Mordred

Who cares what wiki states it's never written by a professor in the field involved. It's never been nor will ever be an authority in physics or any other science. Garbage not even close to being accurate regardless if your describing LET or SR/GR.. Tell me do even understand what an inertial frame is as opposed to a non inertial frame ? It's no different in LET and you cannot even describe LET correctly if you don't know the difference. Tell me many more pages will it take before you realize that you never convince anyone that you are correct when you cannot describe the theories under discussions without being full of errors? Everyone is literally forced to correct your errors to the point where discussing the Pros and Cons between the two theories simply isn't happening.

This is the FLRW metric \[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2\] \[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\] This is the redshift equation(cosmological) that gets used at all ranges as it takes the evolution of matter, radiation and Lambda. \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\]

The redshift has little to do with gravitational constant and we have means of testing redshift by understanding the processes involved. We can for example examine hydrogen which is one of most common elements in our universe and using spectrography. There is nothing random that isn't cross checked by numerous means involving redshift. We don't even rely on it as our only means of distance calculation. Quite frankly no one method works for every distance range. A huge portion of papers can be found studying the accuracy of redshift at different ranges and those cross checks using other means such as interstellar parallax. Same applies to luminosity distance. By the way the redshift formula you find in textbooks is only useful at short distances cosmological scale.

This evening when I get home I will be able to run the formulas for you. Yes you can calculate the vacuum energy density per cubic metre. For that one can get a decent estimate using the critical density formula. (Assuming Lambda is the result of the Higfs field) one line of research. The calculations differ for the quantum harmonic oscillator contributions however that results in the vacuum catastrophe but I also have the related calculations for that as well.

As light climbs in and out of a gravity well it will blue shift or redshift. For example an outside observer looking at infalling material at the EH of a blackhole will see infinite redshift but an observer at the EH will see infinite blue shift. This is due to gravitational redshift The path will be determined by the Principle of least action which correlates the Potential energy and kinetic energy terms. What most ppl don't realize is that the path is never truly straight. That's just the mean average. If you consider all the little infinitesimal changes in direction (sometimes up/down left right etc) then it becomes much easier to understand. As Markus the geodesic equations are the extremum of all the miniscule deviations

How does a coupling constant appear smaller ? If you apply \(F=G\frac{m_1m_2}{r^2}\) the coupling constant remains constant what changes is the force exerted by the coupling between two masses as a function of radius. Not the coupling constant itself. We describe our observable universe itself in the FLRW metric we know the universe extends beyond that it could be finite or infinite as we can never measure beyond that we deal with what we can Observe and measure. (Region of shared causality)

yes I did understand that but I'm trying to ascertain your eventual goals with this to provide direction for improvement. If you think about we do much the same with the use of the scale factor under the FLRW However a key point is that G is a constant under the FLRW so your going to have to explain why you feel G would change as a result of change in radius ?

