SI World and Planck World - And an IDEA

I was thinking about adding this to the back of the last post, Coupling Constants, but it was already pretty long.  To understand the following though, it is probably useful to read that post first.

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To make my point, I am going to compare the same activity conducted in two different worlds.  The first world I am going to call SI world, in which SI units are used.  The second world is Planck world, in which – unsurprisingly – only Planck units are used.  In Planck world, everyday people are used to talking in terms of “p-mass”, “p-length”, “p-time”, “p-charge” and all of the derived units, such as “p-force” (instead of newtons), “p-energy” (instead of joules), “p-current” (instead of amperes) and so on.  Because they live in Planck world, they know that all the fundamental physical constants are unity (see the table at the end of Coupling Constants) and they do everything in terms of Planck.

The activity is considering the force on a medium sized pineapple at sea level on an Earth-like planet.

There’s the simple way:

And then there’s the complex way, noting that we wouldn’t use the complex way other than as a method to extract a value which we can use to compare the strength of forces:

The complex way provides us with a couple of things, the value of n₁n₂/r², which could be reused if were talking about a conglomeration of charged massive particles (like protons) and, via a little side track, the coupling constants (which for Planck world is, of course, unity).  Note however that the magnitude of the gravitational coupling constants relates only to the square of the mass divided by Planck mass, so the side track isn’t really necessary in either world.

The same sort of process applies if we consider charged objects, imagining the same number of items (protons) for each object and the same distance, and noting that the resultant force will be one of repulsion:

Note that α_Ep is the fine structure constant, α.  Again, it is just the square of the value (the elementary charge) divided by the related Planck value (Planck charge).

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There’s no real reason to use the complex method to calculate the force.  And as mentioned above, there’s no need to use the side track in either case (gravitational or electromagnetic).  If you want to know the strength of coupling associated with any reference mass or charge, just divide the reference value (mass or charge) by the related Planck value (mass or charge) and square the result.  Simple.

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Just out of curiosity, I plugged in different values, namely two objects with a Planck unit each (mass in the first case and charge in the second).  These are the results:

Unsurprisingly, from one perspective at least, the values end up the same.  The question then arises: what is the Planck charge?  One way to think of it as the charge on the smallest possible charged (but non-rotating) black hole.

Such a black hole has a radius of r_s=2GM/c²=l_Pl.  So, M=c²l_Pl/G/2=m_Pl/2. It also becomes “extremal” when 2r_Q=r_s, where r²_Q=Q²G/(4πε₀c⁴), so:

2r_Q=2Q.√(G/(4πε₀c⁴))=l_Pl

Q=l_Pl.c².√(4πε₀/G)/2

Noting that l_Pl=√(ħG/c³) and

Q=√(ħG/c³).c².√(4πε₀/G)/2=√(ħc.4πε₀)/2=q_Pl/2

This means that the ratio of charge to mass in an extremal Reissner–Nordström black hole is unity, and half a Planck charge is the maximal charge on a minimal black hole (of half a Planck mass and radius of one Planck length).  This does at least gives some sort of meaning to the values I used, although it should be noted that it would not be possible to have such masses or charges at a separation of only one Planck length.  Halve them and you might be in business.

Generally, gravitation is labelled as a weak force, while electromagnetism is considered to be strong(er).  This is because, when considering atomic scales, we are talking about quite small masses and multiples of elementary charge.  Consider a hydrogen ion, which is a proton (the other isotopes of hydrogen which have neutrons have the special names deuterium [²H] and tritium [³H]).  This has a mass of 1.673×10^-27 kg, or 7.69×10^-20 units of Planck mass, and one elementary charge of 1.602×10^-19 C, or 0.0854 units of Planck charge.  So, yes, gravitational force between two protons is miniscule in relation to their mutual electromagnetic force of repulsion, but only because the charge is enormous by comparison.

What we could say, perhaps, is that subatomic (and elementary) particles can carry significantly more charge than mass.  Why is that?  I don’t know if anyone has even asked the question before.  There is fact that particles have relativistic mass, which varies between frames with different speeds, but not relativistic charge.

So, the question that comes to me is, what speed (in a given frame) would a proton need to reach to have the same magnitude of Planck mass as it has in Planck charge.

