Mordred

Another theorem that's a good study is density wave theorem this will cover the flattening effect due to mass rotation. it also covers metallicity distributions as it applies to star formations. as well as how the spiral arms come about. The theory also applies to the rings of Saturn, as the momentum terms apply the same in both cases.

Its an apparent horizon where the radius can alter depending on observer. Another term commonly used is a coordinate horizon or coordinate singularity. Mathius Blau if I recall also discusses this in his lecture notes on General relativity on arxiv. It is more often described in literature as an "artifact of coordinate choice". If I recall the Kruskal-Szekeres coordinates will eliminate the R=0 singularity condition at EH. Page 182 covers it in Lecture Notes on General Relativity Sean M. Carroll https://arxiv.org/pdf/gr-qc/9712019.pdf LMAO I guess one could say the frozen can be thawed with a different choice of coordinates

Just so your aware I'm still looking into this as I get the time, part of the problem is the separating topological rigidity problem that is apparently more common than I initially realized lol. The local conditions often fit the criteria to establish invariance. Or in the case common in particle physics the c.m. (center of mass frame) or lab frame. Gauge theory by criteria must be Lorentz invariant, however not all groups are Lorentz invariant particularly with the first order QM operators. The Schrodinger equation for example isn't Lorentz invariant which is one of the reasons QFT uses the second order Klein Gordon equation. Granted another reason directly applies to the conservation laws which are symmetric relations in closed systems. You have via the continuum equation \[\frac{\partial\rho}{\partial t}+\nabla\cdot j=0\] for a conserved quantity. QM,QFT,QED, Navier Stokes and fluid dynamics have variations of this equation but in each instance can be shown the equivalent via the related mathematical proofs. Noether's theorem shows the symmetry relations to a conserved quantity being symmetric. from wiki "In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups)." does that help address the reason for local vs global. https://en.wikipedia.org/wiki/Gauge_theory edit: @joigus Have you considered the above in terms if the rigidity like properties in regards to the topological spaces you have studied ? just food for thought but there is a good likelihood it will relate.

Why would that make any difference the moons angular momentum is far too low to even make any difference however even then those same lasers can still test for it. You still seem to not understand that inertial mass is identical to gravitational mass for all practical tests. If you ever plan on ever getting any form of funding you will require explicit mathematics to present your case. No one invests in haphazard guesses an conjecture.

As Markus mentioned the question is rather meaningless. We can only measure differences in potential from one geometric location to another. There is technically no limit to the amount of difference involved. Even if you took the entire mass of the universe and calculated that mass within a single infinitesimal and compared that to a region of zero mass density you still would not have a determination of an upper limit.

Ok I do have tchnically 2 threads ongoing with the related mathematics of Higgs inflation. https://www.scienceforums.net/topic/128412-musings-of-a-mad-scientist-inflation-as-cosmological-constant/ the first part is just the FLRW metric as I needed to get the equations of state for comparison for a single scalar Higgs field. The second part details the standard Higgs inflation mathematics. I'm still working on this part in greater detail in this thread I have in speculation but its rather random as I research each portion. https://www.scienceforums.net/topic/128332-early-universe-nucleosynthesis/ anyways if the Higgs inflation is correct and for the record the equations of state are identical to the standard inflaton field. A quick run down is as follows, the initial hot dense state had an extremely high kinetic energy term that when applied to the equations of state exceeded the critical density at that time. As the kinetic energy exceeded the potential energy by such an extreme expansion is a consequence to begin with. So your maximal false state is roughly \[10^{16\rightarrow..19^2}.. GeV\] As the expansion is already underway, you get the subsequent cooling (the slow roll stage) quark gluon plasma state where all particles are still in thermal equilibrium. At the volume curvature is still meaningless and in that thermal equilibrium state the effective degrees of freedom is a mere 2... hence the low entropy beginning. The two degrees of freedom is the polarities of the photon. As the effective degrees of freedom re the result of temperature with the photon as the mediator. At this stage you apply the Goldstone bosons correlations to the Higg's field. (invariant massless field). The consequence of the expansion allowed a sufficient drop in temperature (you an apply the photon redshift relations to the cosmological redshift formula) for a rough estimate Symmetry breaking occurs, this correlates with the rapid descent of the Sombrero hat. (number of e-folds in excess of 60 to fit observational data. The result of the rapid expansion leads to supercooling. Now this stage is important to understand prior to inflation the universe curvature didn't particularly matter due to the small volume in essence. However inflation of 60 e-folds later left a very close to critical density so k=0, approximate. the effective equation of state is the ultra relativistic radiation. The supercooling and critical density value both apply to the slow roll stage. Inclusive in this is the addition of a mass term to the particle species that have dropped out of thermal equilibrium. The "Friction is a correlation to that additional mass term". Another important detail is the transition from the false vacuum state to the true vacuum state need not be smooth you can have numerous other semi states and still match observational data. This detail likely be the result of various particle species decoupling, the result of each decoupling alters the effective expansion rate. quote from above Further, if you make a real world model of the sombrero potential and use a marble as the universe's potential, you find that the marble oscillates across the brim, before coming to a complete stop at the lowest point. Could the same oscillations be occurring to the universe's potential, and account for periods of increased and decreased expansion rates ? Or, am I reading too much into the model ? Not really due to the equations of state for radiation relation to the Hubble volume which relates to how the E-folds are calculated. If the kinetic energy term can exceed to Hubble radius in a given short time frame as defined by the E-fold logarithmic function quickly decreases but as it stabilizes it essentially merges with the radiation equation of state rather it becomes dominates by the radiation equation of state we do not know if were currently at minimal there is some conjecture we may also be in a semi stable stage.