SU(2) \[{\small\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline Field & \ell_L& \ell_R &v_L&U_L&d_L&U_R &D_R&\phi^+&\phi^0\\\hline T_3&- \frac{1}{2}&0&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&0&0&\frac{1}{2}&-\frac{1}{2} \\\hline Y&-\frac{1}{2}&-1&-\frac{1}{2}&\frac{1}{6}&\frac{1}{6}& \frac{2}{3}&-\frac{1}{3}&\frac{1}{2}&\frac{1}{2}\\\hline Q&-1&-1&0&\frac{2}{3}&-\frac{1}{3}&\frac{2}{3}&-\frac{1}{3}&1&0\\\hline\end{array}}\] \(\psi_L\) doublet \[D_\mu\psi_L=[\partial_\mu-i\frac{g}{\sqrt{2}}(\tau^+W_\mu^+\tau^-W_\mu^-)-i\frac{g}{2}\tau^3W^3_\mu+i\acute{g}YB_\mu]\psi_L=\]\[\partial_\mu-i\frac{g}{\sqrt{2}}(\tau^+W_\mu^-)+ieQA_\mu-i\frac{g}{cos\theta_W}(\frac{t_3}{2}-Qsin^2\theta_W)Z_\mu]\psi_L\] \(\psi_R\) singlet \[D_\mu\psi_R=[\partial\mu+i\acute{g}YB_\mu]\psi_R=\partial_\mu+ieQA_\mu+i\frac{g}{cos\theta_W}Qsin^2\theta_WZ_\mu]\psi_W\] with \[\tau\pm=i\frac{\tau_1\pm\tau_2}{2}\] and charge operator defined as \[Q=\begin{pmatrix}\frac{1}{2}+Y&0\\0&-\frac{1}{2}+Y\end{pmatrix}\] \[e=g.sin\theta_W=g.cos\theta_W\] \[W_\mu\pm=\frac{W^1_\mu\pm iW_\mu^2}{\sqrt{2}}\] \[V_{ckm}=V^\dagger_{\mu L} V_{dL}\] The gauge group of electroweak interactions is \[SU(2)_L\otimes U(1)_Y\] where left handed quarks are in doublets of \[ SU(2)_L\] while right handed quarks are in singlets the electroweak interaction is given by the Langrangian \[\mathcal{L}=-\frac{1}{4}W^a_{\mu\nu}W^{\mu\nu}_a-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}+\overline{\Psi}i\gamma_\mu D^\mu \Psi\] where \[W^{1,2,3},B_\mu\] are the four spin 1 boson fields associated to the generators of the gauge transformation \[\Psi\] The 3 generators of the \[SU(2)_L\] transformation are the three isospin operator components \[t^a=\frac{1}{2} \tau^a \] with \[\tau^a \] being the Pauli matrix and the generator of \[U(1)_\gamma\] being the weak hypercharge operator. The weak isospin "I" and hyper charge \[\gamma\] are related to the electric charge Q and given as \[Q+I^3+\frac{\gamma}{2}\] with quarks and lepton fields organized in left-handed doublets and right-handed singlets: the covariant derivative is given as \[D^\mu=\partial_\mu+igW_\mu\frac{\tau}{2}-\frac{i\acute{g}}{2}B_\mu\] \[\begin{pmatrix}V_\ell\\\ell\end{pmatrix}_L,\ell_R,\begin{pmatrix}u\\d\end{pmatrix}_,u_R,d_R\] The mass eugenstates given by the Weinberg angles are \[W\pm_\mu=\sqrt{\frac{1}{2}}(W^1_\mu\mp i W_\mu^2)\] with the photon and Z boson given as \[A_\mu=B\mu cos\theta_W+W^3_\mu sin\theta_W\] \[Z_\mu=B\mu sin\theta_W+W^3_\mu cos\theta_W\] the mass mixings are given by the CKM matrix below \[\begin{pmatrix}\acute{d}\\\acute{s}\\\acute{b}\end{pmatrix}\begin{pmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{pmatrix}\begin{pmatrix}d\\s\\b\end{pmatrix}\] mass euqenstates given by \(A_\mu\) an \(Z_\mu\) \[W^3_\mu=Z_\mu cos\theta_W+A_\mu sin\theta_W\] \[B_\mu= Z_\mu sin\theta_W+A_\mu cos\theta_W\] \[Z_\mu=W^3_\mu cos\theta_W+B_\mu