Dubbelosix

Yes I have looked into thermal statistics for the model as well. As well as creation and annihilation operators as well. I will take a read of the links before saying anything else. Those papers are interesting, and the last one is helpful for a better investigation into those statistics, which I have looked at in some ... relatively basic forms.

Yes I too have investigated the universe with torsion - today, the torsion is very small to correspond to the small residual rotation left over known as dark flow. I will post up on torsion in my next topic. By the way, the rotation axis is only a conjecture: In the Godel metric, no such axis is distinguishable. In our approach, we assume dark flow is too slow to create a divergence in the average curvature over large distances.

There is nothing wrong, in fact, I have already applied the creation and annihilation operators in some attempts). Certainly, in the diabatic model in my other thread, the creation and annihilation operators will play a role.

New model suggests we don't need dark energy and if that is the case, it may require a modification of the Friedmann equation as an equation of motion. I have provided such an avenue. Certainly not the only one, but an interesting one that does apparently fit data. https://www.sciencealert.com/this-new-model-of-the-universe-needs-no-dark-energy-and-works-just-as-well

I only said general relativity theory lacks it, I didn't mean that it needs it... it may very well need it, one day, but we may not. What was meant is that conservation is not a priori so long as Wheeler de Witts equation is true concerning time. Believe it or not, but there are good reasons to think general relativity will require a definition... one such example is that the scale factor of a universe is explicitly time-dependent. And yes, to have conservation, even in relativity, you need a translation of time. However, just because there is a translation of time, still doesn't mean energy needs to be conserved - that's the troubling thing. All the physics is right in the early universe for short chaotic events. Yes, rotation produces a centrifugal force, an intrinsic one to space time that pushes the dust inside of it in an outwards motion. It's not a real force of course. The great thing about the input and output thing, is that this model is outdated for a self-contained universe - field theory, for instance, in curved spacetimes, may ensure particle production happens in irreversible ways. In this sense, things are changing intrinsically, without any external source to a universe. Carrol argues it in another way, he says if the universes metric varies as it expands, then energy must also change.

I noticed later, that the theory did have formal similarities to the Ekpyrotic theory - except for two crucial points 1) There is no bounce in this theory, because the phase change from the super cool state into the warm radiation vapor stage is an irreversible process in the universe. Extended, this has serious implications, because it would mean in the far-reaching sense, information isn't conserved in the early universe when curvature was dominant. 2) There is no need for other universes, something the Ekpyrotic theory relies on as a mechanism... it is also something inflation theories lead to, which is why Steinhardt, one of the original creators of the theory, is now a vocal opponent. Also we avoid singularities, because of the negative pressure. This was Motz' original motivation for looking at a modified Friedmann equation, which we have only partially taken. We have formulated an entirely modern view of his theory which has been adapted in some crucial ways.

Earlier I didn't have much time, so I rushed in answering.... the rotation decays due to the Hoyle-Narlikar process. They are quite famous for their paper, which shows that rotational properties become exponentially dampened as it contains increasing linear acceleration. Additionally, the equation of state for the Friedmann can already be established. I tend to take there is no conservation - In a Friedmann equation, to explain that there needs to be a third derivative in time, and we also take that the equation of state does not equal zero. It was always an unfounded, but arguably, understandable assumption that energy must remain the same, always in a universe. It may still be true, but we must understand there are processes of exchanging the early primordial bulk energy into the rotational properties of a universe, which [may] explain the vacuum energy discrepancy. It was understandable Friedmann assumed constant energy (because he may have been just aware) of the work by Noether. Later, we find out from general relativity that cosmological conservation is not a priori - and in fact, to do so requires a cosmological definition of time, something general relativity within its framework, lacks. I am leaning towards some very specific paths - 1) That vacuum energy does not contribute to the observable energy of the universe. A cut off of fluctuations on the Planck frequency may provide reasons why their short existences influence the vacuum, so very little. 2) That energy varies in a number of possible ways - such as a non-conservation process of particle production in curved spaces. Interesting things happen when curvature is in the physics - not only does it imply non-conservation for early particle production, Sakharov points out, that in present day quantum field theory, it is assumed that the energy momentum tensor of fluctuations of the vacuum and the corresponding action which are proportional to divergent integral of the forth power over the momenta of virtual particles are actually equal to zero. However, it has been suggested that gravitational interactions could lead to small disturbances in the equilibrium and thus a finite value of the cosmological constant.That means, he is investigating a model which has a dependence of the action of the quantum fluctuations on the curvature of space. 3) Not all of my physics rests on the rotation, because I know that will only reach out to a select few. But I keep it as an interesting alternative to the controversial inflation theories.

The rotation exponentially decays. There are no non-linearities arising today. And the universe is not perfectly homogeneous. Far from it in fact. Godels metric is outdated. There is no expansion in his metric, so you don't start from there. You start from Friedmann's equations. The rotation today is compatible with dark flow. The standard model cannot answer for it in any way. I read the paper you provided the other day... yes it ended up being related to my work in ways.

I was made aware of some research with some results just out: ''Manipulating rubidium atoms with lasers, scientists led by researchers from Italy gave the atoms an upward kick and observed how gravity tugged them down. To compare the acceleration of normal atoms with those in a superposition, the scientists split the atoms into two clouds, put atoms in one cloud into a superposition, and measured how the clouds interacted. These clouds of atoms behave like waves, interfering similarly to merging water waves. The resulting ripples depend on the gravitational acceleration felt by the atoms.The scientists then compared the result of this test to one where both clouds were in a normal energy state. Gravity, the researchers concluded, pulled on atoms in a superposition at the same rate as the others — at least to the level of sensitivity the scientists were able to probe, within 5 parts in 100 million.'' It seems gravity can affect a wave function!!! This is crucial information, as it tells us gravity does tug on the superpositioned states. This proves that gravity could induce the gravitational collapse. https://www.sciencenews.org/article/key-einstein-principle-survives-quantum-test

It was suggested by Arun (et al.) That rotation enters the Friedmann equation like [math](\frac{\ddot{R}}{R})^2 = \frac{8 \pi G}{3}\rho + \omega^2[/math] [1]. see references It's proposed the correct derivation is not only longer, but in this form, should have a sign change for the triple cross product. It is also apparent, no one has offered in their work a derivation other than the one implied through the Godel model. Really what is in implied by the centrifugal term is a triple cross product, is the use of the cross product terms [math]\ddot{R} = \frac{8 \pi G R}{3}\rho + \omega \times (\omega \times R)[/math] Even though the centrifugal force is written with cross products [math]\omega \times (\omega \times R)[/math] it is not impossible to show it in a similar form by using the triple cross product rule [math]a \times (b \times c) = b(a \cdot c) - c(a \cdot b )[/math] Using [math]\omega \cdot R = 0[/math] because of orthogonality we get [math]\ddot{R} = \frac{8 \pi G R}{3}\rho - \omega^2R[/math] which justifies this form of writing it as well. If the last term is the centrifugal acceleration then the acceleration in the two frame is just, while retaining the cross product definition with positive sign, [math]a_i \equiv \frac{d^2R}{dt^2}_i = (\frac{d^2R}{dt^2})_r + \omega^2 \times R[/math] Expanding you can obtain the asbolute acceleration [math]\ddot{R} = (\frac{d}{dt} + \omega \times)(\frac{dR}{dt} + \omega \times R)[/math] [math]= \frac{d^2R}{dt^2} + \omega \times \frac{dR}{dt} + \frac{d\omega}{dt} \times R + \omega \times \frac{dR}{dt}[/math] [math]= \frac{d^2R}{dt^2} + \omega \times \frac{dR}{dt} + \frac{d\omega}{dt} \times R + \omega \times ([\frac{dR}{dt}] + \omega \times R)[/math] [math]= \frac{d^2R}{dt^2}_r + \frac{d\omega}{dt} \times R + 2\omega \times \frac{dR}{dt} + \omega \times (\omega \times R)[/math] Which is the four-component equation of motion which describes the pseudoforces. This gives us an equation of motion, with using [math] \frac{d^2R}{dt^2}_r \rightarrow \frac{8 \pi G R}{3}\rho[/math], [math]\ddot{R}_i = \frac{8 \pi G R}{3}\rho + \frac{d^2R}{dt^2}_r + \frac{d\omega}{dt} \times R + 2\omega \times \frac{dR}{dt} + \omega \times (\omega \times R)[/math] Or simply as [math]\ddot{R}_i = \frac{8 \pi G R}{3}\rho + a_r + \frac{d\omega}{dt} \times R + 2\omega \times \frac{dR}{dt} + \omega \times (\omega \times R)[/math] In the rotating frame we have [math]\ddot{R}_r = \frac{8 \pi G R}{3}\rho + a_i - \frac{d\omega}{dt} \times R - 2\omega \times \frac{dR}{dt} - \omega \times (\omega \times R)[/math] Since inertial systems are only a local approximation, the inertial frame of reference here may been seen to go to zero leaving us the general equation of motion for a universe [math]\ddot{R} = \frac{8 \pi G R}{3}\rho - \frac{d\omega}{dt} \times R - 2\omega \times \frac{dR}{dt} - \omega \times (\omega \times R)[/math] Using the three standard identities ~ [math]a_{eul} = -\frac{d\omega}{dt} \times R[/math] [math]a_{cor} = -2 \omega \times \frac{dR}{dt}[/math] [math]a_{cent}= -\omega \times (\omega \times R)[/math] we get [math]\ddot{R} = \frac{8 \pi GR}{3}\rho + a_{eul} + a_{cor} + a_{cen}[/math] The concept of rotation in a universe was initially explored by Godel, but we have made some progress since his day and his simple non-expanding metric. The idea the universe has a rotation was also explored by Hawking who admitted they very well could be the kind of model we associate to reality, except the rotation has to be remarkably slow. In the discovery of dark flow, it seems this could be the perfect candidate of a residual primordial rotation. It was proven by Hoyle and Narlikar that any primordial rotation would exponentially decay in an expanding universe. This is an important realization to understand how rotation in the primordial stages was allowed to be large and decayed as the scale factor of a universe increases, leaving behind presumably, something like dark flow in the observable motion of all the systems in the universe - hopefully this will be supported with further mapping of dark flow over larger quantities of systems. Even though technically speaking, the axis could not be discernible in the Godel universe, it becomes a non-problem in a late universe which experiences a decay in rotation. Arguably, dark flow is way too slow to discern any axis of evil. For exact values on how rotation mimics dark energy (or expansion energy) please read the link below. The rotation will give rise to the classical centrifugal force, arising in a universe and pushing systems away. A good question that puzzled me for a while is, if rotation has in fact almost decayed, why is the universe still speeding up in expansion? I have come to realize the universe is exponentially many times the size it is today, so we must reconcile the tug of gravity has been overwhelmed by the acceleration of the universe, which just like Newtons law of inertia, will continue to expand, or continue to accelerate, unless hindered by something. Dark energy, if the substance exists, is believed only to become significant when a universe gets large enough. This means dark energy does not explain how a dense universe was capable of expanding out of the Planck era. Rotation does explain this, very easily and may offer solutions in which vacuum energy can be stolen by the rotational property of the universe - I call the latter, a Bulk to Rotation process, similar to Bulk to Horizon energy transfers in cosmological models. Rotation also explains chirality oreference and can explain why the universe has a ''handedness'' that may be directly related to the antimatter problem. references [1].https://www.researchgate.net/publication/51956326_Primordial_Rotation_of_the_Universe_and_Angular_Momentum_of_a_wide_rangeof_Celestial_Objects https://en.wikipedia.org/wiki/Centrifugal_force

In a universe, a realistic total entropy should be given as [math]dS = dS_{rev} + dS_{irr}[/math] That is, it consists of two parts, one that is the reversible entropy in a universe and the irreversible entropy. Now, if [math]V[/math] is the volume of a sphere, the rate of change of the volume is [math]\frac{dV}{dt} = V(3 \frac{\dot{R}}{R})[/math] The term in the paranthesis is known as the fluid expansion [math]\Theta = 3(\frac{\dot{R}}{R})[/math] The rate of change of its internal energy would satisfy [math]\frac{d}{dt}(\rho V) = \dot{\rho}V + \rho \dot{V} = (\dot{\rho} + 3 \frac{\dot{R}}{R}\rho)V[/math] If the energy density is replaced with the particle number [math]N[/math], you get back the particle production rate [math]\Gamma[/math] [math]\frac{d}{dt}(N V) = \dot{N}V + N \dot{V} = (\dot{N} + 3 \frac{\dot{R}}{R}N)V = N V \Gamma[/math] which is useful to know, because this quantity is a version of the continuity equation, except in the form when irreversible dynamics are involved, leads to a theory that is diabatic in nature. The continuity equation is just [math]\dot{\rho} = (\rho + 3P)\frac{\dot{R}}{R}[/math] It looks messy, but that particle production equation can be simplified, making use of the particle number density [math]\frac{N}{V} = n[/math] as well [math]\dot{n} + 3 \frac{\dot{R}}{R}n = \dot{n} + n\Theta = n \Gamma[/math] The reversible and irreversible parts can be written in terms of the first law of thermodynamics [math]\frac{d}{dt}(\rho V) + P \frac{dV}{dt} = (\frac{dQ}{dt})_{rev} + (\frac{\rho + P}{n} \frac{d}{dt}(nV))_{irr}[/math] *Note, also from this last equation, you could rewrite a Friedmann equation involving an extra term describing the reversible dynamics. Replacing terms we have uncovered for diabatic particle creation we get (and dropping the unecessary reversible and irreversible notation), [math]\frac{d}{dt}(\rho V) + P \frac{dV}{dt} = (\frac{dQ}{dt}) + (\frac{\rho + P}{n})nV \Gamma[/math] Now, let's compare this with earlier work, with a modified Friedmann equation, of various forms. One such for I worked on was [math]\frac{\dot{R}}{R}(\frac{\ddot{R}}{R} + \frac{kc^2}{a}) = \frac{8 \pi G}{3}(\frac{\rho + P}{n})n\frac{\dot{R}}{R}[/math] If we allow the time derivative on the ''almost'' fluid expansion coefficient, it looks identical to the diabatic universe. What you would not have known from the previous equations though, is that this required the third derivative in time, considered as the derivative which leads to non-conservation in the expanding Friedmann universe. The rules have not changed since the original work in my Friedmann model, as it turns out, you can still by definition introduce the heat per unit particle, which would change the thermodynamic law further to suit a Gibbs equation. Heat per unit particle is [math]dq = \frac{dQ}{dN}[/math] which changes the law into [math]d(\frac{\rho}{n}) = dq - qd(\frac{1}{n})[/math] This lead to a Friedmann equation of the form [math]\frac{\dot{R}}{R}(\frac{\ddot{R}}{R} + \frac{kc^2}{a}) = \frac{8 \pi G}{3}[(\frac{\rho}{n}) + 3P(\frac{1}{n})]\dot{n}[/math] What we have learned from the following equation: [math]\frac{d}{dt}(\rho V) + P \frac{dV}{dt} = (\frac{dQ}{dt}) + (\frac{\rho + P}{n})nV \Gamma[/math] Is that it fits the general equation of state one would look for to describe the effective density in diabatic universes. Adiabatic universes, for the particle production equation, would satisfy [math]\dot{n} + \Theta n = 0[/math] Diabatic models satisfy [math]\dot{n} + \Theta n = n \Gamma[/math]. Based on the new perspective of reversible and irreversible dynamics, the Friedmann equation is now [math]\frac{\dot{R}}{R}(\frac{\ddot{R}}{R} + \frac{kc^2}{a}) = \frac{8 \pi G}{3}(\dot{\mathbf{q}}_{rev} + [(\frac{\rho}{n}) + 3P_{irr}(\frac{1}{n})]n\Gamma)[/math] I argue that irreversible dynamics have gone on in the universe - it is possible to view the universe in a pre-big bang model. So the reversible and irreversible investigation is actually part of a much larger investigation into my cosmology studies, which was related to a richer plethora of ideas that circled around a universal primordial rotation term attached to a Friedmann equation. Those early equations are still the basis of my arguments to explain dark energy and even dark flow, including several other topics of interest, such as an interesting relationship to the antimatter problem. In a later post, I will show that work that attempts to view the universe with the all-important Poincare symmetry: This work is still related to those previous investigations, because we deal with a different kind of de Sitter spacetime, a type only I can classify as a pseudo de Sitter spacetime, but what makes it different? Well, a third in time derivative Friedmann equation does not conserve the energy of an expanding universe. In this work, we adopt a solution from Motz and Kraft for a reversible isothermal Gibbs-Helmholtz