Mordred

no problem try to think of it this way any complex problem is composed of little problems. In a sense its similar to programming you break the program into smaller simpler steps to get the final form. Its no different in physics you start with classical physics then you build it up to your complex systems. Every physics theory is comprised of kinematics (equations of motion) even when those equations of motion are waveforms they have vector equivalence.

F is force given in Newtons which is the amount of force to accelerate a 1 kg mass 1 meter per second squared. \[f=ma\] https://www.lehman.edu/faculty/anchordoqui/chapter07.pdf it becomes easier to learn complex physics by first mastering basic classical physics

Here is a handy article showing how phase and amplitude has similarities and equivalence in mathematics to the pendulum simple harmonic motion. https://www.lehman.edu/faculty/anchordoqui/chapter23.pdf as mentioned its an invaluable lesson

energy for E yes its a box with lots of particles hence its a total summation of hitting each box edge ie a probability function of number density striking the surface. \[E=\sqrt{(pc)^2+(m_o c^2)^2}\] start with 1 particle first to learn the math then determine the number density after then do your summation is the steps to learn this ( hint I already provided the formulas for number density above) Edit correction applied to above formula

no problem its why we have a Speculation forum is to help others learn as well as assist in how to properly model a new theory.

take a hypothetical box and place an ensemble of particles in the box. the box boundaries is the boundary conditions equivalence in the mathematics. Calculate how much force is delivered by the momentum term of the particle on the box boundaries. That's how the equations of state are determined ie the pressure term. matter has low momentum so exerts no pressure radiation has high momentum so it exerts pressure

All particle fields in Cosmology have an equation of state association this is analogous to thermodynamic hydrostatic fluid equations that includes DM even though we don't know the cause of DM we can model and determine its equation of state. w=0 meaning it has no pressure term.

Yes that's what I wrote try adding your own statements and I don't need you to rewrite my statements. I know quite well the mathematics for both gravity and the Higgs field. The Higgs field does not account for all the mass terms of the Standard model which means it does not account for all mass involved in gravity only roughly 1 percent of the mass of a single proton for example. So regardless of dimensionality it cannot account for all elements involved in gravity as it does not account for all the mass For example equation 1 will not work with the three primary Higgs field cross sections Higgs cross sections partial width's \[\Gamma(H\rightarrow f\bar{f})=\frac{G_Fm_f^2m_HN_c}{4\pi \sqrt{2}}(1-4m^2_f/m^2_H)^{3/2}\] \[\Gamma(H\rightarrow W^+ W^-)=\frac{GF M^3_H\beta_W}{32\pi\sqrt{2}}(4-4a_w+3a_W^2)\] \[\Gamma(H\rightarrow ZZ)=\frac{GF M^3_H\beta_z}{64\pi\sqrt{2}}(4-4a_Z+3a_Z^2)\] These cross sections above describe all the Higgs field interactions The first term is the interaction between Higgs and other fermions such as leptons. the next two are the bosons W+,W- and Z This should be a strong indication that your math is insufficient for the task as an assist here is the Higgs electroweak couplings. \[(\frac{g}{2}\vec{\tau}\cdot\vec{W}+\frac{\acute{g}}{2} B)\phi_0\] this equation involves wavefunctions and Fourier transformations so include tensors for the associated fields

For starters Higgs is a quartic field the mass couplings differ for W+, W- and Z bosons leaving one field uncoupled. This is confirmed via experiments so equation 1 is incorrect. Your equation for higher dimensions doesn't include any higher dimensions. In physics a dimension is not an alternate reality or other such item but refers to effective degrees of freedom Example {ct, x,y,z) has 5 values that can alter without any dependency on the other variables. I do not know what ChatGPT is placing in for the right right statements I can only assume Dirac Bra-ket notation. For latex on this site use \[latex\*] just remove the * I placed there to prevent activation. As mentioned DO NOT rely on ChatzGPT it's nothing more than a glorified search engine You will not be able to show higher dimensions using the definition for dimension I provided using Newtonian mathematics all your equations would fall under scalar field treatment and do not have the vector and spinor relations for particle field interactions. The closest statement you have to a vector is the inner product in equation 1 but that equation is still wrong

for the above and my previous post notice the usage of an observer watching the two spaceships now consider an observer not on the train. here is 4 coordinate systems reference 3982882af388dae3407906357a419cba_coordsproptime.pdf notice what occurs in each case last case is using rapidity. first two cases are two sperate reference frames under SR. Third and 4th case is using complex conjugations of the first 2 graphs ie dual vectors, co-vectors and contra-vectors are examples of those complex conjugates

One could also argue that in every example they worked from other previous works so that's a tricky argument particularly in the Einstein example. Newton not so much there wasn't a lot of previous works he had access to.

Oh I don't know look at say Feymann or Allen Guth their works has had huge ramifications they are certainly more recent examples than Einstein. Though I would readily place Feymann ahead of Guth on overall contribution

Ok obviously any previous help hasn't worked for you in that your still not really making the connection that any acceleration is not a straight line path A path through the space-time diagram is called a world line. The world line for the acceleration motion is described by a curve, but not a straight line. as trying to tell you using boosts and rotations nor instantaneous velocity treatments done previously albeit there was a ton of cross argument that had absolutely nothing to do with you and should never have occurred to begin with. That is not your fault in any form. The above BOLDED statement is one of the primary reasons why hyperbolic geometry is needed. That hyperbolic geometry is applied for the Minkowskii diagrams. This article has the most straightforward examination of all the applicable mathematics without using a single tensor or matrix. It has the train included in the article including simultaneity https://bingweb.binghamton.edu/~suzuki/ModernPhysics/2_Minkowski_spacetime_diagram.pdf please study this and ask questions if needed. for rapidity which isn't needed but useful the relation to velocity is the inverse hyperbolic function between v and c. \[w=\tanh(\frac{v}{c})\] The article uses a scaling factor k which is one method. One could also apply rapidity as that Tanh function can give a Natural log function for scaling as part of its ladder operation. seen here under inverse hyperbolic functions. https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions Events that occur at two separate places at the same time in the S' frame do not happen at the same time, as viewed in the S frame. previous to that statement the article has a mathematical expression \[\Delta \acute{t}=\frac{\Delta(t)\frac{v}{c^2}\Delta(x)}{\sqrt{1-\frac{v^2}{c^2}}}\] see previous mathematics for where the numerator and denominator gets derived from this is the section under relativity of simultaneity. the next sections will apply that to your train as you can see figure 27 is quite complex as a result with repetitious signals being figure 28 for the record the very reasons I kept mentioning velocity vs acceleration and referred to rapidity was to indicate that with acceleration you are no longer dealing with linear equations of motion but non linear equations of motions which uses hyperbolic geometry to describe hence the use of the tangent in the earlier sections of the article Tangent to the slope which is how one can linearize a non linear graph. The proper time is the point where the tangent intersects the worldline. Anywhere else its coordinate time and it is defined by \[\Delta(\tau)^2=(t_1-t_2)^2+(x_1-x_2)^2+(y_1-x_2)+(z_1-z_2)^2\] or simply \[d\tau^2=dt^2-(x^2+y^2+z^2)\] The article link the following video this can also be applied to the train. I will have to wait a bit till I can get past post merging as any images or videos will force any latex to Rich text format screwing it up. However I was trying to describe a similar example previous post to the observers on the train where I used the curved path to indicate that the worldline null geodesic does not always follow the path of the train to every observer on the train or other observers not on the train.

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.