Slightly different formulas. From thread 277 we have

$$ H = \frac{1}{\pi}\frac{\alpha h^2}{m_e^2 c}n_e $$

While here we have

$$ H = \frac{2}{\pi}\alpha c \lambda_e^2 n_e \quad \mbox{with} \quad \lambda_e = h/(2 m_e c) $$

And so

$$ H = \frac{1}{2\pi}\frac{\alpha h^2}{m_e^2 c} n_e \qquad \mbox{But} \quad \frac{1}{2\pi} \ne \frac{1}{\pi} $$

I am becoming (re)tired of Tired Light in this way.

NTL and STz differ by a factor ~2 in their predictions. (I already reported that on my poster, footnote [b].)

NTL and STz are not the same physical derivation, so they give different answers. What’s the problem?

NTL and STz are not the same physical model, so they give different answers.

That may be so in the eloquent prose as presented by @RedshiftDrift and @tiredlyndon, but mathematically the models are quite similar, if not just the same, as I have demonstrated.

What’s the problem?

The problem is the physical quantity $n_e$ , denoting the number of electrons in a volume of the intergalactic medium (dimensionality = $m^{-3}$ ). In order to be consistent with known values of the Hubble constant, $n_e$ must be $\approx$ 1 electron per cubic metre in one model, while in the other it must be $\approx$ 1 electron per 2 cubic metres. However, there is only one such physical quantity $n_e$ , right? Therefore I must conclude that the $n_e$ as employed by both authors is nothing else than a fuzz parameter which can be chosen as per convenience for calculating their $H$ to be in concordance with observations.

Update.

There must exist independent means to determine the number of electrons $n_e$ per unit volume. A possibility one can think of is to make use of a well known relationship between Hubble parameter $H$ and cricital density $\rho_c$ of our Flat and Static Euclidean Universe.

$$ H = \sqrt{\frac{8}{3}\pi G\rho_c} $$

where $G$ is the gravitational constant. The InterGalactic Plasma is a mixture of protons and electrons. It is neutral as a whole, so the number of positive charges (protons) muat be equal to the number of negative charges (electrons). Protons are much heavier than electrons, so the electron masses can be neglected in

$$ n_e = \rho_c/m_p = \frac{H^2}{(8/3) \pi G} / m_p $$

where $m_p$ is the mass of a proton. Let’s make an estimate.

H := 70;
Mpc := 3.08567758*10^22;
H := H*1000/Mpc;
G := 6.6743*10^(-11);
m_p := 1.67262192*10^(-27);
rho_c := H^2/(8*Pi/3*G);
n_e := evalf(rho_c/m_p);

Resulting in an electron density $n_e \approx 5.5$ , which I consider not bad, given the crude approximations involved with the above. But there is more. When substituting the expression for $n_e$ in the equation for the Hubble parameter one obtains in fact another equation for calculating $H$ .

$$ H = \frac{1}{2\pi}\frac{\alpha h^2}{m_e^2 c} \mbox{ } \frac{H^2}{(8/3) \pi G} / m_p \quad \Longrightarrow \quad H = 2\pi m_e^2 c (8/3)\pi G m_p / (\alpha h^2) $$

q := 1.60217663*10^(-19);
epsilon_0 := 8.854*10^(-12);
h := 6.62607015*10^(-34);
c := 299792458;
alpha := 1/(4*Pi*epsilon_0)*q^2/(h/(2*Pi)*c);
m_e := 9.1093837*10^(-31);
H := 2*Pi*m_e^2*c*8/3*Pi*G*m_p/(alpha*h^2);
H := evalf(H*Mpc/1000);

Resulting in $H \approx 14\mbox{ }km/s/Mpc$ . At least one can say that the order of magnitude is correct, which I consider not bad, given again the crude approximations involved with the above. Any suggestions for improvement are quite welcome, though !!

Edit. A rather obvious improvement is assuming that the IGM which is responsible for the cosmic redshift is extremely hot. The picture below is from ESA. The part on the left can safely be ignored in the usual context of our ACG. The part on the right shows that the medium which is of interest for us is 40% of total ordinary matter. The accompanying math is modified as follows.

$$ n_e = \frac{4}{10}\mbox{ }\rho_c/m_p $$

Resulting in a modest improvement:

$$ H = \frac{5}{2} \times \mbox{(the previous result)} \approx 35\mbox{ }km/s/Mpc $$

A value of 5 times the previous result would feel better.

@RedshiftDrift . There is a factor of two wrong with a formula in the above comment of yours.

$$ \lambda_e = h/(2 m_e c) \approx 1.2 \times 10^{-12}\mbox{ } m \quad \ne 2.4 \times 10^{-12}\mbox{ } m $$

And for the Hubble parameter I calculate (as a consequence):

$$ H = \frac{2}{\pi} \alpha c \lambda_e^2 n_e \approx 2 \times 10^{-18} \mbox{ } n_e \quad \ne 8 \times 10^{-18} \mbox{ } n_e $$

Hi @HanDeBruijn , The formula you came up with by replacing the number of electron in a volume in @RedshiftDrift STz formula is very similar to the formula no 36 in my paper (https://www.scirp.org/journal/paperinformation?paperid=103476) which, according to my model, gives the correct value for H0 (70.74 km/s/Mpc). You said “any suggestions for improvement are quite welcome, though !!”, here is my suggestion ! ;-)

Hi @PatrGT , >The formula you came up with [ .. ] is very similar to the formula no 36 in my paper

Actually I did have a look at your theory for the purpose of finding common ground, without any results worthwile mentioning. Apart from this, the formula you are referring to is wrong with a factor of four, as explained elsewhere in this thread. It must be

$$ H = (\pi/2) m_e^2 c (8/3)\pi G m_p / (\alpha h^2) $$

Giving numerical values that are even more off when compared to the empirical ones. I do not consider anymore the post as relevant.

Thanks for the explanation. I don’t fully understand it but will think about it. I still doubt the QED plane wave model of light.

Also,

What is the velocity of the electron before and after the interaction? Can you show total momentum is the same before and after?

Could you clarify “counter-propagating beam”? Is the angle between the beams exactly 180° or (90°, 180°]?

Does each wavefront have to arrive at the electron at the same time? What if one photon is half a wavelength behind the other?

Could you clarify “counter-propagating beam”? Is the angle between the beams exactly 180° or (90°, 180°]?

Counter-propagating means that the angle between the propagation direction of the photon modes is exactly 180°.

For other questions unrelated to the current thread, please start another thread by adding a new comment e.g. like this one with the new question. This will help keeping the discussion organized for easier access in the future. Thanks.

Thanks for the explanation.

If it works exactly with higher order terms by accounting for the gamma term, and if STz is to be presented more widely, I recommend having those equations prepared.

must have different wavelengths differing by 2λₑ

This is a requirement so stringent that I think such collisions would be rare. What if the wavelength delta is 2.001λₑ or 2.002λₑ?

Combined with the waves having to be exactly 180° head on would make this interaction rare^2.

The interaction transfers a photon-F which disappears from mode-F, to a photon-C which appears in mode-C

Wait what happened to there being a forward moving photon after the interaction? The way it’s explained and the way the momentum calculation was done above, it appears that after the interaction there is no light moving forward!

Pending question here.. What happened to the forward moving photon after the interaction? The momentum calculation you showed doesn’t have any light moving forward post interaction.

I missed this pending question…

must have different wavelengths differing by 2λₑ - @RedshiftDrift

This is a requirement so stringent that I think such collisions would be rare. - @sahil5d

No it won’t since the interaction lasts for a finite time, so the uncertainty principle $\Delta E \Delta t \ge \hbar/2$ on energy allows a finite range of wavelength difference that provides a continuum for the final state. See Fermi’s golden rule for details.

The interaction transfers a photon-F which disappears from mode-F, to a photon-C which appears in mode-C

Wait what happened to there being a forward moving photon after the interaction? The way it’s explained and the way the momentum calculation was done above, it appears that after the interaction there is no light moving forward!

You’re right, the explanation leaves no photon in the forward-propagating beam after one interaction. There is no photon left moving in the forward direction since that photon is now at a longer wavelength moving in the opposite direction (as depicted here).

Take 2: then it would be clearer if your momentum and energy equation accounted for the full interaction and showed that something is still moving forward, and that the electron net p and KE are *lower.

Also there is still the unexplained How is light shooting out of a telescope to a star??

then it would be clearer if your momentum and energy equation accounted for the full interaction…

This is already mentioned in the paper arXiv:2410.02036, section 3.1 “Momentum diffusion…”, where I’ve written the integrals over space and momentum coordinates, and summations over all the photons involved:

$$ \int {\rm d}\vec{r} \int {\rm d}\vec{r}’ \int {\rm d}\vec{p} \int {\rm d}\vec{p}’ \sum_i \sum_j $$

These integrals go in front of Eq. (1) in addition to the time integrals $\int {\rm d}t’ \int {\rm d}t$ already in Eq. (1) that take care of energy conservation. This accounts for the full interaction with one electron at rest. Energy and momentum conservation is implicit in physics and those integrals appear in all QED calculations, so to lighten the notation, they don’t need to be explicitly written.

and showed that something is still moving forward…

Figure 2 clearly shows two Stimulated Emission arrows, one that transfers a photon to beam “C” and one that transfers a photon to beam “F”, the latter is that something moving forward you are looking for.

and that the electron net p and KE are lower

No, the electron takes up the energy of the less-energetic redshifted-photons, so the electron has a higher kinetic energy. This is also in the paper, paragraph on p. 8 that ends with “The electrons gain an energy $\Delta E_{st} = \sigma_{ST}^2/(2m_e) = 2(\hbar k)^2/(m_e)$ which is characterized by an increase of their temperature.”

My point is that your message at the top of this thread professed to be a total momentum and kinetic energy accounting, but it wasn’t.

The ‘lower’ electron p and KE meant lower with respect to the p and KE accounting at the top of this thread, in which no photon was moving forward at the end.

The telescope question was about a requirement of your model, the counter-propagating beam on a line from the telescope to the star: https://github.com/orgs/a-cosmology-group/discussions/281#discussioncomment-11131312. What is the evidence for this?

your message at the top of this thread professed to be a total momentum and kinetic energy accounting, but it wasn’t.

After one transfer, the photon moving forward is gone, but the electron now has that forward momentum. Therefore total momentum is conserved as claimed. The total includes photons and electron together, not photons taken separately from electrons as you seem to think.

The top message reports the total momentum for one transfer, as claimed.

I meant total as in your total tired light mechanism. You showed an interaction in which no light ended up propagating forward, and that was confusing.

If someone at a conference asked you about your tired light mechanism and for an accounting of the total p and KE, and you presented some equations where the light didn’t propagate forward at the end, they’d be confused too.

You showed an interaction in which no light ended up propagating forward

What are you looking at?

On the image shown at the CMC-2 conference, light is propagating forward (toward the right) in beam-F. Beam-F is mostly green (mode-k2 of beam-F has more photon-buckets) when it leaves the left-hand side of the image. After 14 stimulated transfers (arrow up from F-to-C and C-to-F), beam-F is mostly yellow: mode-k3 of beam-F has more photon-buckets when it reaches the right-hand side of the image. The spectrum of beam-F has been redshifted, resulting in yellow light propagating forward. Every stimulated transfer shown on the image satisfies energy and momentum conservation. The electrons have gained momentum and are hot (i.e. they gained energy). Total energy and total momentum of the system are conserved.

All the conditions you require are satisfied: redshift of a beam propagating forward, conservation of energy, conservation of momentum. The redshift of a beam is not caused by a single interaction, it emerges from a large number of stimulated transfers between beams F and C.

I don’t know how to make this clearer. Where is the part that’s confusing you? Can you point at it with an arrow on the image? I could use your advice for the next conference…

$\ $

By the way, using “Kinetic-Energy” muddles the discussion. KE is $mv^2/2$, a term reserved for massive particles only, not photons.

What is conserved is

• total energy (the sum of $mv^2/2$ for each electron and $h\nu$ for each photon), and
• total momentum (a vector that is the sum of $m$ $\vec{v}$ for each electron and $\hbar$ $\vec{k}$ for each photon).

@sahil5d maybe I see where the problem is. You only looked at the equations for a transfer from beam-F to beam-C.

The equations for a transfer from beam-C to beam-F are trivially the same as those given here, just switch the indices $F \rightarrow C$ and $C \rightarrow F$. Then you have the equations of the interaction in which light ends up propagating forward again.

Does that help?

This was the message, at the top of this thread, https://github.com/orgs/a-cosmology-group/discussions/281#discussioncomment-11101415.

By the way, using “Kinetic-Energy” muddles the discussion

Yes, I should’ve written “total p and E”.

are trivially the same as those given here, just switch the indices

Ok. However, if the indices are just switched, doesn’t that mean the electron should get a kick the other way?

If so, the net KE of the electron would go down. Then that affects the hypothesis that the net redshift would be λ + 4 λₑ.

@sahil5d

Ok. However, if the indices are just switched, doesn’t that mean the electron should get a kick the other way? If so, the net KE of the electron would go down. Then that affects the hypothesis that the net redshift would be λ + 4 λₑ.

Yes, kicking the same electron in the other direction reduces its kinetic energy to zero. That’s discussed in Sec. 2.1 of arXiv:2410.02036 which concludes “This force is derivable from a potential and is practically conservative”. This is however not the STz process.

During the STz process, the kicked electron couples with other modes of beam-F and beam-C because its motion causes Doppler shifts. It does not get kicked twice from the same spectral components of the beams. Instead another electron at rest gets the second kick. The way to see this is to add an extra electron and an extra photon-mode (see Sec. 2.2, 3.3 and Fig. 2 in arXiv:2410.02036).

Three photon-modes interact in two steps with two electrons, both initially at rest. The modes are separated by $2\lambda_e$. The initial conditions are:

• A forward propagating mode-B at $\lambda_B = \lambda - \lambda_e$, (“B” for ‘blue’)
• a counter-propagating mode-C at $\lambda_C = \lambda + \lambda_e$, (“C” for ‘counter-propagating’)
• a second forward propagating mode-R in the same forward beam, at $\lambda_R = \lambda_C + 2\lambda_e$, (“R” for ‘red’) (Modes B and R are two spectral components of the same forward propagating beam.)
• two electrons at rest before the transfer, with momentum $p_1 = 0$ and $p_2 = 0$.

In the first interaction, electron-1 transfers a photon from $F_B$ to $F_C$, then electron-1 has momentum $p_1 = 2h/\lambda$. In the 2nd interaction, electron-2 transfers a photon from $F_C$ to $F_R$, then electron-2 has momentum $p_2 = -2h/(\lambda + 2\lambda_e)$.

The momentum before and after each transfer is

$$ h/\lambda_B + 0 = -h/\lambda_C + p_1, \quad {\rm and} \quad -h/\lambda_C + 0 = h/\lambda_R + p_2. $$

The total momentum is then

$$ h/\lambda_B + 0 + 0 = h/\lambda_R + p_1 + p_2. $$

Substituting gives $h(1 + \lambda_e/\lambda)/\lambda \approx h(1 - 3\lambda_e/\lambda)/\lambda + 2h/\lambda - 2h/(\lambda+2\lambda_e)$.

Multiplying both sides by $\lambda/h$ gives $(1 + \lambda_e/\lambda) = (1 - 3\lambda_e/\lambda) + 2 - 2(1 - 2\lambda_e/\lambda)$. Momentum is conserved.

(I neglect second order terms $\mathcal O(\lambda_e^2/\lambda^2)$ since $\lambda_e \ll \lambda$. ¹ Note that the electrons have a small combined momentum $p_1 + p_2$ that is in the forward direction. That momentum is from photon-B in the forward beam that has been redshifted to photon-R, and hence the forward beam lost forward momentum.)

The total energy before and after the transfer is

$$ hc/\lambda_B + 0 = hc/\lambda_R + 2 p_e^2/(2m_e). $$

Substituting and multiplying both sides by $\lambda/(hc)$ gives $\frac{1}{1 - \lambda_e/\lambda} = \frac{1}{1 + 3\lambda_e/\lambda} + \frac{8h}{(2m_e c\lambda)}.$ With a bit of algebra we find $(1 + \lambda_e/\lambda) \approx (1 - 3\lambda_e/\lambda) + 4 \ [h/(m_e c\lambda)],$ or $1 + \lambda_e/\lambda = 1 - 3\lambda_e/\lambda + 4 \lambda_e/\lambda.$ Energy is conserved.

In this case, after two stimulated transfers:

• a photon went from mode-B to mode-R of the forward propagating beam,
• the wavelength is increased by $4\lambda_e$, or $2\lambda_e$ per transfer,
• Beam-C “acts as a catalyst, as it remains unaffected by the stimulated transfers produced by this configuration.” (Sec. 3.3 in arXiv:2410.02036).

By considering two transfers which conserve momentum and energy, I show a process that also conserves momentum and energy, and propagates light forward while being red-shifted.

What I understand is that STz treats the electron like a mini battery, which can store energy from wave-F then transfer it to wave-C.

Such a mechanism seems to be a 1 photon-1 electron interaction and has no dependence on wave-C. The wave-C could arrive very late (very high δ) or even never, and your mechanism should still proceed.

STz does not treat the electron as a battery, this would be incorrect physics. A battery has an internal structure that can store potential energy (i.e. “charged battery”). An electron can only have kinetic energy; it has no internal structure to store potential energy (like an atom would). Therefore, an electron cannot absorb a photon’s energy and momentum - there is no physical solution for that, as you showed yourself in an earlier email (copied below). Forget about the “battery” model of the electron, this will lead you to false conclusions.

Such a mechanism seems to be a 1 photon-1 electron interaction and has no dependence on wave-C.

No, this is not STz. The only physically possible 1-photon-1-electron interaction is a collision where the photon bounces off the electron, changing momentum direction and energy of both the photon and the electron. That process is Compton scattering.

STz is an interaction between an electron and two waves, simultaneously. Only one equation describes momentum conservation, and one equation describes energy conservation which happen in a single step. That’s how STz conserves momentum and energy without changing the direction of light beams.

… and has no dependence on wave-C. The wave-C could arrive very late (very high δ) or even never

A wave does not “arrive”, it is there all the time. Look at the equation of a wave: $\cos(k x - \omega t)$. It has no “time of arrival”, it exists everywhere. Both waves F and C must be present for STz to happen.

16 Oct 2024 23:54:41 (from @sahil5d on another forum) How do you manage that with conservation of energy and momentum?

$f$ is the photon frequency before interaction [with an electron] $f’$ is the photon frequency after interaction

$hf - hf’ = 1/2 m v^2$ $hf/c - hf’/c = m v$

… factor terms…

$h(f - f’) = 1/2 m v^2$ $h/c (f - f’) = m v$

… divide the topleft/bottomleft and topright/bottomright…

$c ==? 1/2 v$

so the velocity of the electron would be 2c ??

