Merged: Why the James Webb Telescope rewrites/doesn't the laws of Physics/Redshifts

[Mike Helland]

An Argument for an Energy Scalar as an Alternative to Redshift

A discussion of redshift, blueshift, energy scaling, and their implications on distance relationships.

1. Redshift

A photon can be described by its wavelength (w), frequency (f), or energy (E), which are all closely related:

c = wf
E = hf
E = hc/w

Where c is the speed of light and h is Planck's constant.

When a photon redshifts, the wavelength, frequency, and energy change. Redshift (z) tells us how these values change from when the photon was emitted (emit) to when it was observed (obs).

1 + z = w_obs / w_emit
1 + z = f_emit / f_obs
1 + z = E_emit / E_obs

Or:

w_obs = w_emit(z + 1)
f_obs = f_emit / (z + 1)
E_obs = E_emit / (z + 1)

As the redshift increases the wavelength observed increases while the frequency and energy observed decrease. An increase in redshift (z) is a decrease in energy (E).

If redshift were to be negative (z<0) that would indicate a blueshift. It seems intuitive to think that blueshift would be redshift times negative one (-1):

blueshift = z * -1

But that is not true. While z may increase to infinity, causing the wavelength to increase to infinity and the frequency and energy to approach zero (0), z may only decrease to negative one (-1) before the wavelength becomes zero (0) and the frequency and energy observed become a divide by zero (0) error.

To illustrate further, when a photon's redshift is z=1, its observed wavelength is 1+z times its emitted wavelength, 1+1=2, so it's double. However, when its redshift is z=-0.5, it's observed wavelength is 1 + -0.5, or 1/2 its emitted wavelength.

A value of 0>z>infinity covers the entire range of redshift, while -1>z>0 covers the entire range of blueshift.

2. Blueshift

Because the redshift (z) and the energy (E) of a photon are inversely related, then blueshift and energy (E) should be directly related. Inverting the formulas of redshift (z) to blueshift (b) gives:

1 + b = w_emit / w_obs
1 + b = f_obs / f_emit
1 + b = E_obs / E_emit

Or:

w_obs = w_emit / (b + 1)
f_obs = f_emit(b + 1)
E_obs = E_emit(b + 1)

With these equations, the situation is different. As blueshift (b) increases, so do frequency (f) and energy (E).

When b=-1, then the frequency and energy are zero (0), and the wavelength is a divide by zero error. Since a photon with a frequency or energy of zero cannot be observed, then b must always be greater than zero (0), and no such error would occur.

Quantified this way, we find that a value of -1<b<0 covers the range of redshift, and 0>b>infinity covers the range of blueshift.

3. Energy Scalar

In redshift and blueshift equations, there is always a plus one (eg, 1 + z). Quantifying our observations as an energy scalar. which I'll call "Q" arbitrarily, is an alternative in which the plus one can be left out.

Q = w_emit / w_obs
Q = f_obs / f_emit
Q = E_obs / E_emit

Or:

w_obs = w_emit/Q
f_obs = f_emit(Q)
E_obs = E_emit(Q)

Due to the absence of a plus one, that means when a photon is observed with the same wavelength, frequency, and energy that it had when emitted, then Q=1. In contrast, that would be quantified as z=0 and b=0.

The energy scalar (Q) of a photon is then 0>Q>1 when redshifted, and 1>Q>infinity when blueshifted.

As the photon's Q approaches zero (0), so does its energy (and frequency) while its wavelength approaches infinity.

4. Distance Relationships

To determine a distance (D) by redshift (z), the formula is:

D = cz/H_0

Where c is the speed of light and H_0 is Hubble's constant. But this only works for very small values of z (z<<1). Because redshift can grow to infinity, when z=1 the distance is one Hubble's length (c/H_0), and when z=10 the distance is ten Hubble's lengths.

To determine a distance (D) by blueshift (b), the formula is:

D = -bc/H_0

The blueshift formula acts differently than the redshift formula. While the maximum redshift when quantified as z is infinite, the maximum redshift when quantified as b is -1. So when b=-0.5, the distance is half a Hubble's length. When b=-1, the distance is one full Hubble's length, and that is the maximum distance allowed by this relationship.

The redshift formula and the blueshift formula are equal where the redshifts are very small, but they quickly diverge with the redshift formula climbing without bounds and the blueshift formula approaching Hubble's length.

