This statement is objectively false. There is no requirement that objects in relative motion must be moving slower than the speed of light.
Then there is the cosmological red-shift itself, which is another mystery. Physicists often talk about the red-shift as a kind of Doppler effect, like the change in frequency of a police siren as it passes by.
It's not a "kind" of Doppler effect. It is the Doppler effect.
But the cosmological red-shift is different because galaxies are stationary in space. Instead, it is space itself that cosmologists think is expanding.
The galaxies are not stationary in space. Both the galaxies and space are moving relative to other objects at other parts of space, and a Doppler shift is predicted in either (and both) cases.
The mathematics that describes these effects is correspondingly different as well,
Nope. Not true.
not least because any relative velocity must always be less than the speed of light in conventional physics.
Nope. Not true.
And yet the velocity of expanding space can take any value.
This is true. It can. Doesn't contradict any known physics. Doesn't require any new mathematics to describe.
One interesting idea is that the red-shifts of distant objects must increase as they get further away.
Already known. Already measured. That's what the Hubble Constant is.
But the evidence is paradoxical. Astrophysicists have measured the linear nature of the Hubble law at distances of a few hundred megaparsecs. And yet the clusters visible on those scales indicate the universe is not homogeneous on the scales.
Nope. No paradox. The large scale clusters are the result of quantum fluctuations that existed in the universe in the time before the first nanosecond. They are the result of Quantum Mechanics, which is more fundamental than General Relativity, and certainly much more fundamental than the "Hubble Law" which depends on assumptions which are not Quantum Mechanical in nature.
And so the argument that the Hubble law’s linearity is a result of the homogeneity of the universe (or vice versa) does not stand up to scrutiny. Once again this is an embarrassing failure for modern cosmology.
Nope. It's not. It's a result of the fact that we don't have a Quantum Mechanical version of General Relativity. In terms of general, rough morphology, it's perfectly adequate, and not contradicted by any information we have.
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Already known. Already measured. That's what the Hubble Constant is.
I saw a graph of Hubble's law once, with a tiny box around the origin and the comment that this was the range of observation that Hubble had originally used to forumlate it.
BTW, I was lookiing at your article and thinking, "Now this guy's making sense!" and then saw your sig ... "Well, if it isn't Fred Z ! Howdy doo!"