Mordred

We are already aware that GR reaches a singularity condition such as you described above ie ds^2=0. However the problem is that GR is incredibly precise at all other velocities where v does not equal c. Curvature is non linear any curve is non linear however one can linearize non linear relations to close approximation. Considering GR high degree of accuracy I wouldn't think of it as flawed but rather has a limit to its accuracy.

Where I disagree with her argument is she isn't letting people know theories such as QFT etc are adaptive to new findings. Same is true for any major theory such as LCDM. Major theories change that's part of all those unusual treatment papers. A good theory has solid foundations so are easily adaptive as a consequence. Good example all the alternative geometry treatments under the EFE alone for flexibility. It's adaptive to different systems and dynamics.

Another khan University lesson I would like you to watch is constructive and destructive interference. https://www.khanacademy.org/science/physics/mechanical-waves-and-sound/standing-waves/v/constructive-and-destructive-interference#:~:text=Constructive interference happens when two,they cancel each other out. This will help to understand Elastic vs inelastic scatterings when two particles meet. https://en.wikipedia.org/wiki/Elastic_scattering https://en.wikipedia.org/wiki/Inelastic_scattering the first link will also help understand wave resonance. https://juddy.com.au/wp-content/uploads/2017/07/Notes-4.1.3.pdf HINT the mechanical elastic PE terms above apply ie a crystal resonating with a frequency those resonations of the atoms will follow the same equations of motion (sound waves are mechanical energy) take k for spring constant now replace with binding energy via the coupling constant of a field. In those Feymann integrals "g" for the EM field g is the fine structure constant https://en.wikipedia.org/wiki/Fine-structure_constant for the Strong force its \[\alpha_s=\frac{g^2}{4\pi}\] Higgs couplings is quite a bit more complex Higgs electroweak couplings below \[(\frac{g}{2}\vec{\tau}\cdot\vec{W}+\frac{\acute{g}}{2} B)\phi_0\] @studiot mentioned another form of mechanical energy that of the pendulum those relations are also useful in QM/QFT it is another way to understand harmonic motion hint a wave has two component or rather polarities Longitudinal and transverse waves. Longitudinal is also called traceless in higher treatments. The pendulum and spring equations of motion will be necessary to understand both Also going to prove useful to understand the E and B fields for EM field. (in regards to the other thread you included on twisted photons) ie Maxwell equations and Lorentz force for E and B fields.\ I would like you to consider the following statement. All physics theories involve equations of motion. So the best stepping stone starting point is to learn the classical systems example mechanical energy above fluidic systems gaseous systems. etc. those equations of motion use vectors including the inner product and cross product provided earlier this thread. So to master any physics theory, master the equations of motion and learn how each theory describes them.

