Mordred

In the future please do no place my questions to you into your AI especially as a direct copy paste. I am having this discussion with you and not AI. However thank you for your reply I will always prefer your reply over any AI generated response. That being said thank you for providing the details relating to your manifold structures with regards to S1 and S2. You do have a distance relation via signal ct. Fine its an interval you have direction so your structure does indeed have vector quantities. Given the above can you mathematically describe Observer A's reference frame and Bobs reference frame ? Are they assigned to coordinate space or are they assigned to your manifold ?

I noted you have the beta function from the beginning. The part Ive been trying to fathom is when do you correlate time dilation. With SR the Interval ( ct) is what allows dimensionality of length. How do you handle distance? How do you handle location of each reference frame. I assume for acceleration specifically change in velocity your using instantaneous velocity. Your math above would certainly work with it. However a change in direction is also a form of acceleration and this is where I do not understand your spacetime = geometry correlation. The energy momentum equation doesn't contain the terms for change in direction. The total energy of the object is unaffected by change in direction. If you have 2 observers and 1 emitter say a spaceship flying by both observers they should be getting different results each for relative velocity etc. Lol if you look back through my posts its not certainly not the first Ive mentioned distance including seperation distance pertaining to the ds^2 line element. Specifically the null geodesic worldline Edit should mention the total energy measured will vary between different observers (variant mass and variant energy as opposed to invariant mass and energy). For other readers assuming x axis is direction of motion the transformations between reference frames is \{\acute{x}= x-vt\} other axis are unchanged with time absolute.

All mathematics is nothing more than a useful tool. To me the mathematical methodology isn't an issue. I don't have strict adherence to any formalism. My comments above relate to versatility of a methodology or ontology. The question is " does your methodology have the same predictive and descriptive ability as that of the entirety of SR or GR" If you can demonstrate that it does and still follow mathematic rules then great. If its lacking then that's something to improve upon.

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.

Interesting article and proposal will be interesting what future findings on this will present itself for those wanting to look through the arxiv article itself https://arxiv.org/abs/2510.19087

This post had nothing to do with your article. I simply thought it was an idea you could make use of. Good luck. You want thinking outside the box its simple any mathematical methodology that can accurately describe a system or state has validity. You dont need tensors to do GR its simply another handy mathematical tool. You dont need to use 4 parameters to describe spacetime you can use parametric equations to reduce them. Thats my view point

Whatever as I mentioned before the rules state that there is no requirement to visit other sites or links and all pertinent information should be here. Im sorry you do not get that policy but its your full pdf on your opening page I have absolutely zero interest in opening up any other of your website links. So good luck with your work . Im done I have better things to do

Here's an interesting trick for you then take the equation of motion for a mechanical spring. If you compare the equation for the quantum harmonic oscillator you will find the precise same relations albeit a change in variables applied. The ratios of change are identical. If you study deep enough you will find a great deal of similarities between the seemingly complex equations have similar relations to many classical physics formulas commonly used in engineering. This is quite a bit more complex first and foremost the conservation rules require a closed system or a closed group. To go into greater detail would be more suitable to a seperate thread and such a discussion can get extensively lengthy. In some treatments involving spacetime one can define a conserved system usually ties into innvariance of a quantity. This is often done under local geometry in some mathematical space or manifolds. Anyways best left for a different thread

When I first started physics one of my earlier goals was to solve DE as an optical illusion. I realized early enough that doing so required mathematics and a good understanding of the FLRW metric and GR. In those studies I learned a valuable lesson. Simply because something doesn't make sense to you doesn't mean its incorrect. Its a lesson I carry on to this very day. More enough enough if one studies why the physics professional community states what they state there is always numerous supportive as well as counter arguments or methodologies supporting one theory or another. To a layman this unfortunately gives a rather daunting task of sorting through. Though the effort of doing so with an open mind can lead one to learn a great deal. The biggest challenge is avoiding any personal bias. Once you form a solid opinion of I feel it should be this way. One tends to close the book on examine other possibilities. This goes for any physics theory or model. The effort to understanding why physics describes something a certain way always has very strong reasoning behind it and those reasons are typically best understood by studying the related mathematical proofs. Its one reason I study not just mainstream models and physics but study numerous potential models or theories. You would be amazed that with enough studying how much something that originally sounded ridiculous starts to make sense. One also discovers how truly interconnected one theory is to numerous others. A simple change in one theory can often have numerous ramifications of dozens of others. Something many of our typical crackpot dont fully recognize. There's a simple consideration no theory or model ever becomes considered mainstream physics without years of rigorous testing and years of sorting the best fit to observational evidence. Eureka moments are typically something that only exists in movies. Theory and model building for a robust theory takes an incredible amount of work and whats often forgotten. One of the most valuable practices is that a good theorist should spend far more time proving his own theory wrong than he/she did in developing it. Little hint here if one studies statistics and statistical mechanics. One discovers that a great deal of the more difficult terminology used in QM and QFT actually originates from classical statistics. Examples being superposition, correlation functions the list goes on. It does help pull a lot of the mystery out of QM.

There is a handy simplification related to the non linearity of the relativistic addition of velocities where the Lorentz transformations matrix comes in handy. Rapidity using rapidity velocity is replaced by rapidity and becomes linearly additive. The method applies the hyperbolic spacetime diagram of the Minkowskii metric its also useful for a constant accelerating object. Not sure if that would interest you or not but its a useful simplification on calculations with regards to Lorentz transformations.

Great so where is the difference between using a unitary basis under GR. Normalization I fully recognize and relate to same goes for dimensionless values. Still doesn't address where dimensionless values isn't appropriate for specific relations. Oh Ive read your article its mannerism of writing is rather scattered but that's another issue so dont call me a liar on that. One of the reasons I had to reread it was I initially thought you declared the Gamna factor being the inverse of the beta function which would be incorrect but your B_y isn't identical to the normal beta function relation. Not that I saw you employing Gamma factor so it was irrelevant to mention. What Im suppose to be convinced by your graphic ? Simply because you employed dimensionless replacements or using normalized units ? Its fairly rudementary to normalize or make some relation dimensionless. Ive come across numerous articles that make \{8 \pi G \} normalized to one good example is the critical density formula nothing new or exciting about that. Do you not want to expand on your article for example The Kerr metric isn't a static solution. Perfect arena for testing your method on a rotating frame. You didn't really go into alot of detail in that section of your article. If you feel your article is a done deal then it amounts to just advertising in which case I lose all interest. Correct me if Im wrong but you assigned E_0 as invariant energy with M_0 being the invariant mass with E being total energy so explain why you have \{E=m_0\} and not \{E_0=m_0\} ? Correct me if Im wrong but your thread title does state testing. So add tests you haven't already done. Solving twin paradox with your methodology might prove a useful challenge as another example. However then you will have to deviate from the symmetry relations of constant velocity to include the rotations involved for acceleration