The relativistic mass equation is (where m₀ is the rest mass in a given frame):

m=m₀/√(1-v²/c²)

Rearranging for v:

v =√(1- m₀²/m²).c

We want to get the magnitude of the mass of the proton (m₀=m_p) in units Planck mass to equal the charge (e) in units of Planck charge (||m||=||e||):

v =√(1- (7.69×10^-20)²/(0.0854)²).c=√(1- 8.09×10^-37).c≈(1-4.05×10^-35).c

In other words, a proton achieves the relativistic mass equivalent in magnitude to its charge only when extremely close to the speed of light.  There is plenty of wriggle room in there, for sure, but the bottom line is that in order that the relativistic mass doesn’t get bigger than the charge (within realistic speeds), the rest mass of a proton must have a magnitude that is significantly lower than the magnitude of the charge. Other subatomic and elementary charged particles are available, but they all have significantly less mass than the proton (and charge within one order of magnitude).

The results in the tables above might make one wonder if the Planck force is a maximum.  It could be, at least in a sense.  It is the force required to accelerate one unit of Planck mass to the speed of light in a period of one Planck time, which would require one unit of Planck energy.  It would basically be instantaneous, especially if Planck time is fundamental, since it would be across a distance of one unit of Planck length.  Remember that one unit of Planck length per unit of Planck time is the speed of light.

The Falcon Heavy rocket has a maximum thrust of 22.8×10⁶ N. NASA’s Space Launch System (Block 1) has a maximum thrust of 39×10⁶ N.  The Starship rocket (still in development) has a projected maximum thrust of 89×10⁶ N.  These are a long way short of 1.21×10⁴⁴ N, but of course we talking about a completely different scale.

For comparison, we can consider the strong force.  Note that in the second paragraph of the related Wikipedia page, the forces are expressed in terms of a proton at an approximate distance of 10^-15 m.  We know the mass and charge of a proton, and can assume that the interaction is between two protons (being bound into a nucleus).  Here are the gravitational and electromagnetic forces involved (just the simple calculations, since it can be seen above that the force and coupling constants don’t change):

The strong force is approximately 100 times stronger than electromagnetic force so, in this case (between two protons), in the order of 20 kN.  See also here (per Bdushaw – I don’t know how accurate this is, or whether it is merely supposed to be representative):

Curiously, this value is within one order of magnitude of the force due of two Planck black holes (m=m_Pl/2) or indeed two Planck masses (m=m_Pl) separated by the same distance – namely 10^-15 m.

The forces are 8 kN in the first instance and 31 kN in the second or, in both cases, approximately 10⁴ N.  Given that this is based on a distance of approximately 10^-15 m and a force that is approximately 100 times stronger than the electromagnetic force, the results being approximately the same is remarkable.

Coincidence?  Probably.  I can’t think of a reason why the maximum strong force at a femtometre should be equivalent to the gravitational force between two individual Planck masses that are separated by a femtometre (and the force distance relationship would certainly not follow the curve shown above).

Well … not a good reason.  Here’s a completely bonkers reason.  Say we had an Intelligent Design and Engineering Agent (IDEA) who has committed to making the background of the universe pretty much like ours, a Planck universe in which the fundamental physical constants resolve to unity and the minimum divisions of space and time are Planck values.  The IDEA has a vague plan for the proton, but hasn’t set a size, mass or charge on it yet, just knowing that protons and neutrons will be bound together in a nucleus by some sort of strong force.  In the worst case, two protons will have to be bound together at a distance of a femtometre, but not for very long (note that ²He is extremely unstable, with a half-life of about 10^-9s).  The IDEA wants to leave open a wide range of values for the mass and charge of the proton (but knows that the proton will be very small), so what value of the strong force at about a femtometre will be required?

In summary:

•    The proton can be any size (smallest being a Planck volume)
•    The proton can be any mass (although the maximum mass in a Planck volume is half a unit of Planck mass)
•    The proton can have any charge (although the maximum charge in a Planck volume is half a unit of Planck charge)
•    Separation between two bound protons in worst case is ~1fm
•    Worst case is maximum charge, minimum mass (effectively zero mass)
•    The IDEA selects, for some reason, the Planck volume as the reference size of the yet to be determined proton

So the IDEA would need a strong force that just about counteracts the repulsive force of two half units of Planck charges:

Therefore, a bit under 10kN.  But the IDEA is not just a designer, it is also an engineer which means that it is going over-engineer the design, so we can easily double that – bump the strong force up to about 20kN.

(Note that, in some explanations this doesn't actually seem to be required.  In those explanations, the apparent factor of 2 appears to be due merely to vague specifications.  For example, the strong force is sometime defined as 137 times stronger [or approximately so] than the electromagnetic force for an elementary charge.  If that were so, the would mean that the strong force is precisely strong enough to account for two charges acting on each other that are q_Pl/e greater than e, or (q_Pl/e)²=1/α stronger.  I have my doubts about this, in part because the strong force as commonly referred to is actually the residual strong force, being the force that is left over from the actual strong force that is holding the constituent quarks together.  It would surely be nice to be able to say that the strong force is has a strength of unity in some sense, but things are not quite that simple.)

This over-engineering would mean that if protons were particles with minimal mass and the maximal charge on a Planck black hole (half a unit of Planck charge), then the nucleus would still hold together with the strong force.  Once this value had been set, the IDEA would be limited to a low proton mass value, because the ²He nucleus would become stable if protons were significantly heavier.

(Stability of ²He nuclei would affect the availability of free protons to be fused into isotopes of hydrogen on the path to other isotopes of helium and maybe prevent stars from forming, meaning that shining stars, planets, pond scum and, ultimately, IDEA worshipping humans would then have to be created using magic and/or miracles rather than physics).

Of course, I don’t think that this is the explanation.  But it is interesting that the forces involved are in approximately the same order of magnitude.

Coupling Constants

I revisited Constants that Resolve to Unity recently and I updated the table there, replicated below.

For the most part these were minor changes, correcting the symbols for current and temperature, fixing the column width, making the subscript notation consistent and so on.  But there was a more significant change, which was a mention of the ratio between the square of elementary charge e and Planck charge that is expressed by fine structure constant α.

I had been thinking about doing something similar with another value, the gravitational coupling constant α_G, but this is a bit of an odd constant as it’s often poorly defined.  I kept niggling at it, trying to get a good explanation of what all these coupling constants are, including the fine structure constant.  I noticed that there is a mathematical similarly between α and α_G, beyond them both using alpha as their notation.  Namely, that α=e²/q_Pl² and α_G=m²/m_Pl² (where m is the mass of whatever you are using as the basis of the gravitational coupling constant, which you can clarify by using a subscript to both the m term and after the G in the α_G term, but they often don’t).

The clearest explanation for these coupling constants that I found was here, but there is an error in it and it is not completely clear, so I will have a hack at it in a way that totally avoids the introduction of that error.

Thanks to Professor Watkins, who has since retired.

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I will start with gravitational coupling since this is the most intuitive and then go back to charge coupling.

We know that a body with mass m₁ that is a distance r away from a second body with mass m₂ will experience a force of gravity F_G given by:

F_G=Gm₁m₂/r²

We could think of m₁ and m₂ being accumulations of unspecified numbers n₁ and n₂ of neutrons that each have mass m_n such that m₁=n₁m_n and m₂=n₂m_n.  This means that the force F_G varies by the number of neutrons involved and the distance and is mediated by a constant value (dubbed a “force constant” by Watkins, with the choice to use φ to denote it being mine).  So:

φ_G=G.m_n²

And:

F_G=Gm₁m₂/r²=φ_G.(n₁n₂/r²)

Watkins then introduces the term ħc and shows, numerically, that:

φ_G/ħc=α_G

where α_G is specifically the gravitational coupling constant for a neutron, so to be more accurate, after rearranging:

α_Gn=φ_Gn/ħc=m_n².(G/ħc)

Note that the value of Planck mass is m_Pl=√(ħc/G).  Therefore, G/ħc=1/m_Pl² and so:

α_Gn=m_n²/m_Pl²

Watkins does not explain the introduction of the term ħc, so I suggest a slightly different approach.

The term φ_G, or rather φ_Gn, can be broken down further into two terms, one that carries the units and one that is dimensionless, noting that the dimensionless term m_Pl²/m_Pl²=1:

φ_Gn=G.m_n².m_Pl²/m_Pl²=(m_n²/m_Pl²).(Gm_Pl²)

The ratio of the two masses is, of course, dimensionless and can therefore be assigned as α_Gn.  We know that m_Pl=√(ħc/G) so the second term becomes:

Gm_Pl²=G.(√(ħc/G))²= G.ħc/G=ħc

So we arrive at φ_Gn expressed as two terms (the second being the ħc that Watkins introduces without explanation):

φ_Gn=α_Gn.ħc

or, rearranged:

α_Gn=φ_Gn/ħc [=m_n².(G/ħc)]

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Note that the derivation above was done in terms of a neutron mass m_n but there was no stage at which the value of m_n needed to be specified.  Therefore, the exact same logic applies irrespective of what mass we use, so we could say:

α_Gx=m_x²/m_Pl²

where x can refer to any mass; that of a neutron, electron, proton, Planck mass and any other random mass you want.  Filling in the first four, we arrive at:

 

The error that Watkins introduced was with the value of α_Gn.  For some reason, Watkins arrives at 3.76915×10^-39 as the ratio between 1.871855×10^-64 and 3.1616×10^-26, which is clearly wrong (1.871855/3.1616=0.5920594, with the small variation in magnitude being due to a difference in our values for G).  This error was irrelevant to the logic of the derivation.

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The same sort of process as above can be used to derive an electromagnetic coupling constant via the formula:

F=q₁q₂/4πε₀r²

Using the same logic, thinking of q₁ and q₂ as unspecified numbers n₁ and n₂ of units of elementary charge e, we see that q₁=n₁e and q₂=n₂e.  There is another force constant φ_E implied here (E for electromagnetic):

φ_E=e²/4πε₀

Watkins just introduces ħc, so that we get:

α_E=φ_E/ħc=e²/(4πε₀.ħc)

And, because q_Pl=√(ħc.4πε₀):

α_E=e²/q_Pl²

The electromagnetic coupling constant associated with the elementary charge isn’t normally called that, but is rather referred to as the fine structure constant, and the subscript E is not used.  So:

α=e²/q_Pl²

As with the gravitational case, the φ_E term can be broken down, this time using q_Pl²/q_Pl²=1:

φ_E=e²/4πε₀.q_Pl²/q_Pl²=(e²/q_Pl²).(q_Pl²/(4πε₀))

The ratio of the two charges is dimensionless and can be assigned as α.  Noting that q_Pl=√(ħc.4πε₀), the second term becomes:

q_Pl²/(4πε₀)=ħc.4πε₀/(4πε₀)=ħc

So:

φ_E= α.ħc

Or, rearranged:

α_E=φ_E/ħc [=e²/(4πε₀.ħc)]

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Note also that k_e=1/4πε₀, so:

α_E=e²/(4πε₀.ħc)=e²/(ħc/k_e)

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Similarly with the gravitational coupling constant derived above, a particular charge was selected with the electromagnetic coupling constant (or fine structure constant) relating specifically to the elementary charge.  Other charges are available, such as the charge on up and down quarks, +2e/3 and -e/3 respectively, or the Planck charge.

Therefore, as per the logic above, we could say that:

α_Ex=q_x²/q_Pl²

where x can refer to any charge.  This gives us, for the options above:

 

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Note that I have included two terms α_GPl and α_EPl.  These could be referred to as the “natural coupling constants” (gravitational and electromagnetic respectively).  For the purposes of adding to the suite of constants that resolve to unity, this means I now have two more.  Doing so demotes the fine structure constant to what it actually is, just the square of the ratio between the charge selected to represent (so, the elementary charge) and the natural unit (the Planck charge).  The same applies to whatever variant of the gravitational coupling constant you would like to talk about.  The natural value is unity, and any other value is merely the square of the ratio of the mass selected and the Planck mass.

Of course, it would no longer be possible, if anyone actually does this, to say that the fine structure constant tells the universe what the relationship between the elementary charge and the Planck charge is.  But it doesn’t, the fine structure constant is merely something that falls out of the fact that the elementary charge has a specific value.

I remain totally unconvinced that the fine structure constant tells us anything more than the ratio between the elementary charge and the Planck charge.  The physical interpretations given at Wikipedia all seem to fall out of this ratio.  (I plan to write something on this in a later post.)

