Mordred

Fair enough, its been awhile since I looked into these. So I will have to do some studying and greater familaritation. Looks like your going to eventually need each type lol. They each have slightly different purposes. Schmidts defines entropy, Von Neumann tests how strongly correlated the entropy is and Shannon entropy applies orthogonality to gain compatibility for qubits Here this gives us the essential distinquishments. https://www.google.ca/url?sa=t&source=web&rct=j&url=http://www.mpmueller.net/seminar/talk2.pdf&ved=0ahUKEwjZ9Nbgt9LWAhUGwGMKHQqEBOs4FBAWCCYwAw&usg=AOvVaw1fFu0JDQWIIg-HTfRdb5JO On digging deeper it looks like Von-Neumann applies the Stirling approximation as per Einstein solid and Boltzmann. Makes sense, still confirming. I can definitely see how this will be useful to you in your modelling goals as the above is applicable.

I think your going to have to be incredibly careful here. The first equation in that link is the standardized Schmidt's decomposition I recall. There is a slew of details you will need to be aware of and account for, to arrive at the equation you posted. Which does involve Von Neumann entropy, however Shannon entropy also involves Renya entropy. (getting the idea on why care is needed). We have numerous different methods to define entropy. So greater care is needed in not mixing up these treatments. Lol the posts by you two is very similar in your approaches and goals, which has made things interesting for me

That is a really bad website that gets that equation completely wrong. That equation is to calculate the Hubble parameter today. The statement it makes of mass density and radiation density being constant is false. Both matter and radiation evolve as the universe expands as per the ideal gas laws. The only value that does not evolve is Lambda. Hubble's constant isn't constant. It is only constant everywhere at a particular time slice, but evolves ovee time.

Its been awhile since I last looked at Schmidts decomposition the above is in the Von Neumann entropy form correct? The Schmidts equation you gave isn't the standard form and looks like a bipartate mixed state form. Where as Schmidt's has a different form for pure states as well. Please clarify as it has been some time since I last looked at Schmidt's (not questioning your usage in the above, just the form posted isn't one I fully recall ). Have you looked at the pure state for Schmidt's decompositions?

I agree lol (to both statementsm now that the latter is stuck in my mind

I like your use of the stream example, it highlights an example of a charged field as opposed to a scalar field. I plan to expand on some of the details you raised in the above. Once RL allows proper focus time lol. The above will make a good example to explain the stress tensor elements. Hence wanting time to concentrate.

My favorite pocket reference has nothing to do with any set theory under physics but instead teaches the tools to understand the mathematics. Mathematical methods for Physicists by Arfken. An essential tool to understand group theory without dealing directly with group theory but covers the basis. It is an invaluable tool in any physics related field covering topics such as Greene's, Louiville, Gaussian, distributions etc focussing on the vectorial aspects. The book is fully applicable in Engineering as well. From personal experience since studying the book I have found a vast majority of physics related topics far easier to understand. (Though much of the book were of topics I was already aware of, in my case. The layout of the book places the pieces together in a clear fashion)

The galaxy rotation curves, in particular spiral galaxies require fairly uniform mass distribution as the volume increases, as per the NFW profile of an isothermal halo distribution that is sufficient to offset the mass of the baryonic distribution in order to avoid the Kepler curve. A Kepler curve will arise on any central potential mass distribution system where the bulk of mass is non uniformly distributed from Com outward. Under Virial there is correlations to the NFW profile. "Elements of Astrophysics is one of my favourite references to NFW profile. (key note NFW is a power law profile)

Yes and depending on the treatment applied the T=0 can be correlated to last scattering or the singularity 10^-43. Though your cosmological horizon singularities also apply under the particulars of the treatment under the FRW metric. ie Hubble Law. does help with forum membership though lol, generates posts of "this doesn't make sense I must reinvent so I can understand it"

valid point here it is a very common question with tons of pop media misdirection

It would be more accurate to express the above as particle wave function boundary confinements. ( just an interjection :p)

The aforementioned treatments will be described by its killing vectors, just an aside note any field treatment will involve killing vectors.

accurate descriptive, however how one defines a singularity depends upon the metrics defining the system and system states involved. Those metrics can often be defined as per a particular observer. Some singularities that show up in the pertaining metric system, may be resolved by a coordinate, metric or observer change. There is often artifacts of a metric to be recognized within the applicable bounds provided by the numerous function cutoffs of a metric. In both GR and particle physics in terms of path integrals of the Feyman scattering diagrams etc. The cutoff treatments of applicability is a prevention of the metric affinities arising. Some examples being a wicks rotation, or Wilson loop. QFT corresponds to displacement via the Hamiltons of action under field treatments. In particular A Quanta of action"

Nicodem Poplowkii uses torsion under GR to try to replace DM. There is others but MOND and Popolwkii's ECT Einstein Cartan based universe were always the two strongest competitors. Coincidentally as I understood it, the same problem that effectively killed MOND also affected Poplowskii. That being the early large scale structure formation.

