Orion1

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  1. Toy model particle chart: [math]\begin{array}{l*{7}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & \neq 0 & \phi \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & \neq 0 & \nu \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} \\ \end{array}[/math] [math]\;[/math] Supersymmetry particle and sparticle chart: [math]\begin{array}{l*{7}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} \\ sf & \text{sneutrino} & 0 & 0 & 1 & 3 & 3 & \neq 0 & \tilde{\nu} \\ b & \text{Higgs} & 0 & 0 & 1 & 1 & 1 & 125.18 \; \text{GeV} & H^{0} \\ sf & \text{higgsino} & +,- & 1/2 & 2 & 1 & 2 & 1.1 \; \text{TeV} & \ddot{H} \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & <1 \; \text{eV} - 10^{15} \; \text{GeV} & \nu \\ sf & \text{neutralino} & +,- & 1/2 & 2 & 4 & 4 & 100 \; \text{GeV} - 1 \; \text{TeV} & \ddot{N}^{0} \\ sf & \text{gravitino} & +,- & 3/2 & 2 & 1 & 2 & 1 \; \text{TeV} & \ddot{G} \\ \end{array}[/math] [math]\;[/math] Bose-Einstein scalar particle dark matter density: [math]\boxed{\rho_{\phi} = \frac{N_{\phi} \pi^2 \left(k_B T_{\phi} \right)^4}{2 C_{\phi} \hbar^3 c^5}}[/math] [math]\;[/math] Bose-Einstein scalar particle dark matter number density: (ref. 1, eq. 7) [math]n_{\phi} = \left(\frac{\zeta \left(3 \right) N_{\phi} \left(k_{B} T_{\phi} \right)^3}{\pi^2 \left(\hbar c \right)^3} \right)[/math] [math]\;[/math] Bose-Einstein scalar particle dark matter mass: [math]m_{\phi} = \frac{\rho_{\phi}}{n_{\phi}} = \left(\frac{N_{\phi} \pi^2 \left(k_B T_{\phi} \right)^4}{2 C_{\phi} \hbar^3 c^5} \right)\left(\frac{\pi^2 \left(\hbar c \right)^3}{\zeta \left(3 \right) N_{\phi} \left(k_{B} T_{\phi} \right)^3} \right) = \frac{\pi^{4} k_B T_{\phi}}{2 C_{\phi} c^{2} \zeta \left(3 \right)} = 3.326 \cdot 10^{-35} \; \text{kg}[/math] [math]\;[/math] Bose-Einstein scalar particle dark matter mass: [math]\boxed{m_{\phi} = \frac{\pi^{4} k_B T_{\phi}}{2 C_{\phi} c^{2} \zeta \left(3 \right)}}[/math] [math]\;[/math] [math]\boxed{m_{\phi} = 3.326 \cdot 10^{-35} \; \text{kg}}[/math] [math]\boxed{m_{\phi} = 18.658 \; \frac{\text{eV}}{c^2}}[/math] [math]\;[/math] Fermi-Dirac neutrino density: [math]\boxed{\rho_{\nu} = \frac{N_{\nu} \pi^2 \left(k_B T_{\nu} \right)^4}{2 C_{\nu} \hbar^3 c^5}}[/math] [math]\;[/math] Fermi-Dirac neutrino number density: (ref. 1, eq. 8) [math]n_{\nu} = \left(\frac{3 \zeta \left(3 \right) N_{\nu} \left(k_{B} T_{\nu} \right)^3}{4 \pi^2 \left(\hbar c \right)^3} \right)[/math] [math]\;[/math] Fermi-Dirac neutrino mass: [math]m_{\nu} = \frac{\rho_{\nu}}{n_{\nu}} = \left(\frac{N_{\nu} \pi^2 \left(k_B T_{\nu} \right)^4}{2 C_{\nu} \hbar^3 c^5} \right)\left(\frac{4 \pi^2 \left(\hbar c \right)^3}{3 \zeta \left(3 \right) N_{\nu} \left(k_{B} T_{\nu} \right)^3} \right) = \frac{2 \pi^{4} k_B T_{\nu}}{3 C_{\nu} c^{2} \zeta \left(3 \right)} = 6.839 \cdot 10^{-38} \; \text{kg}[/math] [math]\;[/math] Fermi-Dirac neutrino mass: [math]\boxed{m_{\nu} = \frac{2 \pi^{4} k_B T_{\nu}}{3 C_{\nu} c^{2} \zeta \left(3 \right)}}[/math] [math]\;[/math] [math]\boxed{m_{\nu} = 6.839 \cdot 10^{-38} \; \text{kg}}[/math] [math]\boxed{m_{\nu} = 0.038 \; \frac{\text{eV}}{c^2}}[/math] [math]\;[/math] Disqualifying dark matter particle candidates for this toy model based upon the predicted mass range, results in the remaining candidates chart. [math]\;[/math] Toy model dark matter particle remaining candidates chart: [math]\begin{array}{l*{7}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & 18.658 \; \text{eV} & \phi \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & <1 \; \text{eV} - 10^{15} \; \text{GeV} & \nu \\ sf & \text{sneutrino} & 0 & 0 & 1 & 3 & 3 & \neq 0 & \tilde{\nu} \\ \end{array}[/math] [math]\;[/math] Would dark matter particle candidates that quantum mechanically decouple before/after neutrinos be a candidate qualifier/disqualifier? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Introduction to Cosmology: Lecture 6 - Thermal history of the Universe: (ref. 1) http://gravitation.web.ua.pt/sites/default/files/migrated2016/Lecture_6.pdf Wikipedia - Scalar boson: (ref. 2) https://en.wikipedia.org/wiki/Scalar_boson Wikipedia - Sterile neutrinos: (ref. 3) https://en.wikipedia.org/wiki/Sterile_neutrino Wikipedia - Dark matter: (ref. 4) https://en.wikipedia.org/wiki/Dark_matter
  2. Toy model particle chart: [math]\begin{array}{l*{7}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & \neq 0 & \phi \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & \neq 0 & \nu \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} \\ \end{array}[/math] [math]\;[/math] Supersymmetry particle and sparticle chart: [math]\begin{array}{l*{7}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} \\ b & \text{quinton} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda^{0} \\ b & \text{higgson} & 0 & 0 & 1 & 1 & 1 & \neq 0 & h^{0} \\ sf & \text{sneutrino} & 0 & 0 & 1 & 3 & 3 & \neq 0 & \tilde{\nu} \\ b & \text{Higgs} & 0 & 0 & 1 & 1 & 1 & \neq 0 & H^{0} \\ sf & \text{higgsino} & +,- & 1/2 & 2 & 1 & 2 & \neq 0 & \ddot{H} \\ sf & \text{gravitino} & +,- & 3/2 & 2 & 1 & 2 & \neq 0 & \ddot{G} \\ \end{array}[/math] [math]\;[/math] A quintessence boson scalar particle is named a 'quinton' for speculative discussion, because no quintessence scalar boson particle is named in quintessence research. [math]\;[/math] A Higgs boson supersymmetry superpartner scalar sparticle boson is named a 'higgson', for speculative discussion. [math]\;[/math] Higgson sparticles generation via nuclear reaction is speculated to be: [math]H^{0} + H^{0} \rightarrow h^{0} + h^{0}[/math] [math]h^{0} + h^{0} \rightarrow H^{0} + H^{0}[/math] [math]\;[/math] Do these sparticle nuclear reactions violate any already known conservation laws? [math]\;[/math] Note that only a quinton and a higgson particle is all that is required to explain all observations in this toy model. [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? Reference: Wikipedia - Quintessence (physics): https://en.wikipedia.org/wiki/Quintessence_(physics) Wikipedia - Lightest Supersymmetric Particle: https://en.wikipedia.org/wiki/Lightest_Supersymmetric_Particle
  3. Affirmative. The quantum-mechanical model boundary conditions for the scalar particle decoupling time versus the [math]\Lambda[/math]CDM model scalar particle decoupling time and a non-zero neutrino mass particle decoupling time and a non-zero sterile neutrino mass particle decoupling time appears to favor a scalar particle for dark matter. [math]\;[/math] Scalar particle decoupling time: [math]T_{u,\phi} = \frac{1}{H_{\phi,t}} = \frac{}{2 \left(k_B T_{\phi,t} \right)^2} \sqrt{\frac{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}{G N_{\phi} \pi^3}} = 0.0255 \; \text{s}[/math] [math]\boxed{T_{u,\phi} = \frac{}{2 \left(k_B T_{\phi,t} \right)^2} \sqrt{\frac{3 C_{\phi} \Omega_{\phi,t} \hbar^3 c^5}{G N_{\phi} \pi^3}}} \; \; \; m_{\phi} \neq 0[/math] [math]\boxed{T_{u,\phi} = 0.0255 \; \text{s}} \; \; \; m_{\phi} \neq 0[/math] [math]\;[/math] [math]\Lambda[/math]CDM universe model semi-emperical temperature-time scale factor: (ref. 2, pg. 18, eq. 2, ref. 3) [math]\left(\frac{T_{\phi}}{T_{\phi,t}} \right)^{2} = \frac{T_{u,\phi}}{T_{u}} = T_{u,\phi} H_0[/math] [math]T_{u,\phi} = \frac{}{H_0} \left(\frac{T_{\phi}}{T_{\phi,t}} \right)^{2} = 0.0166 \; \text{s}[/math] [math]\Lambda[/math]CDM universe model semi-emperical scalar particle decoupling time: [math]\boxed{T_{u,\phi} = \frac{}{H_0} \left(\frac{T_{\phi}}{T_{\phi,t}} \right)^{2}}[/math] [math]\boxed{T_{u,\phi} = 0.0166 \; \text{s}} \; \; \; m_{\phi} \neq 0[/math] [math]\;[/math] Neutrino decoupling time: [math]T_{u,\nu} = \frac{1}{H_{\nu,t}} = \frac{}{2 \left(k_B T_{\nu,t} \right)^2} \sqrt{\frac{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}{G N_{\nu} \pi^3}} = 0.148 \; \text{s}[/math] [math]\boxed{T_{u,\nu} = \frac{}{2 \left(k_B T_{\nu,t} \right)^2} \sqrt{\frac{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}{G N_{\nu} \pi^3}}} \; \; \; m_{\nu} \neq 0[/math] [math]\boxed{T_{u,\nu} = 0.148 \; \text{s}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Sterile neutrino decoupling time: [math]\boxed{T_{u,\nu} = \frac{}{2 \left(k_B T_{\nu,t} \right)^2} \sqrt{\frac{3 C_{\nu} \Omega_{\nu,t} \hbar^3 c^5}{G N_{\nu} \pi^3}}} \; \; \; m_{\nu} \neq 0[/math] [math]\boxed{T_{u,\nu} = 0.372 \; \text{s}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Particle decoupling time summary: [math]\boxed{T_{u,\phi} = 0.0255 \; \text{s}} \; \; \; m_{\phi} \neq 0[/math] [math]\boxed{T_{u,\phi} = 0.0166 \; \text{s}} \; \; \; m_{\phi} \neq 0[/math] [math]\boxed{T_{u,\nu} = 0.148 \; \text{s}} \; \; \; m_{\nu} \neq 0[/math] [math]\boxed{T_{u,\nu} = 0.372 \; \text{s}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Is it possible for the Higgs boson to have a related supersymmetric scalar particle boson with non-zero mass? (ref. 4) Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: WMAP satellite content of the Universe: (ref. 1) http://map.gsfc.nasa.gov/media/080998/index.html Cosmology: nucleosynthesis and inflation: (ref. 2) http://www.uio.no/studier/emner/matnat/astro/AST1100/h07/undervisningsmateriale/lecture25.pdf Wikipedia - Lambda-CDM_model parameters: (ref. 3) https://en.wikipedia.org/wiki/Lambda-CDM_model Wikipedia - Lightest Supersymmetric Particle: (ref. 4) https://en.wikipedia.org/wiki/Lightest_Supersymmetric_Particle
  4. Affirmative, revision complete. Derivation of neutrino mass from neutrino scattering: [math]\;[/math] [math]\theta[/math] - scattered neutrino angle [math]\phi[/math] - electron recoil angle [math]E_{\nu i}[/math] - initial neutrino total energy [math]E_{\nu f}[/math] - final neutrino total energy [math]E_{e}[/math] - electron total energy [math]E_{\nu}[/math] - neutrino total energy Scattered particles rebounding with relativistic momentum have total energy: [math]E_{e} = \gamma_{e} m_{e} c^{2} \; \; \; \; \; \; E_{\nu} = \gamma_{\nu} m_{\nu} c^{2}[/math] Where [math]m_{e}[/math] and [math]m_{\nu}[/math] are the particle rest masses. [math]\gamma_{e}[/math] and [math]\gamma_{\nu}[/math] are the Lorentz factors. (ref. 1) [math]\;[/math] Observational measurements of both particle total energy and velocity, it is possible to calculate the particle rest masses [math]m_{e}[/math] and [math]m_{\nu}[/math]: [math]\boxed{m_{e} = \frac{E_{e}}{\gamma_{e} c^{2}}} \; \; \; \; \; \; \boxed{m_{\nu} = \frac{E_{\nu}}{\gamma_{\nu} c^{2}}} \tag{0}[/math] [math]\;[/math] [math]E_{e} \sin \phi = E_{\nu f} \sin \theta \tag{1}[/math] [math]\;[/math] [math]E_{e} \cos \phi + E_{\nu f} \cos \theta = E_{\nu i} \tag{2}[/math] [math]\;[/math] Isolate [math]E_{e} \cos \phi[/math] from equation (2): [math]E_{e} \cos \phi = E_{\nu i} - E_{\nu f} \cos \theta \tag{3}[/math] [math]\;[/math] Divide equation (1) by equation (3) for an expression for [math]\tan \phi[/math]. [math]\;[/math] [math]\tan \phi = \frac{E_{\nu f} \sin \theta}{E_{\nu i} - E_{\nu f} \cos \theta} = \frac{\sin \theta}{\frac{E_{\nu i}}{E_{\nu f}} - \cos \theta} \tag{4}[/math] [math]\;[/math] Acquire a substitution for [math]\frac{E_{\nu i}}{E_{\nu f}}[/math] to eliminate [math]E_{\nu f}[/math]. Use the Compton equation, which can be rearranged to yield [math]\frac{\lambda_{\nu f}}{\lambda_{\nu i}} = \frac{E_{\nu i}}{E_{\nu f}}[/math] in terms of [math]\lambda_{\nu i}[/math] alone. [math]\;[/math] [math]\frac{\lambda_{\nu f}}{\lambda_{\nu i}} = \frac{E_{\nu i}}{E_{\nu f}} \tag{5}[/math] [math]\;[/math] [math]\frac{\lambda_{\nu f}}{\lambda_{\nu i}} = \frac{E_{\nu i}}{E_{\nu f}} = 1 + \frac{E_{\nu i}}{E_{e}} \left(1 - \cos \theta \right) = 1 + \frac{\gamma_{\nu} m_{\nu} c^2}{\gamma_{e} m_{e} c^2} \left(1 - \cos \theta \right) = 1 + \frac{\gamma_{\nu} m_{\nu}}{\gamma_{e} m_{e}} \left(1 - \cos \theta \right) \tag{6}[/math] [math]\;[/math] Substituting equation (6) into equation (4) and eliminate [math]E_{\nu i}[/math] and [math]E_{\nu f}[/math] in favor of [math]m_{\nu}[/math] alone. [math]\tan \phi = \frac{\sin \theta}{\frac{E_{\nu i}}{E_{\nu f}} - \cos \theta} = \frac{\sin \theta}{1 + \frac{\gamma_{\nu} m_{\nu}}{\gamma_{e} m_{e}} \left(1 - \cos \theta \right) - \cos \theta} = \frac{\sin \theta}{\left(1 + \frac{\gamma_{\nu} m_{\nu}}{\gamma_{e} m_{e}} \right)\left(1 - \cos \theta \right)} \tag{7}[/math] [math]\;[/math] Utilizing a trigonometric identity produces the desired result, specifically: [math]\frac{1 - \cos \theta}{\sin \theta} = \tan \left(\frac{\theta}{2} \right) \tag{8}[/math] [math]\;[/math] Substituting this trigonometric identity into equation (7) results in: [math]\left(1 + \frac{\gamma_{\nu} m_{\nu}}{\gamma_{e} m_{e}} \right) \tan \phi = \cot \frac{\theta}{2} \tag{9}[/math] [math]\;[/math] Solve for neutrino rest mass [math]m_{\nu}[/math]: [math]\tan \phi + \frac{\gamma_{\nu} m_{\nu}}{\gamma_{e} m_{e}} \tan \phi = \cot \frac{\theta}{2} \tag{10}[/math] [math]\;[/math] [math]\frac{\gamma_{\nu} m_{\nu}}{\gamma_{e} m_{e}} \tan \phi = \left(\cot \frac{\theta}{2} - \tan \phi \right) \tag{11}[/math] [math]\;[/math] Electron-neutrino scattering neutrino rest mass: [math]\boxed{m_{\nu} = \frac{\gamma_{e} m_{e} \cot \phi}{\gamma_{\nu}} \left(\cot \frac{\theta}{2} - \tan \phi \right)} \tag{12}[/math] [math]\;[/math] Nuclear-neutrino scattering neutrino rest mass: [math]\boxed{m_{\nu} = \frac{\gamma_{n} m_{n} \cot \phi}{\gamma_{\nu}} \left(\cot \frac{\theta}{2} - \tan \phi \right)} \tag{13}[/math] [math]m_{n}[/math] - nuclear rest mass [math]\;[/math] Electron interaction neutrino scattering angle [math]\theta[/math]: [math]\boxed{\theta = 2 \operatorname{arccot} \left(\frac{\left(\gamma_{e} m_{e} + \gamma_{\nu} m_{\nu} \right) \tan \phi}{\gamma_{e} m_{e}} \right)} \tag{14}[/math] [math]\;[/math] Neutrino interaction electron recoil angle [math]\phi[/math]: [math]\boxed{\phi = \arctan \left(\frac{\gamma_{e} m_{e} \cot \frac{\theta}{2}}{\gamma_{e} m_{e} + \gamma_{\nu} m_{\nu}} \right)} \tag{15}[/math] [math]\;[/math] Nuclear interaction neutrino scattering angle [math]\theta[/math]: [math]\boxed{\theta = 2 \operatorname{arccot} \left(\frac{\left(\gamma_{n} m_{n} + \gamma_{\nu} m_{\nu} \right) \tan \phi}{\gamma_{n} m_{n}} \right)} \tag{16}[/math] [math]\;[/math] Neutrino interaction nuclear recoil angle [math]\phi[/math]: [math]\boxed{\phi = \arctan \left(\frac{\gamma_{n} m_{n} \cot \frac{\theta}{2}}{\gamma_{n} m_{n} + \gamma_{\nu} m_{\nu}} \right)} \tag{17}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Lorentz factor: (ref. 1) https://en.wikipedia.org/wiki/Lorentz_factor Wikipedia - Compton scattering - Derivation of the scattering formula: https://en.wikipedia.org/wiki/Compton_scattering#Derivation_of_the_scattering_formula Physics 253 - Compton Scattering - Patrick LeClair http://pleclair.ua.edu//PH253/Notes/compton.pdf Orion1 - Neutrino mass from Fermi-Dirac statistics...: https://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/ Science News - Neutrinos seen scattering off an atom’s nucleus for the first time: https://www.sciencenews.org/article/neutrinos-seen-scattering-atoms-nucleus-first-time
  5. De Broglie relativistic momentum: (ref. 1) [math]p = \frac{\hbar}{\overline{\lambda}} = \gamma m_0 v[/math] [math]\;[/math] Relativistic energy-momentum relation and relativistic mass particle total energy identity: [math]\boxed{E_{t} = \sqrt{\left(m_{0} c^{2} \right)^{2} + \left(pc \right)^{2}} = \gamma m_0 c^2}[/math] [math]\;[/math] [math]\boxed{E_{t} = \sqrt{\left(m_{0} c^{2} \right)^{2} + \left(\frac{\hbar c}{\overline{\lambda}} \right)^{2}} = \gamma m_0 c^2}[/math] [math]\;[/math] Is this equation an identity for a relativistic mass particle? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Matter wave: (ref. 1) https://en.wikipedia.org/wiki/Matter_wave
  6. Relativistic energy-momentum relation total energy: (ref. 1) [math]E_{t}^{2} = \left(m_{0} c^{2} \right)^{2} + \left(pc \right)^{2} [/math] [math]E_{t} = \sqrt{\left(m_{0} c^{2} \right)^{2} + \left(pc \right)^{2}}[/math] [math]\;[/math] Relativistic mass particle total energy: (ref. 2) [math]E_{t} = \gamma m_0 c^{2}[/math] [math]\;[/math] Relativistic energy-momentum relation and relativistic mass particle total energy identity: [math]\boxed{E_{t} = \sqrt{\left(m_{0}c^{2} \right)^{2} + \left(pc \right)^{2}} = \gamma m_0 c^2}[/math] [math]\;[/math] Is this equation an identity for a relativistic mass particle? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Energy-momentum relation: (ref. 1) https://en.wikipedia.org/wiki/Energy–momentum_relation#Special_relativity Wikipedia - Energy-momentum relation - heuristic approach for massive particles: (ref. 2) https://en.wikipedia.org/wiki/Energy–momentum_relation#Heuristic_approach_for_massive_particle
  7. Derivation of neutrino mass from neutrino scattering: [math]\;[/math] [math]\theta[/math] - scattered neutrino angle [math]\phi[/math] - electron recoil angle [math]p_{i}[/math] - initial neutrino momentum [math]p_{f}[/math] - final neutrino momentum [math]p_{e}[/math] - electron momentum [math]\;[/math] Scattered particles rebounding with relativistic momentum have total energy: [math]E_{e} = \gamma m_{0e} c^{2} = m_{e} c^{2} \; \; \; \; \; \; E_{\nu} = \gamma m_{0\nu} c^{2} = m_{\nu} c^{2}[/math] Where [math]m_{e}[/math] and [math]m_{\nu}[/math] are the relativistic particle masses. [math]\;[/math] Observational measurements of both particle total energy and velocity, it is possible to calculate the rest particle masses [math]m_{0e}[/math] and [math]m_{0\nu}[/math]: [math]\boxed{m_{0e} = \frac{E_{e}}{\gamma c^{2}}} \; \; \; \; \; \; \boxed{m_{0\nu} = \frac{E_{\nu}}{\gamma c^{2}}} \tag{0}[/math] [math]\;[/math] [math]p_{e} \sin \phi = p_{f} \sin \theta \tag{1}[/math] [math]\;[/math] [math]p_{e} \cos \phi + p_{f} \cos \theta = p_{i} \tag{2}[/math] [math]\;[/math] Isolate [math]p_{e} \cos \phi[/math] from equation [math](2)[/math]: [math]p_{e} \cos \phi = p_{i} - p_{f} \cos \theta \tag{3}[/math] [math]\;[/math] Divide equation [math](1)[/math] by equation [math](3)[/math] for an expression for [math](3)[/math] for an expression for [math]\tan \phi[/math]: [math]\;[/math] [math]\tan \phi = \frac{p_{f} \sin \theta}{p_{i} - p_{f} \cos \theta} = \frac{\sin \theta}{\frac{p_{i}}{p_{f}} - \cos \theta} \tag{4}[/math] [math]\;[/math] Acquire a substitution for [math]\frac{p_{i}}{p_{f}}[/math] to eliminate [math]p_{f}[/math]. Use the Compton equation, which can be rearranged to yield [math]\frac{\lambda_{f}}{\lambda_{i}} = \frac{p_{i}}{p_{f}}[/math] in terms of [math]\lambda_{i}[/math] alone. [math]\;[/math] [math]\frac{\lambda_{f}}{\lambda_{i}} = \frac{p_{i}}{p_{f}} \tag{5}[/math] [math]\;[/math] [math]\frac{\lambda_{f}}{\lambda_{i}} = \frac{p_{i}}{p_{f}} = 1 + \frac{E_{\nu}}{E_{e}} \left(1 - \cos \theta \right) = 1 + \frac{m_{\nu} c^2}{m_{e} c^2} \left(1 - \cos \theta \right) = 1 + \frac{m_{\nu}}{m_{e}} \left(1 - \cos \theta \right) \tag{6}[/math] [math]\;[/math] Substituting equation [math](6)[/math] into equation [math](4)[/math], and eliminate [math]p_{i}[/math] and [math]p_{f}[/math] in favor of [math]m_{\nu}[/math] alone. [math]\;[/math] [math]\tan \phi = \frac{\sin \theta}{\frac{p_{i}}{p_{f}} - \cos \theta} = \frac{\sin \theta}{1 + \frac{m_{\nu}}{m_{e}} \left(1 - \cos \theta \right) - \cos \theta} = \frac{\sin \theta}{\left(1 + \frac{m_{\nu}}{m_{e}} \right)\left(1 - \cos \theta \right)} \tag{7}[/math] [math]\;[/math] Utilizing a trigonometric identity produces the desired result, specifically: [math]\frac{1 - \cos \theta}{\sin \theta} = \tan \left(\frac{\theta}{2} \right) \tag{8}[/math] [math]\;[/math] Substituting this trigonometric identity into equation [math](7)[/math] results in: [math]\left(1 + \frac{m_{\nu}}{m_{e}} \right) \tan \phi = \cot \frac{\theta}{2} \tag{9}[/math] [math]\;[/math] Solve for neutrino mass [math]m_{\nu}[/math]: [math]\tan \phi + \frac{m_{\nu}}{m_{e}} \tan \phi = \cot \frac{\theta}{2} \tag{10}[/math] [math]\;[/math] [math]\frac{m_{\nu}}{m_{e}} \tan \phi = \left(\cot \frac{\theta}{2} - \tan \phi \right) \tag{11}[/math] [math]\;[/math] Electron-neutrino scattering neutrino mass: [math]\boxed{m_{\nu} = m_{e} \cot \phi \left(\cot \frac{\theta}{2} - \tan \phi \right)} \tag{12}[/math] [math]\;[/math] Nuclear-neutrino scattering neutrino mass: [math]\boxed{m_{\nu} = m_{n} \cot \phi \left(\cot \frac{\theta}{2} - \tan \phi \right)} \tag{13}[/math] [math]\;[/math] [math]m_{n}[/math] - nuclear mass [math]\;[/math] Electron interaction neutrino scattering angle [math]\theta[/math]: [math]\boxed{\theta = 2 \operatorname{arccot} \left(\frac{\left(m_{e} + m_{\nu} \right) \tan \phi}{m_{e}} \right)} \tag{14}[/math] [math]\;[/math] Neutrino interaction electron recoil angle [math]\phi[/math]: [math]\boxed{\phi = \arctan \left(\frac{m_{e} \cot \frac{\theta}{2}}{m_{e} + m_{\nu}} \right)} \tag{15}[/math] [math]\;[/math] Nuclear interaction neutrino scattering angle [math]\theta[/math]: [math]\boxed{\theta = 2 \operatorname{arccot} \left(\frac{\left(m_{n} + m_{\nu} \right) \tan \phi}{m_{n}} \right)} \tag{16}[/math] [math]\;[/math] Neutrino interaction nuclear recoil angle [math]\phi[/math]: [math]\boxed{\phi = \arctan \left(\frac{m_{n} \cot \frac{\theta}{2}}{m_{n} + m_{\nu}} \right)} \tag{17}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Compton scattering - Derivation of the scattering formula: https://en.wikipedia.org/wiki/Compton_scattering#Derivation_of_the_scattering_formula Physics 253 - Compton Scattering - Patrick LeClair http://pleclair.ua.edu//PH253/Notes/compton.pdf Orion1 - Neutrino mass from Fermi-Dirac statistics...: https://www.scienceforums.net/topic/90189-neutrino-mass-from-fermi-dirac-statistics/ Science News - Neutrinos seen scattering off an atom’s nucleus for the first time: https://www.sciencenews.org/article/neutrinos-seen-scattering-atoms-nucleus-first-time
  8. Please cite which Wikipedia reference? Also, I am not certain which 'discussion' with 'list of references' that is being inferred, please clarify. (ref. 7) 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) Do you agree with this equation for deriving a relativistic lagrangian? Relativistic Lagrangian integration via substitution: [math]\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[/math] Relativistic Lagrangian: [math]\boxed{\mathcal{L} = \sum_{1}^{n} \mathcal{L}\left(n \right) = 0}[/math] Affirmative, revision complete. [math]\;[/math] Einstein's field equations: [math]G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = \frac{8 \pi G}{c^{4}} T_{\mu \nu}[/math] [math]\;[/math] Einstein's field equations in natural units: [math]G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = 8 \pi T_{\mu \nu}[/math] [math]\;[/math] Einstein's field equations Ricci tensor in natural units: [math]\boxed{ R_{\mu \nu} = 8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu}}[/math] [math]\;[/math] Lagrangian equation: [math]\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[/math] [math]\;[/math] General relativity Lagrangian equation: [math]\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}[/math] [math]\;[/math] A massless gravitational wave is still massless on a Planck scale, and the result is still a metric tensor field. [math]\;[/math] Metric tensor field: (ref. 1, pg. 21, eq. 1.68) [math]T^{' \mu \nu} \left(x' \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right)[/math] [math]\;[/math] The general relativity Ricci tensor is a metric tensor field: [math]\boxed{R_{\mu \nu} = T^{' \mu \nu} \left(x' \right)}[/math] [math]\;[/math] Lagrangian equation for a massless Planck graviton: [math]\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}[/math] [math]\;[/math] General relativity and Planck quantum gravity identity: [math]\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)}[/math] [math]\;[/math] General relativity weak field limit spacetime metric: (ref. 2) [math]g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}[/math] [math]\;[/math] [math]\eta_{\mu \nu}[/math] - perturbed nondynamical background metric [math]h_{\mu \nu}[/math] - true metric deviation of [math]g_{\mu \nu}[/math] from flat spacetime [math]\;[/math] [math]h_{\mu \nu}[/math] must be negligible compared to [math]\eta_{\mu \nu}[/math]: [math]|h_{\mu \nu}| \ll 1[/math] [math]\;[/math] General relativity weak field limit spacetime metric and Planck quantum gravity identity 1: [math]\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)}[/math] [math]\;[/math] General relativity weak field limit spacetime metric and Planck quantum gravity identity 2: [math]\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)}[/math] [math]\;[/math] General relativity Ricci scalar curvature: (ref. 3, ref. 4) [math]R = g^{\mu \nu } R_{\mu \nu}[/math] [math]\;[/math] General relativity Ricci scalar curvature and Planck quantum gravity scalar curvature: [math]R = g^{\mu \nu } R_{\mu \nu} = g^{\mu \nu } \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)[/math] [math]\boxed{R = g^{\mu \nu } \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)}[/math] [math]\;[/math] General relativity weak field limit spacetime metric and Planck quantum gravity identity 2 integration via substitution: [math]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)[/math] [math]\;[/math] General relativity weak field limit spacetime metric and Planck quantum gravity identity 3: [math]\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)}[/math] [math]\;[/math] General relativity weak field limit spacetime inverse metric: (ref. 2) [math]g^{\mu \nu} = \eta^{\mu \nu} - h^{\mu \nu}[/math] [math]\;[/math] General relativity weak field limit spacetime metric and Planck quantum gravity identity 4: [math]\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)}[/math] [math]\;[/math] [math]\Lambda^{\mu}_{\alpha} = \frac{\partial \xi^{\mu}}{\partial x^{\alpha}} \; \; \; \; \; \; \Lambda^{\nu}_{\beta} = \frac{\partial \xi^{\nu}}{\partial x^{\beta}}[/math] [math]\;[/math] [math]x^{\alpha} = \left(ct, r, \theta, \phi \right) \; \; \; \; \; \; x^{\beta} = \left(ct, r, \theta, \phi \right)[/math] [math]\;[/math] General relativity weak field limit spacetime metric and Planck quantum gravity identity 5: (ref. 5) [math]\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)}[/math] [math]\;[/math] General relativity stress-energy tensor: [math]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)[/math] [math]\;[/math] In spherical coordinates [math](ct, r, \theta, \phi)[/math] the Minkowski flat spacetime metric takes the form: [math]ds^{2} = -c^{2} dt^{2} + dr^{2} + r^{2} d\theta^{2} + r^{2} \sin^{2} \theta \; d\phi^{2}[/math] [math]\;[/math] General relativity Minkowski flat spacetime metric tensor: [math]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}[/math] [math]\;[/math] General relativity Minkowski flat spacetime metric: [math]\eta_{\mu \nu} = \pm \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}[/math] [math]\;[/math] General relativity Minkowski flat spacetime metric is equivalent to the inverse metric: (ref. 6) [math]\boxed{\eta_{\mu \nu} = \eta^{\mu \nu}}[/math] [math]\;[/math] General relativity Minkowski flat spacetime perturbed nondynamical background metric deviation is equivalent to the inverse metric deviation: [math]\boxed{h_{\mu \nu} = h^{\mu \nu}}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? 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/
  9. Affirmative, is this an identity of the lagrangian Dirac equation with EM interaction? [math]\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}[/math] [math]\;[/math] [math]\mathcal{L} = \overline{\psi} \left(i \gamma^{\mu} D_{\mu} - m \right) \psi - e \overline{\psi} \gamma^{\mu} Q \psi A_{\mu}[/math] [math]\;[/math] [math]\mathcal{L} = \overline{\psi} i \gamma^{\mu} D_{\mu} \psi - \overline{\psi} m \psi - e \overline{\psi} \gamma^{\mu} Q \psi A_{\mu}[/math] [math]\;[/math] [math]\mathcal{L} = \overline{\psi} i \gamma^{\mu} D_{\mu} \psi - e \overline{\psi} \gamma^{\mu} Q \psi A_{\mu} - \overline{\psi} m \psi[/math] [math]\;[/math] [math]\mathcal{L} = \overline{\psi} \gamma^{\mu} \left(i D_{\mu} - e Q A_{\mu} \right)\psi - \overline{\psi} m \psi[/math] [math]\;[/math] [math]\boxed{\mathcal{L} = \overline{\psi} \left[\gamma^{\mu}\left(i D_{\mu} - e Q A_{\mu} \right) - m \right] \psi}[/math]
  10. The scientific author of the original lagrangian equation appears to be modeling a massless and chargeless quantum field interaction. [math]\;[/math] General relativity Lagrangian equation with mass and charge and a GUT quaternion: (ref. 1, ref. 2, pg. 8, eq. 2.8) [math]\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}[/math] [math]\;[/math] 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
  11. 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. [math]\;[/math] Yukawa lagrangian: [math]\mathcal{L} = \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0[/math] [math]\;[/math] 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. [math]\;[/math] GUT quaternion lagrangian equation: (ref. 1) [math]\mathcal{L} = \overline{\psi^{a}} \gamma_{\mu} \left(A_{\mu}^{ab} \psi^{b} + \psi^{a} B_{\mu} \right)[/math] [math]\;[/math] 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. [math]\;[/math] GUT energy scale: [math]\Lambda_{\text{GUT}} \approx 10^{16} \; \text{GeV}[/math] [math]\;[/math] Planck energy scale: [math]E_{P} = 1.221 \cdot 10^{19} \; \mathrm {GeV}[/math] [math]\;[/math] General relativity Lagrangian equation with a GUT quaternion: [math]\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}[/math] 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
  12. Affirmative, revision complete. [math]\;[/math] Einstein's field equations: [math]G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = \frac{8 \pi G}{c^{4}} T_{\mu \nu}[/math] [math]\;[/math] Einstein's field equations in natural units: [math]G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = 8 \pi T_{\mu \nu}[/math] [math]\;[/math] Einstein's field equations Ricci tensor in natural units: [math]\boxed{ R_{\mu \nu} = 8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu}}[/math] [math]\;[/math] Lagrangian equation: [math]\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[/math] [math]\;[/math] General relativity Lagrangian equation: [math]\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}[/math] [math]\;[/math] A massless gravitational wave is still massless on a Planck scale, and the result is still a metric tensor field. [math]\;[/math] Metric tensor field: (ref. 1, pg. 21, eq. 1.68) [math]T^{' \mu \nu} \left(x' \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right)[/math] [math]\;[/math] The general relativity Ricci tensor is a metric tensor field: [math]\boxed{R_{\mu \nu} = T^{' \mu \nu} \left(x' \right)}[/math] [math]\;[/math] Lagrangian equation for a massless Planck graviton: [math]\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}[/math] [math]\;[/math] General relativity and Planck quantum gravity identity: [math]\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)}[/math] [math]\;[/math] General relativity weak field limit spacetime metric: (ref. 2) [math]g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}[/math] [math]\;[/math] [math]\eta_{\mu \nu}[/math] - perturbed non-dynamical background metric [math]h_{\mu \nu}[/math] - true metric deviation of [math]g_{\mu \nu}[/math] from flat spacetime [math]\;[/math] [math]h_{\mu \nu}[/math] must be negligible compared to [math]\eta_{\mu \nu}[/math]: [math]|h_{\mu \nu}| \ll 1[/math] [math]\;[/math] General relativity weak field limit spacetime metric and Planck quantum gravity identity 1: [math]\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)}[/math] [math]\;[/math] General relativity weak field limit spacetime metric and Planck quantum gravity identity 2: [math]\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)}[/math] [math]\;[/math] General relativity curvature scalar: (ref. 3) [math]R = g^{\mu \nu } R_{\mu \nu}[/math] [math]\;[/math] General relativity and Planck quantum gravity curvature scalar: [math]R = g^{\mu \nu } R_{\mu \nu} = g^{\mu \nu } \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)[/math] [math]\boxed{R = g^{\mu \nu } \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta} \left(x \right)}[/math] [math]\;[/math] General relativity weak field limit spacetime metric and Planck quantum gravity identity 2 integration via substitution: [math]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)[/math] [math]\;[/math] General relativity weak field limit spacetime metric and Planck quantum gravity identity 3: [math]\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)}[/math] [math]\;[/math] General relativity weak field limit spacetime inverse metric: (ref. 2) [math]g^{\mu \nu} = \eta^{\mu \nu} - h^{\mu \nu}[/math] [math]\;[/math] General relativity weak field limit spacetime metric and Planck quantum gravity identity 4: [math]\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)}[/math] [math]\;[/math] [math]\Lambda^{\mu}_{\alpha} = \frac{\partial \xi^{\mu}}{\partial x^{\alpha}} \; \; \; \; \; \; \Lambda^{\nu}_{\beta} = \frac{\partial \xi^{\nu}}{\partial x^{\beta}}[/math] [math]\;[/math] [math]x^{\alpha} = \left(ct, r, \theta, \phi \right) \; \; \; \; \; \; x^{\beta} = \left(ct, r, \theta, \phi \right)[/math] [math]\;[/math] General relativity weak field limit spacetime metric and Planck quantum gravity identity 5: [math]\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)}[/math] [math]\;[/math] General relativity stress-energy tensor: [math]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)[/math] [math]\;[/math] In spherical coordinates [math](ct, r, \theta, \phi)[/math] the Minkowski flat spacetime metric takes the form: [math]ds^{2} = -c^{2} dt^{2} + dr^{2} + r^{2} d\theta^{2} + r^{2} \sin^{2} \theta \; d\phi^{2}[/math] [math]\;[/math] General relativity Minkowski flat spacetime metric tensor: [math]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}[/math] [math]\;[/math] General relativity Minkowski flat spacetime metric: [math]\eta_{\mu \nu} = \pm \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}[/math] [math]\;[/math] General relativity Minkowski flat spacetime metric is equivalent to the inverse metric: [math]\boxed{\eta_{\mu \nu} = \eta^{\mu \nu}} \left(ref. 4 \right)[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] 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
  13. Affirmative, revision complete. [math]\;[/math] Einstein's field equations: [math]G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = \frac{8 \pi G}{c^{4}} T_{\mu \nu}[/math] [math]\;[/math] Einstein's field equations in natural units: [math]G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = 8 \pi T_{\mu \nu}[/math] [math]\;[/math] Einstein's field equations Ricci tensor in natural units: [math]\boxed{ R_{\mu \nu} = 8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu}}[/math] [math]\;[/math] Lagrangian equation: [math]\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[/math] [math]\;[/math] General relativity Lagrangian equation: [math]\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}[/math] [math]\;[/math] A massless gravitational wave is still massless on a Planck scale, and the result is still a tensor field. [math]\;[/math] Tensor field: (ref. 1,pg. 21, eq. 1.68) [math]T^{' \mu \nu} \left(x' \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right)[/math] [math]\;[/math] The general relativity Ricci tensor is a tensor field: [math]\boxed{R_{\mu \nu} = T^{' \mu \nu} \left(x' \right)}[/math] [math]\;[/math] Lagrangian equation for a massless Planck graviton: [math]\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}[/math] [math]\;[/math] General relativity and Planck quantum gravity identity: [math]\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)}[/math] [math]\;[/math] General relativity weak field limit spacetime metric: [math]g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}[/math] [math]\;[/math] General relativity weak field limit spacetime metric and Planck quantum gravity identity 1: [math]\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)}[/math] [math]\;[/math] General relativity weak field limit spacetime metric and Planck quantum gravity identity 2: [math]\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)}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Lorentz Group and Lorentz Invariance: (ref. 1) https://gdenittis.files.wordpress.com/2016/04/ayudantiavi.pdf
  14. Affirmative, that is correct. [math]\;[/math] Einstein's field equations: [math]G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = \frac{8 \pi G}{c^{4}} T_{\mu \nu}[/math] [math]\;[/math] Einstein's field equations in natural units: [math]G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = 8 \pi T_{\mu \nu}[/math] [math]\;[/math] Einstein's field equations Ricci tensor in natural units: [math]\boxed{ R_{\mu \nu} = 8 \pi T_{\mu \nu} + \frac{1}{2} R g_{\mu \nu}}[/math] [math]\;[/math] Lagrangian equation: [math]\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[/math] [math]\;[/math] General relativity Lagrangian equation: [math]\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}[/math] [math]\;[/math] A massless gravitational wave is still massless on a Planck scale, and the result is still a tensor field. [math]\;[/math] Tensor field: (ref. 1, pg. 21, eq. 1.68) [math]T^{' \mu \nu} \left(x' \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right)[/math] [math]\;[/math] The general relativity Ricci tensor is a tensor field: [math]\boxed{R_{\mu \nu} = T^{' \mu \nu} \left(x' \right)}[/math] [math]\;[/math] Lagrangian equation for a massless Planck graviton: [math]\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}[/math] [math]\;[/math] General relativity and Planck quantum gravity identity: [math]\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)}[/math] [math]\;[/math] General relativity spacetime metric: [math]g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}[/math] [math]\;[/math] General relativity spacetime metric and Planck quantum gravity identity: [math]\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)}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Lorentz Group and Lorentz Invariance: (ref. 1) https://gdenittis.files.wordpress.com/2016/04/ayudantiavi.pdf
  15. [math]\;[/math] Affirmative, according to Wikipedia, The action S is given by: [math]S = -mc \int ds = \int L dt[/math] [math]\;[/math] Where L is the relativistic Lagrangian for a free particle: [math]L = -mc^{2} \sqrt{1 - \frac {v^{2}}{c^{2}}}[/math] [math]\;[/math] And my solution for the proper time relativistic Lagrangian for a free particle: [math]\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}}}[/math] [math]\;[/math] [math]\;[/math] 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: [math]L = - \frac{m_0 c^2}{\gamma\left( \dot{\mathbf{r}} \right)} - V\left(\mathbf{r}, \dot{\mathbf{r}}, t \right)[/math] [math]\;[/math] [math]\mathbf{v} = \dot{\mathbf{r}} = \frac{d\mathbf{r}}{dt} = \left(\frac{dx}{dt} , \frac{dy}{dt} , \frac{dz}{dt} \right)[/math] [math]\;[/math] 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. [math]\;[/math] Because, It is implied that the Ricci tensor is still the classical theory of general relativity in this form. Affirmative, I think that you have answered your own question. [math]\;[/math] Einstein's field equations: [math]G_{\mu \nu} = R_{\mu \nu} - \frac{R g_{\mu \nu}}{2} = \frac{8 \pi G}{c^{4}} T_{\mu \nu}[/math] [math]\;[/math] Einstein's field equations in natural units: [math]G_{\mu \nu} = R_{\mu \nu} - \frac{R g_{\mu \nu}}{2} = 8 \pi T_{\mu \nu}[/math] [math]\;[/math] Einstein's field equations Ricci tensor in natural units: [math]\boxed{ R_{\mu \nu} = 8 \pi T_{\mu \nu} + \frac{R g_{\mu \nu}}{2} }[/math] [math]\;[/math] Lagrangian equation: [math]\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[/math] [math]\;[/math] General relativity Lagrangian equation: [math]\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}[/math] [math]\;[/math] 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. [math]\;[/math] A massless gravitational wave is still massless on a Planck scale, and the result is still a tensor field. [math]\;[/math] Tensor field: (ref. 1,pg. 21, eq. 1.68) [math]T^{' \mu \nu} \left(x' \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right)[/math] [math]\;[/math] Lagrangian equation for a massless Planck graviton: [math]\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}[/math] [math]\;[/math] General relativity and Planck quantum gravity identity: [math]\boxed{8 \pi T_{\mu \nu} + \frac{R g_{\mu \nu}}{2} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} T^{\alpha \beta}\left(x \right)}[/math] [math]\;[/math] Reference: Lorentz Group and Lorentz Invariance: (ref. 1) https://gdenittis.files.wordpress.com/2016/04/ayudantiavi.pdf