Cosmological redshifts are a physical observation. There is little math to it. One looks for the emission and absorption lines of an element or molecule in the lab, the wavelength that involves the same wave count per wavelength with the proper position of the absorption and emission lines.. Looking through telescopes at distant galaxies one will observe that the wavelengths are longer than those observed in the lab. To determine which element is being observed in the spectra one must identify the amount and position of the omission and absorption lines in their spectra. If this can be determined, then the element or molecule can be identified. Then its total wavelength must be measured and compared to the same element in the lab.

The length measured over its lab length is called its redshifted wavelength. Thia is represented mathematically by 1+z, where one is the laboratory (standard) length of the element or molecule, and the z factor is called its redshift, its extended wavelength thought to be due to the expansion of space over the time that light has traveled over millions or billions of years.

Any other math involves analyzing the absorption and emission lines to determine what element or molecule is being observed.

Dear Forrest Noble

Thank you for your answer to my question. You are a cosmologist so you are best suited to answer my particular question.

What you describe is known, I know how it works. That's not what I asked.

1) "Cosmological redshift are a physical observation."

Cosmological redshift is NOT a physical observation but an INTERPRETATION of this observation.

2) "The length measured over its lab length is called its redshifted wavelength."

Yes, 1 + z = λ'/λ, where λ' is the observed wavelength and λ is the wavelength of the same element/molecule in the laboratory. But this is the same formula as for the Doppler effect. The formula for the Doppler effect shows red/blue shift due to the relative speed between the source of light and the observer.

3) "Any other math involves analyzing the absorption and emission lines to determine what element or molecule is being observed."

I was not looking for the above mathematics, but I am looking for a mathematical model in which one explains/shows how the expansion of the universe affects the wavelength of the light from distant cosmic objects.

4) Say we have a light signal from a distant cosmic object. When the light signal departs from the source, this object is at a distance D from the observer. When the light signal reaches the observer, the distance will increase to D + d. Outgoing wavelength is λ and incoming wavelength is λ'. If the cosmological redshift 1 + z depends on the distance between the source and the observer (changing from D to D + d), how do you calculate 1 + z from D, D + d?

"This happens because of the expansion of the universe."

This is total nonsense: Edwin Hubble: "… if redshift are not primarily due to velocity shift … there is no evidence of expansion, no trace of curvature … and we find ourselves in the presence of one of the principles of nature that is still unknown to us today … whereas, if redshifts are velocity shifts which measure the rate of expansion, the expanding models are definitely inconsistent with the observations that have been made … expanding models are a forced interpretation of the observational results." ("Effects of Red Shifts on the Distribution of Nebulae" by E. Hubble, Ap. J., 84, 517, 1936)

Geoffrey Burbidge: “By 1982, when a conference on cosmology was held at the Vatican, a new approach was taken. The radicals around, such as F. Hoyle, V. Ambartsuminan and this speaker (to mention a few) were not even invited. The conference was confined completely to Big Bang cosmology and its proponents. In fact in the introduction to the published volume of the proceedings of the meeting (Pontifical Academy of Sciences, 1982) it was emphasized that only believers (in the Big Bang) were present; and that there was clearly a deliberate decision of the organizers” : G Burbidge, In “The Universe at Large: Key Issues in Astronomy and Cosmology. Edited by Guido Münch, Antonio Mampaso, Francisco Sánchez; Cambridge University Press 1997.

"Oldest Spiral galaxy BX442 supports Hubble’s belief: Redshift does not mean expansion" : https://cosmologyscience.com/cosblog/spiral-galaxy-bx442-supports-hubbles-belief-redshift-does-not-mean-expansion/

"New Galaxy Redshift Distance Record (z=11) Seriously Threatens Big Bang models" : https://cosmologyscience.com/cosblog/new-galaxy-redshift-distance-record-z11-seriously-threatens-big-bang-models/

Bursting dark energy's bubble :Geoff Brumfiel : https://www.nature.com/articles/news.2007.215

The Greatest Rebuttal: "Ambartsumian, Arp and the Breeding Galaxies": http://redshift.vif.com/JournalFiles/V12NO2PDF/V12N2MAL.pdf

"QUASARS – POINTERS TO AN INFINITE, ETERNAL AND DIALECTICAL UNIVERSE AND A REPUDIATION OF THE BIG BANG COSMOLOGY" : Article QUASARS – RETROSPECT, PROSPECT AND A POINT of DEPARTURE

I am also writing a theory on the formation of galaxies and their growth and expansion. As of today, the observable fact is that each galaxy is growing internally, much like the population increases on Earth without the Earth itself expanding in size. Similarly, the number of stars and planets within galaxies is increasing, but the overall size of the galaxies remains constant.

