joigus

(My emphasis.) One example? Easy enough: An Ising magnet. You get a reasonably good treatment of it with the mean-field approximation, in which you consider each magnet interacting with the average field of every other magnet in the crystal. The approximation gets better and better every time you extend the number of nearest neighbours (1st nearest neighbours, 2nd, etc.) in order to calculate the mean field. Another example? The N-body problem in gravitation. Another one? A Bose condensate. Another one? Superfluid Helium Another one? A proton (3 quarks plus gluons) Do you want more? Studiot already mentioned chemical reactions with 3 molecules. Any gas that's not ideal, and therefore presents phase transitions... I think I've made my point. In fact, there are practically no actual 2-body problems in Nature. 2 bodies is just an approximation that's useful only because it's exactly solvable, and because there are instances in which it's close enough to what's going on that it's worth studying in detail. It almost never happens in reality.

Yes. It's all a big resounding, reverberating mantra, with no physics in it, no mathematics in it, and full of embarrassing contradictions with many things we know --see picture from @Bufofrog above. We've seen it before. The only question is: How long is it going to take? And BTW, I'm still waiting for anything even remotely shadowing "quantitative". Not a chance.

Here's an explanation that's as good as yours: Light is not reflected, there's no reflection of light. Photons know all along where they have to go. There. Now prove I'm wrong.

If a photon is a material point, how do you explain the paradox of partial reflection of light by a thin layer of glass? From QED, the Strange Theory of Light and Matter, by R. P. Feynman. If a photon is a material point, how does this bouncing point, reflecting/not reflecting on the first limit surface of the glass layer react to the existence of a second layer, farther within, so as to reflect/not reflect on the previous surface? Quantum mechanics explains this easily. Unfortunately, physics takes more than the recitation of mantras, no matter how many words from topology or other branches of maths you use in your incantations. You might as well say that everything's made of two elements: mumbo and jumbo.

Stop right there. Avogadro's number is not a fundamental constant of Nature. It depends on two arbitrary choices: 1) The choice of an arbitrary mass unit (the gram.) 2) The average mass of nucleons (protons, neutrons) as to their abundance when participating in making up atoms and molecules in this galaxy. Ignorance, when combined with hubris, is louder than a siren.

You read my mind. I was thinking of the Amur tiger. Korean researcher Sooyong Park spent months in the Siberian and Chinese forests, eating rice, nuts and salt, and removing all traces of his own poo, hiding in a hole underground for years to film them. Yet we have extensive footage of these animals and their family life. And they had been seen and captured before. Also, bigfoot must be the only primate species in like 55 million years that's not highly social, curious, extremely boisterous. Isn't that peculiar? And lastly, we're aware of the existence of a ancient species of ape (the Denisovans) thanks to a tiny fragment of a phalanx from the pinky of one hand, including sequences of their DNA. And we can't obtain definite proof of the existence of a big hulking ape who inhabits the Earth now?

Is the answer not here?:

Good point, zero probability does not imply impossibility if the continuum is a reality. In maths it's to do with the measure problem, as I'm sure you know. The measure of an interval can be zero or not, but the measure of a point is always zero. The measure of any denumerable set in a continuum is zero. Lo and behold! (My emphasis.) The integral of Dirac's delta distribution is finite. If you want the probability of a denumerable set to be non-zero, you must construct a combination of delta distributions. For example p(x)=p1delta(x-x1)+p2delta(x-x2) with p1+p2=1. The derivative of delta makes sense, but the square doesn't... Think of that. It's a subtle business. Very interesting. Fractality makes me wonder so much...

Agreed that the cosmological constant problem is probably the most significant problem faced by modern physics. I'm not so sure it's related with singularities. Singularities seem to have to do with very strong local fields. Vacuum energy seems to have to do with global properties of ST. In any case, I think it's a topic for discussion in a separate thread. Why don't you open such a thread and present your concerns there?

This is liable to be considered a hijack of another post. Why don't you open a new thread instead? https://www.scienceforums.net/guidelines/

Oh, this is a bad analogy on so many levels... It's more like: Somebody owns a wall. Everybody's free to write on the wall, but the wall is not yours. The rules are: You may paint on it if you want, but we reserve the right to keep anything we want. Because you can't be bothered with reading the rules, you write on the wall, and then regret what you wrote. Finally, you whine about your writing not being removed. Whether your name is Van Gogh or Van Morrison is irrelevant. Also, the arguments by Phi and iNow, which give you a practical reason that makes a lot of sense, doesn't it?

(My emphasis.) Oxford Learner's Dictionary: I wouldn't call it 'cruel.' Composing a bad song is not cruel. Publishing a bad song is not cruel. Keeping copies of it is not cruel. OTOH, forcing you to listen to it over and over can be cruel. But that's not what's happening here. It happens to me every single day.

A final embelishment:

Differentiate: \[ f\left(x\right)=\log_{x}\left(\sin x\right) \] Solution: I hope I didn't make any silly mistake. If you find any other fun way to solve it, please let me know.

Actually, you're right. I stand corrected. Hubble --> Expansion; Penzias & Wilson --> CMB Those are different things. Confirmation of CMB is considered the last stepping stone in proving the standard cosmological model (previous to inflation), but the expansion was observed first. Thank you. Nevertheless, the Bondi-Gold-Hoyle model is to do with a steady universe. The idea was, if I remember correctly, that a tiny amount of matter is produced at a constant rate uniformly throughout the universe, compensating for the dilution of matter due to expansion. My details are hazy, as it's been such a long time since it was made irrelevant by experimental observation. We're getting farther and farther away from the topic of singularities though. Maybe a split is in order?

Sure. Very different things. Vacuum energy is very different even from the initial big bang theory before inflation was started by Alan Guth. So it was a contender of the big bang theory. The big-bang theory prevailed after Penzias and Wilson found CMB (1964), and inflation was introduced to solve some of its puzzles. Back in the 40's the idea that the universe was eternal and essentially the same at all times in its history was still popular. That had been Einstein's motivation for introducing a cosmological constant in 1917. He wanted a static universe. He failed at this, as his cosmological solution would have been unstable, not static.

Yes. This is the Bondi steady-state model. I grew up with that theory being a serious contender. But that theory came long before inflation, and actually before Penzias and Wilson's discovery of the expanding universe. Hermann Bondi wanted to account for a stationary universe, which we don't believe in anymore. It purported that the density of matter in the universe is constant. It also implied that the universe is eternal. It sounds similar, but it's quite different.

The term "coupling constant" comes from field theory. In field theory, all matter and radiation is studied in terms of fields. Fields are quantities whose value depends on position in space, as well as time. Fields have kinetic energy. This appears in the equations in the form of time variation of those fields. Fields also have "internal energy" due to spatial gradients of their values. They also have interactions with other fields, that appear as proportional to the product of one field with another. The constant of proportionality is called "coupling constant", and how big it is tells you how strongly they interact with each other. Fields also have self-interactions. These appear as powers of the field itself. Those also have coupling constants.

Thank you, Studiot. Standard cosmology separates different terms for these local densities. I hesitate to use this word "local" on account of how much confusion it's created through the years. People distinguish several terms on the RHS of Einstein's eqs. within the FRWL model according to the different stuff that fills the universe, which leads to the standard "epochs" post-big-bang: radiation dominated, matter dominated, and vacuum-energy dominated (present cosmological epoch): Radiation-dominated: energy density proportional to a-1 Matter-dominated: energy density proportional to a-3 Vacuum-energy-dominated: energy density = constant (a-independent) where a is the scale factor. It's in this sense that I say the vacuum-energy density does not dilute when the universe expands. As everything else would dilute (a-1,a-3), on account of no new matter or radiation being formed, while the vacuum energy density is kept constant (which leads to the exponentially growing solution with time that characterises vacuum-driven universes, also known as De Sitter universes) that's why I say it seems difficult to conceive of a model made of stuff --never mind it coming from negative-energy interiors of BHs. It doesn't seem plausible to me. It's the least I can say.

