Mordred

There's no jumping to conclusion I read your documents you dont have anything relating to a reference frame. That requires geometry. How else do you describe kinematic motion relative to an observer or to multiple observers. How do you relate the angles of one observer motion relative to another. Try for example an emitter in transverse motion to the observer. Unless Im mistaken every single equation you have outputs a scalar value. You don't have any vector addition rules with regards to distance and angle of travel. Relativity involves more than just scalar ratios. So tell me without any geometry how do you apply a Galilean or even a Lorentz transformation between multiple events ? You and I also have difference of opinion of a conserved system. Freefall is a conserved state. There is no external influence such as force acting upon the object in motion. Yet planetary orbits is not a conserved system you have change in direction aka acceleration. In GR this requires the transformation matrix. You dont have one so how do you translate the freefall state to one of acceleration and stay conserved ? A boost ( change in velocity under the Minkowskii metric is just a type of rotation ). How do you relate an observer measuring kinematic motion of that orbiting body without geometry to equate an angle of view ? Aside from the statement closure whats your mathematical proof of closure ? You describe orthogonal projections but in the same breath state there is no geometry yet an orthogonal projection is 90 degrees relative to the axis its projecting from classical example x axis is orthogonal to the y axis. I dont care if your manifolds involve spacetime. Thats not a requirement of a manifold it doesn't even require spatial coordinates if a manifold only requires one parameter to uniquely identify each point that's a 1 d manifold. If the manifold requires 2 or more parameters to uniquely identify each point. The number of parameter required is the dimensionality of that manifold. It doesn't require any coordinate basis the number of required parameters or dimension is the number of effective degrees of freedom. With regards to boosts in Lorentz for the benefit of other readers here's a listing Lorentz group Lorentz transformations list spherical coordinates (rotation along the z axis through an angle ) \[\theta\] \[(x^0,x^1,x^2,x^3)=(ct,r,\theta\phi)\] \[(x_0,x_1,x_2,x_3)=(-ct,r,r^2,\theta,[r^2\sin^2\theta]\phi)\] \[\acute{x}=x\cos\theta+y\sin\theta,,,\acute{y}=-x\sin\theta+y \cos\theta\] \[\Lambda^\mu_\nu=\begin{pmatrix}1&0&0&0\\0&\cos\theta&\sin\theta&0\\0&\sin\theta&\cos\theta&0\\0&0&0&1\end{pmatrix}\] generator along z axis \[k_z=\frac{1\partial\phi}{i\partial\phi}|_{\phi=0}\] generator of boost along x axis:: \[k_x=\frac{1\partial\phi}{i\partial\phi}|_{\phi=0}=-i\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0 \end{pmatrix}\] boost along y axis\ \[k_y=-i\begin{pmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0 \end{pmatrix}\] generator of boost along z direction \[k_z=-i\begin{pmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0 \end{pmatrix}\] the above is the generator of boosts below is the generator of rotations. \[J_z=\frac{1\partial\Lambda}{i\partial\theta}|_{\theta=0}\] \[J_x=-i\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&1\\0&0&-1&0 \end{pmatrix}\] \[J_y=-i\begin{pmatrix}0&0&0&0\\0&0&0&-1\\0&0&1&0\\0&0&0&0 \end{pmatrix}\] \[J_z=-i\begin{pmatrix}0&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&0 \end{pmatrix}\] they obey commutations \[[A,B]=AB-BA\] Does your work do anything to replace the above ? The above applies for the Minkowskii metric essentially SR.

You have parameters that do vary by definition you have effective degrees of freedom. They may or may be independent degrees of freedom. You also have trigonometric relations between your effective degrees of freedom within your article. So direction is inherent in your S^2 manifold regardless of what parameters you use to determine each unique point on said 2d manifold. As mentioned I was curious as to how you would answer. The title of your thread specifically states "Simplifying SR and GR" yet I don't anything relating to observer effects and what different observers will see or measure. Relative motion from one frame of reference to another etc. I find that curious as well evidently its not in the scope of the work .

Edit my memory was way off deuterium corresponds to 7.2 ×10 ^8 Kelvin for deuterium production.

The formulas the calculator uses is on www.physicsforum.com under their insight article section if you go through the links you can get which formulas are employed for each column. I still use it myself regularly as a cross reference and we continually test its accuracy for the particular datasets selected though it does allow a bit of additional adjustments outside of any particular dataset. I should not in your article posted here the Saha equations give a range for example at 6000 kelvin you have 25% the neutral hydrogen at 3000 kelvin its roughly 75% and at 4000 kelvin its roughly 50%. Deuterium is roughly 4500 kelvin if I recall for 75%. I would have to check later on. This pertains to your constraint mentioned in the article at 4000 kelvin. I also question your statement of absolute coordinate time. Please explain as coordinate time is relative to the observer it isnt proper time.