excellent plan and saves me some time as well. Lets start with the spring example Studiot supplied. You may recall earlier I stated Energy is the ability to perform work. Well when it comes to mechanical energy you can seperate two distinct ways a spring can perform work. Via its Potential energy which is the energy due to the springs position and its kinetic energy the work the spring can perform due to its momentum. These are described under Hookes Law, so the total mechanical energy of our system must include both PE and KE energy. To perform work a system requires a force. so the total mechanical energy of the spring is the combination of PE and KE \[E_M=E_{ke}+E_{PE}\] work is \(W=(force *distance)\) or \(W=fx\) now lets focus on just the PE portion as per Hooke's law the work on a spring due to PE is called its elastic potential energy. So when the spring is unstretched \(F_S=0\). When the spring is stretched the force increases \(F_S=kx\). The average applied force over the distance " x "is \(\bar{F}=0+kx/2\) or \(\bar{F}_{average force}=\frac{1}{2}kx\) where k is the spring constant. if you substitute this back to the works equation above \(W=Fx\) gives average work \(W=\bar{F} x\) including the spring constant k \(W=\frac{1}{2} k x\) gives work as \[W=\frac{1}{2}kx^2\] Here is a Khan university article showing graphs of the above. Kinetic energy is due to the springs movement ie its velocity \[E_k=\frac{1}{2} mv^2\] https://www.khanacademy.org/science/physics/work-and-energy/hookes-law/a/what-is-elastic-potential-energy Quantum Harmonic Oscillator Section Now recall I described the harmonic oscillator as a spring ? we can apply those equations above but we have a couple of details to cover first. Now in QM and QFT we look at the momentum terms these momentum terms has quantum equivalence of the PE and KE terms used in the spring total energy using momentum terms of a particle is energy momentum relation \[E^2=\sqrt{(pc)^2+(m_o c^2)^2}\] zero point energy of harmonic oscillator \[E=\frac{\hbar\omega}{2}\] https://en.wikipedia.org/wiki/Zero-point_energy so for kinetic energy the kinetic energy in momentum terms is \[E_k=\frac{P}{2M}\] where M is the mass of object and P is the momentum. So the kinetic energy using the energy momentum relation above is the energy without the mass term \(m_oc^2\) but just its momentum term \(pc)^2\) Now the next caveat is in QM and QFT both are what is termed canonical treatments in math speak. What this really means is its quantized. Now we have two Operators in QM position and momentum. \[[\hat{x},\hat{p}]=i\] i is just an integer x in this form is the position complex conjugate (don't worry about that atm it means it depends on two relations not just one) and the Momentum complex conjugate. All operators are complex conjugates but that's another detail for later on. I'm adding them so you learn to recognize what these symbols mean. Now in QFT we make the field the Operator to include the four momentum of GR. (relativistic). so in non relativistic QM the harmonic oscillator becomes \[\hat{H}=\frac{\hat{p^2}{2m}+\frac{M}\omega^2}{2}\hat{x}^2\] where H is the Hamilton Operator (details for another time you need more math skills in vector associations). We can get two new Operators. These operators are non hermitean (again later on lesson) called the annihilation and creation operators. respectively below \[\hat{a}=\sqrt{\frac{m\omega}{2}}(\hat{x}+\frac{i}{m\omega}\hat{P})\] \[\hat{a}^\dagger=\sqrt{\frac{m\omega}{2}}(\hat{x}-\frac{i}{m\omega}\hat{P})\] given that \([\hat{x},\hat{P}]=i\) gives \([\hat{a},\hat{a}^\dagger]=1\) the Hamilton takes on a useful form \[\hat{H}=\omega(\hat{N}+\frac{1}{2}\] with eugenstates \[\hat{H}|n\rangle=\omega(n+\frac{1}{2}|n\rangle)\] this gives the energy of the state as \[E_n=\omega(n+\frac{1}{2})\] where \(|n\rangle\) is the number states ie the number of particles states. Where the annihilation operator drops \(|n\rangle\) by one and the creation operator increases \(|n\rangle\) by one (ladder Operators). the above for QFT portion can be found in quantum field theory Demystified chapter 6. by David McMahon. So from the above I showed the PE energy relations of the harmonic oscillator by describing the PE relations of a spring. Then explained how energy and Work are related and Force is included. then took the Spring equations and QM/QFT quantization for its Operators to get the creation and annihilation operators to describe how the Quantum harmonic oscillator can produce particles using the Eugen-states of the Hamilton by it non Hermitian creation and annihilation operators. Those equations can also be used to determine the particle NUMBER density of a field via the Bose-Einstein and Fermi-Dirac statistics in QFT equivalence (MUCH later TOPIC). Now think back to @studiot last post where one spring can resonate with another and place a spring at every coordinate. Your field is constantly undergoing oscillations due to resonance and the above action. now common symbols \(\vec{x}\)=vector \(\bar{x}\)=average \(\hat{x}\)=complex vector conjugate \(\hat{a}\)=annihilation operator (complex conjugate) \(\hat{a}^\dagger\)=creation operator what I haven't shown is anti particles which is \(\hat{b}\)=annihilation for anti particles \(\hat{b}^\dagger\)=creation operator for antiparticles Now Recall those Feymann diagrams Operators are the external lines the propagator is the internal wavy lines. Think of propagator as the mediation between the Operators where Real particles are defined by the Operators and the offshell mediator gauge bosons are the propagators

Connecting the term vibration to the harmonic oscillator is perfectly fine. After work I will step you into how that harmonic oscillator is used to formulate into the creation/annihilation operators used in QFT that leads to particle production. Your picking up skills at a good pace so +1 on that.

