Is Poincaré's conjecture related to CMB cosmology?

A

arfa brane

call me arf
Valued Senior Member
The cosmic microwave background, discovered in the 1964 by Penzias and Wilson, shows that the universe is very nearly isotropic at our location. (The deviation is of order 10^−5 and it was only announced in 1992 after 2 years of data from the COBE telescope.) If you accept the Copernican principle that Earth isn't at a special point in space, it means that there is an approximately canonical fibration by time slices, and that the universe, at least approximately and locally, has one of the three isotropic Thurston geometries, E3, S3, or H3.

The Penzias-Wilson result makes it a really good question to ask whether the universe is a 3-manifold with some isotropic geometry and some fundamental group. I have heard of the early discussion of this question was so naive that some astronomers only talked about a 3-torus. They figured that if there were other choices from topology, they could think about them later. Notice that already, the Poincaré conjecture would have been more relevant to cosmology if it had been false!
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--https://mathoverflow.net/questions/9708/poincaré-conjecture-and-the-shape-of-the-universe
 
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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.
 
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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]

48b6399a8e211ba5a7dd5f4192d16794452160f6

. . . 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 .
 
In differential geometry, the Ricci flow (/ˈriːtʃi/, Italian: [ˈritt͡ʃi]) is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric.
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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).
 
arfa brane said:
--https://mathoverflow.net/questions/9708/poincaré-conjecture-and-the-shape-of-the-universe
Click to expand...

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.
 
Another day, another Reiku sockpuppet.
arfa brane said:
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]

48b6399a8e211ba5a7dd5f4192d16794452160f6

. . . 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 .
Click to expand...


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.
 
curvature said:
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.
Click to expand...

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
 
curvature said:
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.
Click to expand...
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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