sin\theta_W\] \[A_\mu=-W^3_\mu\sin\theta_W+B_\mu cos\theta_W\] ghost field given by \[\acute{\psi}=e^{iY\alpha_Y}\psi\] \[\acute{B}_\mu=B_\mu-\frac{1}{\acute{g}}\partial_\mu\alpha Y\] [latex]D_\mu[/latex] minimally coupled gauge covariant derivative. h Higg's bosonic field [latex] \chi[/latex] is the Goldstone boson (not shown above) Goldstone no longer applies after spontaneous symmetry breaking [latex]\overline{\psi}[/latex] is the adjoint spinor [latex]\mathcal{L}_h=|D\mu|^2-\lambda(|h|^2-\frac{v^2}{2})^2[/latex] [latex]D_\mu=\partial_\mu-ie A_\mu[/latex] where [latex] A_\mu[/latex] is the electromagnetic four potential QCD gauge covariant derivative [latex] D_\mu=\partial_\mu \pm ig_s t_a \mathcal{A}^a_\mu[/latex] matrix A represents each scalar gluon field Single Dirac Field [latex]\mathcal{L}=\overline{\psi}I\gamma^\mu\partial_\mu-m)\psi[/latex] under U(1) EM fermion field equates to [latex]\psi\rightarrow\acute{\psi}=e^{I\alpha(x)Q}\psi[/latex] due to invariance requirement of the Langrene above and with the last equation leads to the gauge field [latex]A_\mu[/latex] [latex] \partial_\mu[/latex] is replaced by the covariant derivitave [latex]\partial_\mu\rightarrow D_\mu=\partial_\mu+ieQA_\mu[/latex] where [latex]A_\mu[/latex] transforms as [latex]A_\mu+\frac{1}{e}\partial_\mu\alpha[/latex] Single Gauge field U(1) [latex]\mathcal{L}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}[/latex] [latex]F_{\mu\nu}=\partial_\nu A_\mu-\partial_\mu A_\nu[/latex] add mass which violates local gauge invariance above [latex]\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^2A_\mu A^\mu[/latex] guage invariance demands photon be massless to repair gauge invariance add a single complex scalar field [latex]\phi=\frac{1}{\sqrt{2}}(\phi_1+i\phi_2[/latex] Langrene becomes [latex] \mathcal{L}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+|D_\mu \phi|^2-V_\phi[/latex] where [latex]D_\mu=\partial_\mu-ieA_\mu[/latex] [latex]V_\phi=\mu^2|\phi^2|+\lambda(|\phi^2|)^2[/latex] [latex]\overline{\psi}=\psi^\dagger \gamma^0[/latex] where [latex]\psi^\dagger[/latex] is the hermitean adjoint and [latex]\gamma^0 [/latex] is the timelike gamma matrix the four contravariant matrix are as follows [latex]\gamma^0=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}[/latex] [latex]\gamma^1=\begin{pmatrix}0&0&0&1\\0&0&1&0\\0&0&-1&0\\-1&0&0&0\end{pmatrix}[/latex] [latex]\gamma^2=\begin{pmatrix}0&0&0&-i\\0&0&i&0\\0&i&0&0\\-i&0&0&0\end{pmatrix}[/latex] [latex]\gamma^3=\begin{pmatrix}0&0&1&0\\0&0&0&-1\\-1&0&0&0\\0&1&0&0\end{pmatrix}[/latex] where [latex] \gamma^0[/latex] is timelike rest are spacelike V denotes the CKM matrix usage [latex]\begin{pmatrix}\acute{d}\\\acute{s}\\\acute{b}\end{pmatrix}\begin{pmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{pmatrix}\begin{pmatrix}d\\s\\b\end{pmatrix}[/latex] [latex]V_{ckm}=V^\dagger_{\mu L} V_{dL}[/latex] the CKM mixing angles correlates the cross section between