Simple way to understand spacetime is to treat time as the Interval using (ct). So your spacetime becomes (ct, x,y,z) this gives time dimensionality equivalence to length. Ps you will also find that works with the four momentum equations.

One of my favorite cases of a study that led to development in a completely different field is Parker radiation. Originally Parker radiation was virtual particles formed by curvature terms through expansion. However it found its uses in MRI's Which is where it's primarily used now and is largely completely forgotten about for its Cosmology application which it was originally developed for. Obviously the 2 cases are distinctive but involve similar processes

The funding aspect of papers do help establish expertise on a given topic provided they are well done. Typically ones that are extremely well done will get higher citations. These type of papers would make getting further research grants easier however it's not particularly based on sheer number of articles but rather quality of said article in the potential of advancing a particular field of study. Articles that examine previous written paper by other authors also count. In many ways those articles serve as a portfolio to eventually gaining the funding to get test equipment etc to test a given theory but that takes time and will require establishing expertise which papers can be useful in doing. However papers are not the only means. Work history at research facilities also count for much. In many ways getting grants is much like applying for a job. Research papers and work history makes the process easier by establishing your a good investment. Though both can also hinder through too many poorly written papers (tends to establish the author as a crank) or poor work history record.

So you claim yet have only shown a single equation. Prove it mathematically here go ahead I challenge you to put your mathematics where your mouth is. Instead of claiming mathematically prove your claims. Start by proving it will work under SR first After all we still have to prove it will work in curved spacetime. Come on pit your mathematics skills and your single equation under examination that it will work under GR. I would love to see that but I already know you can't

Not a chance mate your equations are essentially useless as they do not describe how our universe evolves over time. Nor have you really anything particularly useful for physics scale factors are a dime a dozen in numerous theories Every major theory will have some form of scale factor. If your scale factor doesn't include any other related metrics specifically under a geometry treatment applicable to the system its describing then its essentially of little use as a replacement. Generating scale factor simulations is nothing new to physics and they can be widely varied. If your equations don't conform to observational evidence it's insufficient as proof.

Then you better show your equations if it deviates from GR you have your work cut out for you and believe me I'll be able to tell. If you don't understand the EFE and how it applies to the FLRW metric then you really don't understand its true flexibility. Every equation I posted you can be the observer. Even the only one why is the recessive velocity important is simple, velocity as shown by the Lorentz transformations directly apply to how we measure time so using recessive velocity is how we factor in the time time component vs the space space components. Using GR relations I will show how the FLRW metric fits with GR. but first here is an interesting trick simply take \[v_{recessive}=H_oD\] and to get an accurate recessive velocity all the way out to the cosmological event horizon do this substitution \[v_{recessive}=(H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}D\] Now this shows that the rate of change in distance to the Cosmological event horizon is accelerating and that the Hubble value for expansion is also not constant over time. The above substitution calculates how H changes as a function of cosmological redshift. That applies the equations of state and includes both equations of the FLRW metric its geometry previously shown. The second term is the acceleration equation for how radiation and matter energy densities evolve over time expansion relations. That's the portion under the square root including cosmological term. now under GR the above relations would give the following including all others I have already posted such as the FLRW metric. take the EFE (Einstein field equation) which is needed for its field treatments of multipoint coordinates. Any coordinate can be an observer including yourself \[G_{\alpha\beta}=\frac{8\pi G}{c^4}T_{\alpha\beta}\] \(T_{\alpha\beta}\) being the stress energy momentum tensor. \[ds^2=g_{\alpha}{\beta}dx^\alpha dx^\beta\] where \(g_{\alpha\beta}\) is the metric tensor as this is an orthogonal matrix above the non vanishing elements can be given in matrix form for the FLRW metric as below for the metric \[g_{\alpha\beta}=\begin{pmatrix}1&0&0&0\\0&-\frac{a^2}{1-kr^2}&0&0\\0&0&-a^2r^2&0\\0&0&0&a^2r^2\sin^2\theta\end{pmatrix}\] for the stress energy momentum tensor \(T_{00}=\rho c^2,,,T_{11}=\frac{Pa^2}{1-kr^2}\) the left hand side of the Einstein field equation becomes \[G_{00}=3(a)^{-2}(\dot{a}^2+kc^2)\] \[G_{11}=-c^{-2}(a \ddot{a}+\dot{a}^2+k)(1-kr^2)-1\] using above the time evolution of the cosmic scale factor then becomes \[\frac{a}{a}^2+\frac{kc^2}{a^2}=\frac{8\pi G}{3}G\rho\] \[2\frac{\ddot{a}}{a}+\frac{\dot{a}}{a}^2+\frac{kc^2}{a^2}=\frac{8\pi}{3}P\] where \(\rho\) is the energy density and P is the pressure. The overdot 's above the scale factor terms are the velocity for single dot with two dots its the acceleration term. This is shows why we use velocity and acceleration the choice of observer is irrelevant its obviously practical to more often than not treat yourself as the observer. The above also shows that the FLRW metric is a GR solution and its generalized relations. They already include any SR application but under GR field treatment which is better suited for spacetime curvature. Spacetime under GR always include the equations of momentum given by \[E^2=\sqrt{(pc)^2+(m_oc^2)^2}\] which is the full equation for \(e=m_oc^2\) called the energy momentum relation for previous. the above shows a mathematical proof that the substitution below is valid and how its applied. SR works for the first term but only at very close range and it degrades in accuracy due to equation 2 below. Equation two is a product of those relations above including how radiation, matter and the cosmological constant evolve over time in energy density and pressure relations \[v_{recessive}=H_oD\] \[v_{recessive}=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}D\] That should give you a good overview of why commoving coordinates are an essential aspect to an expanding universe. Its the influence of our matter/energy content and how they affect expansion. Yes the above is complex but once you understand it. It is absolutely remarkable how flexible the above is in describing how the scale factor evolves over time and why the affine connection for proper time or cosmic time is tied to the scale factor as given as \(a(t)\).

As I stated that Observer could be you but you are a commoving observer under the equations I posted. Those work just as well expressing you but invariance requires any observer for proper velocity relations for the four momentum. All part of GR requirements also required for SO(3.1) Poincare group = spacetime metric. Simply arguing your the observer so it shouldn't matter doesn't work when the very coordinates your located at are commoving with the universe. hence you would need a different geometry with a different flow of any measurements you take of any particles or objects around you unless you are moving with the coordinates ie fixed coordinate.

THEN define your reference frame mathematically and post the relevant details for discussion as you are required to have a geometry that should be a cinch. observer is measured by you it is commoving as everything in our universe is a commoving coordinate. So you better get to work on your observer. As it doesn't fall under SR nor GR unless your in some absolute frame of reference in which case this thread will belong under Eather based and isn't main stream physics.

I'm describing as measured from the Commoving observer using recessive velocity MISTER.

I already answered that read back prior to Hubble radius v<c after Hubble radius v>c Hubble radius The Observable universe if bigger than the Hubble radius by a significant amount. Hubble radius is defined as the radius where the age of the universe times c. So if our universe is 13.8 Gly the Hubble radius is only 13.8 Gly our observable universe is however 46.3 Gly in radius https://en.wikipedia.org/wiki/Hubble_volume equation from above link \[R_H=\frac{c}{H_0}\] the SR transformations will give ds^=0 at the above Hubble radius our universe is larger precisely what I described in my first couple of posts https://en.wikipedia.org/wiki/Observable_universe The reason why the universe exceeds the Hubble horizon is the following \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\] \(H_0\) is not constant that is todays value but in the past its higher and at its highest at BB. Those equations of state I mentioned