To determine a distance (D) by energy scalar (Q), the formula is:

D = (1 - Q)c/H_0

Here the need for 1 - Q is necessary, because at D=0, there is no redshift or blueshift so Q=1. As the energy scalar approaches 0, D approaches one Hubble's length.

5. Conclusion

Quantifying cosmological redshifts in the traditional manner leads to a distance relationship that is only valid at very small values, and predicts far too large of distances with even moderate redshifts (z=1).

But quantifying them instead as negative blueshifts (or an energy scalar) yields a different distance relationship, on account of the range of negative blueshifts being 0 to -1, and the range of redshifts being 0 to infinity. The distances predicted by this formula never exceed one Hubble's Length.

The energy scalar quantification gives the same distance predictions as the blueshift quantification, and may cause less confusion due to it being color agnostic.

[Mike Helland]

Originally Posted by Mike Helland

A value of 0>z>infinity covers the entire range of redshift, while -1>z>0 covers the entire range of blueshift.

I got those inequalities in the wrong direction. In a couple other places too. Hopefully you get it should be greater than zero and less than infinity.

[Ziggurat]

Originally Posted by Mike Helland

But thinking back to the strong force, it's so much more powerful than the EM force, but it doesn't "work" far from the nucleus.

Maybe that's what redshifted EM is. It makes all atoms stick together. Does it really need to be infinite?

It needs to be infinite if the photon is massless.

Give the photon mass, and electromagnetism falls off faster than 1/r². But how much faster depends on how massive the photon is. A larger mass makes it fall off faster, a smaller mass makes it fall off slower.

We cannot prove the photon is massless, but we can constrain its possible mass. And the largest mass it can have is still very, very tiny. Nor would mass produce the red shift you're suggesting. A massive photon can't rescue your theory.

[Mike Helland]

Originally Posted by Ziggurat

It needs to be infinite if the photon is massless.

That's assuming relativity holds on an infinite scale.

General relativity was already in place before galaxies were discovered.

[Ziggurat]

Originally Posted by Mike Helland

That's assuming relativity holds on an infinite scale.

No, it's not. The connection between massive force carrier particles and short range forces comes from quantum mechanics, not relativity.

[Mike Helland]

Originally Posted by Ziggurat

No, it's not. The connection between massive force carrier particles and short range forces comes from quantum mechanics, not relativity.

It seems more like the connection between a massless particle and an infinite range comes from relativity, but whatever. I'm not trying to give the photon a mass.

According to my equations, negative blueshifts give a different distance relation than redshift equations, which are known to be valid when z<<1.

[Mike Helland]

Originally Posted by Mike Helland

According to my equations, negative blueshifts give a different distance relation than redshift equations, which are known to be valid when z<<1.

Let's say a photon is emitted with 1 MeV (mega electron volt).

But it's observed with 0.5 MeV.

Redshift formula:

1 + z = E_emit / E_obs

1 + z = 1 / 0.5

! + z = 2

z = 1

Redshift-distance formula:

D = zc/H_0

D = (1)c/_H_0

Now let's do it for the blueshift formula:

1 + b = E_obs / E_emit

1 + b = 0.5 / 1

1 + b = 0.5

b = -0.5

Blueshift-distance formula:

D=-bc/H_0

D = -(-0.5)c/H_0

They predict different distances. The redshift formulas predict infinite distances over the inputs. Negative blueshift predicts finite distances over the inputs.

We trade a horizontal asymptote for a vertical one.

[jonesdave116]

Originally Posted by Mike Helland

Redshift-distance formula:

D = zc/H_0

Errr, nope. As mentioned, that is not a valid formula for redshift. It works as an approximation at very low redshift, because the differences are tiny when compared to using the correct, more accurate formula. Once you get to ~ z = > 0.1, it is no longer usable. The higher the redshift, the less accurate, and more useless it becomes. So, you are starting off with the wrong formula.

That was Lerner, et al's schoolboy error. Do not repeat it. It has already been shown why that is a complete train wreck, that is simply unphysical.

[Mike Helland]

Originally Posted by jonesdave116

Errr, nope. As mentioned, that is not a valid formula for redshift. It works as an approximation at very low redshift, because the differences are tiny when compared to using the correct, more accurate formula. Once you get to ~ z = > 0.1, it is no longer usable. The higher the redshift, the less accurate, and more useless it becomes. So, you are starting off with the wrong formula.

What is the correct formula?

