# The Lagrangian equation...

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I request to initiate a thread on what I only know as the Lagrangian equation. The original published scientific paper derivation proof is unknown to me.
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Lagrangian equation:
$\mathcal{L} = \underbrace{\mathbb{R}}_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0$
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This Lagrangian equation appears to represent a massless field tensor fundamental field interaction Lagrangian combination zero summation action in natural units.
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The classical General Relativity Lagrangian tensor appears to describe a Ricci tensor on a smooth spacially flat Ricci maniold $\mathbb{R}$. (ref. 1, ref. 2)
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The massless field tensor Yang-Mills Maxwell Lagrangian term represents at the core of the unification of the electromagnetic force and weak forces $(U(1) \times SU(2))$ and quantum chromodynamics, the theory of the strong force $(SU(3))$ and predicts all the massless spin one Maxwells equations. (ref. 3, ref. 4)
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The massless field tensor Dirac Lagrangian term is a relativistic wave equation that describes all spin one-half particle interactions. (ref. 5)
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The massless field tensor Higgs Lagrangian term describes all spin zero Higgs field interactions. (ref. 6)
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The massless field tensor Yukawa coupling interaction term describes the interaction between a massless spin zero scalar field $\phi$ and a massless spin one-half Dirac field $\psi$ (ref. 7)
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Do you know where the original published scientific paper derivation proof is located?
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Reference:
Wikipedia - Ricci curvature: (ref. 1)
https://en.wikipedia.org/wiki/Ricci_curvature
Wikipedia - Riemannian manifold: (ref. 2)
https://en.wikipedia.org/wiki/Riemannian_manifold
Wikipedia - Maxwells equations: (ref. 3)
https://en.wikipedia.org/wiki/Maxwell's_equations#Formulation_in_SI_units_convention
Stackexchange - derivation of maxwells equations from field tensor lagrangian: (ref. 4)
https://physics.stackexchange.com/questions/3005/derivation-of-maxwells-equations-from-field-tensor-lagrangian
Wikipedia - Dirac Lagrangian: (ref. 5)
https://en.wikipedia.org/wiki/Dirac_equation#Dirac_Lagrangian
Wikipedia - Higgs field: (ref. 6)
https://simple.wikipedia.org/wiki/Higgs_field
Wikipedia - Yukawa Lagrangian: (ref. 7)
https://en.wikipedia.org/wiki/Yukawa_interaction#The_action

Edited by Orion1
source code correction.

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22 minutes ago, Orion1 said:

I request to initiate a thread on what I only know as the Lagrangian equation. The original published scientific paper derivation proof is unknown to me.

This just appears to be the Lagrangians of different interactions. As such, it would likely not be "derived" in any more detail.

In classical (nonrelativistic) mechanics, the Lagrangian is simply L = KE - PE

It's a way of solving problems using energy rather than forces. Which lends itself to QM problems, since you don't really talk about forces and kinematics at the quantum level.

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LOL I created a monster...

Unfortunately I do not have the pdf I got that formula from, I had a copy of the original pdf on my phone which I had lost a while back but had the equation written down. Lately I decided to break it down to identify the various terms in the equation and see how encompassing it is. Here is what I have thus far. I will simply cut and paste what I have however keep in mind much of it will be in a note to myself format including related formulas. (in a real sense personal study notes)

Quote

$\mathcal{L}=\underbrace{\mathbb{R}}_{GR}-\overbrace{\underbrace{\frac{1}{4}F_{\mu\nu}F^{\mu\nu}}_{Yang-Mills}}^{Maxwell}+\underbrace{i\overline{\psi}\gamma^\mu D_\mu \psi}_{Dirac}+\underbrace{|D_\mu h|^2-V(|h|)}_{Higgs}+\underbrace{h\overline{\psi}\psi}_{Yukawa}$

$D_\mu$ minimally coupled gauge covariant derivative. h Higg's bosonic field $\chi$ is the Goldstone boson (not shown above) Goldstone no longer applies after spontaneous symmetry breaking $\overline{\psi}$ is the adjoint spinor

$\mathcal{L}_h=|D\mu|^2-\lambda(|h|^2-\frac{v^2}{2})^2$

$D_\mu=\partial_\mu-ie A_\mu$ where $A_\mu$ is the electromagnetic four potential

QCD gauge covariant derivative

$D_\mu=\partial_\mu \pm ig_s t_a \mathcal{A}^a_\mu$ matrix A represents each scalar gluon field

Single Dirac Field

$\mathcal{L}=\overline{\psi}I\gamma^\mu\partial_\mu-m)\psi$

under U(1) EM fermion field equates to

$\psi\rightarrow\acute{\psi}=e^{I\alpha(x)Q}\psi$

due to invariance requirement of the Langrene above and with the last equation leads to the gauge field $A_\mu$

$\partial_\mu$ is replaced by the covariant derivitave

$\partial_\mu\rightarrow D_\mu=\partial_\mu+ieQA_\mu$

where $A_\mu$ transforms as $A_\mu+\frac{1}{e}\partial_\mu\alpha$

Single Gauge field U(1)

$\mathcal{L}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$

$F_{\mu\nu}=\partial_\nu A_\mu-\partial_\mu A_\nu$

add mass which violates local gauge invariance above

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^2A_\mu A^\mu$ guage invariance demands photon be massless to repair gauge invariance add a single complex scalar field

$\phi=\frac{1}{\sqrt{2}}(\phi_1+i\phi_2$

Langrene becomes

$\mathcal{L}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+|D_\mu \phi|^2-V_\phi$

where $D_\mu=\partial_\mu-ieA_\mu$

$V_\phi=\mu^2|\phi^2|+\lambda(|\phi^2|)^2$

$\overline{\psi}=\psi^\dagger \gamma^0$ where $\psi^\dagger$ is the hermitean adjoint and $\gamma^0$ is the timelike gamma matrix

the four contravariant matrix are as follows

$\gamma^0=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}$

$\gamma^1=\begin{pmatrix}0&0&0&1\\0&0&1&0\\0&0&-1&0\\-1&0&0&0\end{pmatrix}$

$\gamma^2=\begin{pmatrix}0&0&0&-i\\0&0&i&0\\0&i&0&0\\-i&0&0&0\end{pmatrix}$

$\gamma^3=\begin{pmatrix}0&0&1&0\\0&0&0&-1\\-1&0&0&0\\0&1&0&0\end{pmatrix}$

where $\gamma^0$ is timelike rest are spacelike

V denotes the CKM matrix usage

$\begin{pmatrix}\acute{d}\\\acute{s}\\\acute{b}\end{pmatrix}\begin{pmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{pmatrix}\begin{pmatrix}d\\s\\b\end{pmatrix}$

$V_{ckm}=V^\dagger_{\mu L} V_{dL}$

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

$\gamma^5=i\gamma_0,\gamma_1,\gamma_2,\gamma_3$

4 component Minkowskii with above 4 component Dirac Spinor and 4 component Dirac gamma matrixes are defined as

${\gamma^\mu\gamma^\nu}=2g^{\mu\nu}\mathbb{I}$ where $\mathbb{I}$ is the identity matrix.

in Chiral basis $\gamma^5$ is diagonal in $2\otimes 2$ the gamma matrixes are

$\begin{pmatrix}0&\sigma^\mu_{\alpha\beta}\\\overline{\sigma^{\mu\dot{\alpha}\beta}}&0\end{pmatrix}$

$\gamma^5=i{\gamma_0,\gamma_1,\gamma_2,\gamma_3}=\begin{pmatrix}-\delta_\alpha^\beta&0\\0&\delta^\dot{\alpha}_\dot{\beta}\end{pmatrix}$

$\mathbb{I}=\begin{pmatrix}\delta_\alpha^\beta&0\\0&\delta^\dot{\alpha}_\dot{\beta}\end{pmatrix}$

