KJW

So? I don't think so. In fact, that the number of missing sequences is greater than the number of sequences is kind of what Cantor is proving. Although Cantor's diagonal method is only generating a single sequence missing from a list, because the number of sequences in a list is aleph-0, and the number of possible sequences is aleph-1, then the number of sequences missing from a list is also aleph-1. Cantor is proving that aleph-1 is greater than aleph-0.

I actually prefer the term "coordinate transformation" because that's what they are. I think the term "diffeomorphism" is a bit too esoteric for me. There are also "point transformations" which are conceptually distinct from coordinate transformations though mathematically identical, and it's not clear to me to which of these the term "diffeomorphism" actually refers. The OP enquired as to why it is only space that expands and not time. The answer is that the difference between time expanding with space and time not expanding with space is just a coordinate transformation, which means that there is no physical difference. However, as you correctly point out, the time coordinate of the time-expanding metric doesn't correspond to anything, in particular, not a co-moving clock, whereas the spatial coordinates actually do correspond to the co-moving cosmological fluid. No, it doesn't. Only the time coordinates are involved in the coordinate transformation. If the original metric describes a flat three-dimensional space, then the transformed metric will be a scalar function multiple of the Minkowskian metric: guv = ƒ(t) ηuv It depends on how the metric is used. For example, if one is describing light-like trajectories in spacetime, then they have a simple straight-line form.

Methane is a single carbon atom to which are attached four hydrogen atoms in a tetrahedral arrangement.

Except that when you place sx in line 1, you've created a new list, requiring a new application of Cantor's diagonal method, generating a new sequence that is not in the new list (nor in the old list). Given any list of sequences of binary digits, Cantor's diagonal method generates from that list a sequence of binary digits that: differs from the first sequence of that list in the first position, differs from the second sequence of that list in the second position, differs from the third sequence of that list in the third position, ... differs from the n-th sequence of that list in the n-th position, differs from the (n+1)-th sequence of that list in the (n+1)-th position, ... etc and therefore differs from every sequence of that list (that is, not in that list).

A metre is still a metre, and a second is still a second for the two observers at different elevation. Any disagreement between these two observers is the result of how these are compared, noting that the comparison is non-local. But on earth, the surrounding spacetime is approximately stationary and admits a Killing vector field, providing a natural frame of reference in which gravitational time dilation manifests itself as an apparent change in time with respect to elevation. But in spite of the naturalness of this frame of reference, it is still just some frame of reference no more special than other frames of reference in general relativity.

In relativity, we frequently see discussions of quantities that are different in different frames of reference. Less often do we see discussions of quantities that remain the same in different frames of reference (other than the speed of light in a vacuum). I think this leads to confusion about the nature of relativity. I think it is important to note that all observers measuring the same quantity will always obtain the same value regardless of their frame of reference. This means that regardless of the picture relativity seems to paint, relativity does present a truly consistent picture of reality. When it is said that observers in different frames of reference obtain different values for a given measurement, it is because they are actually measuring different things. For example, length contraction is the result of observers in different frames of reference measuring the proper distance between different points in spacetime. The measured width of a river will depend on whether it is measured perpendicularly across or obliquely across, and the same is true for the world-strip of a rod in spacetime. In the case of time dilation, one is comparing the proper time between one pair of points in spacetime with the proper time between another pair of points in spacetime, with some notion of simultaneity between the pairs of points justifying the comparison. But it is when one believes that the different measurements are the same measurement that one can have a confused picture of relativity.

Actually, one can perform a coordinate transformation on the FLRW metric to produce a metric in which time and space expand equally. The resultant metric is a scalar function multiple of the Minkowskian metric, and therefore FLRW spacetime is conformally flat, or as Markus Hanke said above, "a Petrov-type O spacetime". One advantage of such a coordinate system is that it simplifies light-like trajectories as well as the cosmological redshift.

You made a mistake with your arithmetic. You said γs=5/3 whereas it's γs=5/4

There was no need to repost the earlier post. It seems to me that you are not understanding the point I'm making. If the acoustic wave equation is only valid for a stationary medium, then that means a different equation applies to a moving medium. And a different equation for a moving medium means that the acoustic wave equation is not invariant to "A'rentz" transformations. You did make the mathematical error of failing to recognise where the analogy between light and sound fails. It's not a preferred frame of reference. It's two non-equivalent frames of reference. And they are non-equivalent because only the travelling twin accelerates.

The Sagnac effect is in agreement with Special Relativity.

But I'm saying the experiment won't agree with your maths because the acoustic wave equation is only valid for stationary media.

I am saying that your problem is not with the maths but with the physics.

The Shapiro time delay is a confirmation of General Relativity.

Doing the math is not the same as agreeing with experiment. A flaw in the argument you seem to be making is that you seem to be assuming that because your modified wave equation is mathematically invariant to your modified Lorentz transformations, that it is also physically invariant as well. But if the scope of your modified wave equation is limited to frames in which the speed of the medium is zero, then it won't be physically invariant in spite of the apparent mathematical invariance.

Here's a link to E.W. Silvertooth's article titled "Experimental detection of the ether": https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.418.8179&rep=rep1&type=pdf

Yeah, I wasn't aware of the Silvertooth experiment, so I looked it up. Wikipedia doesn't have a page on the experiment. When I Googled "Silvertooth experiment", the only seemingly legitimate article I found was in Forbes titled "The Tale Of A 1986 Experiment That Proved Einstein Wrong" written in 2018. Some of the other Google entries were blocked by my security software, while others were on sites with names such as "Aetherometry", "Spirit-Science", or "viXra". Anyway, the following is an excerpt from the Forbes article (https://www.forbes.com/sites/briankoberlein/2018/04/06/the-tale-of-a-1986-experiment-that-proved-einstein-wrong/?sh=563e9dde3ed3) about a follow-up experiment by Doug Marett in 2012:

I would like to point out that the frame of reference in which the dipole anisotropy of the cosmic microwave background radiation is zero is not an absolute rest frame.

So why is the paper entitled On the electrodynamics of moving bodies ? And why is page 1 of the paper all about Maxwell, electrodynamics and what the paper is going to do with them (which he subsequently does) ? And why is his concluding technical statement "These three relationships are a complete expression for the laws according to which, by the theory here advanced, the electron must move." ? That was more than a hundred years ago. Relativity has moved on since Einstein. The concept of spacetime didn't even exist at the time of Einstein's paper on Special Relativity... that was Minkowski's innovation. And it was an important innovation because without it there would be no General Relativity. Relativity, both Special and especially General, is about spacetime, regardless of historical development.

Relativity is not about electromagnetism, it is about spacetime.

vp = λ f Are you sure about that? After all, differences in phase are observable without contradicting gauge invariance. I see two meanings to "wave" in QM. Firstly, there is "wave" as in "wavefunction", a general term referring to any waveform. But there is also "wave" as in "wave-particle duality" which I see as referring specifically to a sinusoidal waveform. In other words, a particle with definite momentum, and therefore definite wavelength. But also definite energy, and therefore definite frequency. Phase velocity might not be meaningful for all wavefunctions, but that does not make it meaningless for all wavefunctions.

vp = λ f I don't know about that. Earlier, I mentioned time crystals, the time-based analogue of ordinary crystals, but nobody seemed to have noticed. This at least allows frequency to be measured in principle even if not currently in practice.

Is this some form of holography? I tend to agree with this. But ultimately, I was simply justifying the statement that phase velocity can be obtained from the measurements of wavelength and frequency.

It's definitely from x-ray crystallography. Note that only the carbon atoms are showing because hydrogen atoms are rather insensitive to x-rays. Neutron diffraction would be better if one needs to explore bonding to hydrogen atoms.

I don't think so. And? Yes, I did. The velocities of the two frames are relative to the third frame. Perhaps I should have simply said the velocity relative to the third frame of the second frame is equal and opposite that of the first frame. Why would I?

Correct. Although I haven't seen an electron density map of hexamethylbenzene before, I have seen electron density maps of other compounds, so I did know what I was looking at with your picture. Hmmm, perhaps I spoke too soon. The direct image from x-ray crystallography is an arrangement of spots of various intensities. A Fourier transformation of these intensities yields a Patterson function (a convolution of the electron density with its inverse) due to the "phase problem". So how does one obtain a photographic plate image of the electron density? Specifically, what is performing the Fourier transformation? In particular, how is the "phase problem" being solved? I didn't use those as examples as I'm not very knowledgeable of those fields.