The same logic applies to the various options for the gravitational coupling constant.

To ram the point home further, we could establish the gravitational coupling constant in terms of standard duck masses (m_duck), because the equation is, very simply:

α_Gduck=m_duck²/m_Pl²

And the same if we use the standard charge on a post-launch duck (q_duck), in which case the duck centred fine structure constant becomes:

α_duck=q_duck²/q_Pl²

By careful selection of the values of m_duck or q_duck, a sufficiently motivated researcher could come up with startling numbers the coupling constants – much more exciting one than ~1/137.  Want it to equal pi to ten significant figures, just make q_duck=3.3233509704×10^-18C.

The immediate response is (or should be) that this duck centred fine structure constant (α_duck=π) doesn’t mean anything.  It certainly wouldn’t tell us why the standard charge on a post-launch duck (q_duck) had that value, if it did.  The value of the α_duck would simply be a different way of saying that q_duck has the value that it has (q_duck= q_Pl.√α_duck) – a simple truism.

Similarly, none of the variants of the gravitational coupling constant nor the fine structure constant are really telling us anything useful in themselves.

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There are (at least) two other coupling constants (associated with weak and strong interaction), but I don’t really want to get into them.  I don’t think they would relate to Planck units, having done a quick check.  Perhaps someone else with a better knowledge of such things could have a good think about it and let me know.

It is possible that they tell us something, but it is equally possible that these coupling constants are similar to the gravitational and electromagnetic coupling constants in at least that sense and tell us nothing new at all.  My brief and patently inadequate check indicated that while the wording “coupling constant” is used for all four, and the meaning is comparable between the weak and strong interactions, they are not actually the same sort of thing as we talking about with respect to electromagnetic and gravitational force coupling.

That said, it is interesting to note that the strong force is usually thought of having a relative strength of 1.  Maybe you might thing, ah, but that's different.  Note however, that while the relative strength is listed at the link above as being in the order of 10^-3, it's more often listed as being in the order of 10^-2 and frequently more specifically as 1/137 (or more precisely, α).  Sadly, however, if you dig a bit deeper, you will find that the strong force is only approximated as 1, and sometimes has a value closer to 14.5 (or the inverse thereof, the explanation I saw was a bit vague on that point), and - in terms of a nucleus - is only the residual force left over from holding the constituent quarks of a proton or neutron together. 

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Finally, here is the updated table of constants that resolve to unity (noting that I have removed the charge to structure ratio as irrelevant, added some more formulas for clarity and also changed to a font that makes the difference between a large “i” and a small “L” a little more distinct):

And, for ease of reference, a clean version with the greyed-out items removed:

All unity, as far as the eye can see.  Truly beautiful.

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.

The Conservatory - Notes on the Universe

In late February 2024, Sabine Hossenfelder put out a video on energy conservation.  The link there is to the point at 2:43 where she talks about the effect of space expanding on energy.

The key point, for the purpose of this discussion, is that a photon ends up having less energy after the space it is in expands.  That got me thinking and, as a consequence, I want to go through a thought experiment.

Imagine that you are the god of physics, starting off with nothing.  You want to create something, but you have an energy budget of precisely zero (in part because, of course, you don’t exist).

Say then that you create, over a period Δt, a circle of radius r=x.  Effectively, what you have done is stretch, out of nothing in this case, a curve to a length of l=2πx, the circumference of a circle with radius r=x.  We can think of this length having potential energy, because all else being equal, it would want to go the ground state (being nothing).  Note that potential energy can be thought of as negative.

The other way to balance the energy budget is for there to be energy associated with the circumference of the circle, positive energy that is of equal magnitude to the potential energy.

Now imagine that the energy that balances out the negative energy due to the expansion is expressed as if it were carried a single photon. As Sabine mentioned, the energy of a photon is inversely proportional to its wavelength, E=hc/λ.  The lowest possible energy, therefore, relates to the longest possible wavelength associated with the circumference l=2πx.