The redshift distance relationship is what we have, and we know its wrong.

Say you have a photon with 1 MeV.

As z increases, it's energy decreases, when z->infinity, E->0.

Now use the negative blueshift formula to do the same thing.

https://www.desmos.com/calculator/dkvluwna7l

You can see here the redshift and the blueshift are equal when z is really small.

I'm not saying the redshift distance formula is right. We all know its been wrong for a really long time. Slapping z<<1 on it isn't a legitimate fix.

[Mike Helland]

Here's a pretty interesting thread on wavelength and frequency:

https://threadreaderapp.com/thread/1...161364480.html

"Students tend to be justifiably surprised (sometimes upset) when this conundrum first dawns upon them. The obvious expectation is that if we transform from wavelength λ to frequency f=c/λ, the λ-maximum of the spectrum gets mapped to the f-maximum. Well, not so!"

[Mike Helland]

Originally Posted by Mike Helland

Redshift:

1 + z = E_emit / E_obs
E_obs = E_emit / (z + 1)

Blueshift:

1 + b = E_obs / E_emit
E_obs = E_emit(b + 1)

If I may illustrate further, assuming the photon was emitted at 1 MeV, this is how its redshift (x-axis) looks compared to its observed energy (y-axis)



When 0<z<infinity that's redshift, when -1<z<0, that's blue shift.

Now consider the blueshift formula, this is how that looks:



There's a big difference there.

To make things clearer, we can reverse the direction of the x-axis from b to -b and get this:



When overlapped, it looks like this:



We can see from this that in the range the redshift-distance relationship is considered valid (about 0<z<0.1) they match. But then they don't.

Instead of thinking about redshift as a positive value and its relationship with wavelength, we should be thinking about negative blueshifts and its relationship with energy and frequency.

Thoughts?

[Puppycow]

I think this video is appropriate for this thread:

YouTube Video This video is not hosted by the ISF. The ISF can not be held responsible for the suitability or legality of this material. By clicking the link below you agree to view content from an external website.

I AGREE

It's a YouTube channel called Sixty Symbols, and it's an interview with Professor Ed Copeland - a cosmologist, about what he hopes to learn about cosmology from the JWST. It could potentially rewrite maybe not the laws of physics, but teach us about the nature of dark energy and why supermassive black holes exist (how were they formed in the first place and how did they get to be so massive; how massive were they in the early universe and so on).

[a_unique_person]

Why didn't they just form by collecting more and more mass? Isn't that what black holes do?

[RecoveringYuppy]

Originally Posted by a_unique_person

Why didn't they just form by collecting more and more mass? Isn't that what black holes do?

Yeah, but we don't understand how and when they did it so fast.

[Mike Helland]

Originally Posted by RecoveringYuppy

Yeah, but we don't understand how and when they did it so fast.

It will dawn on somebody someday that the universe is far older than we think.

Black holes suck up energy and then let it go.

They are the universe's backup batteries. That's how it avoids heat death.

[jonesdave116]

Originally Posted by Mike Helland

What is the correct formula?

Not that one. That only applies where z << 1.

Try this;

https://ned.ipac.caltech.edu/level5/...eacock3_4.html

In particular, see eq. 3.93. You cannot use a linear relationship at high z. That is what Lerner is trying to do, and it fails.

Also, see here;

https://www.teachastronomy.com/textb...-and-Distance/

"Combining the two results gives

d = z c / H0

Again, this formula is only appropriate if the recession velocity is much less than the speed of light, or if z << 1."

[jonesdave116]

Originally Posted by Mike Helland

It will dawn on somebody someday that the universe is far older than we think.

Why? There is zero evidence to suggest such a thing.

[Crossbow]

Originally Posted by Mike Helland

It will dawn on somebody someday that the universe is far older than we think.

Black holes suck up energy and then let it go.

They are the universe's backup batteries. That's how it avoids heat death.

This absurd statement on the composition of the universe makes about as much sense as your other absurd statements on the composition of the universe.

[Ziggurat]

Originally Posted by Mike Helland

It will dawn on somebody someday that the universe is far older than we think.

Black holes suck up energy and then let it go.

They are the universe's backup batteries. That's how it avoids heat death.

Black holes do not violate thermodynamics. They release energy far too slowly, and they release it as thermal radiation, ie, high entropy. They cannot prevent the heat death of the universe.