Space itself is not expanding; instead, the changes in distances between galaxies are due to their individual motions. Think of space as an ocean and galaxies as large ships. Each ship moves at different speeds and in different directions. As a result, the distances between them change over time. However, this does not imply that the ocean itself is expanding. Similarly, in space, it is the motion of galaxies, not the stretching of space, that accounts for these changes.

I have done extensive graphics for Cosmological Redshift. Please see TrueCosmology.info/animated-universe

and TrueCosmology.info/announcements. The documents there have detailed graphics.

As for the Foundations of Cosmological Relativity I have discovered that all of Quantum Mechanics & QFT have foundations in Number Theory. Specifically, all quantum phenomena can be mapped to the 1-to-1-to-1 relationship of (1-Planck Length) per (1-Planck Time) per each hBar that the universe experiences.

Please give me feedback. -- Thanks.

Hi.

may be this paper can help

Preprint Expansion of the Universe and cosmological redshift

Muhammad Iqbal

Thank you for your comment. But you're not giving me the answer I'm looking for. Your comparison of the universe as the world ocean does not answer why one measures higher and higher redshift the further away the cosmic object is.

John Wsol Ángel Torregrosa Lillo

There are certainly countless articles, books, works about the cosmological redshift. You have to be patient and willing to go through everything and understand how others think.

I am looking for a mathematical model that answers my question:

As light travels through space, space itself is stretched, causing the wavelengths of light to be stretched as well. HOW?

Hi Jan.

Point 2.2 in paper I mentioned has the classical mathematical demonstration. Then, he refused it :-)

May be we can comment it here.

Here is a conceptual illustration of cosmological redshift, showing light waves stretching as they traverse an expanding universe. The image captures the phenomenon visually, with light transitioning from blue to red due to the universe's expansion.

Ijaz Durrani

1) "showing light waves stretching as they traverse an expanding universe"

I aim for a mathematical model, a figure, and not an artistic image. I want to see how adepts of cosmological redshift draw the "stretching" of an electromagnetic wave.

