence of a minimum frequency (f_min).

The big question then arises: How on earth could this value of f_min be the same as that of the Hubble constant?

The answer to that question is simple. The assumption that the Hubble constant is decreasing derives from the ΛCDM conformal model:
https://www.forbes.com/sites/startswithabang/2021/01/05/how-is-the-universe-accelerating-if-the-expansion-rate-is-dropping/?sh=19d6ab984093 [nofollow]
The fact is that the ΛCDM model is flawed and doomed. It fails to predict a value of H consistent with observation. This could be the subject of another thread.

[Origin]

Could you explain that a bit more, I still do not understand why the constants in the numerator were multiplied by 2.

I would be cautious on that question, since I am way out of my comfort zone.

How can it be out of your comfort zone if it is something you put in your own equation?  It sounds like you are saying you combined a bunch of constants and then put a factor of 2 in the equation to get the answer you want.  Assuming that is what you did, that is precisely why I am calling your technique numerology.  I can take random constants and combine them in ways to get about anything.  The problem is this results of this technique are meaningless.

This is not unusual as we can see in the calculation of the kinetic energy : Ek = ½ mv2.
As for the physical meaning of that factor ½ preceding the frequency, I think it may be related to the spin of the charged particles (½).

First of all spin, charge and frequency have nothing to do with Kinetic energy formula.  Of course there are numbers in some the equations for physical properties, but they were not arbitrarily thrown in!  The '1/2' term is a direct result of the derivation of KE.  Energy or work is a force applied over a distance.  So integrating F dr will give you the formula for KE.

What you have done is combined random constants to get a predetermined result.  This is going to give a meaningless equation.  I did not read much of your paper, but right off the bat you incorrectly identified the Hubble constant as a frequency and the led you down a path that yielded more meaningless results. 

If you are interested in physics you need to learn some of the basics before you try to come up with new concepts.

[Bored chemist]

There is no more reference to the Hubble constant in this chart. All values are considered constant, so your question becomes irrelevant.

Then you should ask the Mods to close the thread.
The question has been answered.
"Could the Hubble constant be related to the proton radius?"
No, that was a silly idea.

[Origin]

The big question then arises: How on earth could this value of fmin be the same as that of the Hubble constant?

Are you serious?  The reason they are the same is because you arbitrarily defined the minimum frequency as the Hubble constant!

[Origin]

To get a more general relationship you would need to have the masses equal, as has been mentioned before.

And why so?

Obviously if you wanted to compare the forces since the charges are 1 to 1 you would want the masses to be 1 to 1 for a valid comparison.  Doesn't that make sense to you?