Mordred

+1 I like that descriptive

\[{\small\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline Particle& Spin & g & Q &B&L_e &L_\mu&M (Mev)&\tau\\\hline \gamma&1&2&0&0&0&0&<3*10^{-33}&stable\\\hline e^-&1/2&2&-1&0&1&0&0.511&>2*10^{22}yrs\\\hline e+&1/2&2&1&0&-1&0&0.511&>2*10^{22}yrs\\\hline v_e&1/2&1&0&0&1&0&,5*10^{-5}&stable\\\hline \overline{v}_e&1/2&1&0&0&-1&0&<5*10^{-5}&stable\\\hline \mu^-&1/2&2&-1&0&0&1&105.7&2.2*10^{-6}sec\\\hline \mu^+&1/2&2&1&0&0&-1&105.7&2.2*10^{-5}sec\\\hline v_\mu&1/2&1&0&0&0&1&<0.25&>10^{32}yrs\\\hline \overline{v}_\mu&1/2&1&0&0&0&-1&<0.25&>10^{32}yrs \\\hline p&1/2&2&1&1&0&0&938.3&.10^{32}yrs\\\hline \overline{p}&1/2&2&-1&-1&0&0&938.3&>10^{32}yrs\\\hline n&1/2&2&0&1&0&0&939.6&898 sec\\\hline \overline{n}&1/2&2&&*-1&0&0&939.6&898 sec\\\hline \pi^0+&0&1&1&0&0&0&139.6&1.39*10^{-8}sec\\\hline \pi^-&0&1&0&0&0&0&135.0&8.7*10^{-17}sec\\\hline \pi^+&0&1&-1&0&0&0&139.6&2.6*10^{-8}sec\\\hline\end{array}}\] will have to go through and update these entries table is as follows g is degrees of freedom, electric charge Q, Baryon number B, (note need to add tau and tau neutrino for the lepton family, gauge bosons W,Z,g and Higgs as well as quarks) electron lepton number\[ L_e\] muon lepton number\[L_\mu\]

\[\array{ \mathfrak{g} \times X && \overset{R}{\longrightarrow} && T X \\ & {\llap{pr_2}}\searrow && \swarrow_{\rlap{p}} \\ && X }\] \[{\small\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline Particle& Spin & g & Q &B&L_e &L_\mu&M (Mev)&\tau\\\hline \gamma&1&2&0&0&0&0&<3*10^{-33}&stable\\\hline e^-&1/2&2&-1&0&1&0&0.511&>2*10^{22}yrs\\\hline e+&1/2&2&1&0&-1&0&0.511&>2*10^{22}yrs\\\hline v_e&1/2&1&0&0&1&0&,5*10^{-5}&stable\\\hline \overline{v}_e&1/2&1&0&0&-1&0&<5*10^{-5}&stable\\\hline \mu^-&1/2&2&-1&0&0&1&105.7&2.2*10^{-6}sec\\\hline \mu^+&1/2&2&1&0&0&-1&105.7&2.2*10^{-5}sec\\\hline v_\mu&1/2&1&0&0&0&1&<0.25&>10^{32}yrs\\\hline \overline{v}_\mu&1/2&1&0&0&0&-1&<0.25&>10^{32}yrs \\\hline p&1/2&2&1&1&0&0&938.3&.10^{32}yrs\\\hline \overline{p}&1/2&2&-1&-1&0&0&938.3&>10^{32}yrs\\\hline n&1/2&2&0&1&0&0&939.6&898 sec\\\hline \overline{n}&1/2&2&&*-1&0&0&939.6&898 sec\\\hline \pi^0+&0&1&1&0&0&0&139.6&1.39*10^{-8}sec\\\hline \pi^-&0&1&0&0&0&0&135.0&8.7*10^{-17}sec\\\hline \pi^+&0&1&-1&0&0&0&139.6&2.6*10^{-8}sec\\\hline\end{array}}\]

Any equation where the units on the LHS does not match the units on the RHS is invalid under dimensional analysis. Here for further detail http://web.mit.edu/2.25/www/pdf/DA_unified.pdf

Might help to know that in order for the cosmological constant to stay constant the. W=-1 too far from that value and it will vary over time. The options are still viable for an evolving cosmological constant but so far research is showing non evolving. If Lambda does evolve then you may have an eventual collapse as opposed to a big rip. Still going through the article however they seems to be using as negative pressure however still unclear on that myself till have a chance to better study the math. Yeah looks looks like the second terms in equation one is describing a vacuum scalar field. The positive and negative kinetic energy sign flips directly apply to that same equation. The stress tensor components in the article equation 10. The -T^00 component is the energy density term T^0_i is the mixed covariant/contravariant momentum Flux in the I direction with T^0_j being in the j direction. Just to help you better understand some of the equations in the article. Insofar as he is describing the stress momentum terms of H_{ij}. The majority of the other equations are fairly standard from thevFLRW metric including the related equations of state. Hope that helps. Edit; are you familiar with how the equations of state are derived ? That might help if the answer is no. I should also note that the article is in the Newton limit under GR and does not include quantum field theory itself (QFT equations are second order). Here is an article on first order perturbation theory as applied to QM. https://courses.physics.illinois.edu/phys485/fa2015/web/perturb.pdf

that would be great, I gathered much of details you described from the mathematics of the article I first found still going through it. Always interested in good resources, I tend to collect good examples of different field treatments so enjoy coming across ones I haven't come across before. They are always handy in model building

seems to be a bit of a glitch that the latex structures tend to drop ah well those equations are fairly straightforward to fix up.

The intended purpose will be to eventually migrate this into a full article using current cosmological parameters. Its a project I've been building towards for several years now and has always been a primary focus of my studies. Now I'd like to formalize it however evidently there is issues going on with lengthy edits as I just lost all the latex work above yet again lmao

good plan will do as much as possible

This thread will take me a considerable amount of time as I will be examining various treatments of BB nucleosynthesis and development of an eventual article of processes involved. for the initial stages I will simply be gathering the relevant formulas. Prior to symmetry Break Relevant equations The FLRW metric of the LCDM universe is used by the LCDM model of the Big bang to describe the evolution history of our Observable universe. The model starts at 10^{-43} seconds forward from a low entropy, hot dense state. One plausible explanation of how our universe began prior to that include quantum fluctuations. The model only describes our Observable portion as we do not know what occurs beyond the Cosmological event horizon. The FLRW metric is given as follows \[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]\] \[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\] where k is the curvature term, a is the scale factor both being dimensionless quantities. The contributions of each particle species via their corresponding equations of state is determines how our universe expands. The evolution history can be determines as a function of Cosmological redshift via the following equation \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\] where the standard model may be represented by the covariant derivative form of the Langrangian \[\mathcal{L}=\underbrace{\mathbb{R}}_{GR}-\overbrace{\underbrace{\frac{1}{4}F_{\mu\nu}F^{\mu\nu}}_{Yang-Mills}}^{Maxwell}+\underbrace{i\overline{\psi}\gamma^\mu D_\mu \psi}_{Dirac}+\underbrace{|D_\mu h|^2-V(|h|)}_{Higgs}+\underbrace{h\overline{\psi}\psi}_{Yukawa}\] \[V_{ckm}=V^\dagger_{\mu L} V_{dL}\] 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 \[\begin{pmatrix}\acute{d}\\\acute{s}\\\acute{b}\end{pmatrix}\begin{pmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{pmatrix}\begin{pmatrix}d\\s\\b\end{pmatrix}\] Bose Einstein Statistics \[n_i = \frac {g_i} {e^{(\varepsilon_i-\mu)/kT} - 1}\] Fermi-Dirac statistics \[ n_i = \frac{g_i}{e^{(\epsilon_i-\mu) / k T} + 1}\] Maxwell Boltzmann \[\frac{N_i}{N} = \frac {g_i} {e^{(\epsilon_i-\mu)/kT}} = \frac{g_i e^{-\epsilon_i/kT}}{Z}\] Saha Boltzmann equation (calculate hydrogen decoupling \[\frac{n_i+n_e}{n_i}=\frac{2}{\omega^3}\frac{g_i+1}{g_i}exp[-\frac{(\epsilon_i+1-\epsilon_i)}{k_BT}\]

The ones I find frustrating are the posters that resort to insult tactics after you spend considerable time trying to correct misconceptions, explain why they are wrong, or describe something that you support with material written by professionals in a particular field. It truly does make one feel that they are literally wasting their time. In those circumstances it never seems to matter how much supportive documentations such as professional peer review material, the relevant mathematics etc you present to support your statements. In those cases I have to constantly remind myself that even when you cannot help said individual you invariably help other readers. Eventually though one has to report the individual which the moderation staff does an excellent job of cooling the situation down.

Well that's a first I never heard of Vaidya spacetime any recommended literature on it ? I spotted this article which seems to have a half decent coverage but if you are familiar with better I would appreciate it "Geometry of the Vaidya spacetime" by Armand COUDRAY & Jean-Philippe NICOLAS https://arxiv.org/pdf/2101.06544

Be well

Thread reported I'm tired of your attitude

No it can involve rapidity but is not the only methodology. I do recall that discussion lol I actually enjoyed that discussion.

Sigh (hopefully you don't get insulted again) lets show you how this works. start with the Lorentz transformation. \[\acute{t}=\gamma(\frac{vx}{c^2})\] \[\acute{x}=\gamma(x-vt)\] \[\acute{y}=y\] \[\acute{z}=z\] the two equations most relevant are the first two note the velocity term velocity is the vector rate of change in position and direction. As you doubt everything I state and argue everything I state here https://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity Oh my it specifies inertia frames https://en.wikipedia.org/wiki/Lorentz_transformation you can read it for yourself and don't take my word for it if you choose acceleration is handled via rapidity the equations are also in that link. https://en.wikipedia.org/wiki/Free_fall Newtons laws of inertia https://en.wikipedia.org/wiki/Newton's_laws_of_motion note that the freefall link specifies no force acting upon the body it is in freefall no net force however Newtons laws of inertia still apply. The body is in motion, one can arbitrarily choose any event as the rest frame under SR it could be the twin leaving the planet or the stay at home twin both twins are in inertial frames (not accelerating, constant velocity). the same applies to freefall Alice is the observer on Earth. Bob is the Observer in freefall. The choice of who is the rest frame is arbitrary with the above transformation. Hopefully in the freefall state your know \[m_g=m_i\] equivalence principle gravitational mass is equivalent to inertial mass. if not google Principle of equivalence. we wont get into tidal forces just yet but succinctly if you have two bodies in freefall the vector direction is towards the CoM. the tidal force between the two bodies is \[\frac{d^2x}{dt^2}-\frac{MG}{r^3}x\] now I suggest you study again the above further applies to the addition of velocities \[U=v+\acute{U}\] where v is the velocity between observers U and \[\acute{U}\] is the two observers.

I Strongly suggest you study the terminology you will find those statements are 100% accurate under both SR and GR. You do not need to take my word for it pick up ant SR or GR textbook Seriously you post something for everyone to read where does reading thoughts enter the picture ? This is a forum anyone can respond at any time whether you like it or not. That's is part of the rules on a public forum that you agreed to when you signed up.

If your addressing a specific individual you should state that but when you leave a question on a physics forum. Anyone can answer said question. That is precisely the purpose of this forum

You asked what real waves I directly answered this question and answered it correctly

I don't particularly see describing it as a non causal wave to be an issue here. As long as it's understood to represent individual components of the train separately powered via the engine. I've still been considering the hyperbolic Lorentz transforms in regards to the different acceleration rates.

Will you please stop trying to insult me. I personally couldn't care what your opinion of me is but this is starting to get old real fast. Stick to the forum topic...

Well sorry if you took that as an insult it's a simple statement of fact nothing more. There are after all other readers

If I choose to reply I will do so.

Good analysis +1 Unfortunately no real wave travels faster than c. I will have time to respond further later on RL

Dissipates into the ground and effectively neutralizes.