Here you are Jan Slowak with a succinct response:To represent the stretching of light waves mathematically in the context of cosmological redshift, we need to describe how the wavelength of light changes as the universe expands. The redshift zzz is a direct result of this stretching, and mathematically, it can be related to the scale factor a(t)a(t)a(t) of the universe. Below, I will outline the steps for creating the mathematical model and describing the wave stretching.Mathematical Model of Cosmological Redshift Scale Factor and Redshift Relation The cosmological redshift zzz is related to the scale factor a(t)a(t)a(t) by:1+z=a(tobserved)a(temitted)1 + z = \frac{a(t_{\text{observed}})}{a(t_{\text{emitted}})}1+z=a(temitted​)a(tobserved​)​where:tobservedt_{\text{observed}}tobserved​ is the time the light is observed (current time), temittedt_{\text{emitted}}temitted​ is the time the light was emitted (when the light originated), a(t)a(t)a(t) is the scale factor of the universe, which describes its expansion. Wavelength StretchingThe wavelength λ\lambdaλ of light observed today (at time tobservedt_{\text{observed}}tobserved​) is related to the wavelength λ0\lambda_0λ0​ at the time of emission by:λobserved=(1+z)⋅λemitted\lambda_{\text{observed}} = (1+z) \cdot \lambda_{\text{emitted}}λobserved​=(1+z)⋅λemitted​This shows that as the universe expands, the wavelength of light stretches proportionally. Frequency of LightSince the frequency ν\nuν of light is inversely proportional to its wavelength (ν=cλ\nu = \frac{c}{\lambda}ν=λc​), the frequency decreases as the wavelength increases. This is consistent with the redshift observed:νobserved=νemitted1+z\nu_{\text{observed}} = \frac{\nu_{\text{emitted}}}{1+z}νobserved​=1+zνemitted​​where νemitted\nu_{\text{emitted}}νemitted​ is the frequency when the light was emitted, and νobserved\nu_{\text{observed}}νobserved​ is the frequency when the light is observed. Relating Redshift to the Hubble ParameterThe redshift is often related to the scale factor using the Hubble parameter H(t)H(t)H(t), which describes the rate of expansion of the universe. The relationship between the scale factor and redshift can be expressed as:dzdt=H(t)⋅z1+z\frac{dz}{dt} = H(t) \cdot \frac{z}{1+z}dtdz​=H(t)⋅1+zz​where H(t)H(t)H(t) is given by the Hubble parameter at a particular time ttt.Drawing the Stretching of an Electromagnetic Wave A typical figure that shows the stretching of a light wave due to cosmological redshift would illustrate how the peaks of the wave move apart as the universe expands. The general shape of the figure would have: Two Waves (Before and After Redshift):The initial wave at the time of emission (at a certain scale factor a(temitted)a(t_{\text{emitted}})a(temitted​)). The stretched wave after traveling through expanding space (at a later scale factor a(tobserved)a(t_{\text{observed}})a(tobserved​)). Waveform Comparison:Before Stretching: A series of peaks in the wave at a shorter wavelength. After Stretching: The same wave, but the peaks are farther apart, indicating a longer wavelength. Visualization of the Scale Factor:You might see a simple grid or expanding "stretch" shown behind the waves to visualize the expansion of space. The wavelength increases by a factor of (1+z)(1+z)(1+z). Sample Code to Plot the Stretching of a Wave (using Python and Matplotlib) To visualize this, you can use a basic plot showing the change in wavelength (or stretching of a sine wave representing light). Here is a simple Python code snippet using Matplotlib to plot the before-and-after effect of redshift: pythonCopy codeimport numpy as np import matplotlib.pyplot as plt # Constants c = 3e8 # Speed of light in m/s lambda_emitted = 1e-6 # Wavelength of emitted light (1 micron) z = 1 # Redshift (example value) # Time array t = np.linspace(0, 10, 1000) # Emitted wave (before redshift) wave_emitted = np.sin(2 * np.pi * t / lambda_emitted) # Stretched wave (after redshift) lambda_observed = lambda_emitted * (1 + z) wave_observed = np.sin(2 * np.pi * t / lambda_observed) # Plotting plt.figure(figsize=(10, 6)) plt.plot(t, wave_emitted, label='Emitted Wave (λ₀)', color='blue', linestyle='--') plt.plot(t, wave_observed, label=f'Observed Wave (λ₀ * (1+z))', color='red') plt.title('Stretching of an Electromagnetic Wave Due to Cosmological Redshift') plt.xlabel('Time (t)') plt.ylabel('Amplitude') plt.legend() plt.grid(True) plt.show()This code will plot two sine waves, one representing the emitted light and the other representing the light after being stretched due to redshift. The observed wave will have a longer wavelength compared to the emitted wave, visualizing the "stretching" process.Conclusion This approach provides both a mathematical framework and a visualization for the stretching of light waves in an expanding universe, which is a key feature of cosmological redshift. It also allows for a clearer understanding of the wavelength change and its impact on light observed from distant galaxies.

John-Erik

Ijaz Durrani

Thanks for your comment and explanations. But I must point out that it was difficult to read. You show the following:

- Cosmological redshift using Scaling factor: 1 + z = a(tobserved)/a(temitted​)

- Cosmological redshift using wavelength: 1 + z = λobserved/λemitted

- Cosmological redshift using frequency: 1 + z = νemitted/νobserved

- Cosmological redshift using Hubble's Parameter: dz/dt = H(t) z/(1 + z)

I have translated the Python code into LaTeX and got a picture of it. I am attaching the pdf-file.

I wonder if your visualization showed much the same thing? If so then you are visualizing two waves with two different wavelengths and that is not what I am after in my main question.

John-Erik Persson

1. I am not the one who says that light can be stretched.

2. Of course the light cannot be bent. But its path to us may not have to be a straight line all the way!

3. According to STR, there is no ether. If you talk about ether means you dismiss STR.

But thanks for a clear and concise comment.

Jan Slowak

Yes, you did not say 'stretched'. Sorry I missed the citation marks.

About bending: I meant to that space cannot be bent.

I make errors. I am 89.

I also thank you for a clear answer.

John-Erik

John-Erik Persson

As for the light's "stretching" I have quoted others. The light cannot stretch on the grounds that the properties of the light signal are linked to a point in space. It is not that the light signal IS between the points x and y, but it has its properties at the point x and others at the point y. The only thing that can be said is that the light properties have changed from the point x (eg wavelength λ0 ) to the point y (wavelength λ), from there comes the red/blue shift.

Regarding the bending of space: space cannot be bent, space is not made of matter, matter is IN space. This is a tricky concept to deal with but this explanation makes more sense than any other.