the mass eigenstates and the weak interaction eigenstates. Involves CP violations and chirality relations. Dirac 4 component spinor fields [latex]\gamma^5=i\gamma_0,\gamma_1,\gamma_2,\gamma_3[/latex] 4 component Minkowskii with above 4 component Dirac Spinor and 4 component Dirac gamma matrixes are defined as [latex] {\gamma^\mu\gamma^\nu}=2g^{\mu\nu}\mathbb{I}[/latex] where [latex]\mathbb{I}[/latex] is the identity matrix. (required under MSSM electroweak symmetry break} in Chiral basis [latex]\gamma^5[/latex] is diagonal in [latex]2\otimes 2[/latex] the gamma matrixes are [latex]\begin{pmatrix}0&\sigma^\mu_{\alpha\beta}\\\overline{\sigma^{\mu\dot{\alpha}\beta}}&0\end{pmatrix}[/latex] [latex]\gamma^5=i{\gamma_0,\gamma_1,\gamma_2,\gamma_3}=\begin{pmatrix}-\delta_\alpha^\beta&0\\0&\delta^\dot{\alpha}_\dot{\beta}\end{pmatrix}[/latex] [latex]\mathbb{I}=\begin{pmatrix}\delta_\alpha^\beta&0\\0&\delta^\dot{\alpha}_\dot{\beta}\end{pmatrix}[/latex] Lorentz group identifiers in [latex](\frac{1}{2},0)\otimes(0,\frac{1}{2})[/latex] [latex]\sigma\frac{I}{4}=(\gamma^\mu\gamma^\nu)=\begin{pmatrix}\sigma^{\mu\nu\beta}_{\alpha}&0\\0&-\sigma^{\mu\nu\dot{\alpha}}_{\dot{\beta}}\end{pmatrix}[/latex] [latex]\sigma^{\mu\nu}[/latex] duality satisfies [latex]\gamma_5\sigma^{\mu\nu}=\frac{1}{2}I\epsilon^{\mu\nu\rho\tau}\sigma_{\rho\tau}[/latex] a 4 component Spinor Dirac field is made up of two mass degenerate Dirac spinor fields U(1) helicity [latex](\chi_\alpha(x)),(\eta_\beta(x))[/latex] [latex]\psi(x)=\begin{pmatrix}\chi^{\alpha\beta}(x)\\ \eta^{\dagger \dot{\alpha}}(x)\end{pmatrix}[/latex] the [latex](\alpha\beta)=(\frac{1}{2},0)[/latex] while the [latex](\dot{\alpha}\dot{\beta})=(0,\frac{1}{2})[/latex] this section relates the SO(4) double cover of the SU(2) gauge requiring the chiral projection operator next. chiral projections operator [latex]P_L=\frac{1}{2}(\mathbb{I}-\gamma_5=\begin{pmatrix}\delta_\alpha^\beta&0\\0&0\end{pmatrix}[/latex] [latex]P_R=\frac{1}{2}(\mathbb{I}+\gamma_5=\begin{pmatrix}0&0\\ 0&\delta^\dot{\alpha}_\dot{\beta}\end{pmatrix}[/latex] Weyl spinors [latex]\psi_L(x)=P_L\psi(x)=\begin{pmatrix}\chi_\alpha(x)\\0\end{pmatrix}[/latex] [latex]\psi_R(x)=P_R\psi(x)=\begin{pmatrix}0\\ \eta^{\dagger\dot{a}}(x)\end{pmatrix}[/latex] also requires Yukawa couplings...SU(2) matrixes given by [latex]diag(Y_{u1},Y_{u2},Y_{u3})=diag(Y_u,Y_c,Y_t)=diag(L^t_u,\mathbb{Y}_u,R_u)[/latex] [latex]diag(Y_{d1},Y_{d2},Y_{d3})=diag(Y_d,Y_s,Y_b)=diag(L^t_d,\mathbb{Y}_d,R_d[/latex] [latex]diag(Y_{\ell 1},Y_{\ell 2},Y_{\ell3})=diag(Y_e,Y_\mu,Y_\tau)=diag(L^T_\ell,\mathbb{Y}_\ell,R_\ell)[/latex] the fermion masses [latex]Y_{ui}=m_{ui}/V_u[/latex] [latex]Y_{di}=m_{di}/V_d[/latex] [latex]Y_{\ell i}=m_{\ell i}/V_\ell[/latex] Reminder notes: Dirac is massive 1/2 fermions, Weyl the massless. Majorona fermion has its own antiparticle pair while Dirac and Weyl do not. The RH neutrino would be more massive than the LH