phase change of an all matter liquid degenerate gas into a radiation vapor. It is assumed in our model, that a realistic phase change involves neither reversible or isothermal phase changes. To help explain a diabatic anisothermal phase change from some super cool pre-big bang phase, we introduce a Friedmann equation which has been rewritten in the style of a Gibbs equation - this specific equation can be argued in a number of different ways: The basic way to view it is that the Friedmann equation is related to the entropy of a universe insomuch that it consists of two parts, a reversible and irreversible particle creation dynamics. [math]\frac{\dot{R}}{R}(\frac{\ddot{R}}{R} + \frac{kc^2}{a}) = \frac{8 \pi G}{3}(\dot{\mathbf{q}}_{rev} + [(\frac{\rho}{n}) + 3P_{irr}(\frac{1}{n})]n\Gamma) = \mathbf{k}nT \dot{S}[/math] where [math]S[/math] has dimensions of [math]k_B[/math] and [math]\mathbf{k}[/math] is the Einstein factor. in which [math]nk_B T \dot{S} = \dot{\rho} + (\frac{\rho + P}{n})n \frac{\dot{T}}{T}[/math] which also justifies the following form as a fully thermodynamic interpretation: [math]\frac{\dot{R}}{R}(\frac{\ddot{R}}{R} + \frac{kc^2}{a}) = \frac{8 \pi G}{3}(\dot{\mathbf{q}}_{rev} + [(\frac{\rho}{n}) + 3P_{irr}(\frac{1}{n})]n\frac{\dot{T}}{T}) = nk_B T \dot{S}[/math] [math]P_{irr}[/math] is known as the irreversible pressure, and inside of it, we can talk about the Gibbs-Helmholtz free energy equation for an irreversible phase change from a liquid particle creation phase to vapor for some infinitesimal change in volume, [math](P_{irr}(\frac{1}{n}))\dot{n} = -(\frac{1}{4 \pi R^2}\frac{dU_2}{dR})\frac{\dot{n}}{n} = -(\frac{dU_2}{dV}(\frac{1}{n}))n\Gamma[/math] Where, [math]n[/math] is the particle number density. The pressure is irreversible because particle production through the phases has happened in this case, in an irreversible way in which we take [math]n\Gamma \ne 0[/math]. The irreversibility of particle creation is expected to happen when the term [math]\frac{kc^2}{a}[/math] is large, implying a large cosmological curvature which would have been present during these phase changes; it is a result of particle creation in curved spacetimes that yields possibilities for non-conservation in the universe. Any reversible dynamics [math]\dot{\mathbf{q}}_{rev}[/math] in the universe I speculate as probably a post big bang phenomenon, whereas I expect this conversion from the two phase states as related to chaotic and irreversible dynamics. In this model, the universe is expected to have a cold dominated era (the pre big bang phase) as a degenerate gas of particles in a very high condensed gravity-dominated region with little thermodynamic freedom. Some collapse underwent in the cold dominated region leading to the radiation vapor phase - which can be thought of as the heating of the universe leading to a big bang scenario. We live inexorably, in this vapor phase of the universe. ref. https://www.researchgate.net/publication/286512912_Gravitationally_induced_adiabatic_particle_production_From_Big_Bang_to_de_Sitter https://www.researchgate.net/publication/283986394_General_form_of_entropy_on_the_horizon_of_the_universe_in_entropic_cosmology [notes] Now that we have a definition of the entropy, we can argue the reversible and irreversibility in another type of form, the Clausius entropy production equation will provide such a possible form: [math]N = S - S_0 - \int \frac{dQ}{k_BT}[/math] When [math]N=0[/math], it means a thermodynamic process was reversible and [math]n > 0[/math] for some irreversible process where [math]S[/math] is the final state and [math]S_0[/math] is the initial state of entropy. So yes, it would be possible to write a Freidmann equation to satisfy this equation as well. In fact, this is doing the same thing as our initial approach in a way as it is measuring the irreversible dynamics, if there is any. In the previous form, we looked at the entropy per unit particle and the heat per unit particle. It can be written as [math]\dot{S} = \sum_k \frac{\dot{Q}_k}{T_k} + \sum_k \dot{S}_k + \sum \dot{S}_{ik}[/math] with [math]\dot{S}_{ik} \leq 0[/math] we can see how this is formally similar to how we treat the reversible and irreversible dynamics separately within the same equation. If any terms in the last equation have a subscript of ''i'' indicates it is an irreversible process, so it can be written in the following way to express which parts are reversible and which are not [math]\dot{S} = (\sum_k \frac{\dot{Q}_k}{T_k} + \sum_k \dot{S}_k)_{rev} + (\sum \dot{S}_{ik})_{irr}[/math] Again, the structure of the equation reveals a global entropy must consist of reversible and irreversible dynamics. Irreversible dynamics occur in strongly curved spacetimes. https://en.wikipedia.org/wiki/Entropy_production https://www.researchgate.net/publication/1922810_Thermodynamics_of_Friedmann_Equation_and_Masslike_Function_in_General_Braneworld

