Orion1

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  1. Derivation of neutrino mass from neutrino scattering: [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]p_{e} \sin \phi = p_{f} \sin \theta \tag{1}[/math] [math]p_{e} \cos \phi + p_{f} \cos \theta = p_{i} \tag{2}[/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] Divide equation [math](1)[/math] by equation [math](3)[/math] for an expression for [math]\tan \phi[/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] 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, noting that [math]p = \frac{E}{c}[/math]. [math]\lambda_{f} - \lambda_{i} = \frac{h}{m_{e} c} (1 - \cos \theta) \tag{5}[/math] [math]\frac{\lambda_{f}}{\lambda_{i}} = \frac{p_{i}}{p_{f}} = 1 + \frac{E_{\nu}}{E_{e}} (1 - \cos \theta) = 1 + \frac{m_{\nu} c^2}{m_{e} c^2} (1 - \cos \theta) = 1 + \frac{m_{\nu}}{m_{e}} (1 - \cos \theta) \tag{6}[/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]\tan \phi = \frac{\sin \theta}{\frac{p_{i}}{p_{f}} - \cos \theta} = \frac{\sin \theta}{1 + \frac{m_{\nu}}{m_{e}} (1 - \cos \theta) - \cos \theta} = \frac{\sin \theta}{\left(1 + \frac{m_{\nu}}{m_{e}} \right)(1 - \cos \theta)} \tag{7}[/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] 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] 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]\frac{m_{\nu}}{m_{e}} \tan \phi = \left(\cot \frac{\theta}{2} - \tan \phi \right) \tag{11}[/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] 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]m_{n}[/math] - nuclear mass Electron interaction neutrino scattering angle [math]\theta[/math]: [math]\boxed{\theta = 2 \operatorname{arccot} \left(\frac{(m_{e} + m_{\nu}) \tan \phi}{m_{e}} \right)} \tag{14}[/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] Nuclear interaction neutrino scattering angle [math]\theta[/math]: [math]\boxed{\theta = 2 \operatorname{arccot} \left(\frac{(m_{n} + m_{\nu}) \tan \phi}{m_{n}} \right)} \tag{16}[/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] Any discussions and/or peer reviews about this specific topic thread? 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
  2. 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
  3. 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
  4. 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
  5. Orion1

    The Lagrangian equation...

    I request to initiate a thread on what I only know as the Lagrangian equation. The original published scientific paper derivation proof is unknown to me. [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] This Lagrangian equation appears to represent a massless field tensor fundamental field interaction Lagrangian combination zero summation action in natural units. [math]\;[/math] The classical General Relativity Lagrangian tensor appears to describe a Ricci tensor on a smooth spacially flat Ricci maniold [math]\mathbb{R}[/math]. (ref. 1, ref. 2) [math]\;[/math] The massless field tensor Yang-Mills Maxwell Lagrangian term represents at the core of the unification of the electromagnetic force and weak forces [math](U(1) \times SU(2))[/math] and quantum chromodynamics, the theory of the strong force [math](SU(3))[/math] and predicts all the massless spin one Maxwells equations. (ref. 3, ref. 4) [math]\;[/math] The massless field tensor Dirac Lagrangian term is a relativistic wave equation that describes all spin one-half particle interactions. (ref. 5) [math]\;[/math] The massless field tensor Higgs Lagrangian term describes all spin zero Higgs field interactions. (ref. 6) [math]\;[/math] The massless field tensor Yukawa coupling interaction term describes the interaction between a massless spin zero scalar field [math]\phi[/math] and a massless spin one-half Dirac field [math]\psi[/math] (ref. 7) [math]\;[/math] Do you know where the original published scientific paper derivation proof is located? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Ricci curvature: (ref. 1) https://en.wikipedia.org/wiki/Ricci_curvature Wikipedia - Riemannian manifold: (ref. 2) https://en.wikipedia.org/wiki/Riemannian_manifold Wikipedia - Maxwells equations: (ref. 3) https://en.wikipedia.org/wiki/Maxwell's_equations#Formulation_in_SI_units_convention Stackexchange - derivation of maxwells equations from field tensor lagrangian: (ref. 4) https://physics.stackexchange.com/questions/3005/derivation-of-maxwells-equations-from-field-tensor-lagrangian Wikipedia - Dirac Lagrangian: (ref. 5) https://en.wikipedia.org/wiki/Dirac_equation#Dirac_Lagrangian Wikipedia - Higgs field: (ref. 6) https://simple.wikipedia.org/wiki/Higgs_field Wikipedia - Yukawa Lagrangian: (ref. 7) https://en.wikipedia.org/wiki/Yukawa_interaction#The_action
  6. Orion1

    The Lagrangian equation...

    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/
  7. Orion1

    The Lagrangian equation...

    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]
  8. Orion1

    The Lagrangian equation...

    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
  9. Orion1

    The Lagrangian equation...

    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
  10. Orion1

    The Lagrangian equation...

    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
  11. Orion1

    The Lagrangian equation...

    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
  12. Orion1

    The Lagrangian equation...

    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
  13. Orion1

    The Lagrangian equation...

    [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
  14. Orion1

    The Lagrangian equation...

    Relativistic Lagrangian Lorentz factor: (ref. 1, ref. 2, ref. 6) [math]dt = \gamma\left(\dot{\mathbf{r}} \right) d\tau[/math] [math]\;[/math] [math]\gamma\left(\dot{\mathbf{r}} \right) = \frac{dt}{d\tau} = \frac{1}{\sqrt{1 - \frac{\dot{\mathbf{r}}^2}{c^2}}}[/math] [math]\boxed{\gamma\left(\dot{\mathbf{r}} \right) = \frac{1}{\sqrt{1 - \frac{\dot{\mathbf{r}}^2}{c^2}}}}[/math] [math]\;[/math] [math]\dot{\mathbf{r}} = \frac{d\mathbf{r}}{dt}[/math] [math]\;[/math] Relativistic Lagrangian neutral particle total energy integration via substitution: (ref. 2) [math]E_{t} = m_0 c^2 \frac{dt}{d \tau} = \gamma\left(\dot{\mathbf{r}} \right) m_0 c^2 = \frac{m_0 c^2}{\sqrt {1 - \frac{\dot{\mathbf{r}}^2 \left(t \right)}{c^2}}} = m_0 c^2 + {1 \over 2} m_0 \dot{\mathbf{r}}^2 \left(t \right) + {3 \over 8} m_0 \frac{\dot{\mathbf{r}}^4 \left(t \right)}{c^2} + \cdots[/math] [math]\;[/math] Relativistic Lagrangian neutral particle total energy: (ref. 2) [math]\boxed{E_{t} = \frac{m_{0} c^2}{\sqrt{1 - \frac{\dot{\mathbf{r}}^2 \left(t \right)}{c^2}}}}[/math] [math]\;[/math] Classical neutral particle kinetic energy: [math]E_{k} = \frac{m_{0} v^{2}}{2} = \frac{m_{0}}{2} \frac{ds^{2}}{dt^{2}}[/math] [math]\boxed{E_{k} = \frac{m_{0}}{2} \frac{ds^{2}}{dt^{2}}}[/math] [math]\;[/math] Classical Lagrangian neutral particle kinetic energy: (ref. 4) [math]E_{k} = \frac{m_{0} g_{bc}}{2} \frac{\mathrm{d}\xi^{b}}{\mathrm{d}t} \frac{\mathrm{d}\xi^{c}}{\mathrm{d}t}[/math] [math]\;[/math] Relativistic Lagrangian neutral particle kinetic energy: [math]E_{k} = m_{0} c^{2} \left(\gamma\left(\dot{\mathbf{r}} \right) - 1 \right)[/math] [math]\;[/math] Newtons second law for neutral particle integration via substitution: (ref. 3) [math]\mathbf{F} = \frac{d \mathbf{p}}{dt} = \frac{d(m_{0} \mathbf{v})}{\mathrm{d}t} = m_{0} \frac{d\mathbf{v}}{\mathrm{d}t} = m_{0} \frac{ds}{dt^2}[/math] [math]\;[/math] Newtons second law for neutral particle: [math]\boxed{\mathbf{F} = m_{0} \frac{ds}{dt^2}}[/math] [math]\;[/math] Relativistic Newtons second law for neutral particle: [math]\boxed{\mathbf{F} = \gamma m_{0} \frac{ds}{dt^2}}[/math] [math]\;[/math] General relativity geodesic equation: (ref. 4) [math]\frac{d^{2}x^{\mu}}{dt^{2}} + \Gamma^{\mu}{}_{\alpha \beta} \frac{dx^{\alpha}}{dt} \frac{dx^{\beta}}{dt} = 0[/math] [math]\;[/math] Newtons second law in Lagrangian form for neutral particle: (ref. 5) [math]F^{a} = m_{0} \left(\frac{d^{2} \xi^{a}}{dt^{2}}+ \Gamma^{a}{}_{bc} \frac{d\xi^{b}}{dt} \frac{d\xi^{c}}{dt} \right)[/math] [math]\;[/math] Relativistic Newtons second law in Lagrangian form for neutral particle: [math]\boxed{F^{a} = \gamma\left(\dot{\mathbf{r}} \right) m_{0} \left( \frac{d^{2} \xi^{a}}{dt^{2}} + \Gamma^{a}{}_{bc} \frac{d\xi^{b}}{dt} \frac{d\xi^{c}}{dt} \right)}[/math] [math]\;[/math] Relativistic Lagrangian for a neutral particle: [math]\mathcal{L} = E_{k} - E_{p}[/math] [math]\;[/math] [math]\boxed{\mathcal{L} = m_{0} c^{2} \left(\gamma\left(\dot{\mathbf{r}} \right) - 1 \right) - E_{p}}[/math] [math]\;[/math] 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] [math]\;[/math] Relativistic Lagrangian: [math]\boxed{\mathcal{L} = \sum_{1}^{n} \mathcal{L}\left(n \right) = 0}[/math] [math]\;[/math] The Lagrangian equation integration via substitution: [math]\mathcal{L} = \sum_{1}^{n} \mathcal{L}\left(n \right) = \underbrace{\mathbb{R}}_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0 \; \; \; \; \; \; n = 5[/math] The Lagrangian equation: [math]\boxed{\mathcal{L} = \underbrace{\mathbb{R}}_{GR} - \overbrace{\underbrace{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}}_{Yang-Mills}}^{Maxwell} + \underbrace{i \overline{\psi} \gamma^\mu D_\mu \psi}_{Dirac} + \underbrace{|D_\mu h|^2-V\left(|h| \right)}_{Higgs} + \underbrace{h \overline{\psi} \psi}_{Yukawa} = 0} \; \; \; \; \; \; n = 5[/math] [math]\;[/math] [math]\;[/math] I think that this revision has compensated for the 4 momentum and 4 velocity under general relativity with respect to proper time [math]\tau[/math], through the Lorentz factor. The 4 momentum and 4 velocity is intrinsic to general relativity as required by four-dimensional space-time. [math]\;[/math] I note two general relativity geodesic equation forms, a spacial and a temporal form. [math]\;[/math] General relativity spacial geodesic equation: (ref. 4) [math]\frac{d^{2}x^{\mu}}{ds^{2}} + \Gamma^{\mu}{}_{\alpha \beta} \frac{dx^{\alpha}}{ds} \frac{dx^{\beta}}{ds} = 0[/math] [math]\;[/math] General relativity temporal geodesic equation: (ref. 4) [math]\frac{d^{2}x^{\mu}}{dt^{2}} + \Gamma^{\mu}{}_{\alpha \beta} \frac{dx^{\alpha}}{dt} \frac{dx^{\beta}}{dt} = 0[/math] [math]\;[/math] Do you agree with this mathematical symbolic formalism derivation revision for the formal Lagrangian equation? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - The Lorentz factor: (ref. 1) https://en.wikipedia.org/wiki/Lorentz_factor Wikipedia - Relativistic Lagrangian mechanics: (ref. 2) https://en.wikipedia.org/wiki/Relativistic_Lagrangian_mechanics#Coordinate_formulation Wikipedia - Newtons second law: (ref. 3) https://en.wikipedia.org/wiki/Newton's_laws_of_motion#Newton's_second_law Wikipedia - Geodesics in general relativity: (ref. 4) https://en.wikipedia.org/wiki/Geodesics_in_general_relativity Wikipedia - Newtons second law Lagrangian form: (ref. 5) https://en.wikipedia.org/wiki/Lagrangian_mechanics#From_Newtonian_to_Lagrangian_mechanics Wikipedia - Four-velocity: (ref. 6) https://en.wikipedia.org/wiki/Four-velocity
  15. Orion1

    The Lagrangian equation...

    [math]\;[/math] Please consider Newtons second law and the relativistic Newtons second law for a moment. [math]\;[/math] [math]\gamma[/math] - Lorentz factor [math]\;[/math] Newtons second law: (ref. 1) [math]\mathbf{F} = \frac{d \mathbf{p}}{dt} = \frac{d\left(m \mathbf{v} \right)}{\mathrm{d}t} = m {\frac{d\mathbf{v}}{\mathrm{d}t}} = m \frac{ds}{dt^2}[/math] [math]\boxed{\mathbf{F} = m \frac{ds}{dt^2}}[/math] [math]\;[/math] Relativistic Newtons second law: [math]\boxed{\mathbf{F} = \gamma m \frac{ds}{dt^2}}[/math] [math]\;[/math] Newtons second law in Lagrangian form: (ref. 2) [math]F^{a} = m \left( \frac{d^{2} \xi^{a}}{dt^{2}}+ \Gamma^{a}{}_{bc} \frac{d\xi^{b}}{dt} \frac{d\xi^{c}}{dt} \right)[/math] [math]\;[/math] Relativistic Newtons second law in Lagrangian form: [math]\boxed{F^{a} = \gamma m \left( \frac{d^{2} \xi^{a}}{dt^{2}}+ \Gamma^{a}{}_{bc} \frac{d\xi^{b}}{dt} \frac{d\xi^{c}}{dt} \right)}[/math] [math]\;[/math] Thus including all classical nonrelativistic Lagrangian mechanics into relativistic Lagrangian mechanics? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Newtons second law: (ref. 1) https://en.wikipedia.org/wiki/Newton's_laws_of_motion#Newton's_second_law Wikipedia - Newtons second law in Lagrangian form: (ref. 2) https://en.wikipedia.org/wiki/Lagrangian_mechanics#From_Newtonian_to_Lagrangian_mechanics
  16. Orion1

    Latex test

  17. Orion1

    THE TIME-FLOW FALLACY

    Physics: The branch of science concerned with the nature and properties of matter and energy. The subject matter of physics, distinguished from that of chemistry and biology, includes mechanics, heat, light and other radiation, sound, electricity, magnetism, and the structure of atoms. The physical properties and phenomena of something. Physical: Relating to things perceived through the senses as opposed to the mind; tangible or concrete. "everything physical in the universe" We are defining the Original Posters semantic definitions?
  18. Orion1

    Observable Universe mass...

    Solar star system average planetary mass: (ref. 1) [math]\boxed{M_{ap} = \frac{}{n_p} \sum_{n = 1}^{n_p} M_p\left(n \right)} \; \; \; \; \; \; n_p = 8[/math] [math]\;[/math] Observable Universe total planetary number based upon solar star system: [math]\boxed{N_p = \frac{\Omega_p \pi^3 k_B^4}{12 M_{ap} c^2 \left(\hbar H_0 \right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}} \right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}} \right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}} \right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}} \right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}} \right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\phi,t}}{T_{\phi}} \right]^3 \right)}[/math] [math]\;[/math] Observable Universe total planetary number based upon solar star system: [math]\boxed{N_p = 3.536 \cdot 10^{23} \; \text{planets}}[/math] [math]\;[/math] Wikipedia observable universe total planetary number: [math]\boxed{N_p = \left(2.000 \cdot 10^{23} \rightarrow 3.200 \cdot 10^{23} \right) \; \text{planets}}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Planetary mass: (ref. 1) https://en.wikipedia.org/wiki/Planetary_mass#Values_from_the_DE405_ephemeris
  19. Orion1

    Observable Universe mass...

    This is my equation for the observable Universe mass, based upon the Cosmic Energy Inventory (CEI) parameters and the Hubble Space Telescope (HST) parameters in (SI) units. Universe observable parameters: Universe total observable radius: [math]R_u = 4.408 \cdot 10^{26} \; \text{m} \; \; \; (46.6 \cdot 10^{9} \; \text{ly})[/math] CEI stellar Baryon density: [math]\Omega_s = 0.00205[/math] HST observable stellar number: [math]N_s = 10^{22}[/math] Solar mass: [math]M_{\odot} = 1.989 \cdot 10^{30} \; \text{kg}[/math] Observable Universe mass: [math]\boxed{M_u = \frac{N_s M_{\odot}}{\Omega_s}}[/math] [math]\boxed{M_u = 9.705 \cdot 10^{54} \; \text{kg}}[/math] Any discussions and/or peer reviews about this specific topic thread? Also, please be sure to vote on the thread to rate this topic. Reference: http://en.wikipedia.org/wiki/Lambda-CDM_model http://en.wikipedia.org/wiki/Universe http://en.wikipedia.org/wiki/Observable_universe http://en.wikipedia.org/wiki/Dark_matter http://arxiv.org/pdf/astro-ph/0406095v2.pdf
  20. Orion1

    Observable Universe mass...

    Planck satellite cosmological parameters: (ref. 1, pg. 11) [math]\Omega_{dm} = 0.268[/math] [math]\Omega_{b} = 0.0495[/math] [math]\Omega_{\Lambda} = 0.6825[/math] [math]\Omega_{\nu} = 1.259 \cdot 10^{-3}[/math] [math]\Omega_{\gamma} = 5.012 \cdot 10^{-5}[/math] [math]\;[/math] Symbolic identity key: [math]n_{s}[/math] - spin states total number [math]N_{s}[/math] - species total number [math]N_{n}[/math] - total effective degeneracy number [math]\text{if } n_{s} \geq N_{s} \text{ then } N_{n} = n_{s}[/math] [math]\text{if } n_{s} \leq N_{s} \text{ then } N_{n} = N_{s}[/math] [math]\;[/math] [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] Total stellar class number: (ref. 2) [math]n_c = 7[/math] key: 1 O, 2 B, 3 A, 4 F, 5 G, 6 K, 7 M [math]\Omega_f[/math] - main sequence stars stellar class fraction [math]N_s[/math] - total observable stellar number [math]M_s[/math] - Main-sequence mass [math]\;[/math] Observable Universe average stellar mass: [math]M_{as} = \frac{1}{N_s} \sum_{n = 1}^{n_c} \Omega_f\left(n\right) N_s M_s\left(n\right) = \sum_{n = 1}^{n_c} \Omega_f\left(n\right) M_s\left(n\right) = 0.219 \cdot M_{\odot} \rightarrow 0.595 \cdot M_{\odot}[/math] [math]\boxed{M_{as} = \sum_{n = 1}^{n_c} \Omega_f\left(n\right) M_s\left(n\right)} \; \; \; n_c = 7[/math] [math]\boxed{M_{as} = \left(0.219 \rightarrow 0.595\right) \cdot M_{\odot}}[/math] Observable Universe average stellar mass upper bound limit: [math]\boxed{M_{as} = 1.184 \cdot 10^{30} \; \text{kg}}[/math] Observable Universe average stellar mass: (ref. 3, pg. 20) [math]M_{as} = 0.6 \cdot M_{\odot} = 1.193 \cdot 10^{30} \; \text{kg}[/math] [math]M_{as} = 1.193 \cdot 10^{30} \; \text{kg}[/math] [math]\;[/math] Observable Universe stellar baryon composition: (ref. 4, pg. 3) [math]\Omega_s = \left(\Omega_{ms} + \Omega_{wd} + \Omega_{ns}\right) = 2.460 \cdot 10^{-3}[/math] [math]\Omega_s = 2.460 \cdot 10^{-3}[/math] [math]---[/math] Milky Way galaxy mass: (ref. 5, 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] Redshift parameter at photon decoupling time: [math]z = 1090.43[/math] [math]\;[/math] Cosmic photon background radiation temperature at present time: [math]T_{\gamma} = 2.72548 \; \text{K}[/math] [math]\;[/math] Cosmic photon background radiation temperature at photon decoupling time: [math]T_{\gamma,t} = T_{\gamma} \left(1 + z\right) = 2974.67 \; \text{K}[/math] [math]\boxed{T_{\gamma,t} = 2974.67 \; \text{K}}[/math] [math]\;[/math] Cosmic neutrino background radiation temperature at present time: (ref. 6, pg. 44, eq. 220) [math]T_{\nu} = \left(\frac{4}{11}\right)^{\frac{1}{3}} T_{\gamma} = 1.945 \; \text{K}[/math] [math]\boxed{T_{\nu} = 1.945 \; \text{K}}[/math] [math]\;[/math] Cosmic neutrino background radiation temperature at neutrino decoupling time: (ref. 7) [math]T_{\nu,t} = 1 \cdot 10^{10} \; \text{K}[/math] [math]\;[/math] Observable Universe dark matter scalar particle temperature is equivalent to cosmic neutrino background radiation temperature: [math]\boxed{T_{\phi} = T_{\nu}}[/math] [math]\;[/math] Observable Universe dark matter scalar particle decoupling temperature is equivalent to cosmic neutrino background radiation decoupling temperature: [math]\boxed{T_{\phi,t} = T_{\nu,t}}[/math] [math]\;[/math] Cosmic scalar particle dark energy background radiation temperature: [math]\boxed{T_{\Lambda} = \frac{}{k_{B}} \left(\frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3}\right)^{1/4}}[/math] [math]\boxed{T_{\Lambda} = 35.013 \; \text{K}}[/math] [math]\;[/math] Observable Universe Cosmological Constant: [math]\Lambda_s = \frac{3 \Omega_{\Lambda} H_0^2}{c^2} = 1.180 \cdot 10^{-52} \; \text{m}^{-2}[/math] [math]\;[/math] Observable Universe scalar particle dark energy Cosmological Constant integration via substitution: [math]\Lambda_s = \frac{3 H_0^2}{c^2} \left(\frac{4 G N_{\Lambda} \pi^3 \left(k_B T_{\Lambda}\right)^4}{45 H_0^2 \hbar^3 c^5}\right) = \frac{4 G N_{\Lambda} \pi^3 \left(k_B T_{\Lambda}\right)^4}{15 \hbar^3 c^7} = 1.180 \cdot 10^{-52} \; \text{m}^{-2}[/math] [math]\;[/math] Observable Universe scalar particle dark energy Cosmological Constant: [math]\boxed{\Lambda_s = \frac{4 G N_{\Lambda} \pi^3 \left(k_B T_{\Lambda}\right)^4}{15 \hbar^3 c^7}}[/math] [math]\;[/math] Observable Universe scalar particle dark energy Cosmological Constant: [math]\boxed{\Lambda_s = 1.180 \cdot 10^{-52} \; \text{m}^{-2}}[/math] [math]\;[/math] Dark matter density: [math]\rho_{dm} = \frac{3 \Omega_{dm} H_0^2}{8 \pi G}[/math] Baryonic density: [math]\rho_{b} = \frac{3 \Omega_{b} H_0^2}{8 \pi G}[/math] Dark energy density: [math]\rho_{\Lambda} = \frac{3 \Omega_{\Lambda} H_0^2}{8 \pi G}[/math] [math]\;[/math] Bose-Einstein total dark matter scalar particle distribution constant: [math]\boxed{C_{\phi} = \frac{4 G N_{\phi} \pi^3 \left(k_B T_{\phi}\right)^4}{3 \Omega_{\phi} H_0^2 \hbar^3 c^5}}[/math] [math]\boxed{C_{\phi} = 3.640 \cdot 10^{-4}}[/math] [math]\;[/math] Fermi-Dirac total neutrino distribution constant: [math]\boxed{C_{\nu} = \frac{4 G N_{\nu} \pi^3 \left(k_B T_{\nu}\right)^4}{3 \Omega_{\nu} H_0^2 \hbar^3 c^5}}[/math] [math]\boxed{C_{\nu} = 0.236}[/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 and Fermi-Dirac baryonic photon and neutrino matter density: [math]\boxed{\rho_{b} = \frac{N_{\gamma} \pi^2 \left(k_B T_{\gamma}\right)^4}{30 \hbar^3 c^5} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) + \frac{N_{\nu} \pi^2 \left(k_B T_{\nu}\right)^4}{2 C_{\nu} \hbar^3 c^5} \left(\frac{\Omega_{b}}{\Omega_{b} - \Omega_{\nu}}\right)}[/math] [math]\;[/math] Bose-Einstein scalar particle dark energy density: [math]\boxed{\rho_{\Lambda} = \frac{N_{\Lambda} \pi^2 \left(k_B T_{\Lambda}\right)^4}{30 \hbar^3 c^5}}[/math] [math]\;[/math] Observable Universe total scalar particle dark matter and photon and neutrino co-moving volumes: [math]V_{\phi} = \frac{4 \pi R_{\phi}^3}{3} = \frac{4 \pi}{3} \left(\left(\frac{c}{2 H_{0}}\right) \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]\right)^3[/math] [math]V_{\gamma} = \frac{4 \pi R_{\gamma}^3}{3} = \frac{4 \pi}{3} \left(\left(\frac{c}{2 H_{0}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]\right)^3[/math] [math]V_{\nu} = \frac{4 \pi R_{\nu}^3}{3} = \frac{4 \pi}{3} \left(\left(\frac{c}{2 H_{0}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]\right)^3[/math] Observable Universe scalar particle dark matter and scalar particle dark energy and neutrino total co-moving volumes are equivalent: [math]\boxed{V_{\phi} = V_{\Lambda} = V_{\nu}}[/math] [math]\;[/math] Bose-Einstein scalar particle dark matter total mass: [math]\boxed{\rho_{\phi} V_{\phi} = \frac{N_{\phi} \pi^3 \left(k_B T_{\phi}\right)^4}{12 C_{\phi} c^2 \left(\hbar H_0\right)^3} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3}[/math] [math]\;[/math] Bose-Einstein and Fermi-Dirac baryonic photon and neutrino matter relative composition total mass: [math]\boxed{\rho_{b} V_{b} = \frac{N_{\gamma} \pi^3 \left(k_B T_{\gamma}\right)^4}{180 c^2 \left(\hbar H_0\right)^3} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} \pi^3 \left(k_B T_{\nu}\right)^4}{12 C_{\nu} c^2 \left(\hbar H_0\right)^3} \left(\frac{\Omega_{b}}{\Omega_{b} - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3}[/math] [math]\;[/math] Bose-Einstein scalar particle dark energy total mass: [math]\boxed{\rho_{\Lambda} V_{\Lambda} = \frac{N_{\Lambda} \pi^2 \left(k_B T_{\Lambda}\right)^4}{180 c^2 \left(\hbar H_0\right)^3} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3}[/math] [math]\;[/math] Observable Universe compositional Equation of State and compositional number: [math]\Omega_{dm} + \Omega_{b} + \Omega_{\Lambda} = 1 \; \; \; \; \; \; n_{c} = 3[/math] [math]\;[/math] Observable Universe total critical mass: [math]M_{c} = \sum_{n = 1}^{n_{c}} \rho_n V_n = \rho_{dm} V_{dm} + \rho_{b} V_{b} + \rho_{\Lambda} V_{\Lambda}[/math] [math]\boxed{M_c = \frac{c^3}{16 G H_0} \left(\left(\Omega_{dm} + \Omega_{\Lambda}\right) \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \Omega_b \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3\right)}[/math] [math]\boxed{M_c = 1.179 \cdot 10^{56} \; \text{kg}}[/math] [math]\;[/math] Observable Universe compositional Equation of State and compositional number: [math]\Omega_{\phi} + \Omega_{b} + \Omega_{\Lambda} = 1 \; \; \; \; \; \; n_{c} = 3[/math] [math]\;[/math] Observable Universe scalar particle dark matter and scalar particle dark energy composition total mass: [math]M_{u} = \sum_{n = 1}^{n_{c}} \rho_n V_n = \rho_{\phi} V_{\phi} + \rho_{b} V_{b} + \rho_{\Lambda} V_{\Lambda}[/math] [math]\;[/math] [math]\boxed{M_u = \frac{\pi^3 k_B^4}{12 c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3\right)}[/math] [math]\boxed{M_u = 1.179 \cdot 10^{56} \; \text{kg}}[/math] [math]\;[/math] Observable Universe total energy: [math]E_u = M_u c^2[/math] [math]\boxed{E_u = \frac{\pi^3 k_B^4}{12 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3\right)}[/math] [math]\boxed{E_u = 1.060 \cdot 10^{73} \; \text{j}}[/math] [math]\;[/math] Observable Universe total stellar number: [math]\boxed{N_s = \frac{\Omega_s \pi^3 k_B^4}{12 M_{as} c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3\right)}[/math] [math]\;[/math] Observable Universe total stellar number: [math]\boxed{N_s = 2.453 \cdot 10^{23} \; \text{stars}}[/math] [math]\;[/math] Wikipedia observable universe total stellar number: (ref. 8) [math]N_s = 3.000 \cdot 10^{23} \; \text{stars}[/math] [math]\;[/math] Observable Universe total galaxy number: [math]\boxed{N_g = \frac{\Omega_b \pi^3 k_B^4}{12 M_{mw} c^2 \left(\hbar H_0\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3\right)}[/math] [math]\;[/math] Observable Universe total galaxy number: [math]\boxed{N_g = 2.330 \cdot 10^{12} \; \text{galaxies}}[/math] [math]\;[/math] Wikipedia observable universe total galaxy number: (ref. 9) [math]N_g = 2.000 \cdot 10^{12} \; \text{galaxies}[/math] [math]\;[/math] Observable Universe stars per galaxy average number: [math]\frac{N_s}{N_g} = \frac{\Omega_s M_{mw}}{\Omega_b M_{as}} = 1.053 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math] [math]\boxed{\frac{N_s}{N_g} = \frac{\Omega_s M_{mw}}{\Omega_b M_{as}}}[/math] [math]\boxed{\frac{N_s}{N_g} = 1.053 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}}[/math] [math]\;[/math] Wikipedia stars per galaxy average number: (ref. 8, ref. 9) [math]\frac{N_s}{N_g} = 1.500 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math] [math]\;[/math] Wikipedia Milky Way galaxy total stellar number: (ref. 10) [math]\frac{N_s}{N_g} = 2.500 \cdot 10^{11} \pm 1.500 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math] [math]\;[/math] Observable Universe planetary composition: (ref. 4, pg. 3) [math]\Omega_p = 1 \cdot 10^{-6}[/math] [math]\;[/math] Wikipedia Milky Way galaxy total planetary number: (ref. 10, ref. 11) [math]\frac{N_p}{N_g} = \left(1.000 \cdot 10^{11} \rightarrow 1.600 \cdot 10^{11}\right) \; \frac{\text{planets}}{\text{galaxy}}[/math] [math]\;[/math] Observable Universe average planetary mass: [math]\boxed{M_{ap} = \frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{N_g}{N_p}\right)}[/math] [math]\boxed{M_{ap} = \left(52.974 \rightarrow 84.758\right) \cdot M_{\oplus}}[/math] [math]\;[/math] Solar star system average planetary mass: (ref. 12) [math]\boxed{M_{ap} = \frac{}{n_p} \sum_{n = 1}^{n_p} M_p\left(n\right)} \; \; \; \; \; \; n_p = 8[/math] [math]\boxed{M_{ap} = 55.855 \cdot M_{\oplus}}[/math] [math]\;[/math] Milky Way galaxy total planetary number based upon solar star system: [math]\boxed{\frac{N_p}{N_g} = \frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_p\left(n\right)\right)^{-1}} \; \; \; \; \; \; n_p = 8[/math] [math]\boxed{\frac{N_p}{N_g} = 1.518 \cdot 10^{11} \; \frac{\text{planets}}{\text{galaxy}}}[/math] [math]\;[/math] Wikipedia Milky Way galaxy total planetary number: (ref. 10, ref. 11) [math]\frac{N_p}{N_g} = \left(1.000 \cdot 10^{11} \rightarrow 1.600 \cdot 10^{11}\right) \; \frac{\text{planets}}{\text{galaxy}}[/math] [math]\;[/math] Milky Way galaxy total planetary number based upon solar star system: [math]\boxed{\frac{N_p}{N_g} = \frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_p\left(n\right)\right)^{-1}} \; \; \; \; \; \; n_p = 8[/math] [math]\;[/math] Observable Universe stars per galaxy average number: [math]\boxed{\frac{N_s}{N_g} = \frac{\Omega_s M_{mw}}{\Omega_b M_{as}}}[/math] [math]\;[/math] Observable Universe average stellar mass: [math]\boxed{M_{as} = \sum_{n = 1}^{n_c} \Omega_f\left(n\right) M_s\left(n\right)} \; \; \; n_c = 7[/math] [math]\;[/math] Observable Universe planets per star average number based upon solar star system integration via substitution: [math]\frac{N_p}{N_s} = \left(\frac{N_p}{N_g}\right)\left(\frac{N_g}{N_s}\right) = \left[\frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_p\left(n\right)\right)^{-1}\right]\left(\frac{\Omega_b M_{as}}{\Omega_s M_{mw}}\right)[/math] [math]\frac{N_p}{N_s} = \left[\frac{\Omega_p M_{mw}}{\Omega_b} \left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_p\left(n\right)\right)^{-1}\right]\left(\frac{\Omega_b}{\Omega_s M_{mw}}\right)\left(\sum_{n = 1}^{n_c} \Omega_f\left(n\right) M_s\left(n\right)\right) = \frac{\Omega_p}{\Omega_s} \left(\sum_{n = 1}^{n_c} \Omega_f\left(n\right) M_s\left(n\right)\right)\left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_p\left(n\right)\right)^{-1}[/math] [math]\;[/math] Observable Universe planets per star average number based upon solar star system: [math]\boxed{\frac{N_p}{N_s} = \frac{\Omega_p}{\Omega_s} \left(\sum_{n = 1}^{n_c} \Omega_{f}\left(n\right) M_{s}\left(n\right)\right)\left(\frac{}{n_p} \sum_{n = 1}^{n_p} M_{p}\left(n\right)\right)^{-1}} \; \; \; \; \; \; n_c = 7, n_p = 8[/math] [math]\;[/math] [math]\boxed{\frac{N_p}{N_s} = 1.443 \; \frac{\text{planets}}{\text{star}}}[/math] [math]\;[/math] Wikipedia planets per star average number: [math]\boxed{\frac{N_p}{N_s} = \left(0.667 \rightarrow 1.067\right) \; \frac{\text{planets}}{\text{star}}}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Is there anything else that you want to see quantified based upon this model for observational comparison? [math]\;[/math] Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf Wikipedia - Stellar classification - Harvard spectral classification: (ref. 2) https://en.wikipedia.org/wiki/Stellar_classification Harvard_spectral_classification (insert number symbol) On The Mass Distribution Of Stars...: (ref. 3) http://www.doiserbia.nb.rs/img/doi/1450-698X/2006/1450-698X0672017N.pdf The Cosmic Energy Inventory: (ref. 4) http://arxiv.org/pdf/astro-ph/0406095v2.pdf Mass models of the Milky Way: (ref. 5) http://arxiv.org/pdf/1102.4340v1 PHYS: 652 Cosmic Inventory I: Radiation: (ref. 6) http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_09.pdf Wikipedia - neutrino decoupling: (ref. 7) https://en.wikipedia.org/wiki/Neutrino_decoupling Wikipedia - Observable universe total stellar number: (ref. 8) https://en.wikipedia.org/wiki/Star#Distribution Wikipedia - Galaxy: (ref. 9) https://en.wikipedia.org/wiki/Galaxy Wikipedia - Milky Way Galaxy: (ref. 10) https://en.wikipedia.org/wiki/Milky_Way Space.com - 160 Billion Alien Planets May Exist in Our Milky Way Galaxy: (ref. 11) https://www.space.com/14200-160-billion-alien-planets-milky-galaxy.html Wikipedia - Planetary mass: (ref. 12) https://en.wikipedia.org/wiki/Planetary_mass#Values_from_the_DE405_ephemeris (source code errors corrected on 01-18-2019)
  21. Orion1

    Observable Universe mass...

    Planck satellite cosmological parameters: (ref. 1) [math]\Omega_{dm} = 0.268[/math] [math]\Omega_{b} = 0.0495[/math] [math]\Omega_{\Lambda} = 0.6825[/math] [math]\Omega_{\nu} = 1.259 \cdot 10^{-3}[/math] [math]\Omega_{\gamma} = 5.012 \cdot 10^{-5}[/math] [math]\;[/math] Dark matter scalar particle composition is equivalent to dark matter composition: [math]\boxed{\Omega_{\phi} = \Omega_{dm}}[/math] [math]\;[/math] Planck mass: (ref. 2) [math]m_{P} = \sqrt{\frac{\hbar c}{G}}[/math] [math]\;[/math] Planck temperature (ref. 3) [math]T_{P} = \frac{E_{P}}{k_{B}} = \frac{m_{P} c^{2}}{k_{B}} = \left(\sqrt{\frac{\hbar c}{G}}\right) \frac{c^2}{k_{B}} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}} = 1.417 \cdot 10^{32} \; \text{K}[/math] [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\boxed{T_{P} = 1.417 \cdot 10^{32} \; \text{K}}[/math] [math]\;[/math] Planck radius: (ref. 4) [math]r_P = \sqrt{\frac{\hbar G}{c^{3}}}[/math] [math]\;[/math] Planck volume: (ref. 5) [math]V_{P} = \frac{4 \pi R_{P}^{3}}{3} = \frac{4 \pi}{3} \left(\sqrt{\frac{\hbar G}{c^{3}}}\right)^{3} = \frac{4 \pi}{3} \sqrt{\frac{\left(\hbar G\right)^{3}}{c^{9}}} = 1.768 \cdot 10^{-104} \; \text{m}^3[/math] [math]\;[/math] [math]\boxed{V_{P} = \frac{4 \pi}{3} \sqrt{\frac{\left(\hbar G\right)^{3}}{c^{9}}}}[/math] [math]\boxed{V_{P} = 1.768 \cdot 10^{-104} \; \text{m}^3}[/math] [math]\;[/math] Observable Universe total energy at present time: [math]\boxed{E_u = \frac{\pi^3 k_B^4}{12 \left(\hbar H_{0}\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{E_u = 1.060 \cdot 10^{73} \; \text{j}}[/math] [math]\;[/math] All particle peak radiation temperatures are equivalent to Planck temperature at Planck time: [math]\boxed{T_{\phi,t_P} = T_{\gamma,t_P} = T_{\nu,t_P} = T_P}[/math] Dark energy scalar particle peak radiation temperature is non-zero at Planck time: [math]\boxed{T_{\Lambda,t_P} \neq 0}[/math] [math]\;[/math] Present time radial metric decoupling temperature redshift parameter global quantizations remain the same: [math]\boxed{\ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right] \; \; \; \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right] \; \; \; \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]}[/math] [math]\;[/math] Observable Universe total energy at Planck time integration via substitution: [math]E_{u}\left(T_P\right) = \frac{\pi^3 k_B^4}{12 \left(\hbar H_{0}\right)^3} \left(\frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}\right)^{4} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)[/math] [math]\;[/math] Observable Universe total energy at Planck time: [math]\boxed{E_{u}\left(T_P\right) = \frac{\pi^3 c^{10}}{12 \hbar G^{2} H_{0}^3} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{E_{u}\left(T_P\right) = 8.439 \cdot 10^{199} \; \text{j}}[/math] [math]\;[/math] Observable Universe total mass at Planck time: [math]M_{u}\left(T_P\right) = \frac{E_{u}\left(T_P\right)}{c^{2}} = \frac{\pi^3 c^{8}}{12 \hbar G^{2} H_{0}^3} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)[/math] [math]\boxed{M_{u}\left(T_P\right) = \frac{\pi^3 c^{8}}{12 \hbar G^{2} H_{0}^3} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{M_{u}\left(T_P\right) = 9.390 \cdot 10^{182} \; \text{kg}}[/math] [math]\;[/math] Observable Universe total energy density at Planck time integration via substitution: [math]\epsilon_{u}\left(T_{P}\right) = \frac{E_{u}\left(T_{P}\right)}{V_{P}} = \frac{\pi^3 c^{10}}{12 \hbar G^{2} H_{0}^3} \left(\frac{3}{4 \pi} \sqrt{\frac{c^{9}}{\left(\hbar G\right)^{3}}}\right) \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)[/math] [math]\;[/math] Observable Universe total energy density at Planck time: [math]\boxed{\epsilon_{u}\left(T_{P}\right) = \frac{\pi^{2}}{16 H_{0}^{3}} \sqrt{\frac{c^{29}}{\hbar^{5} G^{7}}} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\;[/math] [math]\boxed{\epsilon_{u}\left(T_{P}\right) = 4.772 \cdot 10^{303} \; \frac{\text{j}}{\text{m}^{3}}}[/math] [math]\;[/math] Planck energy density integration via substitution: [math]\epsilon_{P} = \frac{E_{P}}{V_{P}} = \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)\left(\frac{3}{4 \pi} \sqrt{\frac{c^{9}}{\left(\hbar G\right)^{3}}}\right) = \frac{3 c^{7}}{4 \pi \hbar G^{2}} = 1.106 \cdot 10^{113} \; \frac{\text{j}}{\text{m}^{3}}[/math] [math]\;[/math] Planck energy density: (ref. 5) [math]\boxed{\epsilon_{P} = \frac{3 c^{7}}{4 \pi \hbar G^{2}}}[/math] [math]\boxed{\epsilon_{P} = 1.106 \cdot 10^{113} \; \frac{\text{j}}{\text{m}^{3}}}[/math] [math]\;[/math] Planck vacuum energy density at Planck time: (ref. 6) [math]\epsilon_{vac} = \frac{\epsilon_{P}}{2} = \frac{3 c^{7}}{8 \pi \hbar G^{2}} = 5.531 \cdot 10^{112} \; \frac{\text{j}}{\text{m}^{3}}[/math] [math]\boxed{\epsilon_{vac} = \frac{3 c^{7}}{8 \pi \hbar G^{2}}}[/math] [math]\boxed{\epsilon_{vac} = 5.531 \cdot 10^{112} \; \frac{\text{j}}{\text{m}^{3}}}[/math] [math]\;[/math] Planck time: (ref. 7) [math]t_{P} = \sqrt{\frac{\hbar G}{c^{5}}}[/math] [math]\boxed{t_{P} = 5.391 \cdot 10^{-44} \; \text{s}}[/math] [math]\;[/math] Particle interaction rate is equivalent to Hubble Parameter at Planck interaction time: [math]\boxed{\Gamma_{t} = n \langle \sigma v \rangle = H_{t}}[/math] [math]\Gamma_{P} = H_{P} = \frac{}{t_{P}} = \sqrt{\frac{c^{5}}{\hbar G}} = 1.855 \cdot 10^{43} \; \frac{\text{particles}}{\text{s}}[/math] Planck particle interaction rate at Planck interaction time: [math]\boxed{\Gamma_{P} = \sqrt{\frac{c^{5}}{\hbar G}}}[/math] [math]\;[/math] [math]\boxed{\Gamma_{P} = 1.855 \cdot 10^{43} \; \frac{\text{particles}}{\text{s}}}[/math] [math]\;[/math] Dark matter scalar particle interaction rate at scalar particle Planck time: [math]\boxed{\Gamma_{\phi,t_P} = 2 \left(k_B T_{\phi,t_P}\right)^2 \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t_P} \hbar^3 c^5}}} \; \; \; m_{\phi} \neq 0[/math] [math]\;[/math] Planck temperature: (ref. 3) [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\;[/math] Dark matter scalar particle interaction rate at scalar particle Planck time integration via substitution: [math]\Gamma_{\phi,t_P} = 2 \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)^2 \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t_P} \hbar^3 c^5}} = \frac{2 \hbar c^{5}}{G} \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t_P} \hbar^3 c^5}} = 2 \sqrt{\frac{N_{\phi} \pi^3 c^{5}}{3 C_{\phi} \Omega_{\phi,t_P} \hbar G}}[/math] Dark matter scalar particle interaction rate at scalar particle Planck time: [math]\boxed{\Gamma_{\phi,t_P} = 2 \sqrt{\frac{N_{\phi} \pi^3 c^{5}}{3 C_{\phi} \Omega_{\phi,t_P} \hbar G}}} \; \; \; m_{\phi} \neq 0[/math] [math]\boxed{\Gamma_{\phi,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\phi} \neq 0[/math] [math]\;[/math] Photon interaction rate at photon Planck time: [math]\boxed{\Gamma_{\gamma,t_P} = \frac{4 \left(k_B T_{\gamma,t_P}\right)^2}{3} \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t_P} \hbar^3 c^5}}} \; \; \; m_{\gamma} = 0[/math] [math]\;[/math] Planck temperature: (ref. 3) [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\;[/math] Photon interaction rate at photon Planck time integration via substitution: [math]\Gamma_{\gamma,t_P} = \frac{4}{3} \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)^2 \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t_P} \hbar^3 c^5}} = \frac{4}{3} \left(\frac{\hbar c^{5}}{G}\right) \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t_P} \hbar^3 c^5}} = \frac{4}{3} \sqrt{\frac{N_{\gamma} \pi^3 c^{5}}{5 \Omega_{\gamma,t_P} \hbar G}}[/math] [math]\;[/math] Photon interaction rate at photon Planck time: [math]\boxed{\Gamma_{\gamma,t_P} = \frac{4}{3} \sqrt{\frac{N_{\gamma} \pi^3 c^{5}}{5 \Omega_{\gamma,t_P} \hbar G}}} \; \; \; m_{\gamma} = 0[/math] [math]\boxed{\Gamma_{\gamma,t_P} = 1.230 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\gamma} = 0[/math] [math]\;[/math] Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time: [math]\boxed{\Gamma_{\nu,t_P} = 2 \left(k_B T_{\nu,t_P}\right)^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t_P} \hbar^3 c^5}}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Planck temperature: (ref. 3) [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\;[/math] Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time integration via substitution: [math]\Gamma_{\nu,t_P} = 2 \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t_P} \hbar^3 c^5}} = 2 \left(\frac{\hbar c^{5}}{G}\right) \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t_P} \hbar^3 c^5}} = 2 \sqrt{\frac{N_{\nu} \pi^3 c^5}{3 C_{\nu} \Omega_{\nu,t_P} \hbar G}}[/math] Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time: [math]\boxed{\Gamma_{\nu,t_P} = 2 \sqrt{\frac{N_{\nu} \pi^3 c^5}{3 C_{\nu} \Omega_{\nu,t_P} \hbar G}}} \; \; \; m_{\nu} \neq 0[/math] [math]\boxed{\Gamma_{\nu,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Particle interaction rate summary: Planck particle interaction rate at Planck interaction time: [math]\boxed{\Gamma_{P} = 1.855 \cdot 10^{43} \; \frac{\text{particles}}{\text{s}}}[/math] Dark matter scalar particle interaction rate at scalar particle Planck time: [math]\boxed{\Gamma_{\phi,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\phi} \neq 0[/math] Photon interaction rate at photon Planck time: [math]\boxed{\Gamma_{\gamma,t_P} = 1.230 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\gamma} = 0[/math] Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time: [math]\boxed{\Gamma_{\nu,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Planck particle strong decoupling time: (ref. 7) [math]t_{P} = \sqrt{\frac{\hbar G}{c^{5}}}[/math] [math]\boxed{t_{P} = 5.391 \cdot 10^{-44} \; \text{s}}[/math] [math]\;[/math] Dark matter scalar particle strong decoupling time: [math]\boxed{t_{\phi,t} = \frac{}{2} \sqrt{\frac{3 C_{\phi} \Omega_{\phi,t_P} \hbar G}{N_{\phi} \pi^3 c^{5}}}} \; \; \; m_{\phi} \neq 0[/math] [math]\boxed{t_{\phi,t} = 2.282 \cdot 10^{-47} \; \text{s}} \; \; \; m_{\phi} \neq 0[/math] [math]\;[/math] Photon particle strong decoupling time: [math]\boxed{t_{\gamma,t} = \frac{3}{4} \sqrt{\frac{5 \Omega_{\gamma,t_P} \hbar G}{N_{\gamma} \pi^3 c^{5}}}} \; \; \; m_{\gamma} = 0[/math] [math]\boxed{t_{\gamma,t} = 8.128 \cdot 10^{-47} \; \text{s}} \; \; \; m_{\gamma} = 0[/math] [math]\;[/math] Cosmic neutrino background radiation particle strong decoupling time: [math]\boxed{t_{\nu,t} = \frac{}{2} \sqrt{ \frac{3 C_{\nu} \Omega_{\nu,t_P} \hbar G}{N_{\nu} \pi^3 c^5}}} \; \; \; m_{\nu} \neq 0[/math] [math]\boxed{t_{\nu,t} = 2.282 \cdot 10^{-47} \; \text{s}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Particle strong decoupling time summary: Planck particle strong decoupling time: [math]\boxed{t_{P} = 5.391 \cdot 10^{-44} \; \text{s}}[/math] Dark matter scalar particle strong decoupling time: [math]\boxed{t_{\phi,t} = 2.282 \cdot 10^{-47} \; \text{s}} \; \; \; m_{\phi} \neq 0[/math] Photon particle strong decoupling time: [math]\boxed{t_{\gamma,t} = 8.128 \cdot 10^{-47} \; \text{s}} \; \; \; m_{\gamma} = 0[/math] Cosmic neutrino background radiation particle strong decoupling time: [math]\boxed{t_{\nu,t} = 2.282 \cdot 10^{-47} \; \text{s}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] Is the Planck energy density the maximum energy density limit in the universe? [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 Wikipedia - Planck mass: (ref. 2) https://en.wikipedia.org/wiki/Planck_mass Wikipedia - Planck temperature: (ref. 3) https://en.wikipedia.org/wiki/Planck_temperature Wikipedia - Planck radius: (ref. 4) https://en.wikipedia.org/wiki/Planck_length Wikipedia - Planck volume: (ref. 5) https://en.wikipedia.org/wiki/Planck_units#Derived_units Wikipedia - Vacuum_energy: (ref. 6) https://en.wikipedia.org/wiki/Vacuum_energy Wikipedia - Planck time: (ref. 7) https://en.wikipedia.org/wiki/Planck_time (source code errors corrected on 12-30-2018)
  22. Orion1

    Latex test

    test
  23. Orion1

    Observable Universe mass...

    [math]\text{Planck satellite cosmological parameters:} \; \left(\text{ref. 1}\right)[/math] [math]\Omega_{dm} = 0.268[/math] [math]\Omega_{b} = 0.0495[/math] [math]\Omega_{\Lambda} = 0.6825[/math] [math]\Omega_{\nu} = 1.259 \cdot 10^{-3}[/math] [math]\Omega_{\gamma} = 5.012 \cdot 10^{-5}[/math] [math]\;[/math] [math]\text{Dark matter scalar particle composition is equivalent to dark matter composition:}[/math] [math]\boxed{\Omega_{\phi} = \Omega_{dm}}[/math] [math]\;[/math] [math]\text{Planck mass:} \; \left(\text{ref. 2}\right)[/math] [math]m_{P} = \sqrt{\frac{\hbar c}{G}}[/math] [math]\;[/math] [math]\text{Planck temperature} \; \left(\text{ref. 3}\right)[/math] [math]T_{P} = \frac{E_{P}}{k_{B}} = \frac{m_{P} c^{2}}{k_{B}} = \left(\sqrt{\frac{\hbar c}{G}}\right) \frac{c^2}{k_{B}} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}} = 1.417 \cdot 10^{32} \; \text{K}[/math] [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\boxed{T_{P} = 1.417 \cdot 10^{32} \; \text{K}}[/math] [math]\;[/math] [math]\text{Planck radius:} \; \left(\text{ref. 4}\right)[/math] [math]r_P = \sqrt{\frac{\hbar G}{c^{3}}}[/math] [math]\;[/math] [math]\text{Planck volume:} \; \left(\text{ref. 5}\right)[/math] [math]V_{P} = \frac{4 \pi R_{P}^{3}}{3} = \frac{4 \pi}{3} \left(\sqrt{\frac{\hbar G}{c^{3}}}\right)^{3} = \frac{4 \pi}{3} \sqrt{\frac{\left(\hbar G\right)^{3}}{c^{9}}} = 1.768 \cdot 10^{-104} \; \text{m}^3[/math] [math]\;[/math] [math]\boxed{V_{P} = \frac{4 \pi}{3} \sqrt{\frac{\left(\hbar G\right)^{3}}{c^{9}}}}[/math] [math]\boxed{V_{P} = 1.768 \cdot 10^{-104} \; \text{m}^3}[/math] [math]\;[/math] [math]\text{Observable Universe total energy at present time:}[/math] [math]\boxed{E_u = \frac{\pi^3 k_B^4}{12 \left(\hbar H_{0}\right)^3} \left(\frac{N_{\phi} T_{\phi}^4}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma} T_{\gamma}^4}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu} T_{\nu}^4}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3 + \frac{N_{\Lambda} T_{\Lambda}^4}{15} \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{E_u = 1.060 \cdot 10^{73} \; \text{j}}[/math] [math]\;[/math] [math]\text{All particle peak radiation temperatures are equivalent to Planck temperature at Planck time:}[/math] [math]\boxed{T_{\phi,t_P} = T_{\gamma,t_P} = T_{\nu,t_P} = T_P}[/math] [math]\text{Dark energy scalar particle peak radiation temperature is non-zero at Planck time:}[/math] [math]\boxed{T_{\Lambda,t_P} \neq 0}[/math] [math]\;[/math] [math]\text{Present time radial metric decoupling temperature redshift parameter global quantizations remain the same:}[/math] [math]\boxed{\ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right] \; \; \; \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right] \; \; \; \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]}[/math] [math]\;[/math] [math]\text{Observable Universe total energy at Planck time integration via substitution:}[/math] [math]E_{u}\left(T_P\right) = \frac{\pi^3 k_B^4}{12 \left(\hbar H_{0}\right)^3} \left(\frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}\right)^{4} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)[/math] [math]\;[/math] [math]\text{Observable Universe total energy at Planck time:}[/math] [math]\boxed{E_{u}\left(T_P\right) = \frac{\pi^3 c^{10}}{12 \hbar G^{2} H_{0}^3} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{E_{u}\left(T_P\right) = 8.439 \cdot 10^{199} \; \text{j}}[/math] [math]\;[/math] [math]\text{Observable Universe total mass at Planck time:}[/math] [math]M_{u}\left(T_P\right) = \frac{E_{u}\left(T_P\right)}{c^{2}} = \frac{\pi^3 c^{8}}{12 \hbar G^{2} H_{0}^3} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)[/math] [math]\boxed{M_{u}\left(T_P\right) = \frac{\pi^3 c^{8}}{12 \hbar G^{2} H_{0}^3} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\boxed{M_{u}\left(T_P\right) = 9.390 \cdot 10^{182} \; \text{kg}}[/math] [math]\;[/math] [math]\text{Observable Universe total energy density at Planck time integration via substitution:}[/math] [math]\epsilon_{u}\left(T_{P}\right) = \frac{E_{u}\left(T_{P}\right)}{V_{P}} = \frac{\pi^3 c^{10}}{12 \hbar G^{2} H_{0}^3} \left(\frac{3}{4 \pi} \sqrt{\frac{c^{9}}{\left(\hbar G\right)^{3}}}\right) \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)[/math] [math]\;[/math] [math]\text{Observable Universe total energy density at Planck time:}[/math] [math]\boxed{\epsilon_{u}\left(T_{P}\right) = \frac{\pi^{2}}{16 H_{0}^{3}} \sqrt{\frac{c^{29}}{\hbar^{5} G^{7}}} \left(\frac{N_{\phi}}{C_{\phi}} \ln \left[\frac{T_{\phi,t}}{T_{\phi}}\right]^3 + \frac{N_{\gamma}}{15} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\gamma}}\right) \ln \left[\frac{T_{\gamma,t}}{T_{\gamma}}\right]^3 + \frac{N_{\nu}}{C_{\nu}} \left(\frac{\Omega_b}{\Omega_b - \Omega_{\nu}}\right) \ln \left[\frac{T_{\nu,t}}{T_{\nu}}\right]^3\right)}[/math] [math]\;[/math] [math]\boxed{\epsilon_{u}\left(T_{P}\right) = 4.772 \cdot 10^{303} \; \frac{\text{j}}{\text{m}^{3}}}[/math] [math]\;[/math] [math]\text{Planck energy density integration via substitution:}[/math] [math]\epsilon_{P} = \frac{E_{P}}{V_{P}} = \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)\left(\frac{3}{4 \pi} \sqrt{\frac{c^{9}}{\left(\hbar G\right)^{3}}}\right) = \frac{3 c^{7}}{4 \pi \hbar G^{2}} = 1.106 \cdot 10^{113} \; \frac{\text{j}}{\text{m}^{3}}[/math] [math]\;[/math] [math]\text{Planck energy density:} \; \left(\text{ref. 5}\right)[/math] [math]\boxed{\epsilon_{P} = \frac{3 c^{7}}{4 \pi \hbar G^{2}}}[/math] [math]\boxed{\epsilon_{P} = 1.106 \cdot 10^{113} \; \frac{\text{j}}{\text{m}^{3}}}[/math] [math]\;[/math] [math]\text{Vacuum energy density at Planck time:} \; \left(\text{ref. 6}\right)[/math] [math]\epsilon_{vac} = \frac{\epsilon_{P}}{2} = \frac{3 c^{7}}{8 \pi \hbar G^{2}} = 5.531 \cdot 10^{112} \; \frac{\text{j}}{\text{m}^{3}}[/math] [math]\boxed{\epsilon_{vac} = \frac{3 c^{7}}{8 \pi \hbar G^{2}}}[/math] [math]\boxed{\epsilon_{vac} = 5.531 \cdot 10^{112} \; \frac{\text{j}}{\text{m}^{3}}}[/math] [math]\;[/math] [math]\text{Planck time:} \; \left(\text{ref. 7}\right)[/math] [math]t_{P} = \sqrt{\frac{\hbar G}{c^{5}}}[/math] [math]\;[/math] [math]\text{Particle interaction rate is equivalent to Hubble Parameter at Planck interaction time:}[/math] [math]\boxed{\Gamma_{t} = n \langle \sigma v \rangle = H_{t}}[/math] [math]\Gamma_{P} = H_{P} = \frac{}{t_{P}} = \sqrt{\frac{c^{5}}{\hbar G}} = 1.855 \cdot 10^{43} \; \frac{\text{particles}}{\text{s}}[/math] [math]\text{Planck particle interaction rate at Planck interaction time:}[/math] [math]\boxed{\Gamma_{P} = \sqrt{\frac{c^{5}}{\hbar G}}}[/math] [math]\;[/math] [math]\boxed{\Gamma_{P} = 1.855 \cdot 10^{43} \; \frac{\text{particles}}{\text{s}}}[/math] [math]\;[/math] [math]\text{Dark matter scalar particle interaction rate at scalar particle Planck time:}[/math] [math]\boxed{\Gamma_{\phi,t_P} = 2 \left(k_B T_{\phi,t_P}\right)^2 \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t_P} \hbar^3 c^5}}} \; \; \; m_{\phi} \neq 0[/math] [math]\;[/math] [math]\text{Planck temperature:} \; \left(\text{ref. 3}\right)[/math] [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\;[/math] [math]\text{Dark matter scalar particle interaction rate at scalar particle Planck time integration via substitution:}[/math] [math]\Gamma_{\phi,t_P} = 2 \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)^2 \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t_P} \hbar^3 c^5}} = \frac{2 \hbar c^{5}}{G} \sqrt{\frac{G N_{\phi} \pi^3}{3 C_{\phi} \Omega_{\phi,t_P} \hbar^3 c^5}} = 2 \sqrt{\frac{N_{\phi} \pi^3 c^{5}}{3 C_{\phi} \Omega_{\phi,t_P} \hbar G}}[/math] [math]\text{Dark matter scalar particle interaction rate at scalar particle Planck time:}[/math] [math]\boxed{\Gamma_{\phi,t_P} = 2 \sqrt{\frac{N_{\phi} \pi^3 c^{5}}{3 C_{\phi} \Omega_{\phi,t_P} \hbar G}}} \; \; \; m_{\phi} \neq 0[/math] [math]\boxed{\Gamma_{\phi,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\phi} \neq 0[/math] [math]\;[/math] [math]\text{Photon interaction rate at photon Planck time:}[/math] [math]\boxed{\Gamma_{\gamma,t_P} = \frac{4 \left(k_B T_{\gamma,t_P}\right)^2}{3} \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t_P} \hbar^3 c^5}}} \; \; \; m_{\gamma} = 0[/math] [math]\;[/math] [math]\text{Planck temperature:} \; \left(\text{ref. 3}\right)[/math] [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\;[/math] [math]\text{Photon interaction rate at photon Planck time integration via substitution:}[/math] [math]\Gamma_{\gamma,t_P} = \frac{4}{3} \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)^2 \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t_P} \hbar^3 c^5}} = \frac{4}{3} \left(\frac{\hbar c^{5}}{G}\right) \sqrt{\frac{G N_{\gamma} \pi^3}{5 \Omega_{\gamma,t_P} \hbar^3 c^5}} = \frac{4}{3} \sqrt{\frac{N_{\gamma} \pi^3 c^{5}}{5 \Omega_{\gamma,t_P} \hbar G}}[/math] [math]\;[/math] [math]\text{Photon interaction rate at photon Planck time:}[/math] [math]\boxed{\Gamma_{\gamma,t_P} = \frac{4}{3} \sqrt{\frac{N_{\gamma} \pi^3 c^{5}}{5 \Omega_{\gamma,t_P} \hbar G}}} \; \; \; m_{\gamma} = 0[/math] [math]\boxed{\Gamma_{\gamma,t_P} = 1.230 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\gamma} = 0[/math] [math]\;[/math] [math]\text{Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time:}[/math] [math]\boxed{\Gamma_{\nu,t_P} = 2 \left(k_B T_{\nu,t_P}\right)^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t_P} \hbar^3 c^5}}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] [math]\text{Planck temperature:} \; \left(\text{ref. 3}\right)[/math] [math]\boxed{T_{P} = \frac{}{k_{B}} \sqrt{\frac{\hbar c^{5}}{G}}}[/math] [math]\;[/math] [math]\text{Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time integration via substitution:}[/math] [math]\Gamma_{\nu,t_P} = 2 \left(\sqrt{\frac{\hbar c^{5}}{G}}\right)^2 \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t_P} \hbar^3 c^5}} = 2 \left(\frac{\hbar c^{5}}{G}\right) \sqrt{\frac{G N_{\nu} \pi^3}{3 C_{\nu} \Omega_{\nu,t_P} \hbar^3 c^5}} = 2 \sqrt{\frac{N_{\nu} \pi^3 c^5}{3 C_{\nu} \Omega_{\nu,t_P} \hbar G}}[/math] [math]\text{Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time:}[/math] [math]\boxed{\Gamma_{\nu,t_P} = 2 \sqrt{\frac{N_{\nu} \pi^3 c^5}{3 C_{\nu} \Omega_{\nu,t_P} \hbar G}}} \; \; \; m_{\nu} \neq 0[/math] [math]\boxed{\Gamma_{\nu,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] [math]\text{Particle interaction rate summary:}[/math] [math]\text{Planck particle interaction rate at Planck interaction time:}[/math] [math]\boxed{\Gamma_{P} = 1.855 \cdot 10^{43} \; \frac{\text{particles}}{\text{s}}}[/math] [math]\text{Dark matter scalar particle interaction rate at scalar particle Planck time:}[/math] [math]\boxed{\Gamma_{\phi,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\phi} \neq 0[/math] [math]\text{Photon interaction rate at photon Planck time:}[/math] [math]\boxed{\Gamma_{\gamma,t_P} = 1.230 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\gamma} = 0[/math] [math]\text{Cosmic neutrino background radiation neutrino interaction rate at neutrino Planck time:}[/math] [math]\boxed{\Gamma_{\nu,t_P} = 1.207 \cdot 10^{46} \; \frac{\text{particles}}{\text{s}}} \; \; \; m_{\nu} \neq 0[/math] [math]\;[/math] [math]\text{Is the Planck energy density the maximum energy density limit in the universe?}[/math] [math]\;[/math] [math]\text{Any discussions and/or peer reviews about this specific topic thread?}[/math] [math]\;[/math] [math]\text{Reference:}[/math] Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf Wikipedia - Planck mass: (ref. 2) https://en.wikipedia.org/wiki/Planck_mass Wikipedia - Planck temperature: (ref. 3) https://en.wikipedia.org/wiki/Planck_temperature Wikipedia - Planck radius: (ref. 4) https://en.wikipedia.org/wiki/Planck_length Wikipedia - Planck volume: (ref. 5) https://en.wikipedia.org/wiki/Planck_units#Derived_units Wikipedia - Vacuum_energy: (ref. 6) https://en.wikipedia.org/wiki/Vacuum_energy Wikipedia - Planck time: (ref. 7)
  24. Orion1

    Latex test

    \[ \text{test} \]
  25. Orion1

    Latex test

    \[ F(x) = \int_{0}^{\infty} f(x) dx \] \[ F(x) = \int_{0}^{\infty} f(x) dx \]