joigus

That's exactly what I was thinking. There's a circularity here. g(r) is no problem. It's T(r) what's a pain in the neck. Last thing I've tried is to use the energy equation for a diatomic ideal gas, which is, \[U_{\textrm{int}}=\frac{5}{2}Nk_{B}T\] and then to relate the internal energy of the gas with the gravitational potential energy to eliminate the temperature, but no luck. The density comes back. Maybe I was too tired. You still have the T(r) problem. Even if you use a virial expansion, you still have the T(r) problem.

My apologies again. I made a mistake here (as usual): \[\int_{S}\textrm{curl}\boldsymbol{v}\cdot d\boldsymbol{S}=\oint_{\partial S}\boldsymbol{v}\cdot d\boldsymbol{l}\] The circuit must be the boundary of the surface. I didn't mention it, but it was always on my mind.

I personally don't take offence at the concept of science being wrong, even though I use my leisure time mostly to learn more about it and I've made of it my method to try and understand the world better, like most of us here I would say. I don't think science aims for absolute truth. It's not about being right or wrong beyond any doubt. It's about being more right and certain and less wrong and uncertain, and pushing the limits of doubt and ignorance. Science doesn't provide us with a magic wand to dictate ethics either. It evidences correlations, most of them of statistical nature. It sheds light on plausible causal connections, it refutes previous ill-conceived ideas. If we do that, we are in a better position to take better decisions, diagnose better, tackle evil before it happens. But this can only be achieved by adding to the structure more layers of rational thinking and open discussion. Our understanding is never complete. What kind of philosophy marginalizes individuals? Do you mean something like social Darwinism? It's not a universal trait of philosophy, AFAIK. I'm guessing you've voted that there are good and bad philosophical theories...

On a related note, but much more humbling for us: https://www.newscientist.com/article/dn11330-pubic-lice-leapt-from-gorillas-to-early-humans/#:~:text=A genetic analysis of pubic,about 3.3 million years ago.&text=But they claim it is,from having sex with gorillas. I don't want to draw any conclusions. Just saying... The most interesting thing is whether these hybrids of sturgeon and paddlefish can in turn reproduce. Most admixing results into sterile individuals.

Very good point, as always. +1 More modern criteria of speciation are slowly diluting into fuzzy categories. If plants, fungi, protists, bacteria etc, could talk, they would give us a memorable speech about interbreeding. Wouldn't they?

I didn't!! +1 Please elaborate or give me a reference, if you don't mind. I agree. ----------- I would like to think, and comment, and read more comments, about how all this Stokes' theorem story could have some bearing on the question of time, as integrating on the boundary of a set requires orientation, while integrating on a bulk that is not a boundary, doesn't. Same as time. I googled for the circle, as I wasn't sure. Here's what I found: My apologies, @studiot, because I've been talking all the time about exterior calculus without mentioning, explaining, or giving a proper reference: https://en.wikipedia.org/wiki/Exterior_derivative It's the technique to wrap it all up (Stokes, Green, Gauss, etc.) into one unified description. Thus, for example, Stokes' theorem (or is it Green's?) can be obtained by exterior differenciating a line element (1-form): \[d\left(v_{x}dx+v_{y}dy\right)=\] \[=v_{x,x}dx\wedge dx+v_{x,y}dy\wedge dx+v_{y,x}dx\wedge dy+v_{y,y}dy\wedge dy=\] \[=v_{x,y}dy\wedge dx+v_{y,x}dx\wedge dy=\] \[=\left(v_{y,x}-v_{x,y}\right)dx\wedge dy\] \[\int_{\partial\Gamma}\textrm{curl}\boldsymbol{v}\cdot d\boldsymbol{S}=\oint_{\Gamma}\boldsymbol{v}\cdot d\boldsymbol{l}\] It's powerful because you can do it for any dimensions, surface element, line elements, volume elements, and in general, n-surface and (n-1)-surface elements.

Exactly. It's a non-starter. I couldn't agree more. +1 Bad arguments give you a chance to highlight interesting issues, common misconceptions... Maybe we all discover here that Darwin missed a subtle point, or perhaps that some of us are missing a matter of subtle detail, some loophole, some interesting aside. If not the case, we all refresh the well-known facts, arguments, etc., and our understanding gets refreshed/reinforced, better understood.

Any arguments, or are you after founding a new religion and we're supposed to nod and follow you?

Very poorly thought, IMO. Did the differently-beaked finches interbreed? (Darwin's-time criterion for speciation).

I tested it and it doesn't work for me. Plus there's a considerable amount of looping. Could account for the remaining 5%. I need a clicking tutorial. The thing is it kind of make sense to me that philosophy is an attractor.

Thank you for anticipating your ideas. +1 I'm in no hurry, in case you're wondering. I'm just interested. The problem is quite academic, but interesting nonetheless. Somewhat out of my scope. I think that's the route to solving it. Slicing the tube into infinitesimal slices of constant thermodynamic conditions. And let the gas equilibrate with itself. The only thing I don't see is the temperature. But maybe you've got that into account for later.