I am impressed that at your age you are still struggling with science and at the same time I am grateful that you share your knowledge with the rest of us.

Jan Slowak

One definition of "stretching" can be increased wave length.

I agree with your opinion about "bending".

My old age means that I sometimes use the wrong word, but my interior images seems mostly to be correct anyhow. However, this can also be explained by the fact English is a second language for me.

Regards from ________________ John-Erik

@ Jan Slowak Yes my visualization showed much the same thing as your did. Two waves with two different wave lengths. What exactly is it that you are after ? Could you be clearer? Thanks. IJAZ

@ Jan Slowak: Thank you for sharing the PDF. It provides a visualization of the redshift process, showing light waves before and after redshift, labelled with their respective wavelengths (λ0 and λ0⋅(1+z). Here's how we can address your question and refine the depiction: Observations on the Provided Visualization: Wavelength-Focused Visualization: The graph effectively represents the concept of wavelength stretching during redshift, highlighting the relationship λobserved=λemitted⋅(1+z). It labels the stretching of the wave but doesn't fully encapsulate the other redshift aspects (e.g., scaling factor, frequency, and Hubble's parameter). Amplitude Depiction: The amplitude aspect doesn't necessarily contribute directly to understanding redshift mathematically—it might add complexity without clear purpose. Scaling Factor and Hubble's Parameter: These critical elements are missing, making it harder to correlate the redshift process with cosmic expansion or time evolution. Enhancing the Visualization: To make it more comprehensive and address your mathematical model: Include the Scaling Factor :Depict spacetime gridlines or scale bars showing the relationship between a(t observed) and a( t emitted). Use arrows or gradients to emphasize universal expansion. Link to Frequency: Add a secondary visual showing how frequency decreases (νobserved

John-Erik Persson

"One definition of "stretching" can be increased wave length"

I understand that but then I want to have an equation for this and not as Ijaz Durrani sent: an equation for the wavelength λ0 and one for λ0(1 + z).

"I agree with your opinion about "bending"."

I am so very happy that there is one or two researchers here at RG who have the same point of view on things as I do.

"this can also be explained by the fact English is a second language for me"

No problem, for me English is in fourth place: Slovak --> Romanian --> Swedish --> English. But I used Google Translate, always.

Ijaz Durrani

2) "Thank you for sharing the PDF. It provides a visualization of the redshift process"

I'm sorry but it does NOT visualize the redshift process. It shows two waves with two different wavelengths. It's not the same thing.

I am attaching another file showing a sinusoid starting with wavelength λ and ending with λt. This is how a light signal should be reflected to show the cosmological redshift.

@ Jan Slowak ---shall be back shortly!!

@ Jan Slowak: Here’s a visualization showing the stretching of a single wave due to the cosmological redshift, starting with a shorter wavelength and progressively stretching as it moves through expanding space. Thank you for clarifying your point! You're absolutely right to highlight the distinction: a proper visualization of the cosmological redshift process must depict the continuous stretching of a single wave over time or distance, rather than simply showing two discrete waves with different wavelengths. Regards--IJAZ

Jan Slowak

Det är fördelen med att byta land.: you get more languages. How do you like Sweden?

I do not accept SRT and not the Lorentz transform either. In 1887 the Lorentz transform was derived by assuming transverse light to reduce light speed in conflict with the wave model. The Lorentz transform is not motivated in physics. Instead of time dilation atomic clocks depend on the ether wind as observed in satellites put into orbits.

I do not accept GRT, since it is just a cover up for missing the fact that gravity is not attraction, but something emerging inside 2 bodies, due to the ether. Abolishing the ether made gravity mysterious.

Regards from _________________ John-Erik

John-Erik Persson , Ijaz Durrani

I show a new image of a light wave coming from distant cosmic objects and having both the wavelength and amplitude affected due to the immense distance to the object.

Sorry. I wrote light, but meant space. I am old.

Jan Slowak

From ___________________________ John-Erik

Contrary to "Flat Universe" teachings our universe is spherical hologram, where the radius is NOT space, but rather Cosmic Time measured as an integer number of time quanta since the the Beginning of Time -- thus, the Cosmic Onion Model. Space is the ever-expanding circumference measured in light-years or meters. Humble's perceptions were fundamentally flawed in that it failed to draw a clear distinction between Doppler shift vs. Cosmological redshift. Please see my publications which have extensive graphic illustrations and explanations.

Also the blog posts at TrueCosmology.info/blog address: