Mordred

inflationary gravity waves Weak field limit transverse , traceless components with \(R_{\mu\nu}=0\) \[h^\mu_\mu=0\] \[\partial_\mu h^{\mu\nu}=\partial_\mu h^{\nu\mu}=0\] \[R_{\mu\nu}=8\pi G_N(T_{\mu\nu}-\frac{1}{2}T^\rho_\rho g_{\mu\nu})\] vacuum T=0 so \(\square h_{\mu\nu}=0\) transverse traceless wave equation \[\nabla^2h-\frac{\partial^2h}{c^2\partial t^2}=\frac{16\pi G_N}{c^4}T\] inhomogeneous perturbations of the RW metric \[ds^2=(1+2A)dt^2-2RB_idtdx^i-R^2[(1+2C)\delta_{ij}+\partial_i\partial_j E+h_{ij}]dx^idx^j\] where A,B,E and C are scalar perturbations while \(h_{ij}\) are the transverse traceless tensor metric perturbations each tensor mode with wave vector k has two transverse traceless polarizations. \[h_{ij}(\vec{k})=h_\vec{k} \bar{q}_{ij}+h_\vec{k} \bar{q}_{ij}\] *+x* polarizations The linearized Einstein equations then yield the same evolution equation for the amplitude as that for a massless field in RW spacetime. \[\ddot{h}_\vec{k}+3H\dot{h}_\vec{k}+\frac{k^2}{R^2}h_\vec{k}=0\] https://pdg.lbl.gov/2018/reviews/rpp2018-rev-inflation.pdf

Just to add for acceleration involving change in direction will involve transverse redshift. Just to add some useful relations more for the benefit of any readers not familiar with the types of redshift. \[\frac{\Delta_f}{f} = \frac{\lambda}{\lambda_o} = \frac{v}{c}=\frac{E_o}{E}=1+\frac{hc}{\lambda_o} \frac{\lambda}{hc}\] Doppler shift \[z=\frac{v}{c}\] Relativistic Doppler redshift \[1+z=(1+\frac{v}{c})\gamma\] Transverse redshift \[1+z=\frac{1+v Cos\theta/c}{\sqrt{1-v^2/c^2}}\] If \(\theta=0 \) degrees this reduces to \[1+z=\sqrt\frac{1+v/c}{1-v/c}\] At right angles this gives a redshift even though the emitter is not moving away from the observer \[1+z=\frac{1}{\sqrt{1-v^2/c^2}}\] From this we can see the constant velocity twin will have a transverse Doppler even though the velocity is constant. The acceleration as per change in velocity is straight forward with the above equations as the redshift/blueshift will continously change with the change in velocity term. The equations in this link will help better understand the equivalence principle in regards to gravity wells such as a planet https://en.m.wikipedia.org/wiki/Pound–Rebka_experiment The non relativistic form being \[\acute{f}=f(1+\frac{gh}{c^2})\]

In essence that's correct without going too indepth on the differences between operators and propogators of QFT. You can accurately treat it as a fundamental constant of the Higgs field with regards to how the field couples to other particles for the mass term I really wouldn't trust Chatgp your far better off in this regard studying the standard model via the Lanqrangian equations. For the W boson it's the SU(2) group and U(1) groups for the relevant details with Higgs. It's also why I recommended starting with Quantum field theory Demystified as it's reasonably well explained for the laymen to grasp.

The VeV isn't an issue it's something you observe only during scatterrings via say a particle accelerator it's a local property at each particle such as the W boson simply put a coupling term. The probabilities are much the same as the probabilities associated with Feymann path integrals. It isn't the vacuum energy density itself so it's not anywhere near Like the vacuum catastrophe from QM.

We cross posted Migl but I included a primary missing detail in terms of VEV being a probability value much like a weighted sum in statistics.

Might be easier to understand that in statistical mechanics, QM and QFT the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement. What that statement tells us is that it includes all possible outcomes. It is a probability function. Expectational values is used regularly in statistical mechanics, QM and QFT. Path integrals also have probability weighted sums

If your certain of your equations and it's validity I'm sure your going to want to test them. If you think about it I provided the essential equations to do just that with a given dataset such as Planck. I certainly do when I model build or simply test and cross check any new relations/interactions. Those equations apply LCDM. to the cosmological redshift. As far as a new value of G well all I can say to that is good freaking luck on that score with what you have shown so far. this is a listing of the various types of studies and results form them for variations of G tests for spatial dependence is page 200 onward http://www2.fisica.unlp.edu.ar/materias/FisGral2semestre2/Gillies.pdf

fair enough, something to keep in mind if your looking at cosmological redshift is that the expansion rates are not linear. The equation above shows this as the resultant is to determine the Hubble value at a given Z compared to the value today. The relations under the square root is the evolution of the energy density for matter, radiation and Lambda. You can learn these here. https://en.wikipedia.org/wiki/Equation_of_state_(cosmology) this related to the FLRW acecleration equations. described here https://en.wikipedia.org/wiki/Friedmann_equations that link supplies some very useful integrals with regards to the scale factor the evolution of the scale factor "a" using the above relations gives \[\frac{\ddot{a}}{a}=-\frac{4G}{3}(\rho+3P)+\frac{\Lambda}{3}\] however to get the FLRW metric cosmological redshift equation you will also need the Newton weak field limit treatments as per GR. Particularly for curvature K=0 if your interested in that let me know and I'll provide more details

ok First off you have vacuum energy and vacuum energy density confused. The first case though not a useful form for energy density. The VeV is the vacuum expectation value VeV this isn't the density. This is a term describing the effective action https://en.wikipedia.org/wiki/Effective_action for Higgs the effective action is defined by the equation \[v-\frac{1}{\sqrt{\sqrt{2}G^0_W}}=\frac{2M_W}{g}\] here \(M_W\) is the mass of the W boson and \( G^0_W\) is the reduced Fermi constant. These are used primarily when dealing with Feymann path integrals in scatterings or other particle to particle interactions involving Higgs in particular dealing with the CKMS mass mixing matrix. So its not your energy density more specifically they describe CKMS mixing angles or Weinberg mixing angles. for the above without going into too much detail the mixing angles are \[M_W=\frac{1}{2}gv\] \[M_Z=\frac{1}{COS\Theta_W}\frac{1}{2}gv=\frac{1}{Cos\theta_W}M_W\] more details can be found here. Page three I'm starting to compile the previous pages now if you want the vacuum energy density the FLRW has a useful equation. \[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\] if you take the value of the Hubble constant today and plug it into that formula you will get approximately \(5.5\times 10^{-10} joules/m^3\) if you convert that over you will find your fairly close to 3.4 GeV/m^3 which matches depending on the dataset used for the Hubble constant. The confusion you had was simply not realizing the VeV isn't the energy density. hope that helps. I won't get into too many details of the quantum harmonic oscillator via zero point energy but if you take the zero point energy formula and integrate over momentum space d^3x you will end up with infinite energy. So you must renormalize by applying constraints on momentum space. However even following the renormalization procedure you still end up 120 orders of magnitude too high. There has been resolutions presented to this problem however nothing conclusive enough. Quantum field theory demystified by David Mcmahon has a decent coverage of the vacuum catastrophe edit forgot to add calculating the energy density for the cosmological constant uses the same procedure as per the critical density formula.

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)