Jump to content

Orion1

Senior Members
  • Posts

    191
  • Joined

Everything posted by Orion1

  1. In Einstein's theory of general relativity, the interior metric or interior fluid solution, is an exact solution to the Einstein field equations and Einstein-Maxwell equations that describes the gravitational field and the space-time geometry in the interior of a non-rotating or rotating neutral or charged spherically symmetric body of mass M, which consists of an incompressible fluid and constant density throughout the body and has zero pressure at the surface and that the electric charge and angular momentum of the mass may be zero or non-zero, and the universal cosmological constant is zero. For a non-zero charged mass, the metric takes into account the Einstein-Maxwell field energy of an electromagnetic field within the space-time geometry. The space-time geometry is in Boyer-Lindquist coordinates. "yet still idealized family of solutions would be the Vaidya spacetimes." - Markus Hanke Are these Vaidya metrics mathematically mapped out accurately at this point? "Mass, charge and spin are global properties of the entire spacetime - you cannot localise these quantities at any particular place." - Markus Hanke What if the black hole quantum charge originates from its center core? "We analyze the “vacuum” polarization induced by a quantum charged scalar field near the inner horizon of a charged black hole in quantum states evolving from arbitrary regular in states." "As the formation of charged BHs necessitates the presence of charged matter, it is actually more natural to consider a charged scalar field" "At the event horizon HR, this current is responsible for the discharge of the BH via Hawking radiation." (ref .7) The model in reference 7, appears to be modeling the field interaction between a black hole that is quantum charged at its core, and with the charged matter near the inner horizon, which discharges as hawking radiation. What determines the charge magnitude and polarity of a quantum charged black hole? The black hole core quantum charge cannot extend past the inner event horizon? Any discussions and/or peer reviews about this specific topic thread? "You will do well to expand your horizons." - Fortune Cookie Reference: Wikipedia - Four-gradient As a Jacobian matrix for the SR Minkowski metric tensor: (ref. 1) https://en.wikipedia.org/wiki/Four-gradient#As_a_Jacobian_matrix_for_the_SR_Minkowski_metric_tensor Wikipedia - Schwarzschild radius: (ref. 2) https://en.wikipedia.org/wiki/Schwarzschild_radius Wikipedia - Interