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Let’s look at this whole quantum entanglement business systematically, because I really don’t think it requires 22 pages of discussion and argument to understand this. It may be counter-intuitive, but it really isn’t that complicated. Suppose you have - to begin with - two completely separate particles, which aren’t part of a composite system; their states are thus entirely separate, and denoted by \[|A\rangle ,|B\rangle\] Don’t mind the precise meaning of this mathematical notation; it simply denotes two separate particles being in two separate states, where the outcome of measurements are probabilistic, and not in any way correlated at all. No mystery to this thus far. Now let’s take the next step - we combine the two particles into a composite system. The state function of that composite system is then the tensor product of the states of the individual particles, like so: \[ |\psi \rangle =|A \rangle \otimes |B \rangle \equiv |AB \rangle\] Again, don’t mind the precise definition of these mathematical operations; the idea here is simply that our two particles A and B form a composite system. Let’s, for simplicity’s sake, assume that each particle can only have two states, ‘0’ and ‘1’ - the physical meaning of the tensor product above is then that it combines each possible state of one particle with each possible state of the other, so the overall combined system can have four possible states: \[|00\rangle ,|01\rangle ,|10\rangle ,|11\rangle\] Thus the overall combined state of the particle pair is (I will omit the coefficients here, as the precise probabilities aren’t important): \[|\psi \rangle =|00\rangle +|01\rangle +|10\rangle +|11\rangle\] This is an example of a system that is not entangled - the combined state function can be separated into the individual states of the constituents, and all combinations are possible (though not necessarily with equal probability). Non-entangled states are separable into combinations of states of the individual constituent particles - they are tensor products of individual states - which means physically that there are no correlations between outcomes of measurements performed at the constituent particles. If you get state ‘0’ for a measurement on particle A, then you can get either state ‘0’ or state ‘1’ for a measurement on B, and these outcomes are statistically independent from each other. Mathematically, the tensor product makes no reference to the separation of the particles, ie it is not a function of their position, hence neither is the overall combined state. An entangled 2-particle state, on the other hand, looks like this: \[|\psi \rangle =\frac{1}{\sqrt{2}}\left(|01\rangle +|10\rangle \right)\] Notice three things: 1. Compared to the non-entangled state, two of the possible measurement outcomes are missing; the set of possible outcomes is reduced 2. The combined state cannot be uniquely separated into tensor products of individual states; it is non-separable 3. The form of the combined state does not depend on the spatial (or temporal) position of the particles - it is purely a stochastic statement, not a function of spacetime coordinates. What does this physically mean? Because the set of possible measurement outcomes in the overall state is reduced as compared to the unentangled case, there is now a statistical correlation between measurement outcomes - with emphasis being on the term statistical. There are now only two possible combinations, as opposed to four in the unentangled case. This is the defining characteristic of entanglement - it restricts the pool of possible combinations of measurement outcomes, because the overall state cannot be separated, due to there being extra correlations that weren’t present in the unentangled case. This is purely due to the form of the combined wave function - the outcome of individual measurements on each of the constituents is still purely stochastic, and not (!!!) a function of distant coordinates. Because the outcome (statistical probability) of local measurements is not a function of coordinates or any distant states, it is completely meaningless to say that this situation is somehow non-local, or requires any kind of interaction, be it FTL or otherwise. The entire situation is fully about statistics and correlations, which is not the same as a causal interaction; in fact, any interaction between the constituents (including FTL ones) would change the combined wave function and preclude the possibility of there being a statistical correlation while at the same time maintaining the stochastic nature of the outcomes of individual measurements. This is evident in the fact that the entanglement property of the above state function isn’t encoded in any kind of coordinate dependence, but rather in a reduction of terms, ie in a reduced pool of possible outcomes. This hasn’t got anything to do with locality at all, but is purely a statistical phenomenon. Hopefully the either helps, or possibly it might spark off another 22 pages of discussion

Consider dG = dH - TdS For a column of atmosphere at uniform density & pressure under a gravitational field, a downwards vertical flow is favoured (supporting the argument of @studiot) since the release of gravitational energy increases total enthalpy sufficiently to counter the reduction in entropy due to reduced occupancy of the higher levels of the column. So we have established an equilibrium condition with a vertical density/pressure and entropy gradients much as the atmosphere we see around us. But for further gravitational settling of, say, CO2 to take place, the gravitational potential energy released is now countered not only by the entropy gradient, but also the necessary displacement of an equal volume of lower density gases previously below it generating an adverse temperature gradient and expansion of the lower levels due to both the temperature gradient and the reduced mass of the upper part of the column. In short, while dH is likely not zero for a perfectly uniform gas mixture (constant mole fractions) it becomes so small that it can support only a tiny mole fraction gradient. I therefore suspect that while @exchemist and @Ken Fabian are not quite 100% accurate in their assertions, in practical terms they are very close to measurable reality. It's certainly an approximation I used throughout my working career without a qualm. The 'phosgene' counter argument simply reflects the very low rate of diffusion of high molecular weight gases. The thermodynamic equilibrium remains an (approximately) evenly dispersed mixture. It's just that these cases take their time about reaching equilibrium.

Space chiefs are to investigate whether electricity could be beamed wirelessly from space into millions of homes. The European Space Agency will this week likely approve a three-year study to see if having huge solar farms in space could work and be cost effective. The eventual aim is to have giant satellites in orbit, each able to generate the same amount of electricity as a power station. ESA's governing council is to consider the idea at its Paris HQ on Tuesday. While several organisations and other space agencies have looked into the idea, the so-called Solaris initiative would be the first to lay the ground for a practical plan to develop a space-based renewable energy generation system. Read more https://www.bbc.co.uk/news/science-environment-62982113

I am sorry, this confusion has to do with little differences between languages on how to denominate certain scientific effects, situations or theories.

Side note; maybe related to this discussion? I remembered about gas used in chemical warfare during WWI. Chemical Warfare and Medical Response During World War I, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2376985/ (bold by me)

But again this is not spontaneous separation by density. Diffusion alone will mix gases of different density eventually. This should be obvious if you think what happens to a dense gas released into a vacuum chamber. It does not all collect at the bottom. That shows that molecular speeds are sufficient to far outweigh the effect of gravity on individual molecules. What may cause confusion is that the mean speed of molecules with greater mass is lower, at a given temperature. So their rate of dissipation by diffusion will be lower.

Expansionism to Soviet levels is Putin's MO. We either let him or we don't. The choice seems to be binary, given that, evidently, any opportunity towards a compromise just gives him time to remaneuver. This why Zelenskyy is not interested in talking to him at this point in the conflict It would be militarily and politically foolish to lose any hard earned gains to anything else, given that this conflict is going to be extremely difficult going forward in their winter, which apparently is brutal. Our next task will be helping the civilians and armed forces keep warm enough to endure the likely struggle ahead. Russian conscript losses are going to be an order of magnitude more numerous. The support they have in the field is pitiful. This image of a new influx of conscripts says it all:

A partial annex this year so we can have peace. In a couple of years V Putin tries again, and Ukraine lets him have another piece to keep the peace. Then a couple of years later ... ( there is no Ukraine left and V Putin moves on to the next country ) I wonder if he would have tried this stunt if Ukraine and the West had stood up to him in 2014 when he annexed Crimea ?

It helps those of us who wish to understand entanglement more clearly.

@studiot my point was to clarify for readers that CO2 in mixed air doesn't separate and sink, even under those circumstances. We don't get stratification of the mixed air, we see stratification of pockets/volumes with different CO2 concentrations that have not mixed - yet. Sources will keep it that way but without them the enclosed air will - eventually - homogenize. Or I should say no significant stratification under ordinary Earth gravity; run it through powerful centrifuges and it can become significant. It is a common misunderstanding (whilst not claiming it of you) that CO2, being more dense, will sink to the bottom - and that the higher atmospheric CO2 concentrations nearer ground level are a result of the CO2 separating rather than the sources of CO2 being at ground level and there being a lag time in mixing. At small scale it mixes by diffusion. At larger scales by bulk air movements, ie wind and turbulence. For example thunderstorms will carry air from ground level to the stratosphere in one go, mixing vigorously as it goes.

Sorry; I just recall reading about entangling different color photons. If you can arrange it so that energy doesn’t identify the entangled property, or that the energy otherwise doesn’t matter, such as energy-time entanglement, which depends on the photons being created at the same time. “Each of the photons is directed into its own unbalanced Mach-Zehnder interferometer (see figure 3), giving it a long path (L)and a short path (S)to the detectors. Because the path length difference is much longer than the coherence length of the photons, no interference is observed in the single rates at either of the detectors when the phase in, say, one of the long paths is changed. However, there is interference in the coincidence rate between detectors. The reason is that there are two processes that could lead to such a coincidence count-both photons could have taken their respective long paths or both could have taken their respective short paths” http://research.physics.illinois.edu/QI/Photonics/papers/My Collection.Data/PDF/Hyper-Entangled States.pdf

I think it is best to just keep the situation fully classic, and consider only physical clocks to begin with, rather than wave functions. The question of evolution operators in RQM is complex and very non-trivial, and does little to illuminate this underlying question. Time dilation is a relationship between reference frames, and not something that physically “happens” to a single clock. Asking for a mechanism that “slows down” some clock is thus meaningless - clocks always tick at the same rate within their own frames. So the correct question would be why inertial frames are related via hyperbolic rotations in spacetime - that’s a very valid question, but it isn’t one that any of our present theories can answer. So to make a long story short, we don’t have an explanation of why this happens, only a description of it. That’s not the same thing at all. The length of a world line between given events in Minkowski spacetime is defined to be equivalent to the proper time of a clock travelling between these events that traces out that world line. In other words, it’s simply the total elapsed time that’s physically measured on a clock that travels along a specific spatial path between events. Intuitiveness is not a necessary condition for a mathematical model to be valid and useful. It just needs to be internally self-consistent, and produce results that can be verified using the scientific method. I think you would agree that SR does this quite well. Beside, something being intuitive (or not) is a very subjective measure - many things I find intuitive might appear otherwise to you, and vice versa. I would, by and large, agree with you - though I wouldn’t put into such strong terms. I just think many depictions of physical concepts get the differences between what is an explanation and what is a description muddled up, especially within pop-sci publications. We do not yet know the underlying mechanism of why spacetime is what it is, but we do have an excellent description of its features. To fully understand why spacetime gives rise to the phenomenology we see, we’d have to figure out first how spacetime itself comes to be, and if it can be broken down further into more fundamental concepts. Such attempts are under way, but at present they are just ideas and conjectures. I disagree. Physics makes models of the world around us, but not all of these models purport to be a fundamental explanation in ontological terms. As such, SR is a very good model that is in excellent agreement with experiment and observation. It’s just important to not confuse a model with an (ontological) explanation, because they are not the same.

As of today I am officially retired from government service (29.5 years in total)

Yes, I mixed up KE and momentum. The kinetic energy of its parts contribute to the system's rest mass. In its center-of-momentum frame, the system has net zero momentum, and the total energy of the system is equivalent to its rest mass. This is galaxy PG1211+143 they're observing, about a billion light years away. Wouldn't they compensate for cosmological redshift? Wouldn't they describe the speed of stuff falling into a black hole relative to the black hole, and not the Earth?

Well, when Einstein came up with his ideas, all physicists were also saying that they are already aware of everything and that there are no more things to discover. What if time dilatation and light bending in total are not as negligible as initially thought? Don´t forget that when we are looking at the sky, we are only seeing light! We do not see stars or galaxies, just light. We cannot travel around stars, we cannot touch them. The more the light is bended and the stronger time dilatation is (and the further away galaxies are), the less does the image we see correspond to reality. Maybe stars and galaxies are just playing a game with us and nothing we see is actually there. But there is no way to prove that, unless we travel for a couple of millions of years to at least some neighbor stars. Well, actually, the light is not "converging on us". That´s why we see an Einstein ring. And after passing by the lens, we just have to be lucky enough to have a source galaxy that is at the correct distance (similar to our distance from this galaxy) to see the Einstein ring. But if there was no galaxy behind, the lens effect would exist anyway. Each heavy object is a lens, there are billions of lens surrounding us, each one with a certain angle/strength of bending. And many lens together seem to act as one big lens as well. There are even billions of lens made only of dark matter. But not seeing them does not mean that they are not there. Light might be bending everywhere and stars and galaxies might not be where we would suppose to find them. The same happens with time. And the light is only being bended at the exact moment when it passes by the lens. The distance between lens and source or lens and observer does not affect the angle or strength of the lens. All the effect should be produced in a couple of thousands of light years (or even less).