Yes I know Clifford and most useful lie algebra. Of course, only used Clifford in respect to Dirac equation. As for the interest in the cosmological constant, I too have an interest in it, as I have been studying the Freidmann equation for a while now. I could certainly look at the Dirac equation in terms of the spacetime uncertainty but its not on my first things to do. I appreciate that you have taken interest in the theory - the problem in taking my model to cosmological interpretations of the cosmological constant is for a full understanding of fluctuations and how they contribute to vacuum energy, if at all. For instance, we may find the fluctuations exist, but exist for such a long time their effects are negligible over the vast distance of spacetime. That would certainly seem the case when we try and measure the vacuum energy - the cosmological constant is something like [math]10^{122}[/math] magnitudes too small. I have came to different conclusions for the vacuum energy problem. I may write up on that here soon.

I have noted in the past, if a vacuum is not truly Newtonian and it does indeed expand (new space appearing) then there will be new fluctuations added to spacetime as well. Fluctuations can also act as the seeds of the universe to explain a primordial gravitational clumping, giving rise to the large scale structure, albeit, this uses the notion of some rapid expansion phase. We too have the same phase characterized by the centrifugal force the universe experienced when it was very young from a furiously fast spin. In fact, Wald and Harren have shown it is possible to retrieve the cosmic seeds without inflation. In their model the inhomogeneities of the universe arises while in the radiation phase – their model also requires that all fluctuation modes would have been in their ground state and that the fluctuations are “born” in the ground state at an appropriate time which is early enough so that their physical length is very small compared to the Hubble radius, in which case, they can “freeze out” when these two lengths become equal. It has been noted in literature that there is clearly a need for some process that would be responsible for the so called “birth” of the fluctuations. I have a mechanism in my own model, which we will discuss at the end - today I want to show how you can talk about fluctuations within the context of expanding space, which is required within a sensible approach to unify the cosmic seeds with the dynamics of spacetime. It is possible to construct a form of the Friedmann equation with what is called the Sakharov fluctuation term, which is the modes of the zero point fluctuations [math]m\dot{R}^2 + 2\hbar c R \int k dk = \frac{8 \pi GmR^2}{3}\rho[/math] When [math]R \approx 0[/math](but not pointlike) then the fluctuations are in their ground state. Though inflation is not required to explain the cosmic seeding, there are alternatives themselves to cosmic inflation such as one particular subject I have investigated with a passion; rotation can mimic dark energy perfectly which is thought to explain the expansion and perhaps even acceleration (if such a thing exists). It is possible to expand the Langrangian of the zero point modes on the background spacetime curvatuture in a power series [math]\mathcal{L} = \hbar c R \int k dk... + \hbar c R^n \int \frac{dk}{k^{n-1}} + C[/math] Where C is a renormalizing constant which is set to zero for flat space. It had been believed at one point that the forth power over the momenta of the particles would be zero [math]\hbar c \int k dk^3 = 0[/math] But interesting things happen in the curvature of spacetime, such a condition doesn't need to hold.The anisotropies may arise in an interesting way when I refer back to equations I investigated in the rotating model. An equation of state with thermodynamic definition can be given as: [math]T k_B \dot{S} = \frac{\dot{\rho}}{n} + \frac{\rho + P}{n}\frac{\dot{T}}{T}[/math] The last term [math]\frac{\dot{T}}{T}[/math] calculates the temperature variations that arise, even in the presence of the cosmic seed and we can therefore change the effective density coefficient in the following way: [math]2m\dot{R}\ddot{R} + 2\hbar c R \int k \dot{k} = \frac{8 \pi GmR^2}{3}(\rho + \frac{3P}{c^2})\frac{\dot{T}}{T}[/math] Simplifying a bit and rearranging [math]\frac{\dot{R}}{R}\frac{\ddot{R}}{R} + \frac{\hbar c}{mR} \int k \dot{k} = \frac{8 \pi G}{6}(\rho + \frac{3P}{c^2})\frac{\dot{T}}{T}[/math] ref http://sci-hub.bz/10.1007/BF00637768 Also, read Sivaram and Arun