neutrino, same for the corresponding LH antineutrino and RH Neutrino via seesaw mechanism which is used with the seesaw mechanism under MSM. Under MSSM with different Higgs/higglets can be numerous seesaws. The Majorona method has conservation violations also these fermions must be electric charge neutral. (must be antiparticles of themselves) the CKM and PMNS are different mixing angels in distinction from on another. However they operate much the same way. CKM is more commonly used as its better tested to higher precision levels atm. Quark family is Dirac fermions due to electric charge cannot be its own antiparticle. Same applies to the charged lepton family. Neutrinos are members of the charge neutral lepton family 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. \[J_z=\frac{1\partial\Lambda}{i\partial\theta}|_{\theta=0}\] \[J_x=-i\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&1\\0&0&-1&0 \end{pmatrix}\] \[J_y=-i\begin{pmatrix}0&0&0&0\\0&0&0&-1\\0&0&1&0\\0&0&0&0 \end{pmatrix}\] \[J_z=-i\begin{pmatrix}0&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&0 \end{pmatrix}\] there is the boosts and rotations we will need and they obey commutations \[[A,B]=AB-BA\] SO(3) Rotations list set x,y,z rotation as \[\varphi,\Phi\phi\] \[R_x(\varphi)=\begin{pmatrix}1&0&0\\0&\cos\varphi&\sin\varphi\\o&-sin\varphi&cos\varphi \end{pmatrix}\] \[R_y(\phi)=\begin{pmatrix}cos\Phi&0&\sin\Phi\\0&1&0\\-sin\Phi&0&cos\Phi\end{pmatrix}\] \[R_z(\phi)=\begin{pmatrix}cos\theta&sin\theta&0\\-sin\theta&\cos\theta&o\\o&0&1 \end{pmatrix}\] Generators for each non commutative group. \[J_x=-i\frac{dR_x}{d\varphi}|_{\varphi=0}=\begin{pmatrix}0&0&0\\0&0&-i\\o&i&0\end{pmatrix}\] \[J_y=-i\frac{dR_y}{d\Phi}|_{\Phi=0}=\begin{pmatrix}0&0&-i\\0&0&0\\i&i&0\end{pmatrix}\] \[J_z=-i\frac{dR_z}{d\phi}|_{\phi=0}=\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}\] with angular momentum operator \[{J_i,J_J}=i\epsilon_{ijk}J_k\] with Levi-Civita \[\varepsilon_{123}=\varepsilon_{312}=\varepsilon_{231}=+1\] \[\varepsilon_{123}=\varepsilon_{321}=\varepsilon_{213}=-1\] SU(3) generators Gell Mann matrix's \[\lambda_1=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix}\] \[\lambda_2=\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}\] \[\lambda_3=\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}\] \[\lambda_4=\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}\] \[\lambda_5=\begin{pmatrix}0&0&-i\\0&0&0\\i&0&0\end{pmatrix}\] \[\lambda_6=\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}\] \[\lambda_7=\begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix}\] \[\lambda_8=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}\] commutation relations \[[\lambda_i\lambda_j]=2i\sum^8_{k=1}f_{ijk}\lambda_k\] with algebraic structure \[f_{123}=1,f_{147}=f_{165}=f_{246}=f_{246}=f_{257}=f_{345}=f_{376}=\frac{1}{2},f_{458}=f_{678}=\frac{3}{2}\] with Casimer Operator \[\vec{J}^2=J_x^2+J_y^2+j_z^2\]