This shows that an electron cannot physically absorb a photon’s momentum and energy.

A wave does not “arrive”, it is there all the time. Look at the equation of a wave: It has no “time of arrival”, it exists everywhere.

This is not coherent to me. Even in the QED plane wave interpretation, how does the wave not have a time of arrival?

STz uses the predictions of QED, a theory developed 95 years ago that has remained unchanged after it was presented by Paul Dirac. QED is a mathematical recipe that makes accurate predictions, as supported by thousands of experiments in thermodynamics, atomic physics, condensed matter physics, nuclear physics, etc. and its applications such as transistor electronics, nuclear bombs and reactors, klystrons in microwave ovens, lasers, magnetic resonance imaging, etc.

QED is a reliable tool to calculate the STz process, independently of the interpretation we may want to give it. It is demonstrated to work for processes involving electron-photons interactions. Therefore, I adopt the “get the numbers out” attitude so I can get quantitative predictions for STz. Unless someone finds out that I do not apply QED correctly in my calculation of the STz process, I consider the answers correct.

This discussion is not about the interpretation of quantum mechanics. The wave-particle duality does not make sense to anybody. But even if “nobody understands quantum mechanics”. that can’t be used as an excuse to avoid learning how to use a tool like QED. There are plenty of online resources e.g. explaining QED at a simple level. Just follow Feynman’s advice, relax and enjoy. Only once you know how to use the QED recipe can the discussions about its interpretation begin, but not on this forum.

How does a QED wave not have a “time of arrival”, when there is a speed of light, when light has a speed?

Both light and a QED wave propagate at the speed $c$.

However, a pulse of light is required to physically measure its speed using ‘time of departure/arrival’ measurements. Such a physical light-pulse is described as the square of a sum of QED waves, such as $\big\langle light-beam_F \big| light-beam_F \big\rangle$.

However, a QED wave is not a pulse but the mathematical description of a continuous wave. The propagation speed of the wave $\exp(+ik_F x -i\omega_F t)$ cannot be measured because this wave is on all the time. Therefore, what you describe as the “time of arrival” of a QED wave does not make physical sense because its probability amplitude $\big| \exp(+ik_F x -i\omega_F t) \big|^2$ would be equal to one everywhere and at all times.

@RedshiftDrift . >However, a pulse of light is required to physically measure its speed using ‘time of departure/arrival’ measurements.

E.g. light pulses caused by by passing light through a fan. See this question on Quora: >How do I measure the speed of light at home with material and equipment one can easily find at a hardware store and/or buy online?

Not only a pulse is required for the purpose. From Wikipedia we quote: >The “one-way” speed of light, from a source to a detector, cannot be measured [ independently of a convention as to how to synchronize the clocks at the source and the detector]. What can however be experimentally measured is the round-trip speed (or “two-way” speed of light) from the source to a mirror (or other method of reflection) and back again to detector. [ .. ] Experiments that attempt to directly probe the one-way speed of light independent of synchronization have been proposed, but none have succeeded in doing so.

And there is another issue. From your reference:

$$ \Psi = \exp(+ik_Fx-i\omega_Ft) + \exp(+ik_Cx-i\omega_Ct) \qquad \mbox{Eq.}(1) $$

I think this formula is not OK in some respects. Could it be replaced (?) by

$$ \Psi = A_F\exp\left[+i\frac{2\pi}{\lambda_F}(x-ct)\right] + A_C\exp\left[+i\frac{2\pi}{\lambda_C}(x+ct)\right] $$

where $A=$ amplitude, $\lambda=$ wavelength and $c=$ speed of light.

Edit. Now I see that you have slightly different versions of the same formula dispersed all over. What version is the good one?

@HanDeBruijn the equation are equal for $k=2π/λ$ and $ω=2πc/λ$ which are also used in equations of harmonic oscillators. A = amplitude is not often used in light-wave equations, or you can just set it to 1. Not always entirely accurate but it simplifies things a lot. At the least I would set $A_f$ equal to $A_c$.

, except your waves are counter-propagating (sign difference in front of ct) which Eq(1) appears not to be at a first glance. (neither k nor ω can be negative so it must be intentional). I need to still read all in the reference to see the whole picture. My observation might not be significant.

@Francois-Zinserling . The answer as given by Louis is >Please read the paper instead of having me ( @RedshiftDrift ) copy parts of it on this forum.

Or, as (having me) copied in this comment: >Electrons transfer photons from any of the modes in light-beam-F to any of the modes $\vec{k}_j$ pointing $180^o$ from $\vec{k}_j$ [ .. ] in light-beam-C.

@HanDeBruijn

Now I see that you have slightly different versions of the same formula dispersed all over. What version is the good one?

As @Francois-Zinserling explained here these equations all have the same meaning, but written with a different notation. They’re all good in their respective context!

$\ $

Not always entirely accurate but it simplifies things a lot. At the least I would set $A_f$ equal to $A_c$.

These equations are not complete, they are simplified to show the parts that are relevant to the discussion. @Francois-Zinserling, $A_F = A_C$ is used for the simple derivation of the Hubble constant in arXiv:2410.02036, but in general $A_F$ is more often not equal to $A_C$. For example, for an electron in the intergalactic medium located near a bright galaxy, $A_F > A_C$ and this leads to different results that are not discussed in the paper.

@RedshiftDrift writes something very peculiar: >However it is correct to say that the counter-propagating beam is on the line between our telescope and the star, propagating away from our telescope.

Is that counter-propagating beam redshifted too when it arrives at the same star where the propagating beam has departed?

@HanDeBruijn yes, the process is symmetric and the counter-propagating beam is also redshifted. The labels beam-F and beam-C are used to describe part of the whole process. The same explanation holds true if “F” and “C” are swapped. All light beam-pairs interact with electrons in the same way, independently of their propagation direction.

This means that an astronomer on a distant star would also see a cosmological redshift of the Milky Way.

@RedshiftDrift. In this comment you write >I call $H_z$ the Hubble-Humason constant [ .. ] to distinguish it from the Hubble-Lemaître constant derived empirically from observations.

Surely You’re Joking, Mr. Marmet. What then is the relationship between the two $H$ constants ?

Elsewhere you write about >a value of the electron density consistent with the value obtained from FRB dispersion-measure.

Now that’s truly interesting ! Are you talking about an independent determination of the electron density $n_e$ ? So far, I’ve only seen more or less impressive but what I call vicious circle relationships in ACG, connecting “known” physical constants such as the gravitational constant $G$, the Hubble-whatever parameter $H$ and now some electron density $n_e$ . Yes, it’s obvious that one thing can calculated from other things, in our case as

$$ n_e = \frac{\pi H}{2\alpha c \lambda_e^2} \quad \mbox{from} \quad H = \frac{2}{\pi} \alpha c \lambda_e^2 n_e $$

I fall asleep by this kind of mathematism. Please wake me up, Louis !

@HanDeBruijn

Are you talking about an independent determination of the electron density $n_e$ ?

Of course, otherwise there would not be anything new and people would fall asleep looking at algebraic tautologies. What you incorrectly perceived as a fuzz parameter is obtained independently with FRB observations. (See footnote 9 in arXiv:2410.02036.)

This is why Lyndon and I are so excited about the discovery of FRBs - it breaks what you call vicious circle relationships. $\ $

Surely You’re Joking, Mr. Marmet¹. What then is the relationship between the two H constants ?

$H_0\equiv$Hubble-Lemaître constant is obtained from distance vs redshift observations.

$H_z\equiv$Hubble-Humason constant is based on $n_e$ and calculated with STz theory. The electron density $n_e$ is obtained from an independent measurement of FRB pulse-dispersion vs redshift and calculated using $H_0$.

There is no relationship between $H_0$ and $H_z$, they are obtained from two independent methods: one observational, the other theoretically based on a different set of observations.

Correctly stated (with indices on $H$ because they are not the same $H$)

$$ n_e({\rm obs}) \approx f_{\rm obs}(FRB) \quad {\rm and} \quad H_z \approx \frac{2}{\pi} \alpha c \lambda_e^2 n_e({\rm obs}) \approx H_0 $$

where the function $f_{\rm obs}(FRB)$ describes astronomers pointing their optical- and radio-telescopes toward the sky, getting data, and calculating $n_e$ with derivations that are independent of STz.

This not mathematism but physics, since it relates independent observations $H_0$ and $f_{\rm obs}(FRB)$ with a derivation of $f_{\rm STz}(\alpha c \lambda_e^2, \ n_e)$.

Wake up, Han!

The electron density $n_e$ is obtained from an independent measurement of FRB pulse-dispersion vs redshift and calculated using $H_0$.

The phrase “is obtained from an independent measurement of FRB pulse-dispersion” sounds OK, but the addition “calculated using $H_0$” makes me suspicious again. Elswehere you write: >NTL gets $n_e(FRB|NTL)=0.499/m^3$ from an analysis of raw FRB data and redshift-independent distances.

Sounds better. Why not the same raw FRB data and redshift-independent distances for STz?

Without going into the details of FRB measurements and the calculation of $n_e({\rm FRB})$… A density such as $n_e$ (units [ $m^{-3}$ ] ) is obtained from a ‘number count’ and a ‘volume’ (i.e. $n_e = ({\rm number}/{\rm volume})$ ). To know the volume, one must measure distances, and distances in astronomy can be measured with redshift and calculated from an equation that is proportional to $H_0^{-1}$.

This might at first look like a possible vicious circle in the argumentation! If $H_z({\rm STz})$ depends on $n_e({\rm FRB})$ in the same reciprocal way $n_e({\rm FRB})$ depends on $H_0$, that could cause a degeneracy that would make STz agree with observations for any value of $n_e({\rm FRB})$! Good for you @HanDeBruijn to notice this - you are awake!

There are at least two reasons why this is not a problem:

1. STz predicts a $H_z({\rm STz})$ proportional to $n_e({\rm FRB})$, while $n_e({\rm FRB})$ is proportional to $H_0^3$ (from the calculation of a volume). This different dependence lifts the degeneracy when comparing the values of $H_z$ and $H_0$.
2. There are other ways to measure distances which do not depend on $H_0$. This catalogue lists thousands of these measurements. It is then possible to get $n_e({\rm FRB})$ without $H_0$.

$\ $

Why not [use] the same raw FRB data and redshift-independent distances for STz?

Good idea, this is what was done for NTL. It leads to the same conclusion since $H_0$-derived distances agree with redshift-independent distances. (No surprise, $H_0$ is derived from ‘redshift observations’ and ‘redshift-independent distances’!) The observational accuracy of the two methods is not the same, but using $H_0$ and citing another paper is much less work. (Life is too short to repeat the work someone else has already done!)

$\ $ Perhaps a better way to summarize this would be

$$ H_0({\rm SNe}) \approx f_{\rm obs} \big( SNe, {\rm independent \ of } n_e \big), $$

$$ n_e({\rm FRB}) \approx f_{\rm obs} \big( FRB, {\rm independent \ of } H_0 \big), $$

$$ 0.3 \pm 0.2 /m^3 \approx n_e({\rm STz}) \approx f_{\rm STz} \big( H_0({\rm SNe}); \alpha c \lambda_e^2 \big) \approx n_e({\rm FRB}) \approx 0.2_{-0.1}^{+0.3} /m^3, $$

where $f_{\rm STz}(…)$ represents a calculation using the STz derivation. The arithmetic equivalent of the last equation would be

$$ 54 \pm 30 {\rm km/s/Mpc} \approx H_z(STz) \approx f_{\rm STz}^{-1} \big( n_e({\rm FRB}); \alpha c \lambda_e^2 \big) \approx H_0(SNe) \approx 74 \pm 2 {\rm km/s/Mpc}. $$

$H_0({\rm SNe})$ and $n_e({\rm FRB})$ are independently measured values, and STz and NTL are the only derivations that reliably establish a link between the two with a physical process that has no adjustable parameter.

while $n_e(FRB)$ is proportional to $H_0^3$ (from the calculation of a volume)

I don’t think so.

• An argument against $H_0^3$ is in a (mostly wrong) reply of mine, where a crude estimate of $n_e$ is made with the square of $H_0$ and $G=$ gravitational constant, $m_p=$ proton mass.

$$ n_e \approx \frac{H_0^2}{(8/3) \pi G} / m_p \approx 5.5 / m^3 $$

• Did you mean $H_0^{-3}$ with a minus sign? The distance is indeed calculated from $cH_0^{-1}\ln(1+z)$ but this is only a line-of-sight length. I can’t imagine that the volume needed to determine $n_e(FRB)$ is the third power of such enormous length.

Perhaps with VM the proportionality constant is $H_0^2$, but FRB analyses use a different method to determine $n_e(FRB)$.

$n_e(FRB)$ is

• proportional to the dispersion measure (DM): (number N of electrons)/(square meters), a quantity measured directly from signal timing,
• inversely proportional to the distance:1/(mega-parsecs)

Then we have $n_e(FRB) \propto N/d^3$. Since $d \propto H_0^{-1}$, that makes $n_e \propto H_0^3$.

One could argue that the “square meters” are not measured with the Hubble constant but with the size of the radio-telescope dish. So that dimension is not a cosmological quantity related to $H_0$. This would make $n_e(FRB) \propto H_0$ in which case my argument #1 fails.

That’s ok, argument #2 remains valid since we can measure $n_e(FRB)$ without reference to $H_0$.

Perhaps with VM the proportionality constant is $H_0^2$

No, I’ve used common ΛCDM. With VM the “critical” mass density $\rho_c$ and so the electron density $n_e$ would become somewhat lower. Apart from the fact that there is no expansion with VM and so $\rho_c$ has a different meaning. ( Remember this thread ? ) BTW, the Hubble tension in $H_0^2$ is twice the one for $H_0$ . For $H_0^3$ it becomes three times larger = $3\Delta H_0$ .

$\rho_c$, $w$, $w_0$, $w_a$, etc… are coefficients of a series expansion of an equation that fits data, so they are associated with various powers of $H_0$. But these LCDM coefficients have no physical meaning in a non-expanding universe.

Here are published (GitHub/book) formulas linking the Hubble parameter $H$ to a mean mass density $\rho_c$ of the universe.

$$ H = \sqrt{\chi G \rho_c} \quad \mbox{with} \quad \chi = 8\pi \times \begin{cases} 1/8 & (: \mbox{OI}) \ 1/6 & (: \mbox{Mach}) \ 1/3 & (: \Lambda\mbox{CDM}) \ 0.3883945571 & (: \mbox{UAC}) \end{cases} $$

Three of these formulas - (OI) , (Mach) , (UAC) - are for a steady state model (SEU) and only (ΛCDM) is for an expanding universe. How so no physical meaning in a non-expanding universe?

By physical meaning is understood the model behind the mathematical relationships that you list. An equation has no meaning on its own.

I meant $\rho_c$, $w$, $w_0$, $w_a$ interpreted with ΛCDM (a space expansion model) have no physical meaning in non expanding models because expansion is meaningless in non expansion models!

Of course $\rho_c$ has a (different) physical meaning in non expansion models.

Sigh !! Here we go again. What is “different” and what is “the same”. There is (too) much physics, yes, there are many “my models”, yes, but there is only one Mother Nature. As a consequence, there is good and there is bad and not everything is OK.

Too much physics??? Generally speaking modern cosmological theories contain too much math and not enough physics; they also postulate, out of math-model-dependent necessity, the existence of things that cannot be observed while studiously ignoring things that are there.

Also @HanDeBruijn since you brought it up, the main problem I had with all your rejiggering of the math only to arrive at the same result was that it added nothing to the fundamental point regarding the nature of the redshift - it said nothing about the physics.

In rereading that thread I realize that you have the basic cosmological picture presented by the ESW model correct, particularly the idea that individual galaxies are at the center of their individually unique “universes”. That is the same picture as presented in standard light cone analysis when a galaxy is placed at the apex of the inbound and outbound light cones.

In Newtonian mechanics, kinetic energy is $E = \frac{1}{2} m v^2$. E is energy; m is mass; v is a velocity.

But Einstein’s energy is $E = m c^2$. E is energy; m is mass; c is a velocity, specifically that of light.

Same energy in Joules, same mass in kilograms, same speed in meters per second.

Yet the equations are different. @HanDeBruijn, which one is the correct equation? Without physical meaning, mathematics can’t help you.

and then there is $E=γmc^2$ which adds a whole new meaning with $(γ-1)mc^2 ≈ \frac{1}{2}mv^2$ as above only valid for low velocities with all still the same units.

What gets too much is not the math or physics, but when we take a wrong fork in the physics road and just keep adding more forks instead of returning to the point where we still knew where we were going.

@budrap00 . >the main problem I had with all your rejiggering of the math only to arrive at the same result was that it added nothing to the fundamental point regarding the nature of the redshift - it said nothing about the physics.

Denied. The analytical equivalent always has additional value. It shows for example that redshift in your model really becomes infinite (physically: grows without bound) for a certain radius. While your numerical model can only suggest it, but not prove it.

@RedshiftDrift .

Without physical meaning, mathematics can’t help you.

Oh yes it can. The former is in a series expansion of the latter (as indicated correctly by @Francois-Zinserling).

$$ mc^2 = \frac{m_0c^2}{\sqrt{1-(v/c)^2}} = m_0c^2\left[1+\frac{1}{2}\left(\frac{v}{c}\right)^2+\frac{3}{8}\left(\frac{v}{c}\right)^4+\frac{5}{16}\left(\frac{v}{c}\right)^6+\cdots\right] \approx m_0c^2+\frac{1}{2}m_0v^2 $$

where $m_0=$ rest mass, last term = kinetic energy, first term = typical for nuclear energy.

@RedshiftDrift . Let’s restrict our attention to calculation of the Hubble parameter $H$. According to a previous reply, the difference between NTL and STz then is

$$ H_{STz} = \frac{1}{2\pi}\frac{\alpha h^2}{m_e^2 c} n_e(STz) \qquad ; \qquad H_{NTL} = \frac{1}{\pi}\frac{\alpha h^2}{m_e^2 c} n_e(NTL) $$

Thumbs down because you don’t like the inevitable conlusion ? Namely $H_{STz} = H_{NTL}$ and so

$$ n_e(STz) = 2\times n_e(NTL) $$

However, in the above comment you write

• ‘STz’ with $n_e \approx 0.3 / m^3$
• ‘NTL’ with $n_e \approx 0.5 / m^3$

Leading to a contradiction. Escape proposed:

• ‘STz’ with $n_e \approx 1.0 / m^3$ and ‘NTL’ with $n_e \approx 0.5 / m^3$

I think the outcomes here may be linked to another reply, where it is shown that $H$ as calculated by you is a factor of 4 too large, giving an electron column density which is a factor of 4 too small $\approx 1.0/4 = 0.25 \approx 0.3 / m^3$.