The problem can easily get compounded when using AI to study physics. One example is having AI look up a specific equation. One example being if you were to look for relations specific to a metric or methodology. AI could very well return a relation specific to say a canonical treatment as opposed to a conformal treatment. With the FLRW metric it often confuses conformal coordinates as opposed to commoving coordinates. If the AI user isnt aware of these distinctions to recognize the AI mistakes they could easily get confused as well as get frustrated when they try to apply those equations. As AI looks through literature Ive seen it throw in cosmographic metrics as well and mix them with commoving metrics. (LOL the above can also be used to recognize someone relying too heavily on AI)

Forgot to add I don't see anything particularly wrong in your treatment above at the moment. In so far as the math relations involved. I would be curious though if you agree that direction would be an inherent degree of freedom of any underlying state/system being described. Where one state resides in relation to another obviously is related.

Thank you for the above, its a tremendous help in understanding the purpose of your article. Sorry I was being a bit of a stickler on material needed being presented here. I do have good reasons for that, lol lets just say I've come across one poster in the past that although his ideas were sound. He had dozens of different papers and articles he kept referring to and you literally had to go through them to get any sense of what he was doing in the first place.... That's not the reason of course but its a good extreme example. In the above you have a statement of avoiding any unnecessary complexity. Obviously scalar relations does indeed simplify the mathematics I would argue that requiring "direction of kinematic relations is a necessary complexity". Which direction an interaction (whatever that kinematic interaction represents) is just as important as the scalar relations. Obviously we all know any " Field treatment requires geometry" particularly for any mappings of particle or measured quantity distributions". Depending on what your after those mappings will also give a necessary complexity. Those are two aspects I would consider as being necessary ( for what I do in physics absolutely necessary) So the question of what is "necessary complexity" is something I think should be looked into in greater detail. Side note I will often post added Mainstream relations relating to a thread. I've found in the past this habit is an aid to other readers not involved in the conversation better understand what is being discussed as well as useful for comparisons between methodologies etc

agreed though propogators cannot be directly measured and include probability currents which are mathematical as well

Your list above is fairly accurate though some of the list is fairly broad. Light path deflection for example would include spectography redshifting. For example integrated Sache Wolfe effect as signals pass through the mass variation of DM halos as one example. If I think of anything not already covered on that list I will post it As far as the fine structure constant your methodology from what you described here sounds remarkably similar to whats done in BSBM model (Berkenstein Model ) a version of TeVeS MOND. The problem with coupling the fine structure constant is that you may find you would require a varying fine structure constant as per BSBM as well as the Hubble constant also varies over time. ( it's only constant everywhere at a given time slice. Ie today. If you would like to test it at different Z ranges I can give you the Hubble constant value at any given redshift value. The cosmocalc in my signature which I was involved with developing has the correct second order terms for when the recessive velocity exceeds c for redshift beyond 1.49 ( Hubble Horizon) to the particle horizon. the following below is for other readers to keep others at the same speed. The second order formula I'm referring to is the last formula on the list. The previous formulas is the mathematical proof using the equations of state and how they evolve over the universe expansion history. FLRW Metric equations \[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]\] \[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\] \[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\] \[H^2=(\frac{\dot{a}}{a})^2=\frac{8 \pi G}{3}\rho+\frac{\Lambda}{3}-\frac{k}{a^2}\] setting \[T^{\mu\nu}_\nu=0\] gives the energy stress mometum tensor as \[T^{\mu\nu}=pg^{\mu\nu}+(p=\rho)U^\mu U^\nu)\] \[T^{\mu\nu}_\nu\sim\frac{d}{dt}(\rho a^3)+p(\frac{d}{dt}(a^3)=0\] which describes the conservation of energy of a perfect fluid in commoving coordinates describes by the scale factor a with curvature term K=0. the related GR solution the the above will be the Newton approximation. \[G_{\mu\nu}=\eta_{\mu\nu}+H_{\mu\nu}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}\] Thermodynamics Tds=DU+pDV Adiabatic and isentropic fluid (closed system) equation of state \[w=\frac{\rho}{p}\sim p=\omega\rho\] \[\frac{d}{d}(\rho a^3)=-p\frac{d}{dt}(a^3)=-3H\omega(\rho a^3)\] as radiation equation of state is \[p_R=\rho_R/3\equiv \omega=1/3 \] radiation density in thermal equilibrium is therefore \[\rho_R=\frac{\pi^2}{30}{g_{*S}=\sum_{i=bosons}gi(\frac{T_i}{T})^3+\frac{7}{8}\sum_{i=fermions}gi(\frac{T_i}{T})}^3 \] \[S=\frac{2\pi^2}{45}g_{*s}(at)^3=constant\] temperature scales inversely to the scale factor giving \[T=T_O(1+z)\] with the density evolution of radiation, matter and Lambda given as a function of z \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\] its other purpose was more my work testing the accuracy of the inverse relation to blackbody temperature. I rarely trust literature on any verbatim basis so often like to see how a statement such as temperature being the inverse of the scale factor is determined as being accurate. Sides its good practice lol ( above i had done previously in my Nucleosynthesis thread. ) the last formula the cosmocalc employs though has from version 1 of the cosmocalc well over a decade ago . specifically this formula will provide the Hubble constant value as a function of redshift \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\] should note for others as well the GR statements are for the Newton approximation which the FLRW metric falls under just a side note the FLRW metric is not maximally symmetric where the Minkowskii metric under SR is. The use of the scale factor is one of the key issues with maximal symmetry (You can see this via the Christoffels for the FLRW metric ) or another way to learn this is through the Rayleigh equations.

