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 by 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 of reference, 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=lPl=√(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=EPl/2
where EPl 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 tPl=√(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.
---
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 (as the god of physics) 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, partly because they aren't reflective of a flat universe such as the one we live in. So, let us assume that the amount of potential
energy generated by expanding a circle such that its circumference increases by
2πlPl (that is, the radius increases by one unit of Planck
length) is EPl/2. If you 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
becomes a full unit of Plank energy, EPl. 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.
---
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.
---
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 ETOTAL=ꬱ(t).EPl/tPl/2, where ꬱ(t) is the age of the universe in units of Planck time (tPl) and EPl/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. These variations would impact in turn on the natural units of electrical potential and current and would affect the resolution of the reduced Planck constant to unity.
In summary, the fact that ETOTAL=ꬱ(t).EPl/tPl/2 seems to be telling us that Planck units are fundamental in some sense.
No comments:
Post a Comment
Feel free to comment, but play nicely!
Sadly, the unremitting attention of a spambot means you may have to verify your humanity.