I do feel I have more to do. I have looked at specific parts of the theory for starters - clearly most of the ground work has been done on this hypothesis. I want to leave some work for other people I have come across some interesting questions, like early doubts because time cannot be treated like an operator, and of course, the time enters like an operator in this theory - but I have had to be clear, the spacetime uncertainty while though well named because of the structure of the inequality, is also misnomer, to call it a modification of the usual uncertainty principle [math]\Delta E \Delta t[/math]. This still troubles me. Time isn't an observable, so hopefully you will understand my discontent. What about coordinate independence? This has also troubled me. The interpretation of classical to quantum gravity is bothering me - I also want to take this theory into entanglement space. See if the two can be conjoined: I have already sought after a theory with intrinsic spacetime uncertainty. Just some basic things I have considered. To ask about the Dirac equation, is a bit more difficult, but not impossible. Just requires more reading and a good brain. I certainly think its possible to reformulate a Dirac equation in terms of the spacetime uncertainty principle, since the latter here is just a reinterpretation of the usual energy-time relationship for scattered particles. The interpretation in Planck spacetime is important as well. I wondered whether there are quantum corrections that lead to a quantum redshift [math]\Delta L \Delta t g_{00} \geq \frac{G \hbar}{c^4} = \frac{L^2_P}{c}[/math] This equation involves the dimensionless time-time component of the metric. A metric notation is also [tex]ds[/tex] and we will see use of it later.The Planck length is [math]\sqrt{\frac{G\hbar}{c^3}}[/math] so we clearly have a squared Plankian structure to the spacetime metric. The non-commutation is expressed on the LHS. A formally similar type of spacetime uncertainty is linked in string theory as well which predicts a spacetime uncertainty linked to the Planck scale. A simple rearranging of the metric term, and a second order expansion leads to: [math]\Delta L \Delta t \geq \frac{(\frac{G \hbar}{c^4})}{[\frac{\phi_1}{c^2} - \frac{\phi_2}{c^2}]} = (\frac{\Delta \phi}{c^2})^{-1}\frac{L^2_P}{c}[/math] The expansion is analogous to gravitational corrections of the redshift value. In a way, this equation already suggests a link between some spacetime uncertainty relationship to the Planck structure and redshift of the system. The next form of the equation is solved for the uncertainty in a single length, but also makes use of [math]\Delta s = \sqrt{g_{00} c^2 \Delta t^2 + g_{xx} \Delta x^2}[/math] which is a metric term, like we discussed not long ago and is actually just relativity's way to discuss the square of a spacetime interval. [math]min[\Delta L] \geq \frac{(\frac{G \hbar}{c^3})}{g_{00}c \Delta t} = \frac{(\frac{G \hbar}{c^3})}{\sqrt{\Delta s^2 - g_{xx}\Delta x^2}} [/math] Basically we have some Planck structure weighted by a spacetime interval which depends on uncertainties in the metric. The result on the LHS may in a way, be interpreted to mean even intervals on the Planck length can jiggle due to uncertainty. note You can expand the gravitational field further [math]1 + \phi_1 - \phi_2 - \frac{1}{2}\phi^2_1 - \phi_1 \phi_2 + \frac{3}{2}\phi^2_2 + ....[/math] For higher corrections to the red shift.

Can you expand on what you saying a bit? You mean a Hilbert space in which [math]\mathcal{H} = \mathcal{H}^0 + \mathcal{H}^1[/math] In which you have a 1 particle system in L^2 space, right? Right, I am catching on a bit now... there are ways to do this. I remember reading about bounds in the equations. It's all theoretical of course. Even renormalization, though infinities I usually take as a breakdown in the theory.

So, I have had some time to think about it. So you have given me a bit of time to think about it, and I have come to see it ultimately depends on the convergence implied from the collapse of the system. I'm dealing in [math]L^2[/math] space and the Hilbert space is [math]\mathcal{H} = L^2(\mathcal{R})[/math]. In [math]L^2[/math] space, you can have many seqences but converges on [math]\int_{-\infty}^{\infty} |a_i|^2 < \infty[/math] The collapse of the system depends on the collapse time. It has been suggested that Penroses collapse time is too long! A quicker collapse time model is needed, so why not things happening at plank time? Wheeler suggested quantum foam would occur at a very tiny distance of the Planck scale [math]10^{-33} cm[/math], so it wouldn't actually be implausibe to assume any zero point fluctuations must have a cutoff corresponding to the Planck frequency [math]10^{43} Hz[/math]. Neverthless, the collapse shows you get a finite number, so this is key to the renormalization of this specific theory. That does imply large densities - you might ask why we can't see those large densities and the truth be told, it may all boil down to the lifetime of the fluctuation itself - the short time implies it has only moved a vanishingly small amount and so never becomes anything that is generally visible, at least not with current technology - though, if my spacetime theory of the vacuum is true, that is, that there is an intrinsic spacetime uncertainty that gives rise to particles, it should be possible to create a device to probe and observe spacetime to create those fluctuations. But then you would come to the chicken and egg question - are you creating those fluctuations by looking into the small regions of spacetime, or are they already there?

It's a good question. I don't really have an answer (yet). Let me work on it.

Without giving you a lecture on the dirac notation, because I am sure you know what that means, in the context we have used it in is [math]<\psi|\psi> = \sum_i |c_i|^2 = 1[/math] where [math]|c_i|^2= c*_ic_i[/math] are composed of probability amplitudes, and so the probability to find something in a particular state is given by the mean of the square of the wave function, also known as a quantum collapse. This above, is also known as an expectation value in Hilbert space when acting on an operator, which is the route I investigated. You ask about repeated indices, the rule is that any repeated indices cancel.

