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  1. Planck satellite baryonic cosmological composition parameter: (ref. 1, pg. 11, ref. 2, pg. 3) [math]\Omega_{b} = 0.0495[/math] [math]\;[/math] Black holes cosmological composition parameter: {ref. 2, pg. 3) [math]\Omega_{bh} = 0.00007[/math] [math]\;[/math] Solar mass: (ref. 3) [math]M_{\odot} = 1.9885 \cdot 10^{30} \; \text{kg}[/math] [math]\;[/math] Milky Way galaxy mass: (ref. 4, pg. 1) [math]M_{mw} = 1.260 \cdot 10^{12} \cdot M_{\odot} = 2.506 \cdot 10^{42} \; \text{kg}[/math] [math]\boxed{M_{mw} = 2.506 \cdot 10^{42} \; \text{kg}}[/math] [math]\;[/math] PSR J2215+5135 pulsar Tolman-Oppenheimer-Volkoff observational lower mass limit: (ref. 5) [math]\boxed{M_{bh} \geq 2.27 \cdot M_{\odot}}[/math] [math]\boxed{M_{bh} \geq 4.514 \cdot 10^{30} \; \text{kg}}[/math] [math]\;[/math] Stellar class O upper mass limit: (ref. 6) [math]\boxed{M_{bh} \geq 16 \cdot M_{\odot}}[/math] [math]\boxed{M_{bh} \geq 3.182 \cdot 10^{31} \; \text{kg}}[/math] [math]\;[/math] Toy model black holes per galaxy average number: [math]\frac{N_{bh}}{N_g} = \frac{\Omega_{bh} M_{mw}}{\Omega_b M_{bh}} = \left(1.114 \cdot 10^{8} \rightarrow 7.851 \cdot 10^{8} \right) \; \frac{\text{black holes}}{\text{galaxy}}[/math] [math]\boxed{\frac{N_{bh}}{N_g} = \frac{\Omega_{bh} M_{mw}}{\Omega_b M_{bh}}}[/math] [math]\boxed{\frac{N_{bh}}{N_g} = \left(1.114 \cdot 10^{8} \rightarrow 7.851 \cdot 10^{8} \right) \; \frac{\text{black holes}}{\text{galaxy}}}[/math] [math]\;[/math] Synthetic catalog black holes per galaxy average number: (ref. 7, pg. 1) [math]\frac{N_{bh}}{N_g} = 1.693 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf The Cosmic Energy Inventory: (ref. 2) http://arxiv.org/pdf/astro-ph/0406095v2.pdf Wikipedia - Sun Sol: (ref. 3) https://en.wikipedia.org/wiki/Sun Mass models of the Milky Way: (ref. 4) http://arxiv.org/pdf/1102.4340v1 Wikipedia - Tolman-Oppenheimer-Volkoff limit: (ref. 5) https://en.wikipedia.org/wiki/Tolman-Oppenheimer-Volkoff_limit Wikipedia - Stellar classification - Harvard spectral classification: (ref. 6) https://en.wikipedia.org/wiki/Stellar_classification#Harvard_spectral_classification Synthetic catalog of black holes in the Milky Way: (ref. 7) https://arxiv.org/pdf/1908.08775.pdf
  2. Affirmative, I seem to have reached a reference citation impasse regarding the tensor field functions integration of [math]\Lambda^{\mu}_{\alpha}[/math] and [math]\Lambda^{\nu}_{\beta}[/math]. 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? Any discussions and/or peer reviews about this specific topic thread?
  3. Affirmative. Your nearest college or university book store should have Calculus I and Physics and Astrophysics books available for purchase. Or you could locate the same or similar books at your local city, college or university library available for checkout. If these options are not available due to remote locality, then online purchases may also be available.
  4. Toy model calculation versus observation comparison summary: [math]\begin{array}{l*{3}{c}r} \text{symbol} & \text{quantity} & \text{percent} \\ N_{g} & 2.330 \cdot 10^{12} \; \text{galaxies} & 116.5 \\ N_{g} & 2.000 \cdot 10^{12} \; \text{galaxies} & \\ N_{s} & 2.453 \cdot 10^{23} \; \text{stars} & 81.767 \\ N_{s} & 3.000 \cdot 10^{23} \; \text{stars} \\ N_p & 3.536 \cdot 10^{23} \; \text{planets} & 110.5 \\ N_p & 3.200 \cdot 10^{23} \; \text{planets} \\ \frac{N_s}{N_g} & 1.053 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}} & 70.2 \\ \frac{N_s}{N_g} & 1.500 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}} \\ \frac{N_p}{N_g} & 1.518 \cdot 10^{11} \; \frac{\text{planets}}{\text{galaxy}} & 94.875 \\ \frac{N_p}{N_g} & 1.600 \cdot 10^{11} \; \frac{\text{planets}}{\text{galaxy}} \\ \frac{N_p}{N_s} & 1.443 \; \frac{\text{planets}}{\text{star}} & 135.24 \\ \frac{N_p}{N_s} & 1.067 \; \frac{\text{planets}}{\text{star}} \\ \end{array}[/math] [math]\;[/math] Toy model particle properties chart: [math]\begin{array}{l*{9}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} & \text{decoupling time} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda & >0 \; \text{s} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & 18.658 \; \text{eV} & \phi & 0.0255 \; \text{s} \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & 8.167 \; \text{eV} & \nu_{s} & 0.0255 \; \text{s} \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & 0.038 \; \text{eV} & \nu & 0.148 \; \text{s} \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma & 1.009 \cdot 10^{13} \; \text{s} & \left(3.197 \cdot 10^{5} \; \text{years} \right) \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} & >0 \; \text{s} \\ \end{array}[/math] [math][/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Scienceforums - Orion1 - toy model revision: https://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=1089828 https://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=1091408
  5. Affirmative, in this toy model, sterile neutrinos have equivalent dark matter candidacy as scalar particles. [math]\;[/math] Planck satellite cosmological parameters at present time: (ref. 1, pg. 11) [math]\Omega_{dm} = 0.268[/math] [math]\;[/math] sterile neutrino composition is equivalent to dark matter composition at present time. [math]\boxed{\Omega_{s \nu} = \Omega_{dm}}[/math] [math]\;[/math] Fermi-Dirac total dark matter sterile neutrino distribution constant: [math]\boxed{C_{s \nu} = \frac{4 G N_{s \nu} \pi^3 \left(k_B T_{s \nu} \right)^4}{3 \Omega_{s \nu} H_0^2 \hbar^3 c^5}}[/math] [math]\;[/math] [math]\boxed{C_{s \nu} = 1.109 \cdot 10^{-3}}[/math] [math]\;[/math] Fermi-Dirac sterile neutrino density: [math]\boxed{\rho_{s \nu} = \frac{N_{s \nu} \pi^2 \left(k_B T_{s \nu} \right)^4}{2 C_{s \nu} \hbar^3 c^5}}[/math] [math]\;[/math] Fermi-Dirac sterile neutrino number density: (ref. 3, eq. 8) [math]n_{s \nu} = \left(\frac{3 \zeta \left(3 \right) N_{s \nu} \left(k_{B} T_{s \nu} \right)^3}{4 \pi^2 \left(\hbar c \right)^3} \right)[/math] [math]\;[/math] Fermi-Dirac dark matter sterile neutrino mass integration via substitution: [math]m_{s \nu} = \frac{\rho_{s \nu}}{n_{s \nu}} = \left(\frac{N_{s \nu} \pi^2 \left(k_B T_{s \nu} \right)^4}{2 C_{s \nu} \hbar^3 c^5} \right)\left(\frac{4 \pi^2 \left(\hbar c \right)^3}{3 \zeta \left(3 \right) N_{s \nu} \left(k_{B} T_{s \nu} \right)^3} \right) = \frac{2 \pi^{4} k_B T_{s \nu}}{3 C_{s \nu} c^{2} \zeta \left(3 \right)} = 1.456 \cdot 10^{-35} \; \text{kg}[/math] [math]\;[/math] Fermi-Dirac dark matter sterile neutrino mass: [math]\boxed{m_{s \nu} = \frac{2 \pi^{4} k_B T_{s \nu}}{3 C_{s \nu} c^{2} \zeta \left(3 \right)}}[/math] [math]\;[/math] [math]\boxed{m_{s \nu} = 1.456 \cdot 10^{-35} \; \text{kg}}[/math] [math]\boxed{m_{s \nu} = 8.167 \; \frac{\text{eV}}{c^2}}[/math] [math]\;[/math] WMAP satellite cosmological parameters at photon decoupling time: (ref. 2) [math]\Omega_{dm,t} = 0.63[/math] [math]\;[/math] Sterile neutrino composition is equivalent to dark matter composition at photon decoupling time. [math]\boxed{\Omega_{s \nu,t} = \Omega_{dm,t}}[/math] [math]\;[/math] Fermi-Dirac dark matter sterile neutrino decoupling time: [math]\boxed{T_{u,s \nu} = \frac{}{2\left(k_B T_{s \nu,t} \right)^2} \sqrt{\frac{3 \Omega_{s \nu,t} C_{s \nu} \hbar^3 c^5}{G N_{s \nu} \pi^3}}} \; \; \; m_{s \nu} \neq 0[/math] [math]\;[/math] [math]\boxed{T_{u,s \nu} = 0.0255 \; \text{s}} \; \; \; m_{s \nu} \neq 0[/math] [math]\;[/math] Toy model dark matter particle candidates chart: [math]\begin{array}{l*{8}{c}r} & \text{identity} & \text{state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} & \text{decoupling time} \\ b & \text{scalar} & 0 & 0 & 1 & 1 & 1 & 18.658 \; \text{eV} & \phi & 0.0255 \; \text{s} \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & 8.167 \; \text{eV} & \nu_{s} & 0.0255 \; \text{s} \\ \end{array}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf WMAP satellite cosmological parameters at photon decoupling time: (ref. 2) https://map.gsfc.nasa.gov/media/080998/index.html Introduction to Cosmology: Lecture 6 - Thermal history of the Universe: (ref. 3) http://gravitation.web.ua.pt/sites/default/files/migrated2016/Lecture_6.pdf Wikipedia - Scalar boson: (ref. 4) https://en.wikipedia.org/wiki/Scalar_boson Wikipedia - Sterile neutrinos: (ref. 5) https://en.wikipedia.org/wiki/Sterile_neutrino Wikipedia - Dark matter: (ref. 6) https://en.wikipedia.org/wiki/Dark_matter
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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
  13. 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/
  14. 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]
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