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.

30 minutes ago, chron44 said:

 

So, here we have noticed at least two Standard Model (QFT) anomalies. First the cosmological catastrophe. And secondly an unexplained and unexpected large discrepancy between the VEV (even if being a probabilistic amount) and the observed cosmological constant.

Thus, being the cornerstone in modern physics it is, it obviously can be shaken and giving strong divergences. If so, can we trust the 2012 LHC result at all? Is there a parallel theory and explanation for maybe the Higgs boson? The LHC did though confirm theories sprung from mid 1960's. -Finally, we have to trust the main parts in the Standard Model when proven right so many times.

/chron44

 

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

 

8 hours ago, chron44 said:

Hi,

The average vacuum energy is estimated to about 3 GeV/m^3 (the observed). If relying on this value (when
the calculated in extremely much higher). -And the Higgs field energy VEV, the vacuum expectation value, is
both observed in the LHC experiment and fairly calculated to about 246 GeV. How are these differences
explained in QM physics?

3 GeV versus 246 GeV?

 

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.

 

3 hours ago, externo said:

 

Yet, Wikipedia says that there is a paradox except in the case where we postulate a privileged reference frame.

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.

3 hours ago, externo said:

 

There is no gravitational field in an accelerated frame of reference, we are in flat space-time, the distance between the accelerating object and the stars varies unlike in a gravitational field, the Doppler effect is therefore kinematic, not gravitaionnel

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. 

 

11 minutes ago, chron44 said:

Hi,

The average vacuum energy is estimated to about 3 GeV/m^3 (the observed). If relying on this value (when
the calculated in extremely much higher). -And the Higgs field energy VEV, the vacuum expectation value, is
both observed in the LHC experiment and fairly calculated to about 246 GeV. How are these differences
explained in QM physics?

3 GeV versus 246 GeV?

I understand that the VEV amount is presented without any special volume in mind. But surely the VEV
isn't correlated to the cubic meter volume, though must be estimated to the Planck scale. Far minor
than the m^3 which the vacuum is given with. The VEV is a universal constant thought to fill all
universe with neither any much higher nor any lower energy content. This issue is raised with the
condition of both the vacuum energy volume and the Higgs field are without any elementary particles,
though being absolutely empty of any visible "matter". -Without even one single photon or neutrino
or whatever. The only content is the absolute vacuum itself (3 GeV).

The VEV can be regarded for proven with the Higgs boson discovery in 2012. And the vacuum content
have with the Planck Collaboration project also been verified.

/chron44

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.

3 hours ago, DanMP said:

So, here, on the Earth surface, we also see the light coming from stars straight above us "blueshifting" as long as our accelerometer shows that we are "accelerating" upward?

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 

3 hours ago, geordief said:

This is clearly right as it has been shown experimentally.

Do you have any other (or complementary) intuitive ways  of  sitting this process in our pattern of thoughts?

I  have always tried to think of this as some kind of "work done" as a body travels through both space and time, following different possible paths  but that approach   doesn't satisfy me (and is probably wrong as well)

 

I find no fault with your description but I wonder could there be other ways to describe this using words.

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

\[\vec{J}^2=J_x^2+J_y^2+j_z^2\]

 

 

1 hour ago, ImplicitDemands said:

 

Magnetism - The electrons have opposite spin causing the bodies to be magnetized

 

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.

2 hours ago, ImplicitDemands said:

I know I know. If you are curious what exactly it is I am doing to place a second circle atop the surface of a sphere with r=3 at the top right to get the r=5.121 value, or at what point the two given volumetric surfaces make contact, I assure you it is not a trigometric function that has been discovered yet or it would have been covered in my education

It is the dimensions being used in the mathematics of motion other than calculus which also does this but it misses how relative perspective factors into the parameters governing change over time. One needs to know how to factor (not integrate as if we were doing calculus) in the gravitational variance. As all shapes have a triangle at their core, trigonometry is really the basis of all geometry, a

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.

1 hour ago, swansont said:

 

Because that’s what happens in the Doppler effect. Red shift is shifted toward longer wavelengths and blue shift toward shorter. It’s observed to happen, so there’s no point in denying it.

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.

59 minutes ago, externo said:

 

I can't stop because you really don't understand relativity. You don't even know that the hypothesis of the invariance of the speed of light is a useless hypothesis for the theory, and that it only serves to get rid of eather

 

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 

On 4/4/2024 at 11:32 PM, Mordred said:

Christoffels for the FLRW metric in spherical coordinates.

 

ds2=−c(dt2)+a(t)1−kr2dr2+a2(t)r2dθ2+a2(t)r2sin2dϕ

 

 

gμν=⎛⎝⎜⎜⎜⎜−10000a21−kr20000a2r20000a2r2sin2θ⎞⎠⎟⎟⎟⎟

 

 

Γ0μν=⎛⎝⎜⎜⎜⎜00000a1−(kr2)0000a2r20000a2r2sin2θ⎞⎠⎟⎟⎟⎟

 

 

Γ1μν=⎛⎝⎜⎜⎜⎜⎜⎜0a˙ca00a˙caaa˙c(1−kr2)00001caa˙r200001caa˙sin2θ⎞⎠⎟⎟⎟⎟⎟⎟

 

 

Γ2μν=⎛⎝⎜⎜⎜⎜⎜00a˙ca0001r0a˙ca1r00000−sinθcosθ⎞⎠⎟⎟⎟⎟⎟

 

 

Γ3μν=⎛⎝⎜⎜⎜⎜⎜000a˙c0001r000cotθa˙ca1rcotθ0⎞⎠⎟⎟⎟⎟⎟

 

a˙ is the velocity of the scale factor if you see two dots its acceleration in time derivatives. K=curvature term

Newton limit geodesic

 

drdt2=−c2Γ100

 

Christoffel Newton limit

 

Γ100=GMc2r2

 

Covariant derivative of a vector Aλ

 

∇μAλ=∂μAλ+ΓλμνAν

 

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.

5 hours ago, Eise said:

Nope, did not work

Post what your trying and we can probably help out