Is Poincaré's conjecture related to CMB cosmology?

Discussion in 'Physics & Math' started by arfa brane, Jul 12, 2018.

1. arfa branecall me arfValued Senior Member

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--https://mathoverflow.net/questions/9708/poincaré-conjecture-and-the-shape-of-the-universe

Last edited: Jul 12, 2018

3. someguy1Registered Senior Member

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Wow it's a long long way from the type of math done by Perelman and anything going on in the physical world. And even if there is a relationship, we're a long long way from understanding it.

More confusion of math with physics. There is so much of it going around these days.

Thanks for the link. Definitely a question that's interesting whether or not it's true.

ps -- Wow I read through the MathOverflow page. Very impressive. I think I did find some support in there for my opinion that the question's essentially meaningless. But a lot of smart people take it seriously. It's interesting how they apply higher topology and even algebraic topology to the universe. Now that's amazing.

Last edited: Jul 16, 2018

5. arfa branecall me arfValued Senior Member

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One of the reasons I looked at Poincaré's conjecture was because of heyuhua's posts about Yang's work. The question I asked myself was, what does the value of the Einstein constant have to do with geometry?

I understand you get a factor of 4π in Gauss's formula because the divergence of potential is spherically symmetric (you sum over all directions, i.e. over all the surface of the spherical volume). Now Wikipedia tells me: "the Ricci flow may be defined by the geometric evolution equation[3]

. . . and there's that factor of -2, a point of contention in that (surprisingly uneducating) thread of heyuhua's.

Or perhaps I'm looking at nothing significant .

7. arfa branecall me arfValued Senior Member

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-https://en.wikipedia.org/wiki/Ricci_flow

But anyway, I object to Yang's EFE "solution" Guv = 4πGTuv because the EFE is written in tensor notation (describes a sum of tensor components); there are two indices which is why the constant is 8πG (up to a sign), in the "real" world we have tensors with single indices (i.e. vectors).

8. curvatureRegistered Member

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Yes it will.

The full Poincare group of spatial symmetries refers to the ''playground'' of all relativistic fields. When a scientist thinks about Poincare symmetry, they think nature itself: In other words, these symmetries are expected to exist in nature. For a universe to be in the full Poincare group would require the universe has a rotary property as well. Godel's rotary universe is a bit outdated, we have much better models now that accurately predict the expansion with rotation properties. With it though comes a price, if the rotation was significant, then torsion played a part in the universe as well as an internal centrifugal force and has been suggested as an alternative to dark energy.

9. curvatureRegistered Member

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• Another day, another Reiku sockpuppet.

Ironically, I had searched for a true derivation of this under special relativity and general relativity, to be disappointed to find only articles that have mentioned four dimensional relativistic Ricci flows but no examples. So I took pen to paper and did manage to write the appropriate equations:

The Ricci flow is the heat equation for a Riemannian manifold. Arun and Sivaram suggest an equation of the form:

$\frac{\partial R}{\partial t} = \alpha \nabla^2 R$

In which R is the scalar curvature. Later we will see another form in which I had an idea in which there was a diffusion equation for $\phi$ (the scalar gravitational potential). I came to find that the equation I had proposed already existed in literature, for instance, for the heat flow per unit area we have:

$Q = -k\nabla T = -k \frac{\partial T}{\partial x}$

must also be modified in such a way that:

$Q = -k\Box T = - k \nabla T + \frac{ik}{c} \frac{\partial T}{\partial t} =- k \nabla T + ik \frac{\partial T}{\partial \tau}$

Now… It was possible to construct all those idea's in the context of curved spacetime. The kind of equation I wanted to study was

$\alpha (\eta^{\mu \nu}\partial_{\mu} \partial_{\nu} \phi) = \alpha \Box^2 \phi = \alpha (\frac{\partial^2 \phi}{\partial \tau^2} + \nabla^2 \phi)$

The original form of this equation even considered a mass parameter which took the form of the dispersion, which as a heat equation, would tell you also how that system would have interacted with a certain type of medium... a good example is reflection. Or even absorption. I’ll show that at the end. The idea was simple - I searched for a solution for the flow which has the appearance of

$\frac{\partial \phi}{\partial t} = \alpha \Box^2 \phi = \alpha \partial^{\mu}\partial_{\mu}\phi = \alpha g^{\mu \nu} \partial_{\nu} \partial_{\mu}\phi$

The idea would invoke a relativistic Newton-Poisson equation of the form

$\frac{\partial \phi}{\partial t} = \alpha \Box^2 \phi = - \alpha (\frac{\partial^2 \phi}{\partial \tau^2} - \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}) = \alpha ( \frac{\partial^2 \phi}{\partial \tau^2} + \nabla^2\phi)$

And this equation in curved spacetime looks like:

$\partial_0 \phi \equiv \alpha g^{\mu \nu} \nabla_{\mu} \nabla_{\nu} \phi = \alpha g^{\mu \nu} \nabla_{\mu} (\partial_{\nu} \phi) = \alpha(g^{\mu \nu} \partial_{\mu}\partial_{\nu} \phi + g^{\mu \nu} \Gamma^{\sigma}_{\mu \nu}\partial_{\sigma}\phi) = \alpha \frac{-1}{\sqrt{-g}}\partial_{\mu}(g^{\mu \nu} \sqrt{-g} \partial_{\nu} \phi)$

Which is a diffusion equation with respect to \phi in the language of general relativity.

10. RainbowSingularityValued Senior Member

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i cant follow the math, so im just reading how you describe things.
i am picturing a black hole spinning in the middle of a galaxy or universe while it creates an almost spiral like circular projection of energy, while it also creates (not sure how) an inward spiral rotating energy.
i am trying to mentally visualise a generic centre of something(black hole) that is creating opposing spiral forces that interplay/overlap or exist inside or crossing over each others energy.

adding the orbital/tortion(of rotating galaxys/matter) to that kinda makes a big wibbly wobbly/mish-mash(multi-direction forces) 3D mesh of fields.
at which point my brain says take a moment and hits the pause button

11. NotEinsteinValued Senior Member

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Large parts of this post appears to be remarkably similar to https://physics.stackexchange.com/q...flow-and-nonlinear-relativistic-heat-equation which was made by Reiku. Copy-pasting again without attribution, are we?

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15. KittamaruAshes to ashes, dust to dust. Adieu, Sciforums.Valued Senior Member

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Much obliged. I made a post in the back room to see if anyone had a reason not to dismiss him. Nobody made a peep against it, so away he goes.

exchemist and origin like this.