@HanDeBruijn this is physics, the calculation has many approximations and astrophysical measurements have a lot of uncertainties.

STz and NTL are physically very different:

• NTL is based on condensed matter physics / STz is based on the gradient force,
• NTL has 1 photon + 1 electron in a double step, releasing 2 more photons / STz has 2 photons + 1 electron in a single step,
• NTL electrons are cold on a lattice / STz electrons are hot,
• $H_{STz} \ne H_{NTL}$,
• $n_e(STz) \ne n_e(NTL)$,
• etc.

Your numerical equations are wrong, they do not account for any of the physical differences. A factor of 2 is not a concern right now. There is no contradiction between $0.5 \pm 0.1 /m^3$ and $0.3 \pm 0.2 /m^3$. More precise modeling can be done when better data is available.

The point is to understand the physics. As shown in the paper arXiv:2410.02036, STz explains the Hubble redshift and heating of the solar corona, two physical phenomena that are otherwise apprently unrelated.

$\ $ Edit: so thumbs down for all comparisons of equations that differ by a factor two. This is comparing apples and oranges.

@RedshiftDrift . >this is physics

No, Louis, it is much simpler: this is a typo in your first comment.

$$ \lambda_e = h / (2 m_e c) \quad \mbox{must be} \quad \lambda_e = h / (m_e c) $$

as it is defined indeed in the main paper you are constantly referring to, just above formula (5) and below formula (7). My bad that I didn’t notice, sorry !! An excuse may be that authors also use the reduced Planck constant $\hbar$ instead of $h$ and that I don’t know what the specific habits are in the world of Optics.

A factor of 2 is not a concern right now. There is no contradiction [ .. ] More precise modeling can be done when better data is available.

Okay, due to that typo of yours I’ve made plenty of errors in my above postings. But I’ve decided to leave the replies as they are, simply because it’s too much effort to make appropriate changes everywhere. Two of the necessary changes are reported here though.

$$ H_{STz} = \frac{2}{\pi}\frac{\alpha h^2}{m_e^2 c} n_e(STz) \qquad ; \qquad H_{NTL} = \frac{1}{\pi}\frac{\alpha h^2}{m_e^2 c} n_e(NTL) $$

I am not talking about more precise modeling with better data in the future. Fact is that STz and NTL give outcomes that differ by a factor of two right now. You cannot solve this by silently introducing an error margin for the data. The Hubble parameter is the same as the Hubble parameter and the electron density is the same as the electron density, independent of whatever theory. We are talking about real world physics, remember?

$$ H_{STz} = H_{NTL} \quad \Longrightarrow \quad n_e(STz) = \frac{1}{2} \times n_e(NTL) $$

Numerically, indeed:

$$ n_e(NTL) = 0.5\mbox{ }m^{-3} \quad \Longrightarrow \quad n_e(STz) = 0.25\mbox{ }m^{-3} \approx 0.3\mbox{ }m^{-3} $$

This is comparing apples and oranges.

Some ACG members have a problem with understanding what “being different” means. Whatever. Two contradictory theories being both OK is impossible. That’s why Big Bangers always win. I find it ridiculous that ACG has a dozen or so different Tired Light theories. Why not have a good drink with e.g. Lyndon Ashmore and find common ground?

Edit. I see that you’ve corrected the typo.

@HanDeBruijn indeed there was a typo in the first comment. Thanks for noticing, it is corrected. The paper should have fewer typos. (Factors of two are a notorious problem in optics. This mostly stems from the definition of the field amplitude which varies between authors. Some use ‘peak-to-peak’ amplitude, some use ‘RMS’ amplitude, some use the ‘complex number amplitude’, etc. and errors by factors of $\sqrt 2$ and $2$ are made from time to time…)

Sorry that the typo made it into your earlier posts, but don’t bother changing your text: NTL and STz still give different numerical results.

Two conclusions can be drawn from that:

1. Uncertainties on observational data does not allow us to decide which derivation can be elimitated. That’s what I meant by ‘both are ok’.
2. Both derivations cannot be correct - I agree with you. Given 90 years of history of failed models, it’s statistically more likely that none of BB/NTL/STz is correct (except for STz, of course 😉).

Some ACG members have a problem with understanding what “being different” means. - @HanDeBruijn

“Different” is with respect to the physics involved in the primary process of the derivation. The cosmological redshift is a result of thousands of repetitions of a primary process.

STz is based on a one step process:

• 1. The original photon-1 interacts simulatneously with an electron and a 2nd photon. Photon-1 disappears to produce recoil of the electron and a new photon with a small redshift,
• Stimulated emission prevents image blurring.

NTL is based on a two-step process: (refer to NTL)

• 1. The original photon-1 interacts with an electron. Photon-1 disappears to produce recoil of the electron that emits a low-energy photon.
• 2. A new photon is emitted with a small redshift, the electron returns essentially to rest and another low-energy photon is emitted.
• Electrons sit on a Wigner lattice that prevents image blurring.

All properties of NTL and STz are derived from these primary processes.

The common ground between the derivations is a photon interacting with an electron. Beyond that, the derivations differ and predict different effects. I can’t think of a good drink that would mitigate these differences!

@HanDeBruijn, concerning $H_0(STz)$ and $H_0(NTL)$ vs $n_e(STz)$ and $n_e(NTL)$

I understand your argument where you consider that STz and NTL can agree on the Hubble constant. So you write $H_0(STz) = H_0(NTL)$ and find that $n_e(STz) \ne n_e(NTL)$. That’s correct.

However consider our point of view. When FRBs were discoveded in 2007, they gave us a way to calculate $n_e$ from the “dispersion measure”. This was great for STz and NTL because we could in principle agree on the value $n_e(FRB)$ and calculate the Hubble constant from that. Using recent data, here are the numbers (omitting uncertainties) for two derivations (and one model):

STz takes $H_0(SNe|STz) = 74$ km/s/Mpc from the value of the Hubble constant $H_0(SNe|BB)$ derived from the Big Bang model. STz derives $n_e(SNe|STz) = 0.29/m^3$ from $H(SNe|STz)$ and the STz derivation. STz gets $n_e(FRB|STz) = 0.22/m^3$ from $n_e(FRB|BB) = 0.33/m^3$ derived by Big Bangers and corrected for ‘no-expansion’ and no contribution to DM by galaxies.

NTL gets $n_e(FRB|NTL) = 0.499/m^3$ from an analysis of raw FRB data and redshift-independent distances. (More details here and here.) NTL derives $H_0(FRB|NTL) = 62.5$ km/s/Mpc from $n_e(FRB|NTL)$ and the NTL derivation.

BB gets $n_e(FRB|BB) = 0.33/m^3$ from raw FRB data and the BB model. BB gets $H_0(SNe|BB) = 74$ km/s/Mpc from supernovae observations and the distance ladder. BB gets $H_0(CMB|BB) = 67$ km/s/Mpc from observations of the CMB and modeling of the so-called “early universe”.

The numbers disagree. (Even within the BB model, i.e. the Hubble tension.) Lyndon and I could perhaps agree on a value of $n_e$ so that $n_e(FRB|STz) = n_e(FRB|NTL)$. Another researcher, A. Trinchera, is showing some interest in calculating this and a third independent analysis would be welcome.

However, this would change our estimates of the Hubble constant but not help reduce the disagreement between $H_0(STz)$, $H_0(NTL)$, $H_0(SNe)$, and $H_0(CMB)$.

There is just not enough information to answer the question “Which is right, BB, NTL, or STz?” only based on measurements of $H_0$ and $n_e$.

Other types of observations are necessary to answer this question.

Thanks Carlos @carlosbelcech for reading and understanding the main points of the paper, your summary is quite accurate!

only a cosmic background is necessary at all wavelengths, which, in fact, is the existing one, so that the shift is not cut off by lack of lower energy photons.

Indeed, the background radiation needs to be at essentially the same wavelength as the redshifted radiation, so it’s the visible light of the background that redshifts visible light, the UV light of the background that redshifts UV light, the radio waves of the background that redshifts radio waves, etc. In other words, it’s the existing “spectrum of the universe”¹ that red-shifts itself.

$\ $

traveling waves that propagate in the opposite direction. Just, I do not understand how this explains the time dilation.

Restrictions on the length of the paper did not allow me to add lengthy explanations and answer all the questions posted here. I am working on a ‘supplementary note’ that will give more details on the propagation of light and how the full spectrum is shifted to give the $d = (c/H_z)\ \ln(1+z)$ relationship. (This will answer some questions asked elsewhere by D. Soares, E. Lerner, S. Gupta, A. Trinchera…) Based on that note, I’ll write a ‘supplementary note - 2’ to explain light curve stretch. It’s not time dilation 😄, instead the intensity of the light curve becomes spread over a longer time because of how it interacts with the waves traveling in opposite directions.

I will post these notes here, but that will take time as I need to finish another paper, then do numerical simulations if I can’t solve the equations analytically…

Ok. I’m waiting for your supplementary note. I think it’s a great breakthrough. Also, I think the following is a great idea: It’s not time dilation, but the intensity of the light curve is spread out over a longer time because of how it interacts with waves traveling in opposite directions But frankly. For my taste, the picture should extend to a wider range of ray directions, in the sense that your model is restricted to interactions between rays propagating in opposite directions. I have the opportunity to study QED, for different reasons, but it seems to me that it is possible to show that this kind of stimulated emission, and therefore spectral transfer, can occur between quasi-collinear rays coming from the same source. And that’s where we’ll win the Nobel! Carlos.

your model is restricted to interactions between rays propagating in opposite directions.

This is a property of the interaction Hamiltonian (first Eq. of Sec.2.2 of arXiv:2410.02036), it has nothing to do with what suits your taste or mine! The interference between the waves of one electron and two photons involved in the process integrates¹ to a non-null value only if the light-modes are counter-propagating.

Note that the counter-propagating restriction is in the electron’s frame of reference. Because the electrons are hot, optical aberration allows angles other than 180° in the observer’s frame of reference.

stimulated emission… can occur between quasi-collinear rays coming from the same source.

But stimulated transfers only occur for counter-propagating beams. Also, energy-momentum conservation implies a wavelength shift proportional to $\sin(\theta/2)$ where $\theta$ is the angle between the beams. If the rays come from the same source, $\theta = 0$ and there is no redshift! Moreover, the constant speed of light would imply that light-curves could not be stretched.

Ok. sorry. It is not an issue of taste. I tell you my intuition (I know that it is not an issue of intuition).I repeat you, that your STz model is at the moment the best we have. Including possible limitations, it is better than expansion because it doesn’t broke physics laws. Due to the quasi-isotropic and quasi-uniform features (or almost, I know about redshift anomalies) of the redshift, the cause must be a simple phenomenon. Or, in other words, an one step probabilistic phenomenon. In your proposal, it is necessary two steps for redshift a beam, the first: a photon is transferred, redshifted, to the counter-propagating beam, and then, the redshifted photon is returned to the original beam (two-step redshift). The two steps are probabilistic. I understand that your mathematical development get to demonstrate that the interference among two traveling waves ((the Kapitza-Dirac effect verified in laboratory, needs standing waves, yes?) with free electrons produces the spectral transfer and the electrons heating. It would be like a Kapitza-Dirac effect with traveling waves in the IGM, and no interference patterns are necessary. Points for you.

By another hand, all stellar sources are extended, not punctual, objects, therefore $\theta$ is not zero. At first, $\theta$ can’t be zero never, in order to the interactions can be possible, but can be arbitrarily close to zero.

I have another question. I understand that your model requires some degree of coherence of light, and that STz phenomenon would must cause alterations (an increasing?) in this degree of coherence. Did you see it?

@carlosbelcech your description of STz using the Kapitza-Dirac effect is correct. (P. 9 of my poster on hal-04762937.)

The Kapitza-Dirac effect was first demonstrated in the lab with a standing wave (p. 23, same poster). However, STz requires two interfering travelling waves which do need some degree of coherence, as you noted in the last question. Intergalactic light does not have much optical coherence, but STz is a probabilistic phenomenon (as you correctly identify) in the sense that when an electron is presented with two beams that have the required properties, the transfer occurs. It is the “opportunistic” electron that picks photon-modes that are coherent for the stimulated transfer. If a specific electron cannot redshift a specific light beam, that’s not a problem; there will be another electron on the way that will eventually do a stimulated transfer.

Back to your comment of Nov. 14, the two steps happen independently, so in that sense they are also probabilistic. Out of all beams available in the IGM, one electron interacts with a pair of beams that are close enough to 180° (the finite interaction time allows for some margin, see post). When another electron provides the second step for specific modes of the spectrum, this is also the first step for another pair of photon-modes!

To produce a redshift of all spectral components of a beam, the process is repeated (by different electrons) for each spectral component of the spectrum of that beam (also in a probabilistic way). By repeating all these processes between all available beams, this eventually produces a redshift of all beams in space. This is possible because there are many electrons in the IGM, and intergalactic light distributed over the entire $4\pi$ steradians provides all the wavelengths.

For simplicity, my paper only describes transfers between three photon-modes. However, STz only makes sense when considering that stimulated-transfer processes happen simultaneously on all modes of the spectra of all light beams, and with electrons everywhere in space.

The probabilistic nature of the process that you highlight is important because it describes a thermodynamical process that heats electrons. The term “diffusive interactions” in the title of my paper/poster/talk is the technical way to express this probabilistic nature. This makes the $\exp(-H_0 t)$ property of STz common to that of thermal exchanges, where the temperature varies with time as $\exp(-a t)$ when an object cools down. There is no new physics in STz, this property of heat exchanges was known by Carnot in the 1820s!

this property of heat exchanges was known by Carnot in the 1820s!

Very inspiring, Louis! Followup = Unification of Variable Mass and Tired Light .

Your ‘heat exchanger’ followup is also very inspiring, Han! The countercurrent-flow design has some things in common with the counterpropagating beams of STz!

The redshift rate depends on other things too.

The proportionality constant in equation $H_z[s^{-1}] = 8\times 10^{-18} n_e$ (Eq. (8) in arXiv:2410.02036, $n_e$ in MKS units) is derived from Eq. (5) (here; or Eq. (2) here) that has a term proportional to $\sqrt{n_i n_j}$, where $n_i$ is related to the light flux of radiation that is being redshifted, and $n_j$ is the light flux of the ambient light that is a necessary catalyst to the redshift process.

The constant in Eq. (8) is derived for the simplified condition $n_i = n_j$, where the light being redshifted has roughly the same flux as the counter-propagating (catalyst) light that contributes to the redshift process. This is a valid approximation in the case of a redshift process in the IGM, far away from bright galaxies.

However, inside a galaxy cluster the light flux of nearby galaxies is much brighter than the general background, much more than 100 times larger. For each photon $n_i = 100 n_j$ gives $\sqrt{1 \times 0.01} = 0.1$, and so the proportionality constant of Eq. (8) becomes $8\times 10^{-19}$ for the conditions of a galaxy cluster, ten times smaller than in intergalactic space.

This amply accounts for the factor of five discrepancy. There’s no contradiction with observations.

STz is an optical effect that depends on much more than the electron density; the optical geometry of the system is also very important and cannot be neglected (see the example of Sec. 3.2 in arXiv:2410.02036). Near a bright source, the redshift rate of STz is reduced because the number of catalyst photons coming from outside the source is insufficient to match the larger number of photons emitted by the source. Despite a higher electron density, this lack of catalyst photons accounts for the undectable redshift near or inside galaxies, including our observations made from inside the Milky Way, and accounts for the lower-than-expected redshift of light emitted by galaxy clusters and stars.

As for the “central galaxies in clusters [that] are blue shifted, not redshifted, relative to the average galaxy in the cluster,” a model with the position and brightness of all galaxies would have to be built. Without a model we can make a simple hand-waving argument: since the brightest-cluster-galaxy is often near the centre of the cluster, it’s brightness reduces the redshift rate and the central galaxies appears blue-shifted relative to the rest of the cluster.

This amply accounts for the factor of five discrepancy.

Coincidence or not?

This seems like a joke to me. The telescope would need to be an emitter only in the case of the star and the telescope are at relative rest during all the years of the light journey.

@sahil5d . >How is there a light beam from a telescope to the star?

One thing I can think of is the idea that in geometrical optics light rays are reversible. In popular terms: if you can see me, then I can see you. If the telescope can see the star, then the star can see the telescope. Yesterday I attempted to refresh my > 50 years old rusty knowledge about Laser technology. I vividly remember the moment that they showed us a very heavy apparatus, turned out the lights and proudly turned on that thin red beam of light. Oh well, it quickly turned out that it is impossible for someone like me to become up to date again with the truly amazing developments in that field of interest since those days. Let’s put vivid memories aside and return to nowadays business. If I understand it well, a necessary element of any laser still is sort of a perfectly reflecting mirror on one side (I do not recall what precisely is needed on the other side). This could be an explanation for the counter propagating light beam as required by our Optics professor Louis: I think he needs a mirror (at the telescope) for his cosmic laser to work.

@HanDeBruijn, STz relies on stimulated emission like lasers do, but the similarity ends there. There is no mirror on stars!

@carlosbelcech has a better explanation (but not the one with the telescope that emits light like the eye of Sauron!) The universe is dynamic and our telescope moves relative to stars and galaxies. Light emitted from stars and galaxies reach another galaxy that we will eventually observe. Our telescope (and Earth) blocks the path between two stars for only a few minutes, a short period compared to the millions of years of counter-propagating beams.

But there’s even another way to get counter-propagating light…

@sahil5d, the requirement of counter-propagating beams applies in the electron’s frame of reference in which the electron is at rest. Since electrons are heated by stimulated transfers, they are even more dynamic than stars. They have a large velocity when they cross the line between a galaxy and our telescope. Two light beams can intersect at the electron’s position at an angle that differs from 180° in our frame of reference, but optical aberration will change the angle between the beams to make it 180° in the electron’s moving frame of reference.

Hot electrons therefore transfer photons coming from all directions into a beam propagating away from our telescope. The velocity of the electron is shown in red on the image. Thermal electrons will average light from the full spectrum of the universe (hence my interest in the paper you found on arxiv:1802.03694) and produce a redshift at all wavelengths and in all directions.

(Galaxies, Earth, telescope and electrons are not to scale!)

@HanDeBruijn, this might look like the counter-propagating light will increase in flux like in a laser! But sorry to disappoint, there are more arrows not shown on the image that go both ways… Electrons also stimulate photons out of the counter-propagating beam, and the aliens looking at us with their telescope will see the Milky Way with the same original number of photons, but all of them with a longer wavelength!

the requirement of counter-propagating beams applies in the electron’s frame of reference in which the electron is at rest. Since electrons are heated by stimulated transfers, they are even more dynamic than stars. They have a large velocity when they cross the line between a galaxy and our telescope. Two light beams can intersect at the electron’s position at an angle that differs from 180° in our frame of reference, but optical aberration will change the angle between the beams to make it 180° in the electron’s moving frame of reference.