Allow me to answer, and Markus, feel free to correct mistakes/imprecissions and/or add info as you see fit. The AdS/CFT dualities are more sophisticated. For example: On the inside (bulk) you have a gravitational theory, which has a metric connection (a rule to parallel-transport vectors from a metric). On the boundary you have a gauge theory (a non-metric connection or affine connection, which stands on its own). They are completely different animals, and there is no simple way to relate the degrees of freedom. Also, some solutions of the theory without metric may be completely devoid of meaning as solutions of the theory in the bulk. There are topological aspects in the spectrum of solutions on the boundary that have no easy interpretation (or non at all) in the bulk. And so on... People who work on these type of dualities generally speak of a dictionary (a set of rules to translate boundary conditions, etc. from one theory to the other). Edit: x-posted with @Markus Hanke . I think you have an extra circle there. It'd be, \[\int_{M}d\omega=\oint_{\partial M}\omega\] I personally prefer not to use the circle because there's no simple way to iterate the operation. So, for example, if you want to express Hodge duality's simple result "the boundary of a boundary is empty": \[\int_{M}d^{2}\omega=\int_{\partial^{2}M}d\omega\] \[d^{2}\omega=0\Rightarrow\partial^{2}M=\textrm{Ø}\] The boundary of a boundary has no points. Edit: And the \partial symbol already implies you're looping around. But that's a matter of taste.

Spaces with intrinsic curvature can expand or contract without any coordinate points touching each other, like when you paint dots on a balloon and start to blow. They separate but they never touch. Exactly. "Remindful of", "suggestive of". And that's the reason. +1 Unless anybody comes up with a closer analogy that one theory is like an exterior differential of the other in some sense.

Well, Maldacena's initial idea wasn't motivated by a realistic model of the universe. So the so-called AdS (anti-DeSitter Space) was not meant to represent the real universe. Although the AdS space-time is an exact solution of the Einstein field equations. But exponentially expanding or contracting universes don't have to have a center. Our universe is a DeSitter universe (exponentially expanding) AFAWK and it's not doing it around any particular point. Everything is expanding with respect to everything else.

https://www.youtube.com/watch?v=hBpetDxIEMU&t=381s @ 20' 34'' - 40' 38'' Both hilarious and spot on.

Let me help you. All of them are contained here, \[\int_{M}d\omega=\int_{\partial M}\omega\]

I wasn't aware of this. I must have been sleeping all these years. Thank you. +1 Mmmm... Remindful of, suggestive of, rather than equal. Reminds more of Cauchy's integral theorem of complex calculus. And even more of Stokes' theorem for differential forms. Because we always use analytic functions, things on the inside are determined by things on the surface. But I'm getting hopelessly vague and metaphorical. Although the FTC is a particular case.

Aaaah. Now I understand much better what you're trying to do. Thanks for careful explanation. +1

Oh, my, you're sharp, Hanke! I may be going nowhere, but you understand exactly what I mean. +1 You're worth 10 points here. In fact, there is a kind of non-locality in my view, but it has nothing to do either with space nor with time. It's abstract, internal-space. The functions you're trying to measure are not point-to-point (eigenvalue-to-eigenvalue) functions of one another. What some analysts call "non-local operators". Maybe the expression filtered out from there. Same way x is non-local operator in p-eigenstates (it depends on all the spectrum) and vice-versa.

You can have a plague, massive wiping out of genes, but the smallest sample get amplified by the founder effect later. And what previously was a Charlemagne differential gene (I suppose Charlemagne had genes for cellular respiration too) get amplified to almost universal proportions. It's kind of a mix, filter, mutate and stretch kind of dynamics.

Mmmm. But temperature must go up as you go down the hole, irrespective. Gravitation always heats up any stuff as you go down towards the core. Never mind gravity field going down to zero. Pressure builds up --> temperature goes up. It's not temperature coming from Earth's core transferring it to the gas. It's the gas' own internal energy/volume that does it. Say, I may have misunderstood something. I must confess I'm a bit confused about this one. If you want to solve the problem, the only thing you can do is let the air in the hole acquire temperature from its own pressurization, so to speak, as it builds up weight on top. The Earth can't touch it, either thermally or pressure-wise. I've just done a lookup and the Van der Waals eq. is not good enough to deal with this either.

But heat transport can happen even though the situation does not depend on time. Heat transfer must have reached a stationary regime (I'm not saying heat doesn't flow). Also, they're assuming perfectly isolating walls... Maybe I got the premises wrong.

Agreed. Language of itself can mislead you. Maths too. Experiments without theoretical analysis are devoid of meaning. Sheer observation can be a crook. It's a network of interrelationships, cross checks, that makes it all solid. Narrowing down the chances of being mistaken. Cladking doesn't even seem to know what a cat is. Most people have no problem with this.

The problem with this is that is sounds sooooo much like a particular philosophy... You simply can't escape philosophy. Break down the word into etymological pieces and you'll understand why.

Some days ago I learnt from @Strange that most Europeans are descended from Charlemagne. I've learnt many other things from him. But this one got me thinking (and still is) about the likely regular Jacks and Susans, and Joes and Marys, who were especially successful in the reproductive sense, but not particularly notorious, and got their genes pushed forward in human history.