In Feymann integrals the propogator ( a propogator propagates an operator) with the propogator being the internal lines and operator being the external solid lines ( observables ) ie real particles with internal often associated with virtual particles though its more accurate to just treat the propogator as field. You require one quanta of effective progator action to affect an operator . Thats about the only way one can potentially denotes some form of minimal threshold that I myself am aware of.

The writeup was likely using a Hilbert space common in QM treatments. A Hilbert space being defined from the inner product of a vector field. Its not the only class of wavefunctions. You can have wavefunctions that do not require a Hilbert space nor the inner product. Scalar field spaces being one example. You have no need for vectors nor inner or cross product. However you can still have a wavefunction relating to number density of photons as one example based on the amplitude of the probability current. Just an FYI. Lol one solid clue to keep track of the distinction. A function is a mathematical set of operations. The prefix of wave is simply naming the type of function. Same applies to correlation function for entanglement.

Lets straighten out the wavefunction being not physical. You develop the wavefunction using known properties of the particles state and apply it to the Schrodinger equation or Klein Gordon etc. You can also take into consideration the experimental apparatus, error margins etc. In QFT you can employ a probability current just a side note. Its simply our formulas employed with previous well tested studies of the particle properties, application of the appropriate formulas. Strictly determined via mathematics. Mathematics are not physical even though they may describe a physical state etc. Physical is what you have measured. You measure physical properties the mathematics only describe or predict what you will measure. That's a very important distinction

Well it may help to consider that its not necessarily the galaxy rotation curves themselves that provide the strongest support of DM being a particle. Consider the following if you take the FLRW metric and use the equations of state and apply the FLRW metric acceleration equations. Then remove the DM component and just apply baryonic matter of just 3% then there would never be enough matter in our universe for matter to become the dominant contributor to expansion. Instead of radiation era, matter era the Lambda era. You would only go from radiation directly to Lambda dominant. The Hubble constant would not have the value it does today. Matter radiation equality would never occur ( roughly when the universe is 7 Glyrs old.) Expansion rates themselves and it how it evolves over time would be completely different. Now as expansion occurs radiation diffuses more readily in an increased volume than matter so their densities evolve at different rates. Matter having an equation of state w=0 meaning it exerts no equivalent pressure term. This one can construe as being the primary evidence that influences the research more in favor of a particle constituent. Coupled with the detail that DM halos do cause gravitational lensing helps us confirm the density distributions. In point of detail Hubble telescope often makes use of these DM halos lenses to extend its range. Hope that helps if you like some of the related mathematics I can post them here. Lol wouldn't take any real effort as I have em handy in another thread. Edit correction on above the time frame was for matter lambda equality radiation/equality is sometime prior to Z=1150 depending on dataset used I would have to check later on. Zeq 3387 using Planck 2018+BAO dataset.

Bingo the one point all the crackpots miss lmao. Not stating anyone here is one lol.

Easy way is to consider a classic example if you determine some probability function for simplicity lets just use coin tosses but dropping a collection of coins in a given time frame. This forms a time or time independent wavefunction depending on drop rate. Once you make measurements ie number of coins with heads up as opposed to heads down. The original wavefunction isn't needed you have made determinations through observation and measurement you now have a determined wavefunction as opposed to a probability wave function. Some often refer to the latter as simply waveform to avoid confusion with the probability characteristic of a wavefunction.