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 t_Pl 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.  My logic is that, using these terms, everything fundamental resolves to unity, even (argued in the next post) the coupling constants.

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 1/100^th of Planck units.

Recall the table at Constants that Resolve to Unity:

I modified this to reflect Centiplanck units:

With this change to the units, the reduced Planck constant no longer resolves to unity. 

Since it is one of the least quoted derived Planck value, and in that sense least important, I put the Planck charge back to unity:

This change simply makes things worse as now three fundamental physical constants no longer resolve to unity (because raised permittivity and Coulomb's constant are just the inverse of each other, I don't distinguish between them).

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.

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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 further explanation for their value when that value, across the board, when expressed in a fundamental natural unit scheme, is unity.

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* 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.