Every electron has its own magnetic moment. https://en.m.wikipedia.org/wiki/Electron_magnetic_moment#:~:text=In atomic physics%2C the electron,24 J⋅T−1.

Well as an Engineer you certainly know that your field requires mathematical rigor. It's no different for physicist or mathematical theory. So if it's your goal to present some new trig function and have it gain weight in the Professional circles then you will need a mathematical proof. One that doesn't rely on words /pictures or descriptives. Anything less simply wouldn't cut it. I'm sure you can recognize the need fir that no forum has any particular influence in the Professional circles. Forums are useful but mainly to help others learn . Nothing discussed in a forum will ever alter how the scientific or mathematical community does things. That requires a professional peer review paper examined by other experts. The mathematical proof would be needed for that. For example every single physics formula has a corresponding mathematical proof no formula ever gets accepted without one.

Lol every single spectrograph I've ever examined has some form of redshift. The only time it doesn't is if it was reading some object in the lab. I even had an instructor that was testing the class with a falsified dataset that not only didn't have redshift but also incorrect elements.

I don't know relativity oh my that's a laugh. I would never have have gotten my degrees without knowing let alone past the undergraduate stage. It's literally part of my job dealing with SR on a regular basis lmao. You might want to try again mate For me it's not a hobby or a curiosity but a career requirement

No problem the easiest way I find is to use the command tags \[\frac{1}{2}\.] I put a dot in the last command to to prevent activation. For inline ie on the same line use \(\frac{1}{2}\.) What's handy about these tags is you don't need to type [\math] [.\math] [\latex] [.\latex] the inline for these two commands is imath and ilatex

Correction applied lol

@externo A solid piece of advise. You really need to stop trying to tell us how SR and GR works or describes. We have gone numerous pages with posters correcting your misunderstandings. Which you continue to repeat. I highly suggest that instead of trying to tell us what SR states that instead you start asking questions concerning SR. Use the math and the knowledge of the posters here and try to properly understand SR. This is article was written by a Ph.D that regularly uses forums. He developed this article to provide corrections to all the numerous misconceptions posters regularly have with regards to SR. http://www.lightandmatter.com/sr/ This article describes the basics of SR in a very easy to understand format and explains the reasons behind its mathematics. Relativity: The Special and General Theory" by Albert Einstein http://www.gutenberg.org/files/30155/30155-pdf.pdf It is an archive reprint.

Post what your trying and we can probably help out

Hey @Orion1 welcome back mate. No software I manually type in the latex. Lol thanks for the reminder to keep the metric tensor separate from the Einstein tensor lol

The standard model knows how to deal with the geometry of spheres, cones, or any other volume determined by a shape. Spacetime curvature doesn't describe its volumetric shape. It describes its affect on the geodesic equation for photon paths in regards to redshift or any signals we recieve due to particle paths. Hence it describes spacetime in geometric terms. With invariance under the metric choice the coordinate choice doesn't particularly matter. The use of differentials a huge part of GR as a conformal metric, so your using differentials is nothing new. However one can easily also choose to use integrals as per the QFT related theories such as loop quantum gravity. So that choice doesn't matter either. The FLRW metric already includes the radius for spheres in its metric. That is how the scale factor "a" is determined for the volume element but equally important is that the formula includes the velocity and the acceleration terms to describe expansion rates. However the volume element is the easiest thing to describe mathematically speaking with regards to the Observable universe. The FLRW metric further employs thermodynamics to determine what causes the expansion rates. The math you have posted here simply doesn't have that capability. We already know how to handle spheres, we already know to to ray cast spheres which would have equivalency with cones or even just use cone segments. That stuff is covered in differential calculus which is already employed. Differential geometry is one of the most used tools used in physics right along side with integrals The choice doesn't matter as its trivial to convert between them How does relations you showed here add anything at all we don't already employ where appropriate ? As mentioned we already take into consideration optical physics via differential calculus. It's a huge part of gravitational lensing for example. We even treat under the entire EM spectrum. The techniques involved in that is Huge part of spectrum analysis. Used all the time to for example to determine how much hydrogen is in a region via the 21 cm line in spectrography. Or determining or geometry (null geodesic paths) by looking for distortions caused by any non flat spacetime in the CMB. Once again it's one of the commonly used tools in observation. Nothing in your relations adds anything we don't already know how to do. That obviously includes wavefunctions. Which is a huge part of luminosity which is a useful tool in and of itself. For that we can even check for different expansion rates in a region or difference in gravitational potential of different regions via the Sache Wolfe effect. All that involves optics

So how does this relate to the gravitational constant or redshift ? Why wouldn't I just use a spherical coordinate system and apply a constant of proportionality via the scale factor "a" of the FLRW metric. That metric works regardless of the curvature term. It doesn't matter if spacetime is flat, positive curved or negative curved. I can calculate the proper distance to any object. I can tell you what the CMB blackbody temperature is at any given cosmological redshift value. I can even modify for different expansion rates as new data comes in for matter, radiation density. Calculate the age of the universe, as well as predict the rates of volume change of our observable universe far far into the future provided the cosmological parameters continue to evolve as current data show. The math you have shown here doesn't give me that ability. So where is the advantage of using mathematics if I cannot derive critical functions used in Cosmology ? here is a sample /[{\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline z&T (Gy)&R (Gly)&D_{now} (Gly)&Temp(K) \\ \hline 1.09e+3&3.72e-4&6.27e-4&4.53e+1&2.97e+3\\ \hline 3.39e+2&2.49e-3&3.95e-3&4.42e+1&9.27e+2\\ \hline 1.05e+2&1.53e-2&2.34e-2&4.20e+1&2.89e+2\\ \hline 3.20e+1&9.01e-2&1.36e-1&3.81e+1&9.00e+1\\ \hline 9.29e+0&5.22e-1&7.84e-1&3.09e+1&2.81e+1\\ \hline 2.21e+0&2.98e+0&4.37e+0&1.83e+1&8.74e+0\\ \hline 0.00e+0&1.38e+1&1.44e+1&0.00e+0&2.73e+0\\ \hline -6.88e-1&3.30e+1&1.73e+1&1.12e+1&8.49e-1\\ \hline -8.68e-1&4.79e+1&1.74e+1&1.43e+1&3.59e-1\\ \hline -9.44e-1&6.28e+1&1.74e+1&1.56e+1&1.52e-1\\ \hline -9.76e-1&7.77e+1&1.74e+1&1.61e+1&6.44e-2\\ \hline -9.90e-1&9.27e+1&1.74e+1&1.64e+1&2.73e-2\\ \hline \end{array}}\] this is from redshift z= 1100 to far into the future I kept the options small simply to demonstrate the calculator in my signature is far far more capable in the chart options. I can even graph each column via the same calculator as well as change my range to just after inflation forward into the future. That is an example of the capabilities needed to to be useful for cosmologists.

No you were never wrong to question it not in the slightest. It's also why I chose to work with you to straighten it out. It's all good glad we could work it out. After all its how we all learn I will try to be more accurate and clear in further mentions of rapidity.

Why do you keep declaring Einstein theory does this or does that even after everyone has told you most your misconceptions are wrong. The Earth does not suddenly age different simply because one observer looks at it. There is literally billions of observers on Earth they all do not have any effect on the rate the Earth ages. That is pure nonsense. The Minkowskii metric doesn't even state that. By the way thanks for providing the math I asked for. Unfortunately so far as pointed out all the evidence with regards to c being invariant is something you shouldn't ignore.

Excellent precisely what you should be of it in terms of

Your fairly close to the right idea. Without going into the quantum regime too intensely. In essence the overall electron spin up/ spin down alignments contained in each domain gets altered. Some electrons will switch from spin up to spin down or the overall orientation changes by some angle. So the fields of the magnet is already present even when it's not interacting with another object. So the charge currents are essentially zero (it's never truly zero as there is always some electron exchanges). So one can equate this to the PE term (potential energy) When the nail interacts with the magnet. The interaction of the magnet including the B field provide directivity of the charge current that results from the interaction between the magnet and the nail. We see this directivity in the magnetic field lines. The tighter the field lines the greater the amount of force. So further away the field lines diverge and gets weaker. (1/r^2). So in essence the electrostatic field does the work. The B field interaction in essence provides directivity of the charge current. A charge current is a kinetic energy term.

I like your example +1 in point of detail Amperes law teaches us that all magnetic phenomena is the result of electric charges in motion. Faraday discovered moving magnets generates an electric current. Maxwell and Lorentz in essence put together the final touch that E and B are not separate entities but are inexplicitly intertwined. So even a point charge has E and B fields. Now it takes a charge to produce an electromagnetic field, but just as importantly is that it takes another charge to detect an electromagnetic field. Now when you have an ensemble of charges you use the principle of superposition which tells us the interaction of two charges is unaffected by the presence of others. So you can compute the force resulting from each charge to the test charge and sum up to the total vector sum for total force on the test charge. Now you probably recognize I just described the electrostatic field. However with that field you now have to think in terms of charge density and charge currents. (By the way this applies to QFT as well) including the Feymann path integrals, just an FYI). So in point of detail the force on the test charge results from the sum of force of the individual point charges mediated by the EM field. Now we can further break down this Electrostatic field into surface charge, line charge, continuous distribution and volume charge. Each has has its own integral combined with Coulombs law. for example charge distribution \[E_r=\frac{1}{4\pi \epsilon_0}\int\frac{1}{r^2}\hat{r}dq\] line distribution \[E_r=\frac{1}{4 \pi\epsilon_0}\int \frac{\lambda(\acute{r})}{r^2}\hat{r}d\acute{l}\] surface charge \[E_r=\frac{1}{4 \pi\epsilon_0}\int\frac{\sigma(\acute{r})}{r^2}\hat{r}d \acute{a}\] and volume charge which we use most often. as being the one most referred to with Coulombs law \[E_r=\frac{1}{4 \pi\epsilon_0}\int\frac{\rho(\acute{r})}{r^2}\hat{r}d\acute{\tau}\] So knowing that according to Amperes law magnetism is the result of electric charges in motion. One has to ask well how does a permanent magnet work. What materials are more likely to make a magnet which materials would make a stronger magnet? To better understand that one has to understand how readily a material accepts domain realignment via a process called hysteresis. However it should be more clear that the charge distributions described by the formulas above directly relates to the sum of coulomb force to the test charge "d" is domain while the identifier after it is the domain type. the "r" with the hat is the distance from the domain to the test charge. So ferromagnets has domains with domain walls the walls are potential difference separations each domain has its own hysteresis. Histeresis describes a phenomena that when you pass a magnet near a ferrous material the alignments of the point charges do not return to the original configuration. (ever have a screw driver that you often use to work on an electric circuit eventually become a permanent magnet ? ) its due to hysteresis. hope that helps better understand the electrostatic field and ferromagnetism So now you should be able to answer the question :" Where does the energy come from" in the permanent magnet case...think domain charge densities and hysteresis due to the magnet interacting with the nail. This will also help when you look at things like Currie temperature and how it effectively it can be used to realign domains The domain alignments has potential energy there is no outside interaction so no current flow but you still have a charge density. When you place the nail near the magnet to interact the interaction exchange results in a charge current flow. This describes a kinetic energy term mediating the force. Now unfortunately a lot textbooks teach flow of electrons in a copper wire etc. It isn't the flow of electrons, its the flow of charge. Electrons could not flow through a medium fast enough for one thing. However the flow of charge can as charge is mediated by photons. It serves as the momentum carrier to alter the spin alignments of the electron ensemble edit forgot to add the primes (I tend to use acute ) are the source coordinates of the given domain for example \(d \acute{a}\). The symbols \(\lambda, \sigma, \rho\) is charge per unit (length, area, volume). The above also helps better understand induction. Your inducing charge current.

It might help to consider even in atoms electrons never stay still. In permanent magnets those electrons are moving around within the atoms of the magnet as well as the environment. However magnetism isn't a force nor is it a form of energy. A common analogy is to think of it as a translator with the E field. It results from the E field current and can thus be used to affect the E field through induction. lol you also run into articles etc stating permanant magnets have no E field but that wouldn't be true. The atoms have electrons and is held together by the EM field. So when you move the magnet to the nail your really just inducing electric current in a field already present which does the work via electromagnetic induction. (keep in mind I'm keeping the mediator photons out of the equation for this discussion) ie keeping it classical rather than quantum lol