[Mike Helland]

Originally Posted by jonesdave116

You cannot use a linear relationship at high z. That is what Lerner is trying to do, and it fails.

Understood. Because z grows without bounds and therefore so does d.

That's why I'm suggesting an alternative to z, which is blueshift b.

Negative blueshifts only reach -1.

Then the formula d = -bc/H₀ can produce a maximum distance of c/H₀.

[Mike Helland]

Originally Posted by jonesdave116

Why? There is zero evidence to suggest such a thing.

https://darkmattercrisis.wordpress.c...arly-universe/

"Finally, the existence of massive galaxies just 200-500 Myr after the Big Bang implies that structure formation is much more enhanced at high z, or that the Universe is much older than predicted by the standard model of cosmology."

[Mike Helland]

Originally Posted by Ziggurat

Black holes do not violate thermodynamics. They release energy far too slowly, and they release it as thermal radiation, ie, high entropy. They cannot prevent the heat death of the universe.

https://www.cfa.harvard.edu/news/wev...shredding-star

‘We’ve Never Seen Anything Like This Before:’ Black Hole Spews Out Material Years After Shredding Star

[Ziggurat]

Originally Posted by Mike Helland

https://www.cfa.harvard.edu/news/wev...shredding-star

‘We’ve Never Seen Anything Like This Before:’ Black Hole Spews Out Material Years After Shredding Star

Oh, I thought you were talking about Hawking radiation, since that's really the only way that black holes ever release energy from within.

Your example is not stuff which ever went into the black hole. It's spitting out less energy than it swallowed. It's not violating thermodynamics. It's not decreasing entropy. And it's not splitting heavier elements back into hydrogen. Nothing about that case will in any way reverse the progress towards the eventual heat death of the universe.

You're grasping at straws.

[Mike Helland]

Originally Posted by Ziggurat

Your example is not stuff which ever went into the black hole.

Did it just hang around for a few years and then decide to jet off?

In any case, what do you think about quantifying observations as redshift vs blueshift leading to two completely different distance predictions?

[jonesdave116]

Originally Posted by Mike Helland

https://darkmattercrisis.wordpress.c...arly-universe/

"Finally, the existence of massive galaxies just 200-500 Myr after the Big Bang implies that structure formation is much more enhanced at high z, or that the Universe is much older than predicted by the standard model of cosmology."

Yeah, right. That's Kroupa. I take him about as seriously as Lerner. He's a MONDist, last I heard. How's that going, Pavel? On its death bed, last time I looked. Fails miserably at large scales. When he can explain the colliding cluster lensing observations without invoking dark matter, I might listen. McGaugh and the rest of the MONDists can't.

[RecoveringYuppy]

Originally Posted by Mike Helland

Did it just hang around for a few years and then decide to jet off?

Why do you think asteroids wait millions of years before deciding to hit a planet?

[jonesdave116]

Originally Posted by Mike Helland

Understood. Because z grows without bounds and therefore so does d.

That's why I'm suggesting an alternative to z, which is blueshift b.

Negative blueshifts only reach -1.

Then the formula d = -bc/H₀ can produce a maximum distance of c/H₀.

Errrm, we aren't seeing blueshifts at high z. It is all redshifted, and the formula Lerner used, d = cz/H₀, fails tragically, and obviously. If he is going to invent some version of tired light, mechanism unknown, then that equation doesn't cut it. There are none that do. That is why nobody takes tired light woo seriously these days. Ben m explained this very succinctly in the posts that I am sure I have linked before;

http://www.internationalskeptics.com...&postcount=145

&

http://www.internationalskeptics.com...&postcount=163

[jonesdave116]

Originally Posted by Mike Helland

Did it just hang around for a few years and then decide to jet off?

No, he isn't saying that. The material in the jets never entered the black hole. Obviously. The star is torn apart, and the infalling material, which will be ionised, gets thrown out in jets caused by the tortured magnetic fields around the black hole. That is what black hole jets are.

[Ziggurat]

Originally Posted by Mike Helland

Did it just hang around for a few years and then decide to jet off?

My guess (and it’s only a guess) is that it created an instability in the accretion disk which eventually exploded and threw a bunch of stuff out of the disk. But at this scale, the details don’t matter. It does nothing to reverse heat death.

Quote:

In any case, what do you think about quantifying observations as redshift vs blueshift leading to two completely different distance predictions?