It might seem intuitive to say that that wavelength is λ=2πx, but note that the distance between the nodes (null points) of a wave is actually half a wavelength:

So, the longest wavelength associated with any segment is double the length of that segment or, in this case, λ=4πx.  We will call this longest associated wavelength the “fundamental wavelength” from here on.  Note that it is equivalent to the circumference of a circle with a radius of 2x.

The minimum energy carried by our hypothetical photon would, therefore, be E=hc/4πx=ħc/2x.  For ease, I am going to call this hypothetical photon a “carrier photon” in reference to that fact that it is “carrying” the mass-energy.  Being hypothetical, it should not be thought of as a real photon.

Note that the energy of a photon is proportional to its angular frequency E=ħω, so in this case, ω=c/2x.  Angular frequency is a measure of the number of wavelengths in a given time and, in this case, can be thought of at the number of wavelengths made possible during the period Δt.  Looking at the fundamental, we can see that the minimum is one half of a wavelength per Δt.

Let us now give x a value.  As a god of physics, you are not only non-existent, but also very lazy, so your effort at creation is the minimum possible, meaning that the radius of your circle is as small as possible.  As per my last post (Why I Like Planck), I will assume that this is one unit of Planck length, x=l_Pl=√(ħG/c³), making the fundamental wavelength λ=4πx=4π.√(ħG/c³.  This means that the energy of the carrier photon would be:

E=ħc/(2.√(ħG/c³))=√(ħc⁵/G)/2=E_Pl/2

where E_Pl is one unit of Planck energy.

The related angular frequency is inversely related wavelength, such that ħω=c/λ, or ω=c/ħλ.  If the wavelength λ=4πx=4π.√(ħG/c³), then the angular frequency is ω=c/2x=c/(2.√(ħG/c³))=√(c⁵/ħG)/2.  Given that t_Pl=√(ħG/c⁵), the angular frequency of the carrier photon is one half of the inverse of one unit of Planck time.  Remember that we are considering angular frequency, in this case, to be the number of wavelengths in the period Δt. You made one half of a wavelength, which means Δt is one unit of Planck time.

The only way to increase the angular frequency, associated with a circle with a radius of one unit of Planck length, and thus increase the energy of the carrier photon, would be to decrease the value of Δt, but its value is already the minimum possible.

This means that energy involved, at one half of one unit of Planck energy, is both the maximum and the minimum possible.

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Right, now we have to go back to the notion of potential energy.  By expanding a circle out of nothingness to a radius of one unit of Planck length, you generated negative energy.  Since the only energy that a carrier photon on the circumference of the circle can have is one half of one unit of Planck energy, we have three options:

The carrier photon’s energy exactly matches the potential energy of the expansion, leading to a balanced energy budget of zero.

The carrier photon’s energy is greater than the potential energy of the expansion, making the circle explode.

The carrier photon’s energy is less than the potential energy of the expansion, leading to the collapse of the circle.

The final two options aren’t very interesting, especially if we want to drive the thought experiment further.  So, let us assume that the amount of potential energy generated by expanding a circle such that its circumference increases by 2πl_Pl (that is, the radius increases by one unit of Planck length) is E_Pl/2.  If we expand the circle again, over a period of one unit of Planck time, by another unit of Planck length, then the total energy of the carrier photon become a full unit of Plank energy, E_Pl.  And so on.

This is, of course, reminiscent of the FUGE model so the question becomes – what about in our universe?

Our universe has been expanding for 13.787 billion years, give or take.  This corresponds to 4.351×10¹⁷ seconds and 8.070×10⁶⁰ units of Planck time.  We want to know how many wavelengths are being brought forward each unit of Planck time (right now).  Given that one additional half of a wavelength has been created per unit of Planck time of expansion, that cumulative number is now 4.035×10⁶⁰ per unit of Planck time.

Converting that into SI units, this is equivalent to a ridiculously high angular frequency of 7.486×10¹⁰³Hz.

A carrier photon with this angular frequency has an energy of 7.895×10⁶⁹J which corresponds to 8.784×10⁵²kg.  Which is in the ballpark of estimates of the mass of the observable universe.

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So, what am I saying?  Basically, I am saying that energy is conserved.  The whole reason that there is much more energy in the universe today than there was back 13.787 billion years ago is that we are in a situation with 7.895×10⁶⁹J of potential energy that is being balanced precisely by the energy introduced by expansion.