Sterile neutrinos are still in consideration as they are predicted by the standard model. I have been looking into the literature to find examples of what the projected cross section would like. However as we have never observed sterile neutrinos it's all naturally conjective

I will have to look closer at the affine connections however I understand where your coming from will closer as to how the operators are handled

The common feeling is that DM doesn't interact with the strong or EM field. It may interact with itself or other weakly interactive particles. All particles obviously interact with gravity

you appear to have solved it. LOl yeah doesn't everyone find 800 pages an article ??? lol

Early Universe Cross section list Breit Wigner cross section \[\sigma(E)=\frac{2J+1}{2s_1+1)(2S_2+1)}\frac{4\pi}{k^2}[\frac{\Gamma^2/4}{(E-E_0)^2+\Gamma/4)}]B_{in}B_{out}\] E=c.m energy, J is spin of resonance, (2S_1+1)(2s_2+1) is the #of polarization states of the two incident particles, the c.m., initial momentum k E_0 is the energy c.m. at resonance, \Gamma is full width at half max amplitude, B_[in} B_{out] are the initial and final state for narrow resonance the [] can be replaced by \[\pi\Gamma\delta(E-E_0)^2/2\] The production of point-like, spin-1/2 fermions in e+e− annihilation through a virtual photon at c.m. \[e^+,e^-\longrightarrow\gamma^\ast\longrightarrow f\bar{f}\] \[\frac{d\sigma}{d\Omega}=N_c{\alpha^2}{4S}\beta[1+\cos^2\theta+(1-\beta^2)\sin^2\theta]Q^2_f\] where \[\beta=v/c\] c/m frame scattering angle \[\theta\] fermion charge \[Q_f\] if factor [N_c=1=charged leptons if N_c=3 for quarks. if v=c then (ultrarelativistic particles) \[\sigma=N_cQ^2_f\frac{4\pi\alpha^2}{3s}=N_cQ^2_f\frac{86.8 nb}{s (GeV^2)}\] 2 pair quark to 2 pair quark \[\frac{d\sigma}{d\Omega}(q\bar{q}\rightarrow \acute{q}\acute{\bar{q}})=\frac{\alpha^2_s}{9s}\frac{t^2+u^2}{s^2}\] cross pair symmetry gives \[\frac{d\sigma}{d\Omega}(q\bar{q}\rightarrow \acute{q}\acute{\bar{q}})=\frac{\alpha^2_s}{9s}\frac{t^2+u^2}{t^2}\]

I've studied a lot of literature on Feymann integrals and have usually found them lacking or simply don't describe the steps to solving them in great detail. I recently came across a reference that I am thoroughly enjoying the scope of how it details the integrals in a wide range of related theories. (warning extremely math intense). If anyone wants a good solid reference I highly recommend this article. Feynman Integrals by Stefan Weinzierl https://arxiv.org/pdf/2201.03593.pdf

Good examination +1

Each topological space is invariant and thus rigid in any of its geometric degrees of freedom. Any variations of say length generates a different topological space. The transformations between topological space will vary between observers. When it comes to amplitudes in the vector space there is no propagation as H=0 within that space \[\Sigma^{d^{n-1}}\] however you will have non trivial tunneling amplitudes between spaces \[\Sigma_0^{d^{n-1}},,, \Sigma_1^{d^{n-1}}\] through an intervening manifold M \[\partial M=\Sigma_0^\ast \cup \Sigma_1\] hope that helps answer that question a simplistic descriptive is that there is no local degrees of freedom (local defined by the space propagation is on the global topography

I don't think you fully understand Swansont's question. All particle creation must obey all conservation laws. That list includes the following. (charge, lepton number, linear and angular momentum, isospin, color, flavor, mass) you must apply all the applicable conservation laws in your examination. You cannot choose one and ignore others. That would be rather tricky to do with your proposal once you consider the experiment has also been done using quantum dot emitters (single quanta aka photons)

You can eliminate any issue with action at a distance by applying the mathematics of the Euler Langrangian.