Seems pointless. As far as I can tell, you’re just substituting one variable for another, but that cannot change what’s actually happening. Redshift vs distance doesn’t come from measurements of z. It comes from independent measurements of distance. Your substitution won’t change them.

[Ziggurat]

Originally Posted by jonesdave116

No, he isn't saying that. The material in the jets never entered the black hole. Obviously. The star is torn apart, and the infalling material, which will be ionised, gets thrown out in jets caused by the tortured magnetic fields around the black hole. That is what black hole jets are.

The specific example he’s referring to really is weird. The problem isn’t that a star got torn up and material ejected, but that the ejection happened so long after the star got torn up. The details of why are a mystery at the moment, so it is a genuine puzzle. But it’s also still irrelevant to our exchange since whatever those details are, it cannot rescue an infinitely old universe from heat death.

[jonesdave116]

Originally Posted by Ziggurat

The specific example he’s referring to really is weird. The problem isn’t that a star got torn up and material ejected, but that the ejection happened so long after the star got torn up. The details of why are a mystery at the moment, so it is a genuine puzzle. But it’s also still irrelevant to our exchange since whatever those details are, it cannot rescue an infinitely old universe from heat death.

Ahh, I should have read the article. I just assumed it was the usual ejection of infalling material from the accretion disk getting hurled out by magnetic fields.

[Mike Helland]

Originally Posted by jonesdave116

Errrm, we aren't seeing blueshifts at high z. It is all redshifted, and the formula Lerner used, d = cz/H₀, fails tragically, and obviously.

Understood, but I'm using a different formula.

This is what Lerner is using:

1 + z = E_emit / E_obs
d = cz / H₀

Say E_emit is 1 MeV, and E_obs is 0.5 MeV.

1 + z = 1 / 0.5
1 + z = 2
z = 1
d = (1)c / H₀

Here's what I'm using:

1 + b = E_obs / E_emit
d = -bc / H₀

Again, E_emit is 1 MeV, and E_obs is 0.5 MeV.

1 + b = 0.5 / 1
1 + b = 0.5
b = -0.5
d = -(-0.5)c / H₀

Same inputs, different results.

[Mike Helland]

Originally Posted by Ziggurat

Seems pointless. As far as I can tell, you’re just substituting one variable for another, but that cannot change what’s actually happening. Redshift vs distance doesn’t come from measurements of z. It comes from independent measurements of distance. Your substitution won’t change them.

Unless they are really close, distances have to be approximated from redshift.

And when they are really close, the redshift and blueshift values are equal.

[jonesdave116]

Originally Posted by Mike Helland

Here's what I'm using:

d = -bc / H₀

Which is still a linear relationship, n'est-ce pas? And therefore won't work for the same reasons that Lerner's doesn't work.

[Mike Helland]

Originally Posted by jonesdave116

Which is still a linear relationship, n'est-ce pas? And therefore won't work for the same reasons that Lerner's doesn't work.

Assume a photon is emitted with 1 MeV.

Redshift z = E_emit/E_obs - 1:
Blueshift b = E_obs/E_emit - 1
Distance d = cz/H₀
Distance d = -bc/H₀

Observed (MeV):	Redshift z	Blueshift b	Distance(z)	Distance(-b)
1	0	0	0 Bly	0 Bly
0.5	1	-0.5	14 Bly	7.0 Bly
0.33	2	-0.66	28 Bly	9.3 Bly
0.25	3	-0.75	42 Bly	10.5 Bly
0.2	4	-0.8	56 Bly	11.2 Bly

Distance according to z grows and grows. At z=10, that's 140 billion light years. Way too big.

Distance according to -b approaches Hubble's length.

So they're clearly different.

[jonesdave116]

Originally Posted by Mike Helland

Assume a photon is emitted with 1 MeV.

Redshift z = E_emit/E_obs - 1:
Blueshift b = E_obs/E_emit - 1
Distance d = cz/H₀
Distance d = -bc/H₀

Observed (MeV):	Redshift z	Blueshift b	Distance(z)	Distance(-b)
1	0	0	0 Bly	0 Bly
0.5	1	-0.5	14 Bly	7.0 Bly
0.33	2	-0.66	28 Bly	9.3 Bly
0.25	3	-0.75	42 Bly	10.5 Bly
0.2	4	-0.8	56 Bly	11.2 Bly

Distance according to z grows and grows. At z=10, that's 140 billion light years. Way too big.

Distance according to -b approaches Hubble's length.

So they're clearly different.