This like me more (I know that it is not a issue of taste).

STz is based on a one step process:

The original photon-1 interacts simulatneously with an electron and a 2nd photon. Photon-1 disappears to produce recoil of the electron and a new photon with a small redshift, Stimulated emission prevents image blurring.STz is based on a one step process:

And this, even more (but I won’t surrender yet)

STz relies on stimulated emission like lasers do, but the similarity ends there.

Hmm .. from another thread and adapted a bit. Don’t we see a laser here?

@HanDeBruijn, this has nothing to do with STz! See this post

And this, even more (but I won’t surrender yet)

Precisely! Again from another thread : >Can we reconcile any version of TR with intrinsic redshift?

In case you consider to open a new discussion, Louis, I find a possible unification of Tired Light (TL) and Variable Mass (VM) certainly on topic. The same for this reply. It shall be reformulated as follows. TL, VM ( and even Time Dilated Past ) share the same math for distance as a function of cosmic redshift and the other way around:

$$ D = (c/H)\ln(1+z) \quad \mbox{or} \quad 1+z = \exp(H/c.D) $$

with $z=$ redshift, $H=$ Hubble parameters, $c=$ standard lightspeed, $D=$ distance. Let subscripts $VM$ and $TL$ denote Variable Mass/intrinsic and Tired-Light (STz or NTR) respectively. Let $\lambda=$ wavelength. and suppose that we have two subsequent redshifts, namely

$$ z_{VM} = \frac{\lambda_1-\lambda_0}{\lambda_0} = \frac{\lambda_1}{\lambda_0}-1 \quad \mbox{and} \quad z_{TL} = \frac{\lambda_2-\lambda_1}{\lambda_1} = \frac{\lambda_2}{\lambda_1}-1 $$

For the total redshift it follows that

$$ 1+z = \frac{\lambda_2}{\lambda_0} = \frac{\lambda_1}{\lambda_0} \frac{\lambda_2}{\lambda_1}= (1+z_{VM})(1+z_{TL}) $$

$$ (1+z_{VM})(1+z_{TL}) = e^{H_{VM} D/c} . e^{H_{TL} D/c} = e^{(H_{VM}+H_{TL}) D/c} = e^{H_0 D/c} = 1+z $$

And so the total Hubble is the sum of the intrinsic and the tired-light one:

$$ \Large H_0 = H_{VM} + H_{TL} $$

I don’t know if cosmic redshift is TL or VM or both, but anyway the two conjectured phenomena are not mutually exclusive and maybe can be unified. I sincerely hope so.

Melia’s $R_h = ct$ cosmology, another space expansion cosmology, shares the same redshift-distance relationship $D = (c/H_0) \ \ln ⁡(1+z ).$ However, the space expansion hypothesis of “ $R_h = ct$ “ makes it incompatible with STz.

STz also predicts the luminosity-distance for supernovae to be $D_L = ( c / H_0 ) \ (1+z) \ \ln ⁡ ( 1 + z )$, which I believe to be incompatible with VM.

VM uses the hypothesis of varying mass, which contradicts the basic physics STz is built on. VM is not compatible with STz.

$\ $ VM may be compatible with other TL models, but that’s not on topic for this STz discussion.

VM may be compatible with other TL models, but that’s not on topic for this STz discussion.

That’s correct. I have found the same final verdict on Variable Mass (VM) vs. two versions of Tired Light (TL): Stimulated Transfer redshift (STz) and New Tired Light (NTL). VM is not compatible with STz / NTL . It is one or the other, but not both.

Thanks HanDeBruijn, You say

That’s correct. I have found the same final verdict on Variable Mass (VM) vs. two versions of Tired Light (TL): Stimulated Transfer redshift (STz) and New Tired Light (NTL). VM is not compatible with STz / NTL . It is one or the other, but not both.

Well, let me assure you that Variable Mass theory is definitely not compatible with NTL so it must be STz The plot of dispersion measure v ln(1+z) as predicted by NTL has a gradient within 0.14% of that predicted by NTL of mc/2hr. If ‘m’ varied it would no longer be a straight line Cheers Lyndon

Well, let me assure you that Variable Mass theory is definitely not compatible with NTL so it must be STz.

Little misunderstanding, I guess (or a joke).

VM is not compatible with STz / NTL

can be interpreted in two ways. Your interpretation:

$$ (\lnot (VM \land NTL)) \lor STz \quad \equiv \quad \mbox{(not (VM and NTL)) or STz} $$

My interpretation / what I meant:

$$ \lnot (VM \land (NTL \lor STz)) \quad \equiv \quad \mbox{not (VM and (NTL or STz))} $$

Yep I misunderstood your post. Sorry.

VM is not compatible with STz / NTL

means it is either VM or STz / NTL? So, lets look on the bright side. NTL predicted a relationship between DM and z in a static universe - ergo DM =mc/2hr (7.32x10^25 m^-2) seven year ago. Now we have data from 33 FRB’s and, when we remove the weighting for expansion, the observational data gives a gradient of 7.33x10^25 m^-2. A difference of 0.14% from that predicted by NTL The gradient of the graph is constant - in the VM theory it should ‘vary’. It does not.

VM is not compatible with STz / NTL

So the positive from this is that last week we had 100 tired light theories, now we only have 99? Cheers Lyndon

Yep I misunderstood your post. Sorry.

I’ve posted an ambiguous statement, so the Sorry is entirely mine.

There are many ways.

Some parts of the process leading to redshift have already been demonstrated. For example, see Fig. 3 in this paper (https://www.nature.com/articles/s41566-022-01104-w) which shows red-shifted UV light by 4 nm caused by stimulated emission in an electron laser.

There are also ways to detect the predicted heating associated with the redshift interaction using an atom interferometer. I am collaborating with Prof. Kumar at York University to develop measurement techniques, e.g. this work, leading to a measurement using an atom interferometer for Precision Measurements of Atomic Recoil Frequency. This is a difficult experiment because the effect is very small, but the apparatus is being upgraded to increase its sensitivity.

From a theoretical point of view, the calculations in my paper (which are standard in QED) can be compared with those of this paper Accardi et al. - 1995 - A third hypothesis on the origin of the redshift .pdf ) The paper uses sunlight as a catalyst for the redshift of radio waves, but the calculation is very similar to mine.

Predictions made by model for effects ‘apparently’ unrelated to the cosmological redshift are also good tests. My paper on arXiv:2410.02036 gives an estimate of the coronal heating caused by stimulated transfers. (My next paper will show how Milgrom’s acceleration $a_0$ in the phenomenological MOND model can be calculated based on stimulated transfers.)

@RedshiftDrift .

A Seeded free-electron laser driven by a compact laser plasma accelerator ? It escapes me completely how the $\Delta \lambda_{\mbox{FEL}}$ in there is comparable with your Stimulated Transfer $z$ redshift.

Same problem or worse with A third hypothesis on the origin of the redshift: Application to the Pioneer 6 data. Relevant quote: Eq. ( 19) shows that the frequency of a monochromatic photon interacting with a gas of charged particles is shifted by a negative quantity, that is directly proportional to the integral of the product of the density and temperature of the gas calculated along the trajectory of the photon. We call this result the redshift theorem.

But then we read this. $B_\omega$ is a dimensionless constant, depending on a divergent integral 3. This is not surprising, since we have performed a perturbative energy shift calculation that is affected by the usual QED divergences. If and how is it possible to remove the divergence will be the subject of a future work. Sure, and that future work meanwhile is lasting for about a century !! How comes, Louis, that your work would not be suffering from the usual QED divergences, while you are employing the same QED? Am I missing something?

(My next paper will show how Milgrom’s acceleration $a_0$ in the phenomenological MOND model can be calculated based on stimulated transfers.)

Beware !! In der Beschränkung zeigt sich erst der Meister ( Goethe ).

• For someone with only a Hammer, everything looks like a Nail
• For someone with only Gravity, everything looks like a Black Hole
• for someone with only Plasma, everything looks like a Z-pinch
• for someone with only SED, everything looks like a ZPE
• for someone with only Optics, everything looks like a Laser

How comes, Louis, that your work would not be suffering from the usual QED divergences, while you are employing the same QED? Am I missing something?

Divergences happen for some QED calculations, but not all QED calculations use diverging series!

@HanDeBruijn you’ll be glad to know that STz calculations do not use any diverging series.

$\ $

In der Beschränkung zeigt sich erst der Meister

Genau! That’s why I didn’t limit myself to using a hammer. I’ve learned electronics, computer science, chess, philosophy, astronomy, number theory, game theory, classical mechanics, calculus, relativity, atomic physics, quantum optics, cosmology, climate science, plasma physics, and above all I’ve been doing experimental physics for over 55 years, including mechanics, astrophotography, condensed matter and crystallography, low temperature physics, nonlinear optics, ion spectroscopy, vacuum techniques, gas and dye lasers, single photon detection, quantum beats, atomic chaos, and precision time measurements. I know nothing about everything - ich bin kein Meister.

I’ll send you a draft of the MOND paper if you want to comment on it before I submit it.

I’ll send you a draft of the MOND paper if you want to comment on it before I submit it.

Yes, Louis, I would very much appreciate !!

Anyway, we are all fishing in the same pool, aren’t we?

@HanDeBruijn here’s an interesting article showing that diverging series in QED are a mathematical problem, not a problem with the physics. The perturbation series in QFT is an “invention of the devil” and this is actually a good thing

It escapes me completely how the $\Delta \lambda_{\mbox{FEL}}$ in there is comparable with your Stimulated Transfer $z$ redshift. - @HanDeBruijn

STz is based on recoil given to an electron that produces Doppler red-shifted photons. This recoil is given by the interacting photons themselves, so the recoil is small for each interaction between low density electrons and incoherent light from stellar sources. This is why it takes millions of years of light propagation in the low electron density of the IGM to produce something that is observable.

Compare this with the free-electron laser in the paper, which uses an undulator magnet that is equivalent to a standing wave for the electrons passing through it at relativistic speeds (see Ref. 4). The undulator transfers a huge momentum recoil to a large number of electrons and stimulates red photons in a coherent laser beam. This increases the efficiency by tens of orders of magnitudes, producing a redshift $z = 0.015$ in the few meters of propagation through the system.

Figure 3 could not be more explicit: Blue light IN, red light OUT, and it’s all done with Stimulated emission and momentum Transfer to electrons, the same physical processes that are at the root of STz.

My old brain is beginning to make sense of this.

• Is energy still ultimately transfered to the electron or does the process leave the electron relatively ‘standing still’?
• If not, what prevents all electrons from eventually being blown away.
• Is the undulator above the absorber of the excess energy? What is the undulator in your model; more electrons in a lattice if I recall? Does the continued absorption not eventually blow it up?
• From Mansuripur and Pfeiffer et al. (separate researchers) I have learned that the photon only transfers momentum during transit. All momentum is recovered when the photon exits. (Scattering thus also excluded of course). Is your mechanism different from theirs? (I mean it’s hard to argue with your experimental proof but how is it different? The undulator?) . Then I have an honest question: If we were to replace your photons with Annila et al. ’s photon pairs (rem E=0 and B=0 but $S_z=2$). And we replace your electron with ‘any particle of mass’ (and this would include all dense masses too, not just IGM). Would your theory work the same on these, or are there conditions that disallow this? I hope you see where I’m going with this.

the same physical processes that are at the root of STz.

One of the problems I have been working on incidentally as an engineer has been the design of scale models with internal fluid flow. The reason is simple: instead of testing a full scale heat exchanger filled with liquid sodium, it’s easier to test a small scale model filled with water or even cheaper (though less accurate) with air. But scaling with fluid flow is not simply making things smaller and thinking that everything is “the same” then. One has to be compliant to Modeling and Scaling Laws. And that requires some knowledge about dimensionless numbers, especially the Reynolds number. Based on this experience, I do not believe beforehand that redshift in cosmos can be simply scaled down to redshift on earth. I wonder if people in Optics show the same discipline as people in Fluid Flow. Are there Modeling and Scaling Laws in Laser technology too? What are the dimensionless “Reynolds” numbers in STz?

article showing that diverging series in QED are a mathematical problem, not a problem with the physics.

A few remarks.

• In my world, good Mathematics is not separate from good Physics. Unfortunately though, both suffer from a 100 Year Wrong Turn, much like in Cosmology.
• There are at least two other sources available on the internet, providing quite “different” information about Quantum ElectroDynamics.
• Is it possible to make common sense of seemingly incompatible “the same” formulations of the Dyson series in QED?
• Could this be called Renormalization?

@Francois-Zinserling I need clarification: are the first two questions above about STz in intergalactic space?

@RedshiftDrift yes, the 4 dotted paragraphs are questions about STz. The whole post including the last question was for you. Sorry, should have tagged you.

@Francois-Zinserling first three questions from this post:

Is energy still ultimately transfered to the electron…

All the energy lost by a photon is transferred to the electron. See here, point 2

“Energy lost by radiation is transferred to electrons as kinetic energy […] causing a temperature increase of the electrons. The energy increase per stimulated transfer with a photon of wavelength $\lambda$ is $\Delta E_{ST} = 2 h^2/(m_e \lambda^2)$.”

$\ $

… or does the process leave the electron relatively ‘standing still’?

First of all, IGM electrons are hot; they are not standing still.

But in a frame of reference moving with an electron (where it is initially standing still by definition), there is a momentum kick equal to $\Delta p_e = 2h/\lambda$ given to the electron. After a stimulated transfer, the electron is no longer standing still in that frame of reference.

(The electron recoils at 2.6 km/s after a stimulated transfer of a green photon $\lambda = 550$ nm.)

$\ $

If not, what prevents all electrons from eventually being blown away.

Nothing prevents electrons from being “blown away”. They are all blown away in every directions by the STz process.

Energy lost by radiation is transferred to electrons as kinetic energy […] causing a temperature increase of the electrons. The energy increase per stimulated transfer with a photon of wavelength $\lambda$ is $\Delta E_{ST} = 2h^2/(m_e\lambda^2)$.

Written otherwise, with frequency instead of wavelength:

$$ \lambda = c/\nu \quad \Longrightarrow \quad \Delta E_{ST} = \frac{2(h\nu)^2}{m_e c^2} = \frac{2(\mbox{energy of photon})^2}{(\mbox{energy of electron})} $$

Applying a formula in The Derivation of the Planck Formula on top of page 11, for making a crude estimate:

$$ k_B \Delta\Theta = \Delta E_{ST} $$

where $\Theta=$ temperature, $k_B=$ Boltzmann constant.

nu := 5.45*10^(14); # green
h := 6.62607015*10^(-34);
m_e := 9.1093837*10^(-31);
c := 299792458;
dE := 2*(h*nu)^2/(m_e*c^2);
k_B := 1.380649*10^(-23);
dTheta := dE/k_B;

Outcome: $d\Theta \approx 0.23$. Thus $\approx 10^{12}$ of such photons would be needed to raise the electron’s temperature to $10^{11}$ K . Is that realistic, @RedshiftDrift ? How many photons are “kissing” one elecron?

Update.

Here is a modest attempt to answer my own last question >How many photons are “kissing” one elecron?

with help of The Derivation of the Planck Formula page 14: the number density of photons in the frequency interval $d\nu$.

$$ N_\nu d\nu = \frac{8\pi\nu^2 d\nu}{c^3} \frac{1}{e^{h\nu/kT}-1} $$

And so the overall number density $N$ of these photons is

$$ N = \int_0^\infty N_\nu d\nu = \frac{8\pi}{c^3}\left(\frac{kT}{h}\right)^3 \int_0^\infty \frac{x^2 dx}{e^{x}-1} $$

A temperature T(heta) has to be assumed; take the Cosmic Microwave Background for this.

h := 6.62607015*10^(-34);
c := 299792458;
k_B := 1.380649*10^(-23);
Theta := 2.726; # CMB
f := int(x^2/(exp(x)-1),x=0..infinity); # 2.404113806
N := 8*Pi*(k_B*Theta/(h*c))^3*f;
evalf(%);

Outcome = 410,952,936 photons / $m^3$, a lot more than I expected !!

The (dis)advantage of an Update is that it goes relatively unnoticed ..

@Francois-Zinserling this third set of questions from your post is not about STz but related to a free electron laser. So I won’t give too much details.

$\ $

Is the undulator above the absorber of the excess energy?

No, the undulator absorbs a negligible amount of energy because its mass is much larger than the mass of an electron. As with STz, it is the electrons that absorb energy.

$\ $

What is the undulator in your model; more electrons in a lattice if I recall?

“A lattice”???!!! Big faux pas 😮 The lattice is in another model discussed elsewhere. Don’t confuse the two models.

With STz, electrons are too hot to form a lattice.

In STz, the momentum is given by two counter-propagating beams that interfere and produce a modulation of the potential energy.¹ These two light beams take the role of the undulator in a FEL.

Compare the image in my paper, Fig. 2 with that of the “undulator”. They are similar because they produce a similar (but not the same) transfer of momentum to electrons.

$\ $

Does the continued absorption not eventually blow it up?

The undulator does not absorb energy from the UV beam. Continuous absorption of momentum won’t move it either - it is bolted to the floor of the lab! You can test this at home by shining a laser pointer on a wall. The momentum from photons bouncing off the wall won’t move anything.

From Mansuripur and Pfeiffer et al. […] the photon only transfers momentum during transit. All momentum is recovered when the photon exits. […] Is your mechanism different from theirs? - @Francois-Zinserling

Mansuripur, Pfeiffer, etc. describe light transiting through a medium with a varying index of refraction. Apart from the word ‘momentum’, this has nothing to do with STz.

In STz, a net momentum is given to the electron, energy is given to the electron, and light is transferred to a counter-propagating beam. None of that happens in an inhomogeneous dielectric material.

$\ $

If we were to replace your photons with Annila et al. ’s photon pairs ($E=0$, $B=0$ but $S_z = 2$), replace your electron with ‘any particle of mass’ (and this would include all dense masses too, not just IGM). Would your theory work the same on these, or are there conditions that disallow this? I hope you see where I’m going with this.

I know nothing about Annila’s photon pairs except for what I read in your paper. Based on that:

1. If we replace STz photons with Annila’s photon pairs:

• because $E = 0$, there is no ponderomotive potential and STz disappears, (from Eq. (5) empty photon modes $| n_i=0, n_j=0 \rangle$ produce nothing)
• both Annila-photons propagate in the same direction, but STz requires counter-propagating photons, so STz disappears.

2. And we replace your electron with ‘any particle of mass’ (and this would include all dense masses too, not just IGM).

• STz is inversely proportional to the mass of the particle/object it interacts with. Stimulated transfers with a proton produce a redshift 1836 times smaller than with an electron. Anything heavier will produce even smaller effects.
• Any collision between particles destroy the coherence for STz to work see XI here. A dense material does not support stimulated transfers. Even the plasma in the Earth’s ionosphere is already too dense to produce any redshift.
• STz works only with charges, there is no redshift produced by neutrons or a dielectric medium.

I hope you see where I’m going with this.

Maybe… You want to apply stimulated transfers to the mechanism of electrostatic attraction and repulsion? If that’s the case, there are at least five conditions unfavorable to STz that give $0 \times 0 \times (1/1836) \times 0 \times 0$.

While Newton’s equation $F = \dot{p}$ is a common point between our papers, the mechanisms we describe are completely different.

@HanDeBruijn

How many photons are “kissing” one elecron [each second]?

I ❤️ the way you write your question! I never calculated that, but whatever time it takes to heat an electron to $10^{11}$K, that time is certainly shorter than the age of a ‘non-expanding’ universe!

Your calculation is on the right track. One would have to check that other wavelengths don’t produce more heating, e.g. there are fewer green photons but each one produces a lot more electron heat than a CMB photon. (The spectrum of the universe given here can be used for that check.)

Outcome = 410,952,936 photons / $m^3$

… for $T = 2.7$ K, this seems about right.

From Hubble’s constant $H_0 \approx 2 \times 10^{-18}$/s, we know that these photons lose an energy $2 \times 10^{-18} h \nu$ each second. That energy heats the electrons.

Since there is about only 0.3 electron in a $m^3$, it’s easy to calculate how much heating an electron gets and the time to go from cold to $10^{11}$ K.

@HanDeBruijn, I did a bit of research and posted a better evaluation of the maximum temperature the IGM can reach:

$$ T_{IGM} \approx 1.2 \times 10^{10} {\rm K}. $$

The number $\approx 10^{11}$ K was an upper limit obtained from the binding energy of iron, $-8.8 {\rm MeV}$ per nucleon, but some nuclear reactions need less energy! I listed these reactions (hopefully without mistake 😓 )

If any nuclear physicist is interested, a more complete calculation would be a good project to work on.

@HanDeBruijn: you can now answer your question with this more accurate value:

How long does it take to raise the electron’s temperature to $1.2 \times 10^{10} {\rm K}$?

Not that I am convinced that LENR is valid, there are indications that we are missing some things about nuclear reactions. Elements Beyond the Stars: A Missing Link in Nucleosynthesis? - See the Pattern; YouTube

Physics related to LENR:

• Edo Kaal - ICCF25 - Fusion and Fission in LENR
• Fundamental Problems with our Understanding of the Atom
• The Explanatory Power of the Structured Atom Model (SAM)
• SAM - Structured Atom Model

The Structured Atom Model is founded on the intuitive notion, supported by observations, that the atomic nucleus should have structural properties. It is a visual tool that permits understanding of physics at a detailed level without having to master difficult mathematics or resorting to quantum mechanics.

Mathematics related to SAM:

• Platonic solid
• Linear isoparametrics with dual finite elements

I never saw the usefulness of these “geometric models”. They only highlight correspondences between mathematical regularities and observations of the universe, without providing an ‘understanding’ of what is really happening.

how do you arrive then at $10^{11}$ Kelvin… - @HanDeBruijn

Already discussed above, $T \approx 1.05 MeV \approx 1.2 × 10^{10}$ K .

… without thermodynamic equilibrium?

There is a power equilibrium that is not a thermodynamic equilibrium. Think of the speed of a car for a given position of the acceleration pedal: the velocity is in “equilibrium” because the “power of burning fuel” (a hot gas) is the same as the “power lost by wind resistance” (wind = cold gas), but it’s not a thermodynamic equilibrium.

In a similar way, the CMB equilibrium occurs because the Sunyaev-Zel’dovich (S-Z) blue-shift (power given to light by hot electrons) has the same magnitude as STz redshift (power lost by cold photons), as already discussed here.

I sent this draft S-Z-effect-STz-resdhift-CMB.pdf some time ago (I forget to whom…) with a “STz − S-Z” balance description that shows how $T_{CMB} = 2.7$ K and $T_e = 10^{10}$ K can coexist. Essentially, the ratio $T_{CMB}/T_e$ will be a function of the ratio of cross sections of the STz and S-Z mechanisms.

I have not completed this calculation yet, I don’t know of a relativistic extension of the S-Z effect to high temperatures like $10^{10}$ K!

@HanDeBruijn you will have noticed that $H_0$ is used in the calculation as a redshift rate in step (2), so it might seem like a tautology to get an electron heating time that is proportional to the Hubble time in the final answer since $t_{hot} = k T_{max} / (E_e H_0) \propto 1/H_0$.

However, the Hubble constant can be eliminated from this result to get the equality $k T_{max} = E_e = w_{CMB}/n_e$, which links the physical quantities $T_{max}$, $T_{CMB}$, and $n_e$.

In principle, if we assume that the energy cycle in the universe is limited by the Hubble time, we can use $T_{max}$ (derived from nuclear reactions) and the measured value of $T_{CMB}$ to calculate $n_e$, the density of electrons in intergalactic space.

Turning this around, knowing $T_{max}$ and $n_e$ means that $T_{CMB}$ can be calculated from Stimulated Transfer redshift in electrons and the assumption that the energy cycle is rate-limited by the redshift process.

I see this as a very convincing argument that STz describes someting fundamental to cosmology. Thanks @HanDeBruijn for asking that question!

@HanDeBruijn the relationship given above is is also interesting for another reason. The electron density $n_e$ implies some mass density in the universe that is related to gravitation, the gravitational constant $G$, and the mass of the proton (since there are about as many protons as there are electrons).

STz augmented with a ‘matter-energy cycle cosmology’ then gives a link between the fundamental constants $h$, $m_e$, $\alpha$, $c$, $k_B$, $\sigma = 2\pi^5 k_B^4/(15c^2 h^3)$, $G$, and $m_p$, constants that are involved in the “large number coincidence”, a relationship proposed by @sahil5d. (This is not Dirac’s large number hypothesis although with some transformation, it may be possible to convert it back into the large number coincidence.)

A better understanding of star dynamics and galaxy dynamics is needed to find the relationship with $G$, and we already have such things as the Tully-Fisher relation and MOND phenomenology with $a_0 \approx cH_0/(2\pi)$, both relating gravity with luminosity, mass, the Hubble constant, and STz, so this opens the possibility that the ‘large number coincidence’ is not a coincidence but an understandable property of the universe.

Wow!

@RedshiftDrift As a rule, euphoria is followed by a wakeup. After a good night sleep, it seems to me that point (3) in your main comment is hardly related to the three other points. Anyway point (3) can be missed in the logic chain that (in my own notation) leads to

$$ n_e k_B \Theta_{max} = \frac{4\sigma}{c}\Theta_{CMB}^4 $$

where $n_e=$ electron density, $k_B=$ Boltzmann constant, $\Theta=$ temperature, $\sigma=$ Stefan-Boltzmann constant, $c=$ lightspeed. The physical interpretation thereof is that heating of the electrons to a maximum temperature $\Theta_{max}$ is caused by the Cosmic Microwave Background with (low) temperature $\Theta_{CMB}$. So far so good, but where has the Stimulated Transfer redshift gone? The CMB wavelength spectrum shows NO redshift. At least I cannot discover anything of the kind in your math. Which is due to the fact that the STz part (3) is unrelated to the above final outcome, as derived from (1)-(2)-(4). A fortunate circumstance is that the Variable Mass theory may be consistent with your thus (re)defined energy cycle in the universe.

Update

I have another proposal. Suppose that the CMB is not causing the heating up of electrons in the InterGalactic Medium. Instead the CMB is conjectured to be the source of increasing Variable Mass of electrons. Equilibrium is then defined by

$$ n_e m_e c^2 = \frac{4\sigma}{c}\Theta_{CMB}^4 $$

To see if this is reasonable, a calculation is carried out.

CMB := 2.726; # CMB
n_e := 0.3;
sigma := 5.670374419*10^(-8);
c := 299792458;
w := 4*sigma/c*CMB^4; 
m_e := w/(c^2*n_e);

Resulting in an electron mass $\approx 15.5 \times 10^{-31}\mbox{ }kg$ here and now.

I’m not sure about the electron density. An alternative estimate has been given elswhere in this thread:

H := 70;
Mpc := 3.08567758*10^22;
H := H*1000/Mpc;
G := 6.6743*10^(-11);
m_p := 1.67262192*10^(-27);
rho_c := H^2/(8*Pi/3*G);
n_e := evalf(rho_c/m_p);
m_e := w/(c^2*n_e);

Resulting in an electron mass $\approx 0.844\times 10^{-31}\mbox{ }kg$ here and now. The experimental value is neatly in between : $0.844\times 10^{-31} \lt 9.1093837 \times 10^{-31} \lt 15.5 \times 10^{-31}$. Coincidence or not? ( Note. It is possible to close the gap with values near $n_e \approx 0.5 \mbox{ } m^{-3}$ )

The CMB wavelength spectrum shows NO redshift. - @HanDeBruijn

Incorrect, the CMB-wavelengths are also redshifted, and longer wavelengths are also redshifted. Radio-telescopes observe galaxies at wavelengths that are longer than that of the CMB, and they have a redshift. The CMB-wavelengths coming from a galaxy are redshifted as a function of distance.

The link you are missing in point 3 is that the CMB itself is also blue-shifted to very short wavelengths by the Sunyaev-Seldovich effect. A single Compton event by a hot, relativistic electron scatters a CMB photon into radiation at much shorter wavelength (high energy X-rays, around $\lambda \approx 2 pm$ corresponding to $\approx 10^{10} {\rm K}$ or a blue-shift $z \approx -4\times 10^9$). This happens very infrequently, slowly removing photons from the CMB spectrum, but when the scattering occurs it sends the photon to the far X-ray end of the spectrum which means that it disappears from the CMB radiation. I don’t know the theory of the relativistic Sunyaev-Seldovich effect (if there is even one…), so I haven’t calculated the exact numbers. But we detect high energy photons from space, and the Sunyaev-Seldovich effect is observed as well, so this is not something I invent.

I mention this in my poster, section VI. 4th item.

Variable Mass of the electron… near measured value. Coincidence or not?

I could believe that this is not a coincidence. But why does VM give the mass of the electron? Why not $n_p m_p$, the protons are much heavier and their density is equal to that of the electrons… So is your result 1936 times too small? In any case, both our calculations keep track of energy which we assume is conserved in whatever process is happening out there. But since we don’t know why there are $\approx 0.2 … 0.5$ electrons/cubic meter, it is difficult to study this further.

@RedshiftDrift .

CMB-wavelengths coming from a galaxy

Are CMB-wavelengths coming from a galaxy? I’ve always thought that the CMB is everywhere, always of the same temperature, and not originating from some galaxy.

slowly removing protons from the CMB spectrum

I think you mean photons instead of protons.

But why does VM give the mass of the electron? Why not $n_p m_p$, the protons are much heavier and their density is equal to that of the electrons …

One could ask the same question for STz. Why do your counterpropagating beams not interact with protons instead of electrons? I suppose this question is merely symbolic. Admittedly, I do not know (yet) why the 1836 (not 1936) proportion $(m_p/m_e)$ with VM must be a constant.

Edit. Proceeding from (2) of your main comment. >the power lost by light, and therefore gained by the electron that causes redshift, is $P_e=E_e H_0$ .

Can we then say the following?

$$ P_e = \frac{d E_e}{dt} = E_e H_0 \quad \Longrightarrow \quad E_e(t) = E_e(0).e^{H_0 t} $$

Quite in agreement with Variable Mass ( as measured in atomic time $t$ ):

$$ \frac{mc^2}{m_0c^2} = \frac{m}{m_0} = e^{H_0 t} $$

Edit. On the other hand, the power is lost by the microwave Tired Light radiation and so

$$ P_{CMB} = \frac{d E_{CMB}}{dt} = - E_{CMB} H_0 \quad \Longrightarrow \quad E_{CMB}(t) = E_{CMB}(0).e^{-H_0 t} $$

The Variable Mass and Tired Light models are mutually incompatible except of course at the here and now timestamp.

$$ P_{tot}(t) = P_e + P_{CMB} = H_0.E_e(0)e^{H_0 t} - H_0.E_{CMB}(0)e^{-H_0 t} \quad \Longrightarrow \quad P_{tot}(0) = 0 $$

provided that

$$ E_e(0) = E_{CMB}(0) \quad \Longrightarrow \quad n_e m_e c^2 = \frac{4\sigma}{c}\Theta_{CMB}^4 $$

with here and now (i.e. for $t=0$) values on the right. As has been found before. Do we observe in the math that conservation of energy is local? And that it does not extend to cosmic timespans?

Are CMB-wavelengths coming from a galaxy?

Yes, all galaxies emit CMB-wavelengths (as part of their full spectrum…) Please read carefully, two things are discussed here: the “CMB” and “CMB wavelengths”.

• The CMB itself comes from every direction, producing a uniform background of radiation at the CMB-wavelengths near 3.2 cm.
• CMB-wavelengths (a spectrum of wavelengths around 3.2 cm) are emitted by galaxies and detected by radio telescopes coming from the specific direction of the galaxy. When redshifted by z = 2 for example, the image of a galaxy is then seen at 9.6 cm, on top of the CMB background at 3.2 cm. Therefore, it is experimentally shown that the light emitted at 3.2 cm by galaxies is redshifted, while the CMB is not (as discussed here and in the provided links).

I think you mean photons instead of protons.

Typo is now corrected!

Why do your counterpropagating beams not interact with protons instead of electrons?

Light also interacts with protons via Stimulated Transfers, but the momentum transfer of the photon results in a recoil that is $m_p/m_e = 1836$ times smaller because of the larger mass of the proton, and therefore the redshift-per-transfer is 1836 times smaller. So I neglect the 0.05% difference. (Photons interact with all charged particles to produce Stimulated Transfers, but since the electron is the lightest of all charged particles, electrons cause the largest part of the redshift.)

Can we then say the following? $E_{CMB}(t) = E_{CMB}(0).e^{-H_0 t}$

No because the exponential is the mathematical result of an assumed time varying energy $E_{CMB}(t)$. The energy of the microwave radiation is stationary $E_{CMB} = \mbox{ constant}$ (as explained above). The solution of the differential equation for the energy of an electron is linear in time.

@RedshiftDrift >1936 times smaller.

IMHO it is 1836 times smaller.

Can we then say the following? $E_{CMB}(t) = E_{CMB}(0).e^{-H_0 t}$

No? Thanks! That only makes it easier for me (and hopefully more consistent too). You cannot deny that $P_e = d E_e / dt$ , right? Then whatever further arguments you might utter, we have this differential equation and its solution for the electron. But the $E_{CMB}$ is a constant and therefore simply $E_e(0) = E_{CMB}$.

$$ \frac{d E_e}{dt} = E_e H_0 \quad \Longrightarrow \quad \large E_e(t) = E_{CMB}.e^{H_0 t} $$

The solution of the differential equation for the energy of an electron is linear in time.

No. It shows exponential growth in time.

@HanDeBruijn
“1936” typos corrected.

You cannot deny that $P_e = d E_e / dt$ , right?

The equation is correct, but electrons (plural) are not all at the same temperature! We have $P_e = P_H + P_N$, the heating power $P_H$ from redshifting radiation, and the lost power $P_N$ (a negative number) to nuclear fission (as discussed here). Energy is recycled, so $P_H = -P_N$ and $P_e = 0$, so $E_e$ is a constant over the entire population of electrons that are hot enough i.e. above some threshold for nuclear reactions at $1.2 \times 10^{10} {\rm K}$.

For colder electrons, $P_N = 0$ and $P_e(cold \ electrons) = P_H = d E_e / dt$. Since $P_H$ is a constant (the spectrum of the universe does not change) then the temperature of a cold electron increases linearly until it stops at $1.2 \times 10^{10} {\rm K}$.

Then whatever further arguments you might utter…

Same to you, whatever arguments you might utter are obviously wrong. So discussion about this is over.

@RedshiftDrift . >So discussion about this is over.

Almost.

It is thermodynamically impossible to have two temperatures at the same spot: one very low CMB temperature and one enormously high electron temperature. But there is a way out. I have just proved that the CMB energy combined with a low electron density is capable of creating an electron with that mass-energy:

$$ \large n_e m_e c^2 = \frac{4\sigma}{c}\Theta_{CMB}^4 $$

And now it’s over.

Interesting @HanDeBruijn; you get $n_e = 0.5099 \mbox{ electron/m}^3$ (with $T_{CMB} = 2.72548$ K), a density very close to @tiredlyndon’s value.

I get $n_z = 0.2909 \mbox{ electron/m}^3$ by using a cross section for the two-photon process of STz - rather than the usual thermodynamical interaction that produces $4\sigma\theta_{CMB}/c$. I know my value of the cross section is not that far off because it agrees with FRB data that will appear in another paper I still have to write.

P.S. From a conjecture in my paper (not accepted for publication yet),

$$ n_z = \frac{2\pi h}{m_e c} \big[ 2\pi\zeta(3) \big]^{4/3} \left( \frac{k_B T_{CMB}}{hc} \right)^4, $$

where $\zeta(3) = 1.202057$.

Friendly ping @wangyy19 (paper’s author) feel free to share if you have any thought about the analysis on the mean IGM electron density :)

@RedshiftDrift Great work!! Out of curiosity, can you tell me the ne value with H0=66.8 ?

The equation is given above: $S = n_e / (H_0/c).$

However, the value $H_0 = 66.8 \mbox{ km/s/Mpc}$ is derived from an analysis of data from Planck using a space-expanding model. The model is incompatible with STz, and that value has no meaning in the STz paradigm of a non-expanding cosmology.

This thread is about Stimulated Transfer redshift (STz). Can we stop trying to hijack this thread with expansionist ideas?

The cross agreement between much more diverse one step probes and the fact it can be mathematically derived are conducive evidence for the true hubble constant being arround 68km.

What physical significance can be attributed to 68km/s/Mpc in a non-expanding-universe cosmology?

Thanks. They’re publishing faster than I can write papers! (Strangely, I count 111 FRBs in their table… and FRB230521B at z=1.35 is no longer there.)

This doesn’t change much my analysis, the distribution of FRBs is still above the STz prediction, as expected. Once I start working on my FRB paper I’ll use the latest data.

Shown below: 112 FRBs with the MW contribution to Dispersion Measure (DM) subtracted. To within uncertainties of the MW models, the points are above the STz prediction green-line $= 1185 \mbox{ pc/cm}^3 \ \ln(1+z)$ due to the contributions from the FRB source.

117 FRBs https://arxiv.org/abs/2508.05161

“A fast radio burst from the first 3 billion years of the Universe” z = 2.148 https://arxiv.org/abs/2508.01648

It’s a good idea, but the large number of required FRBs makes this a very-long-term project: “We consider a future specialized FRB survey where the event rate will be generated based on the SKA survey, which is expected to operate for 50 yr and more.”

Their fitting model is based on expanding space, so they use

$$ \mbox{DM} = \int \frac{n_e}{(1+z)} \ dl $$

and

$$ \mbox{DM}_E = \mbox{DM} _{IGM} + \frac{ \mbox{DM} _{host} }{1+z}. \quad \quad \quad \quad (2) $$

This will add a bias on the final result since the correct relations in the non-expanding STz model are

$$ \mbox{DM} _{STz} = \int n_e dl $$

and

$$ \mbox{DM} _{E, STz} = \mbox{DM} _{IGM} + \mbox{DM} _{host}, $$

which are already confirmed with an analysis based on a non-expanding model.

Hi Carlos @carlosbelcech , “Diffusive interactions between photons and electrons: an application to cosmology” is published here https://indico.icranet.org/event/8/contributions/1535/ and an older version is available here https://arxiv.org/abs/2410.02036

I edited the top post of this thread.

@RedshiftDrift for that first link, they ask you to log in to view the PDF

Ok. thank you. I didn’t see it, I looked for the update on RG. Let’s move on!

@ExpEarth

for that first link, they ask you to log in to view the PDF

Thanks for pointing that out, I didn’t know other people could not access it. It may take several months before the MG-17 Proceedings are published, so until then the arXiv version https://arxiv.org/abs/2410.02036 is the official preprint.

By amazing coincidence I met a researcher in Toronto, Romeo Bruni, who recently did a video on this topic. It’s meant to be an educational video. I chastised Romeo for his low-level audio-visual techniques, but his explanations are still concise. Romeo also has a model of the shell universe which is based on Big Bang assumptions. I suggested Romeo join AGC to help refine his work.

https://www.youtube.com/watch?v=qTA7BaTvhi8

@carlosbelcech I have read about the de Broglie - Proca - Maxwell theory and the effects they predict from it, but didn’t follow any of the theoretical derivations in details.

They start by assuming the photon has a mass, then derive properties that follow from that. I trust the math is fine and that under that assumption of a photon mass, there is dispersion (speed of massive-photon is frequency dependent). However, the physics implied by such a hypothesis may be wrong since under the assumption of a “photon mass”, usual equations such as $E=mc^2$ become invalid. As a result, redshift in massive photons causes energy to disappear on its own (I don’t subscribe to ‘energy non-conservation’).

So they modify the Maxwell equations but mistakenly use otherwise “standard physics” in the derivations. This approach is bound to fail. By assuming a photon mass, an entirely new physics would have to be developped, not just a modification of the Maxwell equations. So I reject the hypothesis of photon mass done independently of everything else it affects.

There are many papers about massive photons claiming to explain redshift, but I can only cite one (I don’t have much interest in the hypothesis) A.D.A.M. Spallicci, J.A. Helayël-Neto, M. López-Corredoira, and S. Capozziello, “Cosmology and the massive photon frequency shift in the Standard-Model Extension,” Eur. Phys. J. C, vol. 81, no. 1, p. 4, Jan. 2021, doi: 10.1140/epjc/s10052-020-08703-3.

The hypthesis of “photon mass” is incompatible with the results derived for STz. STz is based on QED (no photon mass). No new assumption is added to QED for the derivation of STz. ‘Redshift’ appears as a consequence of a quantum interaction with electrons, without any new physics (I reference 30-year-old textbooks in my papers).

Fine, thanks. Really, the nonzero rest mass of photons represents a disruptive change in relativity and electrodynamics. After all, the constancy and unattainability of c in vacuum is the cornerstone of the theory and principle of relativity. If the constant c doesn’t hold, we’ll be in trouble. Mine is just an exploratory attitude. But if cosmological time dilation is confirmed, I think this problem merits quick calculations. Incidentally, the problems posed by the nonzero rest mass can be neglected, reducing its value to the minimum necessary. That is, from cosmological time dilation, an upper limit for this could be calculated. By other hand, I have doubts that non zero mass implies non conservation of energy in every theoretical frames.

Based on latest FRBs with the highest dispersion measure, we already have candidates with a redshift of ~3.5 though JWST observations are required

Unfortunately, FRB20240304B has a high host-DM, not too useful for my paper…

Starting with people in the Netherlands - see here , I have had intensive contact with (Young Earth) Creationists for quite some time. Especially Barry Setterfield should be mentioned in this context. John Hartnett too has thoroughly criticised Setterfield’s theories. It’s a small world after all !

Note.

This picture and the corresponding naming in The Cosmic Time Frame discussion is copyrighted by Barry Setterfield.

@HanDeBruijn Interesting! I see in this image the same idea about time presented in the framework of VM. Does VM lead to Setterfield’s c-decay cosmology, or vice-versa, or are the two exclusive but with similar features?

Like me, John Hartnett had a career in time measurement with sapphire clocks and atomic clocks. John Hartnett was building the best cryogenic oscillators in the world (a mechanical oscillator, not an atomic clock). I met him at CCC-2, when I also met @tiredlyndon . John’s physics view on time is very similar to mine (not the biblical part :wink:). He worked on a 5-dimensional theory, three spatial dimensions and two temporal dimensions, but unlike @HanDeBruijn ’s the temporal dimensions were $t_1 =$ physical time and $t_2 =$ biblical time, the universe being only 6000 years old in $t_2$. The model was relativistic based on Carmeli’s 5D theory. Of course, theorists can invent models with any number of dimensions… and John added $t_2$ as he believes the bible is as true as physics.

John eventually came to his senses and rejected the Carmeli cosmology. He supports tired light that works without a hypothesis like space expansion or other changes of the universe (which of course, was created immutable by God).

So it’s not surprising that John rejects Setterfield’s theory (thanks @HanDeBruijn for the article, I did not know about it). His physical arguments are the same I would use (and he complements that with biblical arguments I do not need nor believe in :wink:)

@RedshiftDrift. >I see in this image the same idea about time presented in the framework of VM.

That’s no coincidence. I’ve been playing around with Young Earth Creationist (YEC) ideas for quite a while. Later on, I discovered similar ideas with other authors, but I still find Setterfield’s nomenclature the best. This table should be augmented with naming in the Time Dilated Past hypothesis by Mike Helland. Which has become somewhat confusing, because Mike has changed his terminology on the fly. I think the correspondence is: Expanded time = Atomic time ; Minkowski time = Orbital time (apart from minor details. Let @mikehelland correct me if I’m wrong).

Does VM lead to Setterfield’s c-decay cosmology, or vice-versa, or are the two exclusive but with similar features?

I’ve been playing around with these ideas as well. See: C-decay Theories and C-decay data fit, with doubtful results. Furthermore, a unification of Variable Mass and Setterfield’s c-decay cosmology inevitably leads to the following.

$$ E = mc^2 = m_0c_0^2 \quad \mbox{in the Orbital time-frame} $$

According to very recent findings this would kill any version of the Variable Mass theory: thus c-decay and VM are incompatible. >where $t_1=$ physical time and $t_2=$ biblical time, the universe being only 6000 years old in $t_2$.

Yes I know. See the last part of Milne’s Formula where I’ve been trying something similar. I do no longer think these attempts of mine are quite relevant, though.

the universe (which of course, was created immutable by God).

Affirmative, for sure. I could call myself an Old Earth creationist.

Edit. Basic formulas of VM are

$$ \frac{m}{m_0} = \left(1+\frac{1}{2}H_0 T\right)^2 \quad ; \quad \frac{dT}{dt} = \sqrt{\frac{m}{m_0}} $$

where $m=$ (elementary particle rest) mass, $H_0=$ Hubble parameter, $T=$ orbital time, $t=$ atomic time, subscript $0=$ here and now. Herewith we find for c-decay $c(T)$ as derived from a distance $r$ and the two time-frames:

$$ c_0 = \frac{dr}{dt} = \frac{dr}{dT}\frac{dT}{dt} = c(T)\sqrt{\frac{m}{m_0}} = c(T)\left(1+\frac{1}{2}H_0 T\right) \quad \Longrightarrow \quad c(T) = c_0/\left(1+\frac{1}{2}H_0 T\right) $$

C-decay (take $T\le 0$) is therefore compatible with Variable Mass. And for mass-energy $mc^2$ we find:

$$ c_0 = c(T)\sqrt{\frac{m}{m_0}} \quad \Longrightarrow \quad \frac{m c^2}{m_0 c_0^2} = \frac{m}{m_0}/\frac{c_0^2}{c^2} = \frac{m}{m_0}/\left(\sqrt{\frac{m}{m_0}}\right)^2 = 1 \quad \Longrightarrow \quad mc^2 = m_0c_0^2 $$

My understanding of your redshift mechanism is a bit rusty, Louis, and sad to say but this abstract does not get me back on the track towards understanding it. The cascade idea does not really work for me. It’s reassuring there is a Schrodinger equation behind it, but the catalytic light field is murky. Since at root is a photon-electron interaction, why not frame it as some form of Compton effect?

You have a photon transferring momentum to an electron, which then transfers it back to a photon moving in the opposite direction. So maybe you could call it a PEP (photon-electron-photon) redshift or PEPS (photon-electron-photon stimulated) redshift.

@ExpEarth There is a lot of history behind this choice of words. Years ago I framed this model as some form of Compton effect. People then argued that the cross section of the Compton effect is 10000000 times too small to have a significant effect, and it blurs images; and so rejected Stimulated-Cascade redshift :disappointed: I have been avoiding the name “Compton” since then.

The name for this effect is actually “ponderomotive force”, but when I used that nomenclature people argued that it only works in free-electron lasers and would not work for the low light conditions of space; and so rejected again Stimulated-Cascade redshift based on a false argument :disappointed: :disappointed:

“photon-electron-photon” also has problems, it is the way to express a two-photon interaction in nonlinear optics. People argue then that the process is quadratic in light power and cannot produce a uniform redshift at different light intensities; and so they reject again Stimulated-Cascade redshift based on an incorrect understanding :disappointed: :disappointed: :disappointed:

Despite mentions of quantum mechanics since my first paper, people think I am pulling this idea out of a hat. Stimulated Light Cascade (what I earlier called Stimulated Transfer) is a solution to the Schrödinger equation, not an ‘made up’ model.

Without mention of the ‘two-interaction cascade’, members on this forum did not understand how the $180^\circ$ reversal doesn’t make light disappear completely in the forward direction (see https://github.com/orgs/a-cosmology-group/discussions/281#discussioncomment-11198970).

I thought “stimulated emission into the catalytic light field” was quite clear, since anybody who has seen a laser knows what stimulated emission is, and catalytic reactions are part of all basic chemistry sets.

What could capture the essential properties of this redshift mechanism without having to dig into long technical details? How can this be explained better?

Wow, that is quite a lengthy history indeed! And reviewing the discussion you linked to, it seems that our AGC members were having difficulties. All I can really say is that your new name for it does not quite convey what you want to convey. When I think of something cascading, it is kind of like an intensification of a flow or something. That’s hard to connect to a weakening magnitude like redshift.

It also occurred to me that your system might not be in energy balance. If the first photon strikes the electron and loses momentum by transferring it to a photon moving in the reverse direction, then both photons are being redshifted. Are the photons of the catalytic field different in some respect? If not, then all photon energy would be slowly diminishing. This is a problem of the BB model also. Where does the energy go in your system? In your first paper that you linked to, you suggest that the interstellar medium is being heated up by this process. Is that it? If so it would just get hotter and hotter and we don’t see that. Sorry if you have covered this before.

@ExpEarth the image of a ‘cascade’ was meant to convey a loss of energy in a step wise manner, going down at each interaction, like water going down the steps of a cascade. I was thinking of ’loss of potential energy“ but I understand you see this as “gain of kinetic energy”… Ideas?

If the first photon strikes the electron and loses momentum by transferring it to a photon moving in the reverse direction

No, the incident photon disappears. The text says “an incident photon is replaced by a new photon of longer wavelength…” The long explanation is: The incident photon of energy $E^+$ is replaced with a photon of longer wavelength, i.e. less energy $E^-$, stimulated into the catalyst beam (which also has an energy $E^-$ for stimulated emission to occur). None of the photons in the catalyst beam are modified. However, an additional photon is added to the catalyst beam with an energy $E^-$.

Where does the energy go in your system?

The text says “… via momentum recoil and the associated energy transfer to a localized electron.” The energy goes to the electron!

…

A cartoon might help! Ambient light has a full spectrum and propagates in every direction. Before: the incoming blue photon is propagating toward the right before reaching a localized electron. After: the incoming photon has disappeared, a new red photon is stimulated (appears), propagating toward the left; the electron has been kicked and has gained kinetic energy. This is one interaction. Two such interactions in a cascade produce a red-shifted photon in the original photon’s direction.

It seems like there could be conservation of energy for the one-photon interaction shown, but I’m not sure about the two-photon interaction, the “cascade”. Is that second photon moving in the same direction as the original one? Does the electron acquire even more kinetic energy? What about the Doppler shift? Seems like a complicated scenario.

Another issue pending. I asked Google AI : can a single electron have a temperature ?

No, a single electron cannot have a temperature, because temperature is a macroscopic property of a collection of particles, not a property of a single particle. It is a statistical concept that requires many interacting particles in thermodynamic equilibrium to be meaningful.

Why Temperature Applies to a System, Not a Single Particle

• Averaging Kinetic Energy: Temperature is a measure of the average kinetic energy per particle within a system. For a single particle, there is no statistical variance of kinetic energy to define an average.
• Macroscopic Concept: Temperature is a macroscopic property, meaning it describes the behavior and properties of large-scale systems, not individual components.
• Thermodynamic Equilibrium: For a quantity to have a temperature, the system must be in thermodynamic equilibrium, which involves a collection of particles interacting and sharing energy.

Analogy

Just as you can’t determine the “average speed” of a single car on a highway without reference to other cars, a single electron does not have a temperature without the context of a larger, interacting group of particles.

It seems like there could be conservation of energy for the one-photon interaction shown

@ExpEarth this is hopeless… the image shows a two photon interaction.

… but I’m not sure about the two-photon interaction, the “cascade”.

The cascade involves three photons… Clearly my text is not clear, or you haven’t read it. If one interaction conserves energy, two such interactions will conserve energy.

Is that second photon moving in the same direction as the original one?

Read up on stimulated emission https://en.wikipedia.org/wiki/Stimulated_emission, the answer is in the second sentence.

Does the electron acquire even more kinetic energy?

The cascade involves two electrons… Clearly my text is not clear, or you haven’t read it. The second electron is kicked like the first one was, and gains kinetic energy while the stimulated 3rd photon has an even lower energy.

What about the Doppler shift?

It is mentioned here: “… an incident photon is replaced by a new photon of longer wavelength via momentum recoil.”

@HanDeBruijn there is more than one electron in intergalactic space, so there is no ‘temperature’ issue.

Since ‘cascade’ brings up the wrong image, and ‘catalyst’ doesn’t improve clarity, perhaps Stimulated Transfer redshift was better.

Forget the description above, how about this one:

“The Stimulated Transfer redshift (STz) mechanism introduces a new category of cosmological redshift based on stimulated emission. It is distinct from both spacetime metric models and all traditional Tired-Light models. At its core is an interaction with an electron, where the incident photon disappears and is replaced by a new photon of longer wavelength, with the leftover energy transferred to the electron via momentum recoil. Critically, the transfer involves stimulated emission into a counter-propagating light field. This fundamental process is described by a second-order time-dependent Schrödinger equation (TDSE), as detailed in previous work. …”

Second half to be posted later…

@RedshiftDrift . >there is more than one electron in intergalactic space, so there is no ‘temperature’ issue.

Let’s repeat. The Cosmic Background Radiation has a temperature of 2.725 Kelvin (K). It has been argued that photons of the CMB are kissing the IGM electrons, such that their temperature rises to a mass-energy equivalent, which is the incedibly high $\approx 10^{11}$ Kelvin.

$$ \large n_e m_e c^2 = \frac{4\sigma}{c}\Theta_{CMB}^4 $$

where $n_e=$ electron density, $m_e=$ electron mass, $c=$ speed of light in vacuum, $\sigma=$ Stefan-Boltzmann constant, $\Theta=$ temperature.

According to the second law of Thermodynamics, thermal energy cannot spontaneously flow from a colder to a hotter object. Therefore the temperature of electrons in the InterGalactic Medium, if that would make sense, cannot be else than the same 2.725 Kelvin (K) of the CMB. Contradiction. The only way out is to recognize that IGM electrons have no temperature at all. More in general I do not believe anymore that a medium containing less that one particle in a cubic metre can have any interesting physical properties. In my humble opinion it’s just empty space, nothingness.

Sorry, Louis, if I’ve put you on the wrong foot.

@RedshiftDrift It’s true my understanding of your model is incomplete - and maybe getting worse! The Wikipedia page you linked to is not helping much, as I’m not sure that process really applies here. Taking out ‘cascade’ is good though. I will refrain from more comment until you have your abstract completed. It should be possible from the abstract to get a basic understanding of your model.

According to the second law of Thermodynamics, thermal energy cannot spontaneously flow from a colder to a hotter object. - @HanDeBruijn

Thermodynamics applies only to systems in equilibrium.

However, the CMB is not at equilibrium with another medium at the same temperature, and stimulated emission produces a population inversion (like in lasers) which renders useless the concept of thermodynamic temperature because there is no equilibrium.

@ExpEarth no worries, you are not the only one who doesn’t understand “stimulated emission” :wink: Unfortunately, without this understanding it is impossible to understand the STz mechanism.

The first half of the abstract should contain everything needed to understand the mechanism (the second part will describe ‘applications’ of this mechanism). Even without an understanding of stimulated emission, you could help answering this (if you wish to do so).

STz is not a model, it is a physical mechanism (or an effect) obtained as a solution to the Schrödinger equation. It describes an interaction between light and electrons. (I am responsible for the confusion… I have frequently called this “a model” and it is even presented in the GitHub section “Redshift Models”, although I list it as a “mechanism” here. Even Lyndon and Brynjolfsson sometimes slip and call their mechanism a model!)

In the short abstract, I try to tell the reader that STz is a calculable mechanism by writing “The … STz mechanism introduces a new category of cosmological redshift … It is distinct from both spacetime metric models and all traditional Tired-Light models. … This fundamental process is described by a second-order time-dependent Schrödinger equation.”

Here’s an updated version: “The Stimulated Transfer redshift (STz) introduces a new category of cosmological redshift mechanisms [effects] based on stimulated emission. It is distinct from both spacetime metric theories and all traditional Tired-Light models. At its core is an interaction with an electron, where the incident photon disappears and is replaced by a new photon of longer wavelength, with the leftover energy transferred to the electron via momentum recoil. Critically, the transfer involves stimulated emission into a counter-propagating light field. This mechanism [effect] is described by a second-order time-dependent Schrödinger equation, as detailed in previous work.”

From this, everything else can be deduced. So it’s important to make it clear. What do you think, does it work better? Should I use “effect” instead?

@RedshiftDrift It seems that by saying “mechanism” you want to cast it as an established mechanism in electronic processes. This makes me think that you might better accomplish that by leaving cosmology out of the discussion for now. If you can establish the redshift mechanism and publish that with a physics journal, you could apply it to cosmology later.

But the points of confusion continue to multiply. Keep in mind I haven’t looked at your paper for awhile and so the setup is hazy for me. So, looking at what you have said just now, at first I thought you were talking about a photon-electron interaction. That’s why I thought it could be a Compton effect and you said you had tried that. But then in explaining your mechanism further you linked to the Wikipedia page on stimulated emission, a process in lasers. This naturally suggested that the photon interaction is with atoms rather than isolated electrons. If I were to conflate these two ideas and also add in your ’catalytic field, I might suppose the catalytic field somehow adopts the role of the atom in stimulated emission. Mention of the Schrodinger equation actually becomes a bit scary here. In short, your abstract needs work!

Another thought. One thing I have noticed when working on a model for a long time, as in years, it can get harder to put on paper what you actually meant to say - and what you might have said better earlier. You can have the impression that you have it all down for readers, when in fact big important chunks had mistakenly been edited out long ago!

I don’t think it makes too much difference whether you use models, mechanisms or effects.

which renders useless the concept of thermodynamic temperature because there is no equilibrium.

@RedshiftDrift , how do you arrive then at $10^{11}$ Kelvin (= “infinity”) ? And how are you going to manage nuclear reactions to happen with a laser temperature that is “negative” and without thermodynamic equilibrium ?

It seems that STz becomes more complicated with every new door in its escape room.

… when working on a model for a long time, as in years, it can get harder to put on paper what you actually meant to say - and what you might have said better earlier. - @ExpEarth

That’s true :wink: My first abstract did not assume any prior knowledge because the idea had never been published! So it doesn’t suffer from the assumption that people have read my previous papers.

“The redshift observed in astronomy is explained by optical forces on electrons and atoms present in the intergalactic medium. The forces occur as a result of momentum exchange between light and matter. This produces an exchange of energy where the intergalactic medium is heated and the radiation is redshifted. The spectra of different light rays interact in the intergalactic medium to produce a spectral transfer redshift. Two mechanisms are examined: the dipole force which acts on atoms, and the ponderomotive force which acts on electrons. The dipole force produces a redshift which is wavelength dependent and can be distinguished from a Doppler redshift. The ponderomotive force produces a redshift which is independent of the wavelength and mimics a Doppler redshift. Both forces arise from a stimulated emission process which preserves the direction of propagation of one of the two interacting photons, a necessary condition to avoid blurring of images. This property distinguishes the spectral transfer redshift from all other redshift mechanisms proposed so far.”

I will actually re-use that abstract (with minor language improvements), since

• it explains STz as it is,
• the formal language is used: “dipole force”, “ponderomotive force”, and “stimulated emission”,
• it contains every essential elements of the mechanism, (almost; also needs “… medium is heated by momentum diffusion and …”)
• it shows the idea was already complete 16 years ago, (“momentum diffusion” was solved later with a full quantum treatment)
• it is still true today that ‘stimulated emission’ sets STz apart from all other proposed redshift mechanisms,
• it answers your questions @ExpEarth, and
• it shows that I am not making up new things while clearing up misunderstandings (right @HanDeBruijn?)

Thanks for the suggestion @ExpEarth.

… stimulated emission, a process in lasers. This naturally suggested that the photon interaction is with atoms rather than isolated electrons. - @ExpEarth

Yes, ‘stimulated emission’ occurs in atoms. ‘Stimulated emission’ also occurs in ‘free electron lasers’! When I say “I own a car”, does it naturally suggest it’s a Ford because that’s the most popular car in Canada?

Mention of the Schrodinger equation actually becomes a bit scary here.

That’s another strange comment! It’s as if I told a librarian “Mention of the Dewey Decimal System in your abstract actually becomes a bit scary here.” The Schrödinger equation, like Maxwell’s and Newton’s equations, is fundamental in physics, and this scares you away?

… all photon energy would be slowly diminishing. This is a problem…

Another strange comment. It’s not a problem, it’s a feature :smile: The STz mechanism diminishes the energy of all light in the universe, redshift is universal!

@RedshiftDrift, in defense of @ExpEarth.

That’s another strange comment! [ .. ] The Schrödinger equation, like Maxwell’s and Newton’s equations, is fundamental in physics, and this scares you away?

It’s not so strange! When scanning through the information I have access to, I cannot discover anywhere in your papers and replies a form of the Schrödinger equation that I’ve been teached at the university, or can be retrieved from e.g. Wikipedia.

Speaking in general, STz indeed has become scaring. It is very much lacking simplicity, mathematical as well as physical. Every clearing up “misunderstandings” in favor of your Cosmos as a Laser Show model makes it more complicated. It will be increasingly difficult to build something on top of it. And there is clearly a strong bias that prevents you from seriously looking at any non-optical alternatives.

0. For someone with only a hammer, everything looks like a nail
1. for someone with only Gravity, everything looks like a Black Hole
2. for someone with only Plasma, everything looks like a Z-pinch
3. for someone with only SED, everything looks like a ZPE
4. for someone with only Optics, everything looks like a Laser

I cannot discover anywhere in your papers and replies a form of the Schrödinger equation that I’ve been teached at the university, or can be retrieved from e.g. Wikipedia. - @HanDeBruijn

Since everybody knows the Schrödinger equation, I don’t repeat it in my papers.

Wikipedia gives the density-matrix representation of the Schrödinger equation you have been taught in university,

$$ i\hbar \frac{\partial \tilde\rho}{\partial t}=[\tilde{H},\tilde{\rho }].$$

In my MG-17 paper posted on arXiv:2410.02036, I expand in Sec. 2 the Hamiltonian $\tilde H$ as

$$\tilde H = \frac{\tilde p^2}{2m_e} + \sum_i \hbar\omega_i \ \tilde a_i^\dagger \tilde a_i + \tilde V(t).$$

In Sec. 2.2, I expand the interaction potential for an electron interacting with two travelling waves as

$$\tilde V(t) = \frac{e}{m_e} \boldsymbol A \cdot \boldsymbol p = \frac{e}{m_e} \int \mbox{ d}\boldsymbol p \sum_i \mbox{ i} \sqrt{\frac{\hbar}{2\varepsilon_0 \omega_i L^3}} \left[ \tilde a_i \mbox{ e}^{+\mbox{ i}\vec k_i \cdot \vec r} +\tilde a_i^\dagger \mbox{ e}^{-\mbox{ i} \boldsymbol k_i \cdot \vec r} \right] \boldsymbol{\hat\epsilon}_i \cdot \boldsymbol p.$$

This interaction potential is substituted into the Hamiltonian above, and then that Hamiltonian into the density-matrix Schrödinger equation.

I expand that Schrödinger equation to second order in time, which gives

$$\boxed{ \Delta \tilde\rho(t) = \frac{1}{i\hbar} \int_{t_0}^{t_0+\Delta t} \mbox{ d}t \left[ \tilde{V}(t), \tilde\rho(t_0) \right] -\frac{1}{\hbar^2} \int_{t_0}^{t_0+\Delta t} \mbox{ d}t’ \int_{t_0}^{t’} \mbox{ d}t \left[ \tilde{V}(t’), \left[ \tilde{V}(t), \tilde\rho(t_0) \right] \right] },$$

where $\rho(t)$ is the density matrix which represents the wavefunction $|\Psi \rangle$.

This, is the time-dependent Schrödinger equation for an electron interacting with travelling waves. Its solution is

$$\frac{\Delta \rho(t)}{\Delta t} = \frac{\alpha c \lambda_e \lambda}{\pi} \sum_{i, j} \frac{\sqrt{n_i n_j}}{L^3} \rho(t_0).$$

from which constants and properties discussed here are derived.

On your queries, the link to “free electron masers” says specifically that the process is not stimulated emission! - @ExpEarth

Not quite, it says that the FEL employs stimulated emission in electrons rather than in atoms or molecules. (The wikipedia article is ambiguous.)

What Einstein called ‘stimulated emission’ in 1917 applied to atoms because they have internal degrees of freedom, that is, atoms can absorb energy and retain it internally for a certain time - the atom is in an “excited state”. When an incident photon interacts with an excited atom, the atom releases its energy by creating a second photon which is an exact copy of the first one. The incident photon has stimulated the emission of a second photon. The incident photon has been ‘amplified’ by the excited atom to two photons. Repeat the process and you get light amplification in a laser.

Electrons don’t have internal degrees of freedom, they cannot hold internal energy. However, when the electron is in an external field, like the “undulator field” of a free-electron laser, then the electron has access to the energy of the external field. When an incident photon interacts with a polarized electron, the electron converts the energy of the field into a second photon which is an exact copy of the first one. The incident photon has stimulated the emission of a second photon. The incident photon has been ‘amplified’ by the electron+field to two photons. Repeat and you get light amplification in a free-electron laser.

The wikipedia page should read: “A FEL functions much as a laser but employs relativistic electrons in an undulator as a gain medium, instead of using atoms or molecules as a gain medium. Amplification in the gain medium is produced by stimulated emission.

Remember, “Free-Electron LASER” means Free-Electron Light Amplification by Stimulated Emission of Radiation.

The text here is clearer:

A free-electron laser is a relatively exotic type of laser where optical amplification is achieved in an undulator, fed with high energy … electrons from an electron accelerator. … the undulator … forces the electrons to radiate … Both spontaneous and stimulated emission occur, allowing for optical amplification in a certain wavelength range.

Einstein did not know about this in 1917, the maser was invented in 1953, and the Free-electron laser was invented in 1971. The term ‘stimulated emission’ is not commonly used for electrons because Einstein applied to atoms. But when an external field is present, an additional photon is stimulated by an electron in the same way Einstein described it for atoms.

@RedshiftDrift Thanks for the clarification! That ‘undulator’ looks like a pretty ordered structure though - all those magnets perfectly aligned. What plays the undulatory role in your model?

That ‘undulator’ looks like a pretty ordered structure though - all those magnets perfectly aligned. What plays the undulatory role in your model? - @ExpEarth

The wavelength of the counter-propagating light field plays the role of the undulatory field of the magnets.

It’s reassuring there is a Schrodinger equation behind it - @ExpEarth

On the Schrodinger, it made me nervous because the new abstract was so vague. It read like the equation was just being added to make it sound scientific. - @ExpEarth

These two comments are not consistent. STz is based on a solution to second-order ¹ of the Schrödinger equation for the problem of a photon interacting with an electron whose motion is polarized by a light field. That’s why the Schrödinger equation is mentioned.

The wavelength of the counter-propagating light field plays the role of the undulatory field of the magnets.

Can those fields match this? How?

@RedshiftDrift I noticed an article that says the quantum vacuum can act as the undulator, but it seems to be able to do this on the photons alone (without electron involvement). In that case, maybe my postulated photonic filament structure of spacetime could also serve in that role.

J T Mendonça 2009. New effects in quantum vacuum: photon undulator and transition radiation. J. Phys. A: Math. Theor. 42 375403 DOI 10.1088/1751-8113/42/37/375403

Can those fields [counter-propagating light fields] match this [undulator field]? How? - @ExpEarth

Yes they can, and they do. See my CMC-2 poster.

Counter-propagating light waves create a standing wave, exactly like the standing wave produced by an undulator.

This is shown on this image (extracted from the CMC-2 poster). The shaded grey rectangle represents the standing wave that produces redshift.

The interference pattern on my image is actually borrowed from another article that shows the undulator producing a redshift on the radiation (by a process very similar to, but distinct from STz).

$DM_{Halo}=0$ I like that. It sets the tone outright and throughout. I will steal that a little bit.

A fine and convincing read, Louis.

EDIT: what do you use to make your charts and graphs?

Thanks @Francois-Zinserling! $DM_{halo} = 0$ is somewhat controversial in $\Lambda{\rm CDM}$ because it could imply that dark-matter halos don’t exist, and therefore dark matter doesn’t exist. But there are too many things we don’t understand about FRBs to make a convincing argument against dark matter based on no measured contribution to the DM by halos.

Graphs made with LibreOffice.

@RedshiftDrift . From the paper at 5. Conclusions : >Using the electron density dependence of the STz effect, we link the cosmological redshift to the dispersion observed in fast radio bursts (FRBs) through the STz cross-section [ .. ] Critics have argued that the STz effect fits the Hubble-Lemaître Law of cosmological redshift by coincidence. However, we demonstrate that the STz effect also quantitatively describes the Macquart relation of FRB dispersion measure.

Using Google AI we get a rather precise description of “dispersion measure astronomy” : >In astronomy, Dispersion Measure (DM) quantifies how much a radio signal (like from a pulsar or Fast Radio Burst) slows down and spreads out (disperses) passing through free electrons in space [ .. ] Lower frequencies travel slower, creating a detectable pulse delay

I’m confused. If dispersion applies to FRB’s then it must hold for other EM waves as well. Could it be then that cosmological redshift is not the same for different frequencies, and that either dispersion OR cosmological redshift may be explained by STz but not both ?

If dispersion applies to FRB’s then it must hold for other EM waves as well. - @HanDeBruijn

Yes, it holds for all electromagnetic waves (up to gamma rays ~511 keV), as per Eq. (5).

Could it be then that cosmological redshift is not the same for different frequencies, - @HanDeBruijn

No, the cosmological redshift $\Delta \lambda/\lambda$ is the same for all frequencies.[1-3] This was tested by measuring redshift with radio telescopes and comparing with the spectroscopical redshift obtained with optical telescopes.

… and that either dispersion OR cosmological redshift may be explained by STz but not both ? - @HanDeBruijn

STz explains redshifts. Dispersion is explained by plasma physics. My choice of the word ‘describes’ may not be the best.

[1] I. Ferreras and I. Trujillo, “Testing the wavelength dependence of cosmological redshift down to $\Delta z \sim 10^{-6}$,” ApJ, vol. 825, no. 2, p. 115, July 2016, doi: 10.3847/0004-637X/825/2/115. [2] N. Maddox, K. M. Hess, S.-L. Blyth, and M. J. Jarvis, “Comparison of HI and optical redshifts of galaxies - The impact of redshift uncertainties on spectral line stacking,” Monthly Notices of the Royal Astronomical Society, vol. 433, no. 3, pp. 2613–2625, Aug. 2013, doi: 10.1093/mnras/stt934. [3] A. E. Lilley and E. F. McClain, “The Hydrogen-Line Red Shift of Radio Source Cygnus a.,” The Astrophysical Journal, vol. 123, p. 172, Jan. 1956, doi: 10.1086/146144.

@tiredlyndon come out of lurking anytime if you’re bringing comments like those. Probability of interaction linked to wavelength makes a lot of sense!

Probability of interaction linked to wavelength makes a lot of sense! - @Francois-Zinserling

Indeed, this must be predicted by the redshift mechanism. What @tiredlyndon writes in words

So a photon with twice the wavelength makes twice as many photon-electron interactions

is already written above and expressed as Eq. (6) in arXiv:2410.02036

$$ \Gamma(n_e) = \frac{\alpha c \lambda_e \lambda}{\pi}n_e. \quad \quad \quad \quad \quad \quad {\rm (6)} $$

as being proportional to the wavelength $\lambda$.

Hi Carlos, doing well but having difficulties to publish :smile: In any case, I’m still trying!

You ask:

Why did you refuse the bichromatic and not collinear stimulated emission scenario?

I wrote the answer to this is in a paper that is not published yet. I edited my draft and left only the part about STz; it is attached below.

Conservation laws implies that the spatial integral of the density operator, which represent the stimulated transfer rate, is non-null only when:

• one photon is paired with exactly one other photon,
• their wavelengths differ by $2h/(m_ec)$, and
• the two photon-modes propagate in opposite directions.

You can read that on line 82 of the attached draft. draft-stz-carlos.pdf

$\ $

Do you think that same process happens to the whole EM spectrum? That is, from nanometers to meters of wavelength?

Yes, the stimulated transfers occur at all wavelengths. However, there are limitations on the applicability of the calculations. At long wavelengths (tens of meters) there could be some collisions between electrons which could destroy the effect. Also, collective motion of electrons could imply a larger effective mass, decreasing the recoil. I don’t know how to calculate that limit yet. At short wavelengths (gamma rays) there will be electron-positron pair productions when the photons reach 511 keV or 0.0024 nm. At that wavelength and shorter, the STz effect will certainly not work because random pair-production will destroy the coherence needed for redshift.

STz works for $0.0024 nm < \lambda < 10 m$, but I don know for sure outside that range.

$\ $

Currently, you haven’t written something about time dilation. I think that in your two-step STz proposal, there is a delay in the transference of photons from the shorter wavelength to the longer one of a given spectrum, and this could be thought like a refraccion index of the IGM plasma. This could be the cause of time dilation.

At this moment, I’ve the idea of performing a possible paper about this, based on invariant embedding, triyng of shown the displacement of spectra and time dilation. What do you think?. I’m not sure.

It would be best to wait until I have a published paper about “light curve stretching”. A refraction index would only cause a time ‘delay’, not stretching!

In another paper that is not published yet (preprint coming soon :open_mouth:) I wrote this. Note the last paragraph about the 511 keV limit and light curve stretching.

A formal description of time-dependent STz is still work in progress. It is expected that a result can be derived from wave propagation in vacuum (Maxwell’s equations) with the addition of the quantum properties described in Sec. 2.

Time-dependent STz can be described in the frequency domain or in the time domain. Since stimulated emission maintains coherence between all frequency compo- nents of a spectrum, STz can be described with a Fourier transform, Eq. (1), where the redshift is coherent between every component of the spectrum. Stimulated transfers maintain the correlations between frequency components of the spectrum as they are all shifted toward the red. Not only the optical frequency is reduced, but the interval between these frequencies is also reduced. The long time variation of the light curve, encoded as a modulation in the Fourier components, is therefore stretched by the same 1 + z factor.

From the Fourier transform argument, the luminosity distance for a transient event is expected to be $d_{Lt}(z ) = (1 + z ) D_z ln[1 + z ] \quad \quad \quad \quad \quad \quad \quad \quad \quad (18)$ for supernovae light-curves and fast radio bursts. This is a result of the a constant photon number stretched over a longer duration.

Confirmation or falsification of Eq. (18) will be obtained from the formal derivation of the time-dependent STz.

I must stress that in the STz framework, there are two distance measures: the luminosity distance d_{L0} (z) given by Eq. (16) for sources of constant luminosity such as galaxies, and the luminosity distance d_{Lt}(z) given by Eq. (18) for time-dependent sources such as SNe and FRBs.