You mean this? [a,b]? This is notation for commutation. It asks whether the process of multiplying in a binary way, two variables is commutative. The notation [a,b] is simply in this context, (ab-ba). Classically speaking, many things never commuted anyway. It wasn't just a special feature of quantum mechanics. When you do speak about commutation and Von Neumann algebra, you end up dealing in what we call the phase space.

Abstract I explore the anticommutating spacetime relation in context of gravity by seeing commutation happen in two connections of the gravitational field, one concerned with space, the other time. We learn nothing revolutionary this time around, but we do explore it in a finite dimensional, Hilbert space-context. I do offer though, in a new context, an equation proposed by Anandan, which can describe the difference of geometries directly related to the [math]L^2[/math] space we explore as a Cauchy Schwarz spacetime. My ultimate hope is that it will catch on that the latter gives a natural mechanism for fluctuations in spacetime, as we relate fluctuations to the energy time relationship [math]\Delta E \Delta t[/math]. The commutation relationship known as the spacetime uncertainty is established to satisfy a direct interpretation into the antisymmetric tensor, [math]R_{\mu,\nu} = [\nabla_x, \nabla_0] = \nabla_x \nabla_0 - \nabla_0 \nabla_x \geq \frac{1}{\ell^2}[/math] That is, a space [math]x[/math] and [math]0[/math] (time) notation. I worked out the Christoffel symbols and I calculate in the normal way as two connections: [math][\nabla_i,\nabla_j] = (\partial_i + \Gamma_i)(\partial_j + \Gamma_j) - (\partial_j + \Gamma_j)(\partial_i + \Gamma_i)[/math] [math]= (\partial_i \partial_j + \Gamma_i \partial_j + \partial_i \Gamma_j + \Gamma_i \Gamma_j) - (\partial_j \partial_i + \partial_j \Gamma_i + \Gamma_j \partial_i + \Gamma_j \Gamma_i)[/math] [math]= -[\partial_j, \Gamma_i] + [\partial_i, \Gamma_j] + [\Gamma_i, \Gamma_j][/math] From here, we reinterpreted this in terms of the Cauchy Schwarz inequality to give spacetime an ''instrinsic relationship'' to the uncertainty principle - which may serve as an origin to fluctuations in spacetime - at least this was my motivation - I later discover through more investigation this makes it part of [math]L^2[/math] space and thus, a finite dimensional Hilbert space. The expectation of the uncertainty is the mean deviation of curvature in the system is: [math]\sqrt{|<\nabla_i^2>< \nabla_j^2>|} \geq \frac{1}{2} i(< \psi|\nabla_i\nabla_j|\psi > + <\psi|\nabla_j\nabla_i|\psi>) = \frac{1}{2} <\psi|[\nabla_i,\nabla_j]|\psi> = \frac{1}{2} <\psi | R_{ij}| \psi > [/math] [math] = \frac{1}{2} < \psi |- [\partial_j, \Gamma_i] + [\partial_i, \Gamma_j] + [\Gamma_i, \Gamma_j]| \psi >[/math] I also speculate in terms of the spacetime uncertainty, anticommutation will exist in the Bianchi identities. The Bianchi identity is true up two three Cyclic Christoffel symbols: [math]R_{\sigma \rho [i j]}g^{\sigma \rho} = \partial_i \Gamma_{j} - \partial_j \Gamma_{i} + \Gamma_{i} \Gamma_{j} - \Gamma_{j} \Gamma_{i}[/math] [math]R_{\sigma i[j \rho]}g^{\sigma i} = \partial_j \Gamma_{\rho} - \partial_{\rho} \Gamma_{j} + \Gamma_{j} \Gamma_{\rho} - \Gamma_{\rho} \Gamma_{j}[/math] [math]R_{\sigma j [\rho i]}g^{\sigma j} = \partial_{\rho} \Gamma_{i} - \partial_i \Gamma_{\rho} + \Gamma_{\rho} \Gamma_{i} - \Gamma_{i} \Gamma_{\rho}[/math] You can write these three relationships out in the Bianchi identity, we can write the commutation again, on the indices [math]R_{\sigma \rho[ i j]} + R_{\sigma i [j \rho]} + R_{\sigma j [\rho i]} = 0[/math] Again, the last two indices reveal antisymmetric properties. I worked out a static model for superpositionng will not satisfy the fundamental spacetime relationship! Using J. Anandan's equation which I investigated: [math]E = \frac{k}{G} \Delta \Gamma^2[/math] I noted the equation confused me early on, but it seems it is constructed in the following way [math]<\Delta \Gamma^2> = \sum <\psi| (\Gamma^{\rho}_{ij} - <\psi |\Gamma^{\rho}_{ij}| \psi>)^2|\psi >[/math] I think I realized what was implied by Anandans first equation by noticing his missing constant of proportionality is [math]c^4[/math]. Then an integral of the volume yields the energy [math]E = \frac{c^4}{G} \int \Delta \Gamma^2\ dV[/math] We have argued, that the squared component of the connection can be interpreted in terms of the curvature tensor in Anandan's equation. This is related to the energy of the difference of geometries