The expression "Nothing is impossible" definetely applies in this instance hehe

I do recall coming across dark flow models as rotation numerous times in the past. I never saved an of them except one related. That being by Nicodem Poplowskii, who applies the the ECT I mentioned above. I have a copy somewhere and will post it once I find it. He attempts to connect particle spin to torsion via field treatments, though he primarily set hiw bounds in regards to the UV limits in his metrics. IE cosmological singularities. EH etc..He also applied the above in particle physics in terms of helicity and variations in the particle states due to torsion. (wonder if he still has his website, he used to keep copies of all his papers on it) Once you get a chance to study most the material I have been providing, (yes takes time to absorb lol) I hope you will come to realize that the metrics I have been referencing are used in a vast majority of your different toy universe rotating models.. I haven't targeted any specific rotating model, but provided some examples using those essential equations for a robust theory. edit yep his website is still up. You will find these incredibly useful in modelling a rotating universe http://math.newhaven.edu/poplawski/publications.html mainly in the techniques, applicable formulas etc

So here is my question, if your familiar with Virial theorem then you have some familiarity with how it applies to equations up to page 6 of this paper we got into debating about. https://arxiv.org/pdf/0902.4575.pdf the first section is specifically applying virial theorem, with the correlations under GR. in terms of the four momentum etc, it has applied a cosmological term to those equations without identification of cause of the terms. It gives the applicable geodesic equations in B.1 and b.2. which is applicable to redshift data. Why do you have a problem with the boundary conditions as applied to this paper? or rather that the establishing of a boundary condition via CMB data wouldn't be a viable option under cosmology? Your dealing with far higher densities in fact its much easier to measure potential rotation aspects under higher density (too bad about the opacity limits, make room for GW cosmology...once we develop the technical infrastructure lol). ie the fields are more strongly coupled so under GR changes take time to propogate throughout the field. Hence development of field anistropies, which will get increasingly chaotic with constructive/destructive interferences in your higher orders of approximations You can't avoid the fact that multiparticle systems under rotation, do behave nonlinearly, they develop their own hydrodynamics in terms of flux and vorticity, which affect \rho under the stress tensor. So does spin under field theories. When we started this thread this was the first concern raised and has been throughout this particular thread. The question of how slow a rotation is required to avoid detection of rotation is of primary concern. Can you show something wrong with the methodology itself in the above paper? If not then why ignore its potential of an upper boundary? edit forgot to add: You not only have a rotation of a field, but also under commoving coordinates of an adiabatic and isentropic fluid under expansion. This will have additional affects to the rates of information exchange under the above hydrodynamics. The paper above already included this, You have an object roughly 1 Mpc in radius. I am using just an example rather than actual CMB size, but I could get a specific example. How how would it take a signal in a vacuum state to reach the center from the outer edge ? How long for that same signal under the medium like properties of spacetime, as the density increases or decreases? Why would this not apply to a multiparticle system under rotation ? The above paper demonstrates that signal speed etc does apply to the CMB under rotation ( side note, the paper does not reference a specific moment of the CMB but its history, ie the CMB still exists today....)

Its a good policy to understand what we can measure first. On a side note, I always find it somewhat amusing that we are all taught in high school the above definition. Yet when we start thinking cosmology, GR and QFT. Everyone forgets them

correct, but then most people don't understand energy either lol

higher density for one just prior to the surface of last scattering the mean free path for photons being less than 10^-32. We are dealing with hydrodynamic equations, just on a more cosmological scale. Have you ever worked with hydrodynamic equations for astrophysics/cosmology applications? ie have you studied Virial theory and Jeans instability ? Here is the thing thee boundary paper I posted which is what this discussion is currently about, involves how thermodynamic characteristics of a rotating body behave using the affine connections of your Christoffels to the Kronecker delta./Levi-Cevita. This reflects upon the Baryon acoustic oscillation data with its correlation to the Sache-Wolfe effect. (Sache-Wolfe is incredibly handy to measure expansion rates) in particularly anistropies of the early and late times (two different set of equations). The above papers have all been in regards to the connections I have been referring to.... I mentioned galaxy rotations curves in reference to the hydrodynamic equations under GR, etc etc (every field model has a correlation).

I disagree on this hence my suggestion to look at the papers regarding boundary conditions applied via the Christoffel connections in regards to Levi_Cevita. I am still looking for a decent coverage of Einstein Cartan but quite frankly too often they are modification papers upon it. I may end up using Elements of astrophysics to get into the hydrodynimic equations of rotating bodies. Then show Cartan kk that makes more sense,

this is the statement I am primarily discussing, please define I may be misreading your previous post I'm asking if you think that, I know they have and do.

Bingo

Well the spin 2 statistics of GR is contained in numerous textbooks of GR and cosmology. ( usually in the introductory textbooks. Later when I have time I will post the series of calcs from Matt Roose. However the spin 2 arises from the two linear polarizations of a GW wave. (H+ and H×) under Noether ( uses the Gell Mann matrixes ) The electromagnetic field is dipolar one linear field with two polarity states.( uses the Pauli matrixes)

ok measure an electric field with no potential difference between DMM leads. Now apply that to a field, with absolutely zero difference in field values. Now add a potential difference in field values lo and behold suddenly an energy value arises and that field is more capable of performing work. Energy is the ability to perform work. Define potential energy under physics. "the energy possessed by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors" One factor is anistropy in field values. Now with those definitions firmly implanted read "On the zero energy universe" https://arxiv.org/abs/gr-qc/0605063 see the conservation of energy/momentum juggling act with the potential and kinetic terms? I realize this paper is in GR formalism.