Here's what I have so far:
Photon Velocity
The redshifts in this hypothesis are not caused by the expansion of space, or other interactions as in tired light theories (discussed in a later section). The redshifts are interpreted as their own phenomenon that is as fundamental to nature as inertia. Light from distant galaxies is observed to be redshifted because light redshifts with distance.
The decelerating photon hypothesis is that cosmological redshifts indicate new physics for a photon:
A photon loses frequency as it travels cosmological distances, but with no change to wavelength, resulting in a loss of speed, according to v=c/(1+D/H)^2
The energy of a photon absorbed by matter is emitted as a new photon with D=0, and therefore v=c and an elongated wavelength.
In this interpretation of the redshifts, the frequency decreases while the photon is in flight, and the wavelength increases at the beginning of a new photon's journey. This is different from the expanding theory where frequency and wavelength change together without affecting the photon's velocity.
This poses a number of conflicts with existing physics. One of the most illustrative issues involves the Hubble Space Telescope (HST) and Snell's law.
The hypothesis says that the photon travels at less than c, and then after interacting with matter, new photons that travel at c are emitted. If decelerated photons reach a mirror and change speed, the photon's motion can be compared to that of light changing mediums.
This would result in a change to the angle at which decelerated light reflected off a mirror, which would be relevant in the case of the HST observing highly redshifted galaxies.
One would expect then, that if redshifted photons are traveling at less than c, the HST would not be able to resolve them at the expected angles with any clarity. Since the HST does resolve such galaxies, light must be reflecting normally, which means the photons have to be moving at c, invalidating the hypothesis.
To demonstrate this, I built two models of light reflecting off a mirror. The first using Fermat's least time principle. The second using Feynman path integrals. These demonstrated the problem.
Photons and Reflection
For the purpose of solving the reflection problem, imagine a non-relativistic space. In this space there are observers who experience time according to one master clock. In this space there are also photons, and each photon has its own clock. Vaugley similar to relativity, but not quite. Every photon's clock begins synced with the master clock, but falls off with distance, at a rate of 1/(1 + D/H)^2 which is equal to 1/(1 + z). This is derived in a later section.
Previously, the code for the model of the decelerating photon looked like this:
Code:
photon.dx = c / Math.pow(1 + photon.x / H, 2)
The hypothesis was directly changing the velocity based on distance. So solve our problems, we'll approach this slightly differently. The photon needs a clock that stats at zero, and this is what the hypothesis directly changes, like so:
Code:
// find out the time slice for a photon at this distance
photon.dt = 1 / Math.pow(1 + photon.x / H, 2)
// add that time slice to the photon's clock
photon.clock += dt
// move the photon at the speed of light for the time slice
photon.dx = c * dt
The photon will actually always be traveling at c according to its own clock. But since that clock gradually runs slower than the master clock, the photon appears to decelerate according to the observers in the model.
Now when we run the problem demonstration of Fermat's least time principle, rather than choose the photon that arrived first according to the master clock, we choose the photon whose clock has the lowest reading.
Lastly, when the photon hits the mirror, it will be emitted as a new photon with D=0, causing dt=1, causing the v=c, and creating an elongated wavelength.
See: Reflection in classical physics
https://mikehelland.github.io/hubbles-law/other/reflection3.htm
When run, photons reflect at the same angle as photons moving at c in a vacuum, no longer demonstrating a problem with the hypothesis. This works because the least time principle always favored the photons that hit the mirror first when a speed increase was involved. If a photon was the first to reflect and gain a higher speed, it had a considerable advantage over the other photons traveling at the slower speed. In the updated demonstration, the time is measured by each photon's clock, which changes speed when the photon hits the mirror. For the photon that hits the mirror first, the speed boost no longer provides an advantage, because the photon's clock speed will have increased too, canceling out any advantage being first to the region once held.
A similar resolution can be added to Feynman's path integral in quantum electrodynamics. Each photon has a clock, which provides the time slice to be used. Photons that are redshifted have slow clocks dt<1, when they are reflected as fresh photons, their clock speed returns to dt=1.
Accounting for the difference in dt before and after the reflection, the problem demonstration again shows there is no problem at all.
https://mikehelland.github.io/hubbles-law/other/reflection_nm.htm