and that is given now as [math]\Delta E = \frac{c^4}{8 \pi G} \int < \Delta R_{ij}>\ dV =\frac{c^4}{8 \pi G} \int <\psi|(R_{ij} - <\psi |R_{ij}| \psi>)|\psi>\ dV[/math] This is actually related to the difference found in Penrose's model of an induced gravitational collapse in a superpositioned system - albiet, ours is quantum geometry related directly to the Riemann tensor. You may have noticed, the energy equation that describes the difference in superpositioned geometry ~ [math]\Delta E = \frac{c^4}{8 \pi G} \int < \Delta R_{ij}>\ dV =\frac{c^4}{8 \pi G} \int <\psi|(R_{ij} - <\psi |R_{ij}| \psi>)|\psi>\ dV[/math] Shares the difference between two expectation values of the system: [math]\sqrt{|<\nabla_i^2>< \nabla_j^2>|} \geq \frac{1}{2} i(< \psi|\nabla_i\nabla_j|\psi > + <\psi|\nabla_j\nabla_i|\psi>) = \frac{1}{2} <\psi|[\nabla_i,\nabla_j]|\psi> = \frac{1}{2} <\psi | R_{ij}| \psi > [/math] That coefficient of [math]\frac{1}{2}[/math] may indeed attach to that energy, just like a kinetic energy term. So really, when you saw this object: [math]<\psi |R_{ij}| \psi>[/math] as we have shown, we had already calculated this identity very early on in the work. So the energy equation is compatible in a Cauchy-Schwarz interpretation of spacetime. How do you vary the expectation value in the equation? [math]<\Delta E> = \frac{c^4}{8 \pi G} \int <\Delta R_{ij}>\ dV =\frac{c^4}{8 \pi G} \int (<\psi|R_{ij}| - <\psi| R_{ij}| \psi>)|\psi>\ dV[/math] The total variation will split each two terms, [math]<\psi|R_{ij}| - <\psi| R_{ij}| \psi>)|\psi>[/math] into [math]<\delta_{a} R_{ij}> = <\psi| R_{ij}| \delta_{a}\psi> + <\delta_{a} \psi |R_{ij}|\psi>[/math] [math]<\delta_{b} R_{ij}> = <\psi| R_{ij}| \delta_{b}\psi> + <\delta_{b} \psi |R_{ij}|\psi>[/math] where the subscript of [math]\delta_{ab}[/math] denotes a ''two particle system.'' So it will become a four-component equation with variations in each term of the wave function. In the main work, we also discussed shortly, my model being related to Penrose's model for the collapse of gravity in a superpositioned state. Penrose has suggested a graviational self energy related to a collapse time model [math]T \approx \frac{\hbar}{E}[/math] And in the Penrose model, the energy is given as [math]E = \frac{1}{4 \pi G} \int (\nabla \phi' - \nabla \phi)d^3x[/math] We can derive a more general case that can be used to measure the density variations of spacetime. Deriving the gravitational binding between any coherent gravitational superpositioning state can be given the following way: The gravitational field inside a radius [math]r = r(0)[/math] is given as [math]\frac{dM}{dR} = 4 \pi \rho R^2[/math] and the total mass is [math]M_{total} = \int 4 \pi\rho R^2 dR[/math] and so can be understood in terms of energy (where [math]g_{tt}[/math] is the time-time component of the metric), [math]\mathbf{M} = 4 \pi \int \frac{\rho R^2}{g_{tt}} dR = 4 \pi \int \frac{ \rho R^2}{(1 - \frac{R}{r})} dR[/math] The difference of those two mass formula is known as the gravitational binding energy: [math]\Delta M = 4 \pi \int \rho R^2(1 - \frac{1}{(1 - \frac{R}{r})}) dR[/math] Distribute c^2 and divide off the volume we get: [math]\bar{\rho} = \rho c^2 - \frac{ \rho c^2}{(1 - \frac{R}{r})}[/math] Were we have used a notation [math]\bar{\rho}[/math] for the energy density. Fundamentally, the equations are the same, just written differently. Notice that [math]\nabla^2 \phi = 4 \pi G \rho[/math] from Poisson's formula, in which we notice the same terms entering [math]\Delta M = 4 \pi \int \rho R^2(1 - \frac{1}{(1 - \frac{R}{r})}) dR[/math] [math]E = \frac{1}{4 \pi G} \int (\nabla \phi' - \nabla \phi)d^3x[/math] So while Penrose suggests calculating the binding energy directly from the gravitational potential [math]\phi[/math] there are ways as shown here, to think about it in terms of the gravitational energy density and the gravitational binding between the two. http://sci-hub.bz/10.1007/BF02105068 Note* It is also possible to write a version of Anandan's equation like the following [math]E = \frac{c^4}{G} \int (\nabla \Gamma)^2\ dV = \frac{c^4}{G} \int \frac{1}{R^2} \frac{d\phi}{dR} (R^2 \frac{d\phi}{dR})\ dV[/math] This part [math]\frac{1}{R^2} \frac{d\phi}{dR} (R^2 \frac{d\phi}{dR})[/math] Is just another way to write a squared product [math]\frac{d\phi}{dR} \cdot \frac{d\phi}{dR}[/math]. And of course, this is just [math]\nabla^2 \phi^2[/math]. We've stated this identity before in an equation - note also, [math]\phi[/math] is dimensionless.