Lorentz group identifiers in $(\frac{1}{2},0)\otimes(0,\frac{1}{2})$

$\sigma\frac{I}{4}=(\gamma^\mu\gamma^\nu)=\begin{pmatrix}\sigma^{\mu\nu\beta}_{\alpha}&0\\0&-\sigma^{\mu\nu\dot{\alpha}}_{\dot{\beta}}\end{pmatrix}$

$\sigma^{\mu\nu}$ duality satisfies $\gamma_5\sigma^{\mu\nu}=\frac{1}{2}I\epsilon^{\mu\nu\rho\tau}\sigma_{\rho\tau}$

a 4 component Spinor Dirac field is made up of two mass degenerate Dirac spinor fields U(1) helicity

$(\chi_\alpha(x)),(\eta_\beta(x))$

$\psi(x)=\begin{pmatrix}\chi^{\alpha\beta}(x)\\ \eta^{\dagger \dot{\alpha}}(x)\end{pmatrix}$

the $(\alpha\beta)=(\frac{1}{2},0)$ while the $(\dot{\alpha}\dot{\beta})=(0,\frac{1}{2})$

this section relates the SO(4) double cover of the SU(2) gauge requiring the chiral projection operator next.

chiral projections operator

$P_L=\frac{1}{2}(\mathbb{I}-\gamma_5=\begin{pmatrix}\delta_\alpha^\beta&0\\0&0\end{pmatrix}$

$P_R=\frac{1}{2}(\mathbb{I}+\gamma_5=\begin{pmatrix}0&0\\ 0&\delta^\dot{\alpha}_\dot{\beta}\end{pmatrix}$

Weyl spinors

$\psi_L(x)=P_L\psi(x)=\begin{pmatrix}\chi_\alpha(x)\\0\end{pmatrix}$

$\psi_R(x)=P_R\psi(x)=\begin{pmatrix}0\\ \eta^{\dagger\dot{a}}(x)\end{pmatrix}$

also requires Yukawa couplings...SU(2) matrixes given by

$diag(Y_{u1},Y_{u2},Y_{u3})=diag(Y_u,Y_c,Y_t)=diag(L^t_u,\mathbb{Y}_u,R_u)$

$diag(Y_{d1},Y_{d2},Y_{d3})=diag(Y_d,Y_s,Y_b)=diag(L^t_d,\mathbb{Y}_d,R_d$

$diag(Y_{\ell 1},Y_{\ell 2},Y_{\ell3})=diag(Y_e,Y_\mu,Y_\tau)=diag(L^T_\ell,\mathbb{Y}_\ell,R_\ell)$

the fermion masses

$Y_{ui}=m_{ui}/V_u$

$Y_{di}=m_{di}/V_d$

$Y_{\ell i}=m_{\ell i}/V_\ell$

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

CKM is also a different parametrisation than the Wolfenstein Parametrization in what way (next study)[/quote]

I haven't as of yet included the numerous Langrene's for scalar, spin 1/2 fermion fields, in either two component or 4 component though I do have those Langrene's handy I am checking the source against other papers to ensure the versions I have a fairly generic in their format. As mentioned much of this is still in a study mode layout.

However from what I have garnished the formula would be applicable to only the standard model of particles with a single Higg's boson under the minimal standard model. I does not include the terms for the MSSM supersymmetric. As mentioned it appears valid to a sense after the spontaneous symmetry break in so far as it doesn't include the Goldstone bosons. (assuming those bosons are a requirement there is some debate on that )

lol I screwed up on the quote section ah well the details are there. Anyways in a sense I have been trying to in essence reverse engineer the equation.

Some of the links you have posted may come in handy in my endeavor

4 hours ago, swansont said:

This just appears to be the Lagrangians of different interactions. As such, it would likely not be "derived" in any more detail.

I have to agree with this, in so far as to locating any proof for the equation itself.

Edited by Mordred

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On 2/8/2019 at 6:05 AM, swansont said:

In classical (nonrelativistic) mechanics, the Lagrangian is simply L = KE - PE   ﻿﻿

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Please consider Newtons second law and the relativistic Newtons second law for a moment.
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$\gamma$ - Lorentz factor
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Newtons second law: (ref. 1)
$\mathbf{F} = \frac{d \mathbf{p}}{dt} = \frac{d\left(m \mathbf{v} \right)}{\mathrm{d}t} = m {\frac{d\mathbf{v}}{\mathrm{d}t}} = m \frac{ds}{dt^2}$
$\boxed{\mathbf{F} = m \frac{ds}{dt^2}}$
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Relativistic Newtons second law:
$\boxed{\mathbf{F} = \gamma m \frac{ds}{dt^2}}$
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Newtons second law in Lagrangian form: (ref. 2)
$F^{a} = m \left( \frac{d^{2} \xi^{a}}{dt^{2}}+ \Gamma^{a}{}_{bc} \frac{d\xi^{b}}{dt} \frac{d\xi^{c}}{dt} \right)$
$\;$
Relativistic Newtons second law in Lagrangian form:
$\boxed{F^{a} = \gamma m \left( \frac{d^{2} \xi^{a}}{dt^{2}}+ \Gamma^{a}{}_{bc} \frac{d\xi^{b}}{dt} \frac{d\xi^{c}}{dt} \right)}$
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Thus including all classical nonrelativistic Lagrangian mechanics into relativistic Lagrangian mechanics?
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Reference:
Wikipedia - Newtons second law: (ref. 1)
https://en.wikipedia.org/wiki/Newton's_laws_of_motion#Newton's_second_law
Wikipedia - Newtons second law in Lagrangian form: (ref. 2)
https://en.wikipedia.org/wiki/Lagrangian_mechanics#From_Newtonian_to_Lagrangian_mechanics

Edited by Orion1
source code correction.

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going straight from Newton to a relativistic Langrangian by simply adding gamma isn't sufficient by itself. You must also preserve Lorentz invariance which will require the use of proper time given as $\tau$. Recall different observers will measure the variant quantities differently this includes time. Even then its not complete, as you will need to include the 4 momentum and 4 velocity.

See here, it mentions some of the issues I didn't

Another related reference here

A primary goal is to ascertain from the Euler-Langrangian the geodesic equation.

This wiki link has a good breakdown of how to employ the Euler Langrangian to derive the geodesic

Edited by Mordred

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Relativistic Lagrangian Lorentz factor: (ref. 1, ref. 2, ref. 6)
$dt = \gamma\left(\dot{\mathbf{r}} \right) d\tau$
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$\gamma\left(\dot{\mathbf{r}} \right) = \frac{dt}{d\tau} = \frac{1}{\sqrt{1 - \frac{\dot{\mathbf{r}}^2}{c^2}}}$
$\boxed{\gamma\left(\dot{\mathbf{r}} \right) = \frac{1}{\sqrt{1 - \frac{\dot{\mathbf{r}}^2}{c^2}}}}$
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$\dot{\mathbf{r}} = \frac{d\mathbf{r}}{dt}$
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Relativistic Lagrangian neutral particle total energy integration via substitution: (ref. 2)
$E_{t} = m_0 c^2 \frac{dt}{d \tau} = \gamma\left(\dot{\mathbf{r}} \right) m_0 c^2 = \frac{m_0 c^2}{\sqrt {1 - \frac{\dot{\mathbf{r}}^2 \left(t \right)}{c^2}}} = m_0 c^2 + {1 \over 2} m_0 \dot{\mathbf{r}}^2 \left(t \right) + {3 \over 8} m_0 \frac{\dot{\mathbf{r}}^4 \left(t \right)}{c^2} + \cdots$
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Relativistic Lagrangian neutral particle total energy: (ref. 2)
$\boxed{E_{t} = \frac{m_{0} c^2}{\sqrt{1 - \frac{\dot{\mathbf{r}}^2 \left(t \right)}{c^2}}}}$
$\;$
Classical neutral particle kinetic energy:
$E_{k} = \frac{m_{0} v^{2}}{2} = \frac{m_{0}}{2} \frac{ds^{2}}{dt^{2}}$
$\boxed{E_{k} = \frac{m_{0}}{2} \frac{ds^{2}}{dt^{2}}}$
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Classical Lagrangian neutral particle kinetic energy: (ref. 4)
$E_{k} = \frac{m_{0} g_{bc}}{2} \frac{\mathrm{d}\xi^{b}}{\mathrm{d}t} \frac{\mathrm{d}\xi^{c}}{\mathrm{d}t}$
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Relativistic Lagrangian neutral particle kinetic energy:
$E_{k} = m_{0} c^{2} \left(\gamma\left(\dot{\mathbf{r}} \right) - 1 \right)$
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Newtons second law for neutral particle integration via substitution: (ref. 3)
$\mathbf{F} = \frac{d \mathbf{p}}{dt} = \frac{d(m_{0} \mathbf{v})}{\mathrm{d}t} = m_{0} \frac{d\mathbf{v}}{\mathrm{d}t} = m_{0} \frac{ds}{dt^2}$
$\;$
Newtons second law for neutral particle:
$\boxed{\mathbf{F} = m_{0} \frac{ds}{dt^2}}$
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Relativistic Newtons second law for neutral particle:
$\boxed{\mathbf{F} = \gamma m_{0} \frac{ds}{dt^2}}$
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General relativity geodesic equation: (ref. 4)
$\frac{d^{2}x^{\mu}}{dt^{2}} + \Gamma^{\mu}{}_{\alpha \beta} \frac{dx^{\alpha}}{dt} \frac{dx^{\beta}}{dt} = 0$
$\;$
Newtons second law in Lagrangian form for neutral particle: (ref. 5)
$F^{a} = m_{0} \left(\frac{d^{2} \xi^{a}}{dt^{2}}+ \Gamma^{a}{}_{bc} \frac{d\xi^{b}}{dt} \frac{d\xi^{c}}{dt} \right)$
$\;$
Relativistic Newtons second law in Lagrangian form for neutral particle:
$\boxed{F^{a} = \gamma\left(\dot{\mathbf{r}} \right) m_{0} \left( \frac{d^{2} \xi^{a}}{dt^{2}} + \Gamma^{a}{}_{bc} \frac{d\xi^{b}}{dt} \frac{d\xi^{c}}{dt} \right)}$
$\;$
Relativistic Lagrangian for a neutral particle:
$\mathcal{L} = E_{k} - E_{p}$
$\;$
$\boxed{\mathcal{L} = m_{0} c^{2} \left(\gamma\left(\dot{\mathbf{r}} \right) - 1 \right) - E_{p}}$
$\;$
Relativistic Lagrangian integration via substitution:
$\mathcal{L} = \sum_{1}^{n} E_{k}\left(n \right) - \sum_{1}^{n} E_{p}\left(n \right) = \sum_{1}^{n} \mathcal{L}\left(n \right) = 0$
$\;$
Relativistic Lagrangian:
$\boxed{\mathcal{L} = \sum_{1}^{n} \mathcal{L}\left(n \right) = 0}$
$\;$

The Lagrangian equation integration via substitution:
$\mathcal{L} = \sum_{1}^{n} \mathcal{L}\left(n \right) = \underbrace{\mathbb{R}}_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0 \; \; \; \; \; \; n = 5$

The Lagrangian equation:
$\boxed{\mathcal{L} = \underbrace{\mathbb{R}}_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0} \; \; \; \; \; \; n = 5$
$\;$

19 hours ago, Mordred said:

going straight from Newton to a relativistic Langrangian by simply adding gamma isn't sufficient by itself. You must also preserve Lorentz invariance which will require the use of proper time given as τ . Recall different observers will measure the variant quantities differently this includes time. Even then its not complete, as you will need to include the 4 momentum and 4 velocity.  ﻿

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I think that this revision has compensated for the 4 momentum and 4 velocity under general relativity with respect to proper time $\tau$, through the Lorentz factor. The 4 momentum and 4 velocity is intrinsic to general relativity as required by four-dimensional space-time.

19 hours ago, Mordred said:

A primary goal is to ascertain from the Euler-Langrangian the geodesic equation.﻿﻿

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I note two general relativity geodesic equation forms, a spacial and a temporal form.

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General relativity spacial geodesic equation: (ref. 4)
$\frac{d^{2}x^{\mu}}{ds^{2}} + \Gamma^{\mu}{}_{\alpha \beta} \frac{dx^{\alpha}}{ds} \frac{dx^{\beta}}{ds} = 0$
$\;$
General relativity temporal geodesic equation: (ref. 4)
$\frac{d^{2}x^{\mu}}{dt^{2}} + \Gamma^{\mu}{}_{\alpha \beta} \frac{dx^{\alpha}}{dt} \frac{dx^{\beta}}{dt} = 0$
$\;$

On 2/8/2019 at 6:05 AM, swansont said:

This just appears to be the Lagrangians of different interactions. As such, it would likely not be "derived" in any more detail.

Do you agree with this mathematical symbolic formalism derivation revision for the formal Lagrangian equation?
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$\;$
Reference:
Wikipedia - The Lorentz factor: (ref. 1)
https://en.wikipedia.org/wiki/Lorentz_factor
Wikipedia - Relativistic Lagrangian mechanics: (ref. 2)
https://en.wikipedia.org/wiki/Relativistic_Lagrangian_mechanics#Coordinate_formulation
Wikipedia - Newtons second law: (ref. 3)
https://en.wikipedia.org/wiki/Newton's_laws_of_motion#Newton's_second_law
Wikipedia - Geodesics in general relativity: (ref. 4)
https://en.wikipedia.org/wiki/Geodesics_in_general_relativity
Wikipedia - Newtons second law Lagrangian form: (ref. 5)
https://en.wikipedia.org/wiki/Lagrangian_mechanics#From_Newtonian_to_Lagrangian_mechanics
Wikipedia - Four-velocity: (ref. 6)
https://en.wikipedia.org/wiki/Four-velocity

Edited by Orion1
source code correction.

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1 hour ago, Orion1 said:

Do you agree with this mathematical symbolic formalism derivation revision for the formal Lagrangian equation?

If you're trying to check and see if you're correct, you should check this against your references.

1 hour ago, Orion1 said:

You cited a wikipedia reference that has discussion and also a list of references. I would check there.

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Yes the four momentum is instrinsic in GR however it requires the use of indices that follow the Einstein summation for covariant and contravariant terms.

see here to see how this can apply in the above

the remainder of the equation that you are trying to fill in the relativity section is in the 4 momentum, 4  force and 3 velocity format. This way you have the same coordinate relations throughout the equation. Though it is assumed direction of motion would be in the $x^1$ direction this isn't always the case.

Now I am going to ask a related question :How much affect does gravity play on the path integral relations that the first equation describes in its particle to particle scattering ? (side note the other portions are already Lorentz invariant).

see here for further details on the Lorentz group and Lorentz invariance with ( you will see that the RHS of the $\underbrace{\mathbb{R}}_{relativity}$ is already Lorentz invariant in the terms. (side note QFT does this via the Klein Gordon equation as opposed to the Schrodinger )

Which brings to mind another question one can ask. Why was the $\underbrace{\mathbb{R}}_{relativity}$ left unfilled ?

personally my feelings on this is that we haven't got a  working quantum theory of gravity that doesn't suffer the renormalization problem sufficient to extrapolate the hypothetical graviton interactions (the graviton isn't yet part of the standard model of particles).

Although if such a graviton could be possible with the spin 2 being the more likely (under research) spin 0 is also plausible. This in turn affects the degrees of freedom required to determine the gauge group representations. One must account for all effective degrees of freedom.

see here for an example equation 36 for an example of the QCD Langrangian.

This articles has pertinent details to understand a large portion of the equation in the OP. (the last article should provide a sufficient answer to the question of whether your Langrangian attempts suffice). Keep in mind the line from the introductory,

Quote

It is a quantum ﬁeld theory described by a deceptively simple Lagrangian

the remainder of the article should hone in on how complex it really is....

In final note an effective Langrangian for the relativity portion should correlate to all the effective degrees of freedom that define the spin 2 statistics. A source of observational evidence is the graviton waves (quadrupolar ).

So I would start here.

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$

if you research these groups you will find much of the work in terms of the Langrenians is already done.

see this example on massive spin 2. gives other spin examples also

https://arxiv.org/pdf/hep-th/0609170.pdf equation 21. Which reflects that the $\underbrace{\mathbb{R}}_{relativity}$ will correspond to something similar to this.

$\underbrace{|D_{\mu} D_{\nu}|\upsilon_{\alpha}}_{relativity}$ ROUGH EXAMPLE ONLY>>>>one lack being massless particles...

Edited by Mordred

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Anyways that should provide a good direction of approach. I've watched your other thread though I have to catch up to where your at with it when I get time lol to know how serious you approach a topic and have a measure of your skills in that topic. Quite frankly its fun to study with all the relevancy to the topic.

Hope this helps...

note to above yes graviton as a gauge boson would likely be massless. One still need to correlate the possibility under group. One also sees the $D_\mu$ above this is the Differential matrix , and it correlates the difference between the covariant and contravariant terms. It will change in values as per the application applied in particular in different field treatments.

Edited by Mordred

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On 2/11/2019 at 6:18 PM, Mordred said:

Yes the four momentum is instrinsic in GR however it requires the use of indices that follow the Einstein summation for covariant and contravariant terms.

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Affirmative, according to Wikipedia, The action S is given by:
$S = -mc \int ds = \int L dt$
$\;$
Where L is the relativistic Lagrangian for a free particle:
$L = -mc^{2} \sqrt{1 - \frac {v^{2}}{c^{2}}}$
$\;$

And my solution for the proper time relativistic Lagrangian for a free particle:

$\boxed{\mathcal{L} = - \frac{m_{0} c^{2}}{\gamma\left(\dot{\mathbf{r}} \right)} = -m_{0} c^2 \sqrt{1 - \frac{\dot{\mathbf{r}}^2 \left(t \right)}{c^2}}}$
$\;$

On 2/11/2019 at 6:18 PM, Mordred said:

How much affec﻿t does gravity play on the path integral relations that the first equation describes in its particle to particle scattering ?

$\;$
A free particle that encounters a gravity field potential will always form a closed path in x,y,z, and time dilation t, and invoke Keplers laws:
$L = - \frac{m_0 c^2}{\gamma\left( \dot{\mathbf{r}} \right)} - V\left(\mathbf{r}, \dot{\mathbf{r}}, t \right)$
$\;$
$\mathbf{v} = \dot{\mathbf{r}} = \frac{d\mathbf{r}}{dt} = \left(\frac{dx}{dt} , \frac{dy}{dt} , \frac{dz}{dt} \right)$
$\;$
Except in the case for a hyperbolic trajectory with escape velocity, where the path integral is inflection curved at the point source for the gravitational field potential.
$\;$

On 2/11/2019 at 6:18 PM, Mordred said:

Why was the Rrelativity left unfilled ?

Because, It is implied that the Ricci tensor is still the classical theory of general relativity in this form.

On 2/11/2019 at 6:18 PM, Mordred said:

we haven't got a  working quantum theory of gravity that doesn't suffer the renormalization problem sufficient to extrapolate the hypothetical graviton interactions (the graviton isn't yet part of the standard model of particles).

$\;$
Einstein's field equations:
$G_{\mu \nu} = R_{\mu \nu} - \frac{R g_{\mu \nu}}{2} = \frac{8 \pi G}{c^{4}} T_{\mu \nu}$
$\;$
Einstein's field equations in natural units:
$G_{\mu \nu} = R_{\mu \nu} - \frac{R g_{\mu \nu}}{2} = 8 \pi T_{\mu \nu}$
$\;$
Einstein's field equations Ricci tensor in natural units:
$\boxed{ R_{\mu \nu} = 8 \pi T_{\mu \nu} + \frac{R g_{\mu \nu}}{2} }$
$\;$
Lagrangian equation:
$\mathcal{L} = \underbrace{ \mathbb{R} }_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0$
$\;$
General relativity Lagrangian equation:
$\boxed{\mathcal{L} = \underbrace{8 \pi T_{\mu \nu} + \frac{R g_{\mu \nu}}{2}}_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0}$
$\;$

Conventional gravitational waves that are quantized below the Planck radius with a total Planck energy would be indistinguishable from what scientists refer to as gravitons. Absent a total Planck energy available to generate them, scientists will never observe them to add them to the standard model.
$\;$
A massless gravitational wave is still massless on a Planck scale, and the result is still a tensor field.

$\;$
Tensor field: (ref. 1,pg. 21, eq. 1.68)
$T^{' \mu \nu} \left(x' \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right)$
$\;$
Lagrangian equation for a massless Planck graviton:
$\boxed{\mathcal{L} = \underbrace{ \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right) }_{QG} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0}$
$\;$
General relativity and Planck quantum gravity identity:
$\boxed{8 \pi T_{\mu \nu} + \frac{R g_{\mu \nu}}{2} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right)}$
$\;$
Reference:
Lorentz Group and Lorentz Invariance: (ref. 1)
https://gdenittis.files.wordpress.com/2016/04/ayudantiavi.pdf

Edited by Orion1

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excellent much better, the only thing I would now look into is the technicality of formalism matching and sign convention matching so that the equation is the same in both throughout.

The equations to the RHS of the relativity underbrace is using conformal as opposed to a canonical treatment with metric signature (+,-,-,-). It is also in Spinor representation form.

Other than that I can see no other main issues as a standalone what you have under the QG underbrace is certainly accurate and includes the spin 2 statistics. So the above is just making sure we match the methodology to the remainder of the equation. Hence why I chose to use the Differential operators from that link.

You might however consider staying in the weak field limit for particle to particle interactions $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ this is mainly due to gravity being such as weak force on the particle to particle regime. Also the bulk of QFT operates in the Newton limit regime.

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13 hours ago, Mordred said:

the technicality of formalism matching and sign convention matching so that the equation is the same in both throughout.

The equations to the RHS of the relativity underbrace is using conformal as opposed to a canonical treatment with metric signature (+,-,-,-). It is also in Spinor representation form.

Affirmative, that is correct.
$\;$
Einstein's field equations:
$G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = \frac{8 \pi G}{c^{4}} T_{\mu \nu}$
$\;$
Einstein's field equations in natural units:
$G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = 8 \pi T_{\mu \nu}$
$\;$
Einstein's field equations Ricci tensor in natural units:
$\boxed{ R_{\mu \nu} = 8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu}}$
$\;$
Lagrangian equation:
$\mathcal{L} = \underbrace{ \mathbb{R} }_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0$
$\;$
General relativity Lagrangian equation:
$\boxed{\mathcal{L} = \underbrace{8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu}}_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0}$
$\;$
A massless gravitational wave is still massless on a Planck scale, and the result is still a tensor field.
$\;$
Tensor field: (ref. 1, pg. 21, eq. 1.68)
$T^{' \mu \nu} \left(x' \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right)$
$\;$
The general relativity Ricci tensor is a tensor field:
$\boxed{R_{\mu \nu} = T^{' \mu \nu} \left(x' \right)}$
$\;$
Lagrangian equation for a massless Planck graviton:
$\boxed{\mathcal{L} = \underbrace{ \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right) }_{QG} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0}$
$\;$
General relativity and Planck quantum gravity identity:
$\boxed{8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)}$
$\;$
General relativity spacetime metric:
$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$
$\;$
General relativity spacetime metric and Planck quantum gravity identity:
$\boxed{8 \pi T_{\mu \nu} + \frac{1}{2} R \left(\eta_{\mu \nu} + h_{\mu \nu} \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)}$
$\;$
$\;$
Reference:
Lorentz Group and Lorentz Invariance: (ref. 1)
https://gdenittis.files.wordpress.com/2016/04/ayudantiavi.pdf

Edited by Orion1
source code correction.

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Woah wait a minute, you have the correct metric signature {+,-,-,-} however you have the Lorentz group tensor field irrep $\Lambda_\alpha^\mu \Lambda_\beta^\nu T^{\alpha\beta} (x)$ as General Relativity spacetime metric and Planck quantum gravity identity yet the proof included in the reference you provided for that term doesn't include the quantum regime in terms of the Planck HUP. That formula is the macro scale not at the quantum regime ?

Also describe $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ as the GR weak field limit it is a specific class of solution 1 of three main class of solutions under GR. You have the vacuum, the weak field and the Schwartzchild metric as the main categories. The vacuum is your scalar field solution, the weak field is the Newton approximation format. Applies well in particle physics as gravity is weak at that scale however under strong relativistic effects the Schwarzschild metric is the better approximation. Now all these are inclusive under the tensor field representation $\Lambda_\alpha^\mu \Lambda_\beta^\nu T^{\alpha\beta} (x)$ However you require additional terms not in this group to handle the quantum harmonic oscillator and HUP.

The three solutions above will fall under

$g_{\mu\nu}=g_{\mu\nu}+h_{\mu\nu}$ this is what you can place under the title General relativity spacetime metric as it includes the full range of solutions.

Anyways its Valentines day so will look at this further tomorrow.

edit: never mind the concern I had on $\Lambda_\alpha^\mu \Lambda_\beta^\nu T^{\alpha\beta} (x)$. Found the terms $\Lambda=exp(-iw_{\mu\nu}J^{\mu\nu}/2$ dang arbitrary matrix $\Lambda$

Let me look at this further you have the right approach but I need to rehash the spin 2 connections, This will become critical to maintain the chirality basis with the remainder of the equation. The iw term corresponds to the QFT operators so I'm good on that lol dang $\Lambda$.

The detail that the remaining portion of the original equation is already Poincare/Lorentz invariant so all this is already accounted for in their derivatives. I can't shake the feeling that all we need to apply is the spin connection itself common format  $D_\mu V_\nu^\alpha$ see here

Edited by Mordred

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Adding  some relevant relations pertaining to the derivatives above for potential references and time saving. In a sense reminders...

"A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch-Gordan coeﬃcients " see arxiv next link under group irrep.

Hermitean operators are self adjoint operators.

I need to make a correction on my Conformal statement in the above post, the equations to the right of the Poincare group are under canonical basis not conformal.

Edited by Mordred

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Ah found a beautiful treatment with the $D_\mu$ operators under the U(1), SU(3) and SU(3) gauge fields. The layout representations of the Langrangian under each group irrep is excellently done. Several of these equations I am going to latex out as I have time now and I know I will be referring to them to break down the QED,QCD, and Higgs portions of the OP equation. They will all be from the same Dissertation paper.

U(1) Langrangian single component fields.

$\mathcal{L}=\overline{\Psi}(i\gamma^\mu \partial_\mu-m)\Psi$

$U(1)_{em}$ fermion gauge field transforms

$\Psi\rightarrow\acute{\psi}=e{i\alpha(x)Q}\Psi$

$\partial_\mu \rightarrow D_\mu ieQ A_mu$

$A_\mu \rightarrow A_\mu +\frac{1}{e\partial_mu \alpha}$

under covariant derivative terms ($D_\mu$ massless

$\mathcal{L}=\underbrace{\overline{\Psi}(i\gamma^\mu \partial_\mu-m)\psi}_{free fermion}-\underbrace{e\overline{\Psi}\gamma^\mu Q\Psi A_\mu}_{interaction}$

Field strength tensor

$F_{\mu\nu}=\partial_\mu \alpha_\nu-\partial_\nu \alpha_\mu$

$\mathcal{L}=\underbrace{\overline{\Psi}(i\gamma^\mu \partial_\mu-m)\Psi}_{free fermion}-\underbrace{e\overline{\Psi}\gamma^\mu Q\Psi A_\mu}_{interaction}-\underbrace{\frac{1}{4}F_{\mu\nu}F^{\mu\nu}}_{k.e. of A_\mu}$

Electroweak

4 component

two component on Majarona spinor basis later post for  neutrino chirality Right hand rule terms further reference to CKM under particle data group treatment. RHS reminder signature (+,-,-,-).

see equations 2.9 to 2.22 to arrive at QCD Langrangian. (Glashow) quaternion/verbian\tetrad under Gell Mann matrix treatments). Pertains to the Weyl Dirac and Majarona spinors. (recall Majorona must be charge neutral under CPT.)

will latex in equations 2.22 later on (lol want to maximize future cut and paste operations in this thread)

need a break lmao

Edited by Mordred

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On 2/14/2019 at 9:51 PM, Mordred said:

GR weak field limit it is a specific class of solution 1 of three main class of solutions under GR.﻿

Affirmative, revision complete.
$\;$
Einstein's field equations:
$G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = \frac{8 \pi G}{c^{4}} T_{\mu \nu}$
$\;$
Einstein's field equations in natural units:
$G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = 8 \pi T_{\mu \nu}$
$\;$
Einstein's field equations Ricci tensor in natural units:
$\boxed{ R_{\mu \nu} = 8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu}}$
$\;$
Lagrangian equation:
$\mathcal{L} = \underbrace{ \mathbb{R} }_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0$
$\;$
General relativity Lagrangian equation:
$\boxed{\mathcal{L} = \underbrace{8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu}}_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0}$
$\;$
A massless gravitational wave is still massless on a Planck scale, and the result is still a tensor field.
$\;$
Tensor field: (ref. 1,pg. 21, eq. 1.68)
$T^{' \mu \nu} \left(x' \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right)$
$\;$
The general relativity Ricci tensor is a tensor field:
$\boxed{R_{\mu \nu} = T^{' \mu \nu} \left(x' \right)}$
$\;$
Lagrangian equation for a massless Planck graviton:
$\boxed{\mathcal{L} = \underbrace{ \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right) }_{QG} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0}$
$\;$
General relativity and Planck quantum gravity identity:
$\boxed{8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)}$
$\;$
General relativity weak field limit spacetime metric:
$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$
$\;$
General relativity weak field limit spacetime metric and Planck quantum gravity identity 1:
$\boxed{8 \pi T_{\mu \nu} + \frac{1}{2} R \left(\eta_{\mu \nu} + h_{\mu \nu} \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)}$
$\;$
General relativity weak field limit spacetime metric and Planck quantum gravity identity 2:
$\boxed{8 \pi T_{\mu \nu} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right) - \frac{1}{2} R \left(\eta_{\mu \nu} + h_{\mu \nu} \right)}$
$\;$
$\;$
Reference:
Lorentz Group and Lorentz Invariance: (ref. 1)
https://gdenittis.files.wordpress.com/2016/04/ayudantiavi.pdf

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Don't see anything wrong with the above I will still be editing last post, I want to keep the equations from that reference I am referencing in the same post. Want to fill as many of the connections to the OP equation. (will help ensure the accuracy of our derivatives).

On a personal note I would like to take our examinations of the various Langrangian's to the SO(10) regime under Pati-Salam in the particle data group restricted to the extended SM (MSM) models with minimal correlations to the MSSM supersymmetric groups with the above. I would like this thread to eventually include the SO(10) terms, with examinations of each group. (its a good study thread) .

personal reminder note (Clifford algebra on double cover Poincare group with operator $\mathbb{Z}^2)$

Edited by Mordred

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On 2/16/2019 at 1:22 AM, Mordred said:

Don't see anything wrong with the above I will still be editing last post, I want to keep the equations from that reference I am referencing in the same post. Want to fill as many of the connections to the OP equation. (will help ensure the accuracy of our derivatives).

Affirmative, revision complete.
$\;$
Einstein's field equations:
$G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = \frac{8 \pi G}{c^{4}} T_{\mu \nu}$
$\;$
Einstein's field equations in natural units:
$G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = 8 \pi T_{\mu \nu}$
$\;$
Einstein's field equations Ricci tensor in natural units:
$\boxed{ R_{\mu \nu} = 8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu}}$
$\;$
Lagrangian equation:
$\mathcal{L} = \underbrace{ \mathbb{R} }_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0$
$\;$
General relativity Lagrangian equation:
$\boxed{\mathcal{L} = \underbrace{8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu}}_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0}$
$\;$
A massless gravitational wave is still massless on a Planck scale, and the result is still a metric tensor field.
$\;$
Metric tensor field: (ref. 1, pg. 21, eq. 1.68)
$T^{' \mu \nu} \left(x' \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right)$
$\;$
The general relativity Ricci tensor is a metric tensor field:
$\boxed{R_{\mu \nu} = T^{' \mu \nu} \left(x' \right)}$
$\;$
Lagrangian equation for a massless Planck graviton:
$\boxed{\mathcal{L} = \underbrace{ \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right) }_{QG} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0}$
$\;$
General relativity and Planck quantum gravity identity:
$\boxed{8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)}$
$\;$
General relativity weak field limit spacetime metric: (ref. 2)
$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$
$\;$
$\eta_{\mu \nu}$ - perturbed non-dynamical background metric
$h_{\mu \nu}$ - true metric deviation of $g_{\mu \nu}$ from flat spacetime
$\;$
$h_{\mu \nu}$ must be negligible compared to $\eta_{\mu \nu}$:
$|h_{\mu \nu}| \ll 1$
$\;$
General relativity weak field limit spacetime metric and Planck quantum gravity identity 1:
$\boxed{8 \pi T_{\mu \nu} + \frac{1}{2} R \left(\eta_{\mu \nu} + h_{\mu \nu} \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)}$
$\;$
General relativity weak field limit spacetime metric and Planck quantum gravity identity 2:
$\boxed{8 \pi T_{\mu \nu} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right) - \frac{1}{2} R \left(\eta_{\mu \nu} + h_{\mu \nu} \right)}$
$\;$
General relativity curvature scalar: (ref. 3)
$R = g^{\mu \nu } R_{\mu \nu}$
$\;$
General relativity and Planck quantum gravity curvature scalar:
$R = g^{\mu \nu } R_{\mu \nu} = g^{\mu \nu } \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)$
$\boxed{R = g^{\mu \nu } \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)}$
$\;$
General relativity weak field limit spacetime metric and Planck quantum gravity identity 2 integration via substitution:
$8 \pi T_{\mu \nu} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right) - \frac{1}{2} g^{\mu \nu} \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right) \left(\eta_{\mu \nu} + h_{\mu \nu} \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right) \left(1 - \frac{1}{2} g^{\mu \nu}\left(\eta_{\mu \nu} + h_{\mu \nu} \right) \right)$
$\;$
General relativity weak field limit spacetime metric and Planck quantum gravity identity 3:
$\boxed{8 \pi T_{\mu \nu} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right) \left(1 - \frac{1}{2} g^{\mu \nu}\left(\eta_{\mu \nu} + h_{\mu \nu} \right) \right)}$
$\;$
General relativity weak field limit spacetime inverse metric: (ref. 2)
$g^{\mu \nu} = \eta^{\mu \nu} - h^{\mu \nu}$
$\;$
General relativity weak field limit spacetime metric and Planck quantum gravity identity 4:
$\boxed{8 \pi T_{\mu \nu} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right) \left(1 - \frac{1}{2} \left(\eta^{\mu \nu} - h^{\mu \nu} \right)\left(\eta_{\mu \nu} + h_{\mu \nu} \right) \right)}$
$\;$
$\Lambda^{\mu}_{\alpha} = \frac{\partial \xi^{\mu}}{\partial x^{\alpha}} \; \; \; \; \; \; \Lambda^{\nu}_{\beta} = \frac{\partial \xi^{\nu}}{\partial x^{\beta}}$
$\;$
$x^{\alpha} = \left(ct, r, \theta, \phi \right) \; \; \; \; \; \; x^{\beta} = \left(ct, r, \theta, \phi \right)$
$\;$
General relativity weak field limit spacetime metric and Planck quantum gravity identity 5:
$\boxed{8 \pi T_{\mu \nu} = \left(\frac{\partial \xi^{\mu}}{\partial x^{\alpha}} \right)\left(\frac{\partial \xi^{\nu}}{\partial x^{\beta}} \right) T^{\alpha \beta} \left(x \right) \left(1 - \frac{1}{2} \left(\eta^{\mu \nu} - h^{\mu \nu} \right)\left(\eta_{\mu \nu} + h_{\mu \nu} \right) \right)}$
$\;$
General relativity stress-energy tensor:
$T_{\mu \nu} = \pm \left(\begin{matrix} -\rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{matrix} \right)$
$\;$
In spherical coordinates $(ct, r, \theta, \phi)$ the Minkowski flat spacetime metric takes the form:
$ds^{2} = -c^{2} dt^{2} + dr^{2} + r^{2} d\theta^{2} + r^{2} \sin^{2} \theta \; d\phi^{2}$
$\;$
General relativity Minkowski flat spacetime metric tensor:
$T^{\alpha \beta} \left(x \right) = \pm \begin{pmatrix} -c^{2} dt^{2} & 0 & 0 & 0 \\ 0 & dr^{2} & 0 & 0 \\ 0 & 0 & r^{2} d\theta^{2} & 0 \\ 0 & 0 & 0 & r^{2} d\phi^{2} \end{pmatrix}$
$\;$
General relativity Minkowski flat spacetime metric:
$\eta_{\mu \nu} = \pm \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$
$\;$
General relativity Minkowski flat spacetime metric is equivalent to the inverse metric:
$\boxed{\eta_{\mu \nu} = \eta^{\mu \nu}} \left(ref. 4 \right)$
$\;$
$\;$
Reference:
Lorentz Group and Lorentz Invariance: (ref. 1)
https://gdenittis.files.wordpress.com/2016/04/ayudantiavi.pdf
Wikipeda - General relativity - linearized gravity: (ref. 2)
https://en.wikipedia.org/wiki/Linearized_gravity
Wikipeda - General relativity: (ref. 3)
https://en.wikipedia.org/wiki/General_relativity
Wikipeda - Lorentz covariance: (ref. 4)
https://en.wikipedia.org/wiki/Lorentz_covariance

Edited by Orion1
source code correction.

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Posted (edited)
On 2/16/2019 at 1:22 AM, Mordred said:

examinations of the various Langrangian's to the SO(10) regime under Pati-Salam in the particle data group restricted to the extended SM (MSM) models with minimal correlations to the MSSM supersymmetric groups with the above. I would like this thread to eventually include the SO(10) terms, with examinations of each group.

There does not appear to be a scientific consensus for the lagrangian equation for GUT Pati-Salam models. The integration strategy appears to involve modeling both gauge symmetry and particle hierarchy into the Yukawa lagrangian.
$\;$
Yukawa lagrangian:
$\mathcal{L} = \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0$
$\;$
However, the GUT quaternion lagrangian equation appears to exhibit both gauge symmety and particle heirarchy and matrix parameters which could be compatible with the SO(10) regime under Pati-Salam.
$\;$
GUT quaternion lagrangian equation: (ref. 1)
$\mathcal{L} = \overline{\psi^{a}} \gamma_{\mu} \left(A_{\mu}^{ab} \psi^{b} + \psi^{a} B_{\mu} \right)$
$\;$
Because the GUT energy scale is well below the Planck energy scale, the general relativity Lagrangian equation may still be utilized without the introduction of a graviton.
$\;$

GUT energy scale:
$\Lambda_{\text{GUT}} \approx 10^{16} \; \text{GeV}$
$\;$
Planck energy scale:
$E_{P} = 1.221 \cdot 10^{19} \; \mathrm {GeV}$
$\;$
General relativity Lagrangian equation with a GUT quaternion:
$\boxed{\mathcal{L} = \underbrace{8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu}}_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{\overline{\psi^{a}} \gamma_{\mu} \left(A_{\mu}^{ab} \psi^{b} + \psi^{a} B_{\mu} \right)}_{\text{GUT quaternion}} = 0}$

Reference:
Wikipedia - Grand Unified Theory - Symplectic groups and quaternion representations: (ref. 1)
https://en.wikipedia.org/wiki/Grand_Unified_Theory#Symplectic_groups_and_quaternion_representations

Edited by Orion1
source code correction.

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Posted (edited)
On 2/16/2019 at 12:11 AM, Mordred said:

see equations 2.9 to 2.22 to arrive at QCD Lagrangian. (Glashow) quaternion﻿/verbian/tetrad under Gell Mann matrix treatments). Pertains to the Weyl Dirac and Majarona spinors. (recall Majorana must be charge neutral under CPT.)

The scientific author of the original lagrangian equation appears to be modeling a massless and chargeless quantum field interaction.
$\;$
General relativity Lagrangian equation with mass and charge and a GUT quaternion: (ref. 1, ref. 2, pg. 8, eq. 2.8)
$\boxed{\mathcal{L} = \underbrace{8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu}}_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{\overline{\psi} \left(i \gamma^{\mu} D_{\mu} - m \right) \psi}_{Dirac} - \underbrace{e \overline{\psi} \gamma^{\mu} Q \psi A_{\mu}}_{EM \text{ } interaction} + \underbrace{|D_{\mu} h|^2 - V\left(|h| \right)}_{Higgs} + \underbrace{\overline{\psi^{\alpha}} \gamma_{\mu} \left(A_{\mu}^{\alpha \beta} \psi^{\beta} + \psi^{\alpha} B_{\mu} \right)}_{GUT \text{ } quaternion} = 0}$
$\;$
Reference:
Wikipedia - Grand Unified Theory - Symplectic groups and quaternion representations: (ref. 1)
https://en.wikipedia.org/wiki/Grand_Unified_Theory#Symplectic_groups_and_quaternion_representations
Search For The Standard Model Higgs Boson In Leptons Plus Jets Final States: (ref. 2)
https://www-d0.fnal.gov/results/publications_talks/thesis/nguyen/thesis.pdf

Edited by Orion1
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On 2/14/2019 at 6:06 AM, Mordred said:

the technicality of formalism matching and sign convention matching so that the equation is the same in both throughout.

Affirmative, is this an identity of the lagrangian Dirac equation with EM interaction?
$\mathcal{L} = \underbrace{\overline{\psi} \left(i \gamma^{\mu} D_{\mu} - m \right) \psi}_{Dirac} - \underbrace{e \overline{\psi} \gamma^{\mu} Q \psi A_{\mu}}_{EM \text{ } interaction}$
$\;$
$\mathcal{L} = \overline{\psi} \left(i \gamma^{\mu} D_{\mu} - m \right) \psi - e \overline{\psi} \gamma^{\mu} Q \psi A_{\mu}$
$\;$
$\mathcal{L} = \overline{\psi} i \gamma^{\mu} D_{\mu} \psi - \overline{\psi} m \psi - e \overline{\psi} \gamma^{\mu} Q \psi A_{\mu}$
$\;$
$\mathcal{L} = \overline{\psi} i \gamma^{\mu} D_{\mu} \psi - e \overline{\psi} \gamma^{\mu} Q \psi A_{\mu} - \overline{\psi} m \psi$
$\;$
$\mathcal{L} = \overline{\psi} \gamma^{\mu} \left(i D_{\mu} - e Q A_{\mu} \right)\psi - \overline{\psi} m \psi$
$\;$
$\boxed{\mathcal{L} = \overline{\psi} \left[\gamma^{\mu}\left(i D_{\mu} - e Q A_{\mu} \right) - m \right] \psi}$

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Posted (edited)
On 2/11/2019 at 5:45 AM, swansont said:

You cited a Wikipedia reference that has discussion and also a list of references. I would check there.

Please cite which Wikipedia reference? Also, I am not certain which 'discussion' with 'list of references' that is being inferred, please clarify. (ref. 7)

Quote

Server guideline rule Section 2 (7): Posting: members should be able to participate in the discussion without clicking any links or watching any videos.

Server guideline rule Section 2 (7) prohibits me from directing discussion to another server. My apology if my stringent interpretation of this rule is overtly strict or interpreted as a deflection or deferment, please clarify. (ref. 7)

On 2/8/2019 at 6:05 AM, swansont said:

This just appears to be the Lagrangians of different interactions. As such, it would likely not be "derived" in any more detail.

In classical (nonrelativistic) mechanics, the Lagrangian is simply L = KE - PE   ﻿

Do you agree with this equation for deriving a relativistic lagrangian?

Relativistic Lagrangian integration via substitution:
$\mathcal{L} = \sum_{1}^{n} E_{k}\left(n \right) - \sum_{1}^{n} E_{p}\left(n \right) = \sum_{1}^{n} \mathcal{L}\left(n \right) = 0$

Relativistic Lagrangian:
$\boxed{\mathcal{L} = \sum_{1}^{n} \mathcal{L}\left(n \right) = 0}$

On 2/14/2019 at 9:51 PM, Mordred said:

Let me look at this further you have the right approach bu﻿t I need to rehash the spin 2 connections, This will become critical to maintain the chirality basis with the remainder of the equation. The iw term corresponds to the QFT operators﻿

Affirmative, revision complete.
$\;$
Einstein's field equations:
$G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = \frac{8 \pi G}{c^{4}} T_{\mu \nu}$
$\;$
Einstein's field equations in natural units:
$G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = 8 \pi T_{\mu \nu}$
$\;$
Einstein's field equations Ricci tensor in natural units:
$\boxed{ R_{\mu \nu} = 8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu}}$
$\;$
Lagrangian equation:
$\mathcal{L} = \underbrace{ \mathbb{R} }_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0$
$\;$
General relativity Lagrangian equation:
$\boxed{\mathcal{L} = \underbrace{8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu}}_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0}$
$\;$
A massless gravitational wave is still massless on a Planck scale, and the result is still a metric tensor field.
$\;$
Metric tensor field: (ref. 1, pg. 21, eq. 1.68)
$T^{' \mu \nu} \left(x' \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right)$
$\;$
The general relativity Ricci tensor is a metric tensor field:
$\boxed{R_{\mu \nu} = T^{' \mu \nu} \left(x' \right)}$
$\;$
Lagrangian equation for a massless Planck graviton:
$\boxed{\mathcal{L} = \underbrace{ \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right) }_{QG} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0}$
$\;$
General relativity and Planck quantum gravity identity:
$\boxed{8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)}$
$\;$
General relativity weak field limit spacetime metric: (ref. 2)
$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$
$\;$
$\eta_{\mu \nu}$ - perturbed nondynamical background metric
$h_{\mu \nu}$ - true metric deviation of $g_{\mu \nu}$ from flat spacetime
$\;$
$h_{\mu \nu}$ must be negligible compared to $\eta_{\mu \nu}$:
$|h_{\mu \nu}| \ll 1$
$\;$
General relativity weak field limit spacetime metric and Planck quantum gravity identity 1:
$\boxed{8 \pi T_{\mu \nu} + \frac{1}{2} R \left(\eta_{\mu \nu} + h_{\mu \nu} \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)}$
$\;$
General relativity weak field limit spacetime metric and Planck quantum gravity identity 2:
$\boxed{8 \pi T_{\mu \nu} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right) - \frac{1}{2} R \left(\eta_{\mu \nu} + h_{\mu \nu} \right)}$
$\;$
General relativity Ricci scalar curvature: (ref. 3, ref. 4)
$R = g^{\mu \nu } R_{\mu \nu}$
$\;$
General relativity Ricci scalar curvature and Planck quantum gravity scalar curvature:
$R = g^{\mu \nu } R_{\mu \nu} = g^{\mu \nu } \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)$
$\boxed{R = g^{\mu \nu } \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)}$
$\;$
General relativity weak field limit spacetime metric and Planck quantum gravity identity 2 integration via substitution:
$8 \pi T_{\mu \nu} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right) - \frac{1}{2} g^{\mu \nu} \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right) \left(\eta_{\mu \nu} + h_{\mu \nu} \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right) \left(1 - \frac{1}{2} g^{\mu \nu}\left(\eta_{\mu \nu} + h_{\mu \nu} \right) \right)$
$\;$
General relativity weak field limit spacetime metric and Planck quantum gravity identity 3:
$\boxed{8 \pi T_{\mu \nu} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right) \left(1 - \frac{1}{2} g^{\mu \nu}\left(\eta_{\mu \nu} + h_{\mu \nu} \right) \right)}$
$\;$
General relativity weak field limit spacetime inverse metric: (ref. 2)
$g^{\mu \nu} = \eta^{\mu \nu} - h^{\mu \nu}$
$\;$
General relativity weak field limit spacetime metric and Planck quantum gravity identity 4:
$\boxed{8 \pi T_{\mu \nu} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right) \left(1 - \frac{1}{2} \left(\eta^{\mu \nu} - h^{\mu \nu} \right)\left(\eta_{\mu \nu} + h_{\mu \nu} \right) \right)}$
$\;$
$\Lambda^{\mu}_{\alpha} = \frac{\partial \xi^{\mu}}{\partial x^{\alpha}} \; \; \; \; \; \; \Lambda^{\nu}_{\beta} = \frac{\partial \xi^{\nu}}{\partial x^{\beta}}$
$\;$
$x^{\alpha} = \left(ct, r, \theta, \phi \right) \; \; \; \; \; \; x^{\beta} = \left(ct, r, \theta, \phi \right)$
$\;$
General relativity weak field limit spacetime metric and Planck quantum gravity identity 5: (ref. 5)
$\boxed{8 \pi T_{\mu \nu} = \left(\frac{\partial \xi^{\mu}}{\partial x^{\alpha}} \right)\left(\frac{\partial \xi^{\nu}}{\partial x^{\beta}} \right) T^{\alpha \beta} \left(x \right) \left(1 - \frac{1}{2} \left(\eta^{\mu \nu} - h^{\mu \nu} \right)\left(\eta_{\mu \nu} + h_{\mu \nu} \right) \right)}$
$\;$
General relativity stress-energy tensor:
$T_{\mu \nu} = \pm \left(\begin{matrix} -\rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{matrix} \right)$
$\;$
In spherical coordinates $(ct, r, \theta, \phi)$ the Minkowski flat spacetime metric takes the form:
$ds^{2} = -c^{2} dt^{2} + dr^{2} + r^{2} d\theta^{2} + r^{2} \sin^{2} \theta \; d\phi^{2}$
$\;$
General relativity Minkowski flat spacetime metric tensor:
$T^{\alpha \beta} \left(x \right) = \pm \begin{pmatrix} -c^{2} dt^{2} & 0 & 0 & 0 \\ 0 & dr^{2} & 0 & 0 \\ 0 & 0 & r^{2} d\theta^{2} & 0 \\ 0 & 0 & 0 & r^{2} \sin^{2} \theta \; d\phi^{2} \end{pmatrix}$
$\;$
General relativity Minkowski flat spacetime metric:
$\eta_{\mu \nu} = \pm \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$
$\;$
General relativity Minkowski flat spacetime metric is equivalent to the inverse metric: (ref. 6)
$\boxed{\eta_{\mu \nu} = \eta^{\mu \nu}}$
$\;$
General relativity Minkowski flat spacetime perturbed nondynamical background metric deviation is equivalent to the inverse metric deviation:
$\boxed{h_{\mu \nu} = h^{\mu \nu}}$
$\;$

Reference:
Lorentz Group and Lorentz Invariance: (ref. 1)
https://gdenittis.files.wordpress.com/2016/04/ayudantiavi.pdf
Wikipeda - General relativity - linearized gravity: (ref. 2)
https://en.wikipedia.org/wiki/Linearized_gravity
Wikipeda - General relativity: (ref. 3)
https://en.wikipedia.org/wiki/General_relativity#Einstein's_equations
Wikipeda - General relativity: (ref. 4)
https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)#Einstein's_equations
Wikipeda - General relativity - Metric tensor - Local coordinates and matrix representations: (ref. 5)
https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)#Local_coordinates_and_matrix_representations
Wikipeda - Lorentz covariance: (ref. 6)
https://en.wikipedia.org/wiki/Lorentz_covariance
Science Forums - Guidelines: (ref. 7)
https://www.scienceforums.net/guidelines/

Edited by Orion1
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9 hours ago, Orion1 said:

Please cite which Wikipedia reference? Also, I am not certain which 'discussion' with 'list of references' that is being inferred, please clarify. (ref. 7)

It's a month later. I don't recall which reference.

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On 2/14/2019 at 9:51 PM, Mordred said:

Let me look at this further you have the right approach but I need to rehash the spin 2 connections

Affirmative, I seem to have reached a reference citation impasse regarding the tensor field functions integration of $\Lambda^{\mu}_{\alpha}$ and $\Lambda^{\nu}_{\beta}$. All of the references that have been cited do not demonstrate a functions integration past this initial point for a tensor field. Is this approach at least mathematically and symbolically correct to this point?

Any citations or recommendations?

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NotReference one is a proof of Lorentz invariance using the inner products of the four momentum which is the expression you have above however those are the two transformation matrices of each of the two four momentum components as seen from two reference frames. S and S prime

Now inner products are symmetric  a lot of literature express this symmetry by this expression

$\mu\cdot\nu=\nu\cdot\mu=\eta$

the author expressed the two four momentum to show this invariance instead. He then showed that one reference frame

has the same equation in the other reference frame equating the axiom the laws of physics is the same in all reference frames.

Arriving  at the equation with the Kronecker delta relations

g_{\mu\nu}\lambda^\mu_\alpha\Lambda^\nu_\beta=g_{\alpha\beta}

Where Lambda is the rotation matrix. Often called a transformation matrix but a boost is a type of rotation. More on that equation can be found here

In essence this is one proof of Lorentz invariance of the Lorentz group which is a subgroup of the Poincare group. Other papers can show the Lorentz invariance by other vector components even using different vectors with different vector components.

Though for Kronecker delta the components are typically i,j,k. This is a common route via Calculus 1.

These being unit vectors. They are unitary =1. They typically use these to further describe the covariant and contravariant relations. The covariant vectors are perpendicular perdendicular projections on M

The contravariant vectors are tangent to the local axis. Or parallel projections on a manifold M

In tensors they are reciprocal.

To get a rank two tensor you need the dyad product of a vector.

A tensor of rank zero is a scalar

A tensor of rank one is a vector

A tensor of rank two is a dyad of a vector

A tensor of rank three is a triad of a vector and so forth.

For tensors rank (m,n) for GR study this guide

PS you will also  want to be familiar with Levi Cevitta coefficient for GR.

Edited by Mordred