The reason I am happy with the field excitation view is that I have yet to encounter any form of particle interaction that QFT cannot adequately explain. Things such as electron spin, the distribution of spin statistics in the particle view would require superluminal angular momentum, where as in the field view with a greater effective radius this is easily accounted for. Particles popping in and out of existence is easily described through the creation/annihilation operators. Other factors tricky to describe with particle view where its easily described in the QFT view include Quantum tunneling, Bose and Fermi condensate, electron creation using photons (has been experimentally done) rather interesting in the method. The above is just some examples however even given the above I still feel its wrong to consider fields as being fundamental regardless of its accuracy and range of predictive ability

which term didn't you understand Field: as set of values under a geometric treatment. energy; the ability to perform work potential energy the energy (ability to perform work) due to location. (geometry) example gravitational potential energy at sea level vs top of Mt Everest the graph I posted is the potential energy and how it evolves prior to electroweak symmetry breaking and after electroweak symmetry breaking. Prior to electroweak symmetry breaking elementary particles did not have mass. After symmetry breaking leptons and neutrinos gain mass due to the Higgs field potential. https://en.wikipedia.org/wiki/Higgs_boson perhaps this might help its a very straightforward FAG by Professor Matt Strassler https://profmattstrassler.com/articles-and-posts/the-higgs-particle/the-higgs-faq-2-0/

The bowl your referring to is the potential energy levels that correspond to the vacuum expectation value https://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking equation 2 on this link in a plot lloks like this without the full 3d rotation https://www.wolframalpha.com/input?i=plot+V(\phi)%3D-5|\phi|^2%2B|\phi|^4 the high point is called the false vacuum potential prior to electroweak symmetry breaking. As the potential rolls from the top point to the potential at the bottom either left or right lower this is the current Vacuum expectation value of 246g GeV today that gives rise to the mass term. It is a potential energy graph so nothing is within it

I agree on that at times I see numerous posters on various multimedia preferring no probability or statistics. Unfortunately they take it to such extremes that they refuse any theory that involves probability. May even be as simple as the preference for easy mathematics that are more readily understandable.

While I don't particularly bother with attempting to define a fundamental reality, I do support the view of particles being field excitations. At one time this would not have been the case. I used to be an avid supporter of the particle view. That view gradually changed as I studied QFT and various researches into the subject. However even then I recognize that the field excitation isn't fundamental either. In truth I cannot name anything truly fundamental. The thing is wave particle duality does describe what we observe, depending upon examination the point-like or wavelike characteristics will be involved. Sometimes the point-like is better suited other times the wave-like. The reason I don't feel fields are fundamental is that a field is simply a set of values under a geometric treatment. This describes all forms of fields, including physical vs mathematical. Many often forget that physical has the meaning of any measurable property as one of its numerous definitions. By that definition any measurable quantity can be described as physically real. You often see that argument used in the distinction between real vs virtual particles, a real particle must be measurable and hence have a quanta of action. as far as I can tell that's likely as close to fundamental as will ever be possible. Though we allow for virtual particles many consider them to be more a mathematical convenience than actuality. In QFT they don't typically refer to the term particle but rather a field state so virtual particles are simply field permutations that cannot be localized with well defined boundaries.

As you stated that's to the perspective of the observer at infinity. The infalling observer sees no difference. The flipping of the space vs time marks a point where the mathematics breaks down into a mathematical singularity condition.

I've always been curious as to why so many have issues with probability. If you have a system or state where you have more than one possible outcome it's only natural to model all possible outcomes and give the probability of those possible outcomes. This is true in classical as well as quantum mechanics. So why is probability in quantum mechanics an issue?

Janus did the last post but your welcome lol. Good post Janus well detailed +1

String theory uses path integrals. The difference is the strings of string theory are fully mathematically described. If your serious about trying to define your strings I would recommend you study how String theory does so and then apply mathematics to your String.