You are not using the terminology in a standard way. "Conformal" is a property of a transformation, not of a spacetime. What you probably mean to say is that de Sitter space is conformal to Minkowski space, or equivalently that it is conformally flat.
It is in quite general use. Indeed I have used it in several publications with no problem.
Conformal is not really a property of a transformation in GR, it is a relationship between two systems. By conformal one usually means shorthand for related to Minkowski spacetime by a common overall factor. But one can have 'conformal to Einstein coordinates' also. It use does not (directly) imply a transformation, since there is no transformation from Minkowski (written in Minkowski-Lorentz coordinates) to RW with k = -1 or k = +1 (written in conformal coordinates).
That did not seem to be the point of your original post. It sounds as though you have come around to agreeing with me, though, which is good.
I am not sure we have yet reached agreement. Nothing you have said so far has changed my view.
Notice that this is not at all like tired light, where the claim was that light always redshifts with time.
I cannot speak with confidence in general for those who advocate 'Tired Light'. It is my impression that they intend to replace metric-induced 'stretching of light waves' if you will, as for example illustrated in MTW, with some propagation-dependent behavior that reddens light, all the while in a static universe.
My point is that this criterion can be met in a de Sitter spacetime written in static coordinates whilst remaining compatible with GR.
(The MTW picture is a RW k = +1 written in 'RW coordinates', though the conformal coordinate system is discussed also.)
As far as I can tell from Terry Witt's postings, he does not contradict this particular tired-light type interpretation of cosmological redshift (and which is compatible with GR). But I am happy to be corrected on this if you can point to something he said.
There is no physical meaning to that statement [that light red-shifts in de Sitter spacetime], in general, but there certainly is once you specify the trajectory of the source and the receiver.
I assumed all along we were talking about so-called 'fundamental observers'. These are static in the RW coordinate system, and static in the static de Sitter system. If you are talking about observers moving with respect to these systems then we have been talking past each other. The Cosmological Principle, from which the RW metrics are derived, is valid only for fundamental observers.
Now we're back to where we started... that statement is incorrect, as I keep trying to explain to you. It is impossible that there could be any preference towards redshifting in static coordinates (precisely because the coordinates are static). In fact, light redshifts with respect to static time only if it propagates towards the origin. If it propagates away, it blueshifts.
There seems to be a major problem of communication here. As far as I can see, the case is decided and closed, just by consideration of how the exponential phase varies in static de Sitter coordinates. If you do not find that explanation satisfying, please say exactly what is wrong with it.
Use that the de Sitter metric in static isotropic form has line element
[latex]
ds^2 = \left( {\left( {1 + {\bf{x}}^2 } \right)^2 {\text{d}}{\kern 1pt} t^2 - {\text{d}}{\kern 1pt} {\bf{x}}^2 } \right)/\left( {1 - {\bf{x}}^2 } \right)^2
[/latex]
(units are all normalized with respect to the Hubble constant), then the exponential phase factor in a plane-wave decomposition of the potential is
[latex]
A \propto \exp \left( {i\frac{{\left( {1 + {\bf{x}}^2 } \right)^2 }}
{{\left( {1 - {\bf{x}}^2 } \right)^2 }}\omega {\kern 1pt} t - i\frac{1}
{{\left( {1 - {\bf{x}}^2 } \right)^2 }}{\bf{k}}{\bf{.}}{\kern 1pt} {\bf{x}}} \right)
[/latex]
where the dot product is Euclidean, and [latex]\omega ^{\left( 0 \right)} ,{\bf{k}}^{\left( 0 \right)} [/latex] are constants. Hence the frequency changes as it propagates according to
[latex]
\omega \left( {\bf{x}} \right) = \frac{{\left( {1 + {\bf{x}}^2 } \right)^2 }}
{{\left( {1 - {\bf{x}}^2 } \right)^2 }}\omega ^{\left( 0 \right)}
[/latex]
and the wavelength changes according to
[latex]
\lambda \left( {\bf{x}} \right) = \left( {1 - {\bf{x}}^2 } \right)^2 \lambda ^{\left( 0 \right)}
[/latex]
That's why it's important that the universe picks a particular coordinate system (the one in which the matter density depends only on time).
No.
Humans (not the universe) may pick a coordinate system in which matter density depends only on time.
More specifically, the
coordinate density of matter depends on time in the RW coordinate system. But that is not a physical statement. The
proper density of matter is constant in this system - which is a physical statement.
By contrast, in the static de Sitter system, the
coordinate density of matter depends only on distance, not time!
In our universe all matter is close to comoving (i.e. at rest in those special coordinates) and essentially all radiation is emitted from surfaces at rest in those coordinates,
Agreed.
and therefore there IS a physical meaning to the statement that light redshifts
There is a physical meaning to the statement that we observe red-shifted starlight.
- and the redshift occurs because the universe is expanding.
The reason you give is coordinate dependent, and therefore does not have a genuine physical meaning.
There is no way around that.
There is, and I gave it above for the static de Sitter system. Please address that mathematics directly.
) it adds nothing and only subtracts from the exchange.