You really aren't getting this. There is nothing blueshifted with anything like those values. About 1 000 km/s is the highest blueshift observed. And that is from a star ejected from a nearby galaxy.
At cosmological distances, any tiny blueshift is swamped by the cosmological redshift. That is what you need to deal with. Blueshift is not a distance indicator, per se. Cosmological redshift is. Andromeda is approaching us at ~ 300 km/s. There will be other groups of galaxies at, say, z = 0.5, where galaxies within the group are approaching each other. And could also be approaching in Earth's direction. That blueshift is minuscule compared to the cosmological redshift that is taking the group away from us. It is scarcely noticeable. Any observations of the galaxies in that group will show a large redshift.
With cosmological redshift we are dealing with relativistic velocities at high z. Blueshift is never ever close to relativistic.

This may help;

http://hyperphysics.phy-astr.gsu.edu...redshf.html#c1

[Mike Helland]

Originally Posted by jonesdave116

There is nothing blueshifted with anything like those values.

A galaxy of redshift z = 1 has blueshift b = -0.5.

Notice the negative (-).

[Mike Helland]

b = (1 / (1 + z) - 1)

Therefore, this can be used as a redshift-distance relationship:

d = -(1 / (1 + z) - 1) c / H₀

This provides a very close approximation to LCDM lookback times:

z=0 d=0
z=1 d=7
z=2 d=9.333333333333334
z=3 d=10.5
z=4 d=11.200000000000001
z=5 d=11.666666666666668
z=6 d=12
z=7 d=12.25
z=8 d=12.444444444444443
z=9 d=12.6
z=10 d=12.727272727272727
z=11 d=12.833333333333332
z=12 d=12.923076923076923
z=13 d=13
z=14 d=13.066666666666666
z=15 d=13.125
z=16 d=13.176470588235293
z=17 d=13.222222222222221
z=18 d=13.263157894736842
z=19 d=13.299999999999999
z=20 d=13.333333333333332

[jonesdave116]

Originally Posted by Mike Helland

A galaxy of redshift z = 1 has blueshift b = -0.5.

Notice the negative (-).

Nothing has a blueshift that high. Not even close. It is either redshifted or blueshifted. A redshifted galaxy cannot have a blueshift!

[jonesdave116]

Originally Posted by Mike Helland

b = (1 / (1 + z) - 1)

Therefore, this can be used as a redshift-distance relationship:

d = -(1 / (1 + z) - 1) c / H₀

This provides a very close approximation to LCDM lookback times:

z=0 d=0
z=1 d=7
z=2 d=9.333333333333334
z=3 d=10.5
z=4 d=11.200000000000001
z=5 d=11.666666666666668
z=6 d=12
z=7 d=12.25
z=8 d=12.444444444444443
z=9 d=12.6
z=10 d=12.727272727272727
z=11 d=12.833333333333332
z=12 d=12.923076923076923
z=13 d=13
z=14 d=13.066666666666666
z=15 d=13.125
z=16 d=13.176470588235293
z=17 d=13.222222222222221
z=18 d=13.263157894736842
z=19 d=13.299999999999999
z=20 d=13.333333333333332

Sorry, I'm losing the will to live. I have no idea what you are trying to do!

Let me summarise;

Cosmological redshift tells us that the universe is expanding. And at an accelerated rate. People like Lerner say it isn't. So, they need to account for that cosmological redshift. They can't. Tired light fails, so he invokes 'an unknown mechanism' for the redshift. His equation for it fails trivially. I am yet to see any equation that can account for the observed redshift from anyone questioning the mainstream model.

Your 'equation' leads to a negative number. That is blueshift. We do not see blueshift at cosmological distances. I'll try again;

d = 2 >>>>>>>>>>>>>>>> d = 1 >>>>>>>>>>>>>>> d = 0
12eV >>>>>>>>>>>>>>>> ??eV >>>>>>>>>>>>>>> 4eV

d = 1 >>>>>>>>>>>>>>> d = 0
8eV >>>>>>>>>>>>>>> 4eV

That is what a linear equation leads to at high z. Which is obviously nonsense. So, Lerner is wrong. What is your equation, and what does it give as the energy at d = 1 in the first part of that graphic? 6eV or 8eV? Either way, your photon needs to remember how far it has travelled, as the two parts of its journey see it losing different percentages of its energy.
That is what you need to solve, and a linear relationship fails from the get-go.