I am not saying that we live on the circumference of a circle, nor the two-dimensional surface of a black hole in a hologram universe, merely that the mathematics seems to work out – if you think of the conservation of energy in this slightly bizarre way.

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What might not be immediately obvious here is that the above is a very complicated way to arrive at the conclusion that the total amount of mass-energy in the universe is E_TOTAL=ꬱ.E_Pl/t_Pl/2, where ꬱ is the age of the universe in units of Planck time (hence t_Pl) and E_Pl/2 is the amount of energy added to the universe every unit of Planck time, a conclusion I arrived at by considering the critical density of the universe (see here).  Note that energy is not related to time per se, but rather to the speed of light.  Therefore, in a different scheme of natural units, this relationship would not stand unless the natural unit of mass varied in inverse proportion to the variation in the natural unit of time.  This would impact in turn on the natural units of electrical potential and current and would make it impossible for the reduced Planck constant to resolve to unity.  In summary, the fact that E_TOTAL=ꬱ.E_Pl/t_Pl/2 seems to be telling us that Planck units are fundamental in some sense.

Why I Like Planck

Any poor soul who has read all of my stuff, probably an imaginary person, or at the most a bot of some kind, will know that I like Planck units.  I’ve even been accused of assigning more importance to Planck units that one should.

To support an upcoming post, I want to explain why I like Planck.

Key to the FUGE model, making up fully half of the initialism, is granular expansion.  The notion is that the universe is granular at a very small scale.  The question then is precisely what scale most accurately reflects the granularity of the universe.

I prefer the Planck units as natural units for this task, since all but one of the fundamental physical constants resolve to unity when using them.

There is an exception, being the elementary charge, e.  However, it should be noted that the value to which the elementary charge resolves using Planck units is such that α=e², where α is the fine structure constant.  This falls out of the equation α=e²/4πε₀ħc, noting the resolution to unity of the three terms: the reduced Planck constant, ħ; the speed of light, c; and the “raised permittivity”, 4πε₀.  Note also that I consider that q_Pl=4πε₀ħc, because it is more meaningful than the other option (because as a consequence α=e²/ q_Pl²).

Note that α is a dimensionless constant which cannot therefore resolve to unity.  As far as I can tell, the only fundamental physical constant that would change in a universe which had a different value of α would be the elementary charge.

That all said, it is possible that there is another set of natural units that underlie the granularity of the universe.  If so however, each fundamental physical constant when expressed in terms of those natural units would make some sort of sense.  The problem with using an alternative is that the Planck units already make maximal sense of the fundamental physical constants.  Any deviation from them merely adds problems.

There are currently five alternative schemes:

• Stoney units,
• Schrödinger units,
• Atomic units,
• (Atomic) natural units, and
• Strong units

The sixth apparent alternative, “geometrized units”, is really just a subset of any other alternative scheme in which c and G resolve to unity.

The first two, Stoney units and Schrödinger units, are better than the schemes used in atomic physics because in both cases the related energy units are equal to Planck energy.  This might not be immediately apparent for the Schrödinger units, but it must be noted that in that scheme, the speed of light is not 1, but rather 1/α.  These schemes are not suitable for representing the granularity of the universe they both include the fine structure constant α in the definition of dimensional units, which is not dimensional and thus cannot be resolved to unity.  It is introduced via the use of the elementary charge as the basic unit of charge.  The reduction of the natural unit of charge to unity deletes any meaning from the value of α which is, on the other hand, obvious with the Planck scheme (that is, as mentioned above, α=e²/q_Pl²).

The schemes used in atomic physics (atomic units and the unhelpfully named “natural units”) both use the electron mass as a basis and is therefore not suitable for representing the granularity of the universe.  The same applies to strong units used in nuclear physics, which has proton mass as a basis.

If there is a better scheme, I certainly cannot think of one.  If there is any action that takes place in a period of less than one unit of Planck time, I cannot think of one.  If there is any fundamental particle that is shorter than one unit of Planck length or has a wavelength shorter than one unit of Planck length, I cannot think of one.

For that reason, when I think of the shortest possible time or distance, I always think Planck.

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Note that, buried in the next post, is another reason to consider Planck units to be eminently suitable.  It’s just a little difficult to explain.