Sunday 10 March 2024

Return to Constants that Resolve to Unity

When I finished The Conservatory – Notes on the Universe, I went away and began to worry that I should remove the last couple of sentences.  Part of my concern, mulling over it, was due to me leaving out the tP term to not only emphasise that it’s the age of the universe in units of Planck time, but also to both make it work if you try to use different units for the age of the universe and ensure that the temporal term is cancelled out.

The other concern was that perhaps I was being strong in my implication that Planck units are the fundamental units.  I know that there are people out there who will argue differently.  So, I did what I normally do in a situation like this and fiddled with a spreadsheet, and I am now prepared to double down.  Planck units are the fundamental units.  And the terms that I introduced, “charge to structure ratio”, “raised permittivity” and “reduced permeability”, should be used (more) widely.  My logic is that, using these terms, everything fundamental resolves to unity. 

Note that masses of electrons and the other particles that in turn make up protons and neutrons are not, by this definition, fundamental but rather arise out of particle physics processes within spacetime.  Given that there is a suite of such particles, it would be rather difficult to point to any one of them as having a fundamental mass anyway.

I tried to find a different set of natural units that might replace Planck.  First I assumed that we’d need something smaller, so I posited “Centiplanck” units that are all 1/100th of Planck units.

Recall the table at Constants that Resolve to Unity:

I modified this to reflect Centiplanck units (noting that I have also removed raised permittivity, as it is merely the inverse of Coulomb’s constant):


With this change to the units, the reduced Planck constant and the charge to structure ratio no longer resolve to unity. 

Since the charge unit is prime to the charge to structure ratio, I tried a different scheme that brings at least that value back to unity, returning to Planck charge:

This change simply makes things worse as now three fundamental physical constants no longer resolve to unity.

So I tried again by returning mass to its normal Planck value:

This is yet worse, with four fundamental physical constants no longer resolving to unity.

I was finally ready to take the nuclear option, to see what happens when I break the link between temporal and spatial units, knowing that by doing so I would lose the speed of light as a unit that resolves to unity:

This merely shifts the deckchairs around because while one fundamental physical constant now resolves to unity, we only get that by losing the speed of light.  The other option is worse:

None of fundamental physical constants resolve to unity.   There is no point fiddling any further with the unit of charge, because that messes up the charge to structure ratio nor with the unit of mass, because the best that can be achieved is the return of two fundamental physical constants (the Coulomb’s constant and Reduced Planck constant), using an arbitrary unit that we could call the Megaplanck unit of mass, about 2.2g.

---

From the analysis above, it is obvious that there is only one natural unit scheme which allows all fundamental physical constants to resolve to unity.  Therefore, I do not believe that there can be more fundamental natural units – and it would be very strange indeed if Planck units were not woven into the very fabric of the universe*.

That is not to say that there is no value in other “natural units” to make calculations in your field of expertise easier, merely that none of the constants require any explanation for their value when that value, across the board, is unity.

---

* I was waxing poetic here, there is no fabric of the universe per se, and certainly no weaver.  Hopefully the reader understands what I was getting at when I wrote that.