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.