And by the way. It's an illuminating exercise to write down the equations of GR for a completely silly, trivial, flat spacetime, by calculating the Christoffel symbols, the geodesic equation, etc. in curvilinear coordinates. Of course, all the components of the Riemann tensor will be identically zero. But every definition and procedure for curved ST goes through. Here's a suggestion: Try and use your imagination, and write down a set of curvilinear coordinates that cover most of flat space, you can fill this totally dumb, seamless, featureless spacetime with (non-existent) "singularities" that all disappear once you introduce the proper set of (singular) coordinate changes that remove all your (non-existent) "singularities". Your toy model of field equations is the Laplace equation with the obvious boundary condition of all fields vanishing at spatial infinity. The genius of Kruskal was to realise that's kinda similar to what's happening with the Schwarzschild space time during a time when many people working on GR were still just chasing shadows. Stop chasing shadows, please. Now I do rest my case. --- One final caveat. If you insist on thinking of so-called vacuum energy in tems of energy content of some stuff, in the sense that it can be expressed as a density of "something", you're gonna run into problems. We call it "dark energy" for lack of a better word. But it's not really a local energy density. Dark energy does not dilute when you stretch your spacetime. This doesn't bode well with the proposal that it's due to a density of stuff filling in spacetime. And QFT certainly needs not negative energies. Those "cores" of negative energy would be quantum-mechanically unstable, it would violate causality, etc. You need a Hamiltonian bounded below for good reasons.

No. It's our old friend the Cartesian plane \( \mathbb{R}^{2} \). Flat, trivial manifold, with Riemann tensor identically zero, and (0,2) signature. It's totally isomorphic to its tangent space. Yet there are singular charts, and singular changes of different charts. I argued, in a way that you either have been unable or unwilling to understand that those singularities say nothing about the plane. They are singularities in the parametrization. It's plain to see now that you don't understand what a differential manifold is. That's why you're making incorrect statements about Schwarzschild spacetime over and over. And over. And over. And over... Yes, it would.

Polar coordinates on the plane: (r,theta) Change of coordinates: x=r*cos(theta) y=r*sin(theta) Singular @ r=0, as J(x,y/r,theta)=0 there. Is something fishy going on at r=0? No! A plane is a plane is a plane. (Sigh) Why?

I suppose what you mean is that a 2-sphere cannot be mapped with just one chart. I fail to see how this is relevant here. You're still trapped in the mirage of singularities of the coordinate chart. By the same token, you'd probably think there's a problem at the north pole of a sphere... I rest my case.

No. GR's job is to describe gravity. Nobody has ever said it's supposed to describe reality. That's a pretty tall order. That's why Einstein himself immediately started working on unification. This is probably Maxwell's most valuable legacy. So you don't believe in calculus then. I see no other way to interpret what you're saying. What does that mean, "the sphere is not reducible"?

Here's the Kruskal-Szekeres change of charts: It's singular at r=RSchwarzschild=2GM because the Jacobian is zero there. I don't know where you got the 00 problem from. 00 is no problem if you can calculate the limit. The problem is the Jacobian is zero. This is not allowed for good reasons I'm not going to delve into. But:What is a chart? It's an assignment of coordinates: \[ \phi:\mathcal{U}\subseteq\mathcal{M}\rightarrow\mathbb{R}^{4} \] This could be the Schwarzschild chart. And in comes Kruskal: \[ \psi:\mathcal{U}\subseteq\mathcal{M}\rightarrow\mathbb{R}^{4} \] What's a change of charts? In this case from Schwarzschild to Kruskal: \[ \psi\circ\phi^{-1}:\mathbb{R}^{4}\rightarrow\mathbb{R}^{4} \] The initial chart is singular at the 2-surface r=2GM. Now Kruskal introduces a singular change of charts that restores smoothness on that region. It stands to reason that you must do something drastic on r=2GM if you want to restore smoothness. This is a particular feature of analytic continuation in this case. Now "stasis". You do not look at the metric in a particular coordinate set and infer anything. That's a sure way to make mistakes. What you do is what Markus told you: In this case, if one wants to prove that the metric is static, what one does is define a Lie derivative, and from there introduce time-like Killing fields. Then prove that your metric is invariant under those infinitesimal transformations.