Since STz is a photon-electron interaction, the validity of Eq. (18) is limited to photon energies below the electron’s rest-mass energy. With photon energies near or above 511 keV, electron–positron pair production will destroy the coherence needed for stimulated transfers. While expansion predicts time dilation at all wavelengths, STz predicts a reduction, and eventually no redshift and light-curve stretching for high energy Gamma-rays. This possibility for a falsification of STz is discussed further in the following Section.

Conservation laws implies that the spatial integral of the density operator, which represent the stimulated transfer rate, is non-null only when: “one photon is paired with exactly one other photon, their wavelengths differ by 2h/(mec) , and the two photon-modes propagate in opposite directions.”

Okay. I understand this to mean that the interaction of beams (with different angles and energies) would have no net effect on the spectra. However, the phenomenon does exist. Only for beams propagating in opposite directions is there a redshift effect.

I tried to approach this topic from thermodynamics, as in Einstein’s original paper on stimulated emission. That is, by enclosing an electron gas and a radiation bath at different temperatures in a cavity, and proposing pathways for the exchange of energy and momentum between the electrons and the radiation bath. If the temperature is low, spontaneous Compton collisions can be neglected. We would only obtain inelastic interactions through the ponderomotive potential. Ultimately, the bath and the gas must reach an equilibrium temperature. And from this, we would deduce the discrete characteristic of the wavelength shift of light. This is beyond my scope.

But now I remember another question. I was analyzing some of Kozak’s work on electron-photon interactions through the ponderomotive potential. He states that if the speed of the interference pattern of two interacting beams is greater than the speed of the electron, energy is transferred from the radiation to the electrons, and the opposite occurs if the speed of the pattern is lower. In your model, I believe the speeds are measured from a system at rest with respect to the distant stars. The patterns must have zero speed, always less than the speed of the electrons. But, for an electron moving at high speed, we must take into account relativistic corrections for wavelength and direction. Is that correct? On the other hand, you’re talking about a phenomenon different from the standard interaction between two beams of light and electrons. Could the diffraction of electrons by beams propagating in opposite directions have any characteristics related to this wavelength discretization?

The long time variation of the light curve, encoded as a modulation in the Fourier components, is therefore stretched by the same, 1 + z.

Does the same happen with the expansion? Or does it have a different factor?. This could be verified with experiments.

STz works for 0.0024 nm < λ < 10 m I think this is important.

I won’t submit any paper until I have something to say.

But the Big Bang is clashing with observations.

Cheers, and get lucky

Carlos.

@carlosbelcech you are making good progress in understanding STz! Congratulations, and thank you.

A few comments on your last post:

I understand this to mean that the interaction of beams (with different angles and energies) would have no net effect on the spectra. However, the phenomenon does exist. Only for beams propagating in opposite directions is there a redshift effect. - @carlosbelcech

Exactly right.

I tried to approach this topic from thermodynamics, as in Einstein’s original paper on stimulated emission. That is, by enclosing an electron gas and a radiation bath at different temperatures in a cavity, and proposing pathways for the exchange of energy and momentum between the electrons and the radiation bath. If the temperature is low, spontaneous Compton collisions can be neglected. We would only obtain inelastic interactions through the ponderomotive potential. - @carlosbelcech

Yes, this is an extremely important calculation that will eventually have to be done.

Ultimately, the bath and the gas must reach an equilibrium temperature. And from this, we would deduce the discrete characteristic of the wavelength shift of light. This is beyond my scope. - @carlosbelcech

This is also beyond my abilities at the present time. However, we cannot neglect Compton scattering in the thermodynamic calculation. Here’s why: STz produces a redshift of the radiation ($T_{\rm rad}$ decreases) and increases the energy of the electron gas ($T_{\rm e}$ increases). If no other pathway exists between electrons and radiation, then electrons will reach a limit temperature $T_{\rm e}[{\rm lim}]$ and radiation will disappear as $T_{\rm rad} \rightarrow 0$. Obviously this is not possible in thermodynamics: $T_{\rm e}[{\rm lim}]$ and $T_{\rm rad} = 0$ cannot coexist forever. Compton scattering cannot be neglected, it will increase the temperature of the electrons via the Sunyaev-Zeldovich effect. I discuss this in a preliminary way in my submitted paper. Here is an excerpt.

The S-Z effect produces an intensity change of the spectrum with an amplitude that has the form[Birkinshaw97.1999]

$$ \Delta I(\nu) = \frac{2h}{c^2} \int_{-\infty}^\infty P_1(s) \left[ \frac{\nu_0^3}{e^{h\nu_0/k_{\rm B}T_{\rm rad}} - 1} - \frac{\nu^3}{e^{h\nu/k_{\rm B}T_{\rm rad}} - 1} \right] d s, $$

where $T_{rad}$ is the temperature of the blackbody radiation, $s \equiv \ln(\nu/\nu_0)$ is a logarithmic frequency scale, and $P_1(s)/\nu$ is the probability that a scattering occurs from frequency $\nu_0$ to $\nu$ that is a function of the IGM electron temperature. Because the distribution of scattered photon frequencies $P_1(s)$ is asymmetric with an blue-scattering skew, the average photon frequency increases and the S-Z effect produces a net blue-shift.

By comparison, the STz effect transforms a spectrum of temperature $T_{rad}$ to $T’$ as given by Eq. (spectral-radiance-stz). Using the relation $1+\delta z = e^{-s} = \nu_0/\nu$ and substituting in Eq. (sz-intensity-change) allows us to write the S-Z intensity change as

$$ \Delta I(\nu) = \int_{-\infty}^\infty P_1(s) \Big[ I(\nu; T’, \delta z) - I(\nu; T_{rad}, 0) \Big] {\rm d}s, $$

where $T’ = (\nu/\nu_0) T_{rad}$. Equation (s-z-with stz) describes the S-Z effect in terms of an initial spectrum $I(\nu; T’, \delta z)$ that is a STz-shifted function of the final spectrum $I(\nu; T_{rad}, 0)$. - @RedshiftDrift

The interpretation of this equation is that S-Z shifts the MBR spectrum towards the blue with the same function that describes STz shifts toward the red, and therefore Stimulated-Transfer redshifts are exactly cancelled by Sunyaev-Zel’dovich blue-shifts. This is what will give an equilibrium solution, but I haven’t calculated that yet.

You write:

But now I remember another question. I was analyzing some of Kozak’s work on electron-photon interactions through the ponderomotive potential. - @carlosbelcech

I am not familiar with Kozak’s work. I had a quick look at this paper arXiv:2503.02987, is that what you are talking about?

He states that if the speed of the interference pattern of two interacting beams is greater than the speed of the electron, energy is transferred from the radiation to the electrons, and the opposite occurs if the speed of the pattern is lower. In your model, I believe the speeds are measured from a system at rest with respect to the distant stars. The patterns must have zero speed, always less than the speed of the electrons. But, for an electron moving at high speed, we must take into account relativistic corrections for wavelength and direction. Is that correct? - @carlosbelcech

Almost correct! In my calculation, the electron is at rest in the frame of reference of the calculation. (There is no need for some arbitrary cosmological frame at rest like distant stars!) The slowly moving interference pattern of two counter-propagating waves (which differ in wavelength by $2h/(m_e c)$) produces Bragg scattering (and the Bloch solution discussed in [Kozak arXiv:2503.02987)].

In the frame of reference of an electron moving at high speed relative to the stars emitting radiation, Doppler shifts and aberration are the relativistic corrections that can bring beams at different wavelengths and from any direction to counter-propagating beams in the frame of reference of the electron. Doppler shifts and aberration are easier to calculate than a relativistic electron moving in a standing wave, hence my choice of a reference frame following the electron!

On the other hand, you’re talking about a phenomenon different from the standard interaction between two beams of light and electrons. - @carlosbelcech

Yes, I am talking about two independent waves propagating in opposite directions and that differ in wavelength by $2h/(m_e c)$ in the frame of reference of the electron. This is described by the first equation of Section 2.2 in arXiv:2410.02036. This always produces an energy gain for the electron, and a STz energy loss of the photon field.

Kozak’s Eq. (49) in arXiv:2503.02987 describes a quantum standing wave produced by reflecting a laser beam on a mirror to produce a superposition of photon states. This is described by the first equation of Section 2.1 in arXiv:2410.02036. This produces an energy loss or gain of the electron depending on its motion relative to the standing wave, which is a completely different thing. On average, the energy loss and gain of the electron is zero in this case.

The only difference between the two approaches is that Kozak considers a standing wave built from the superposition of one retro-reflected wave, while STz considers a standing wave built from a pair of independent waves. The distinction between the two is very well explained in this article https://opg.optica.org/josab/abstract.cfm?uri=josab-8-4-903

Could the diffraction of electrons by beams propagating in opposite directions have any characteristics related to this wavelength discretization? - @carlosbelcech

Yes, in the frame of reference of the initial electron at rest, the de Broglie wavelength of the STz scattered electron is simply equal to $\lambda/2$, the wavelength of the standing wave! This is Bragg scattering in its simplest form.

The long time variation of the light curve, encoded as a modulation in the Fourier components, is therefore stretched by the same, 1 + z. - @RedshiftDrift

Does the same happen with the expansion? Or does it have a different factor?. This could be verified with experiments. - @carlosbelcech

Supernova light curves and spectra have been measured to be stretched by exactly this factor $(1+z)$. The entire spectrum is shown to be stretched in time in these two papers https://iopscience.iop.org/article/10.3847/1538-4357/abec3c and https://iopscience.iop.org/article/10.3847/1538-4357/abec3b. This is considered by cosmologists to be a proof of time dilation, but STz produces the same effect.

I won’t submit any paper until I have something to say. - @carlosbelcech

Let me know what your ideas are and I will help you as much as I can with the STz effect.

Okay. Yes, I’m referring to this type of article. If you look at Martin Kozac’s RG profile, you’ll find his work on ponderomotive scattering.

I understand that the equilibrium scenario between SZ and STz isn’t the same as in the IGM, since the IG plasma is an open system out of equilibrium. But, in a hypothetical closed system, these processes should lead to equilibrium. Now, I’m wondering if the STz proposal can be presented as a refined approximation to the Kapitza-Dirac effect, whose application to astrophysics should be a consequence or a derivation. I mean, to avoid the “academic restrictions.”

I’ll have to keep studying. I haven’t read your PDFs yet. If I find anything interesting, I’ll let you know.

I’m wondering if the STz proposal can be presented as a refined approximation to the Kapitza-Dirac effect, whose application to astrophysics should be a consequence or a derivation. I mean, to avoid the “academic restrictions.” - @carlosbelcech

Yes, that’s my new year’s resolution: to present STz outside the context of cosmology! I will first present it as a dipole effect, the Kapiza-Dirac diffraction of atoms in the Bragg regime, as I first discovered it in 2006. I now have the theoretical tools to publish the full calculation, and I can also propose an experiment to be done with atoms.

Then I’ll publish the full calculation with electrons (which leads to the results of my paper), but without mentioning applications to cosmology. Kozac’s paper may not help me that much since he uses a standing wave instead of two travelling waves, but I may use his papers to contrast the two different approaches.

Hi again Louis. I found another doubt. Is about the question of redshift happening to only one spectrum of counterpropagating beams. I was calculating for the case of two rest relative sources whose beams suffer spectral transfer among them. Obiously both spectra must be redshifted due the interaction. I dont understand why the background counterpropagating spectrum is not affected by the phenomenon. I’m bearing in mind the idea that the spectra are indistinguishable and therefore the redshift affects to propagating and counterpropaganting beams, in such way that the whole IG photon field is redshifted. But, I was calculating, obtaining that the spectra got mixed and loses their initial features.

Carlos.

Obviously both spectra must be redshifted due the interaction.

Yes, the situation is symmetric as I mentioned here

I was calculating for the case of two rest relative sources whose beams suffer spectral transfer among them. […] I dont understand why the background counterpropagating spectrum is not affected by the phenomenon.

So far I have only calculated the redshift caused by one electron on counterpropagating light, at one position in space. Both spectra are redshifted. I don’t have yet a derivation of the spectral shift as a function of position. However, I think that the quantum calculation shows that both spectra become entangled in one state by the interaction with the electron, so that the resulting redshift is on the entire spectrum.

How is your calculation done? Do you shift one photon at a time? (I have tried this simple calculation in the past and spectroscopic lines were fading as the bulk of the spectra was being shifted… I believe this oversimplification makes the spectra lose their features.)

Perhaps this is what confused me, is the section 3.3 of “Diffusive interactions”, when says “Beam k′ acts as a catalyst, as it remains mostly unaffected by the stimulated transfers produced by this configuration”. In the following I show my rustic calculation. Let suppose a point (x,y,z,t) in the IG region. A propagating radiation flux with spectrum 1, come from source 1 along axis x. The wavelenght spectral function can have any form, supposing a gaussian curve for simplicity, this spectrum is mounted over a background with a constant intensity spectral function. This can be the resultant function: $I_{1}(\lambda,t)=a\ exp(-k_{1} (\lambda-\lambda_{0})^2)+k$, where k is the wavelenght independent background and $\lambda_{0}$ is the maximum. If you draw it for given values of contants can have something like this: We can suppose that counterpropagating flux 2 has the same function with different values for constants, the background can be the same.

Now, we focus on flux 1. Each wavelenght value, can be considered like a “channel”, spectral transfer puts and removes photons of each one while time go ahead (and space). In the “channel” $\lambda$, the intensity at time $t+\delta t$ will be: $I_{1}(\lambda, t+\delta t)=I_{1}(\lambda,t)+I_{2}(\lambda-\delta \lambda,t) \sigma_{21}(t)-I_{1}(\lambda,t) \sigma_{12}(t)$. Where the cross sections have the form: $\sigma_{jk}(t)=c\ \sqrt{I_{j}(\lambda,t) I_{k}(\lambda+\delta \lambda,t)}$. For now, constant c is secondary. We have neglecting distance-dependent variations. Ok. I think there are no mistakes here, but I’m never sure.

If you put the last equation in a for-next loop in any program, so that it runs over time, the spectra shift towards longer wavelengths. Due to time constraints, I did this using Mathematica. If spectra 1 and 2 are the same, they shift while preserving their characteristics, but if they are different, they don’t. I did this in a hurry, so I might be wrong, sorry. Here I show you my for-next loop:

Carlos.

I need to correct this, sorry, it’s wrong.

“Beam k′ acts as a catalyst, as it remains mostly unaffected by the stimulated transfers produced by this configuration.5” - section 3.3 of “Diffusive interactions”

@carlosbelcech this is specific to one configuration. The footnote adds:

5. However, if multiple spectral components are considered for beam k′, the picture is reversed and it is the spectrum of k′ that is shifted toward longer wavelengths.

The effect involves two terms: $$\sqrt{I_j(\lambda,t) I_k(\lambda+\delta \lambda,t)}$$ and $\sqrt{I_k(\lambda,t) I_j(\lambda+\delta \lambda,t)}$. These appear from the summations $\sum_j$ and $\sum_k$ which I have not always made explicit in my papers :disappointed:

In the following I show my rustic calculation. If spectra 1 and 2 are the same, they shift while preserving their characteristics, but if they are different, they don’t.

Excellent @carlosbelcech, I did the same numerical calculation in 2014 and obtained the same (disappointing) results! Unfortunately, the term $\sqrt{I_{j}(\lambda,t) I_{k}(\lambda+\delta \lambda,t)}$ is an incomplete semi-classical description.

It took until 2023 until I understood: the interaction involves one electron interacting with one photon $n_1 =1$ and one catalyst photon $n_2 = 1$. The term $\sqrt{n_1 n_2}$ (Eq. (5) in “Diffusive interactions…”) is therefore always equal to 1. By using the density matrix elements, the equation can be simplified.

More later…

Hola Lyndon. How are you? Yes, it’s something like that, but according to Stz, the result is the same. I wanted to tell you that yesterday I saw a video on the YouTube channel “See the Pattern” about five possible alternative explanations for astronomical redshift, one of which is your “plasma hypothesis.”

I hope you like this group of galactic rebels.

Carlos

@carlosbelcech writes:

a video on the YouTube channel “See the Pattern” about five possible alternative explanations for astronomical redshift, one of which is your “plasma hypothesis.”

I wrote (perhaps in your mailbox): >if you remember the title or URL of this video, I would very much like to see it !!

But meanwhile I found it myself: 5 Alternative Explanations for the Redshift we Observe .

@tiredlyndon , screendump at 9:24 :

Interesting video !!

Hi Louis. I think so too. There should be a lot of accelerated charge radiation. Let’s think about it. If there’s no spectral signature, could there be some other cooling process? What about the IG magnetic field, or interactions with neutral hydrogen or free protons? Perhaps you should revise the article to include the reviewer’s observation, and then maybe they’ll publish it!

Carlos

That’s the only cooling process I can think of.

• Magnetic fields require some bulk motion of charges to increase which is difficult to get from random motion of hot electrons. But once a magnetic field is established, the conductivity of the IGM is so good the field does not dissipate! So that doesn’t seem to be a good candidate for a cooling process.
• Neutral hydrogen only requires 13.6 eV to get dissociated, then it no longer absorbs energy and we are left with an electron and a proton.
• Free protons also dissipate energy but at a slower rate than electrons because of their higher mass.
• Nuclear reactions can also dissipate energy (this is what I was looking for here), but at a higher temperature $T \approx 1$ MeV that is not reached because free electrons limit the temperature at 0.5 MeV.

Unfortunately, the reviewer rejected my article without giving me a chance to explain my point. I am trying to convince the editor to give me a chance to respond, but that only has a small probability to succeed. If that doesn’t work, I’ll submit the article to another journal…

Thanks Carlos for sharing your ideas!

While I thought that Now it’s over, >the referees object that “… the electrons cannot be arbitrarily hotter [than the WHIM at $10^7$ K] without producing observable spectral signatures (e.g., free-free emission, recombination lines) that are not seen.”

Which IMHO is not far off my “Now it’s over” reply.

It catch my attention they doesn’t reject the spectral transfer mechanism itself, instead reject its consequence on the IGM plasma. It’s striking because with just to recalculate parameters like cross sections or densities the trouble could be solved. Is to say, spectral transfer phenomenon is not under discussion.

@carlosbelcech writes: >t catch my attention they doesn’t reject the spectral transfer mechanism itself, instead reject its consequence on the IGM plasma. It’s striking because with just to recalculate parameters like cross sections or densities the trouble could be solved. Is to say, spectral transfer phenomenon is not under discussion.

I can think of at least one, two, three reasons why the spectral transfer mechanism itself IS under discussion.

So why mercy !? In science, someone is guilty until the contrary has been proven. Example given (e.g.) : does the IGM plasma überhaupt exist?

@HanDeBruijn, Ja, und wie, the IGM plasma exists! The electrons of the IGM plasma are detected by the dispersion of fast radio bursts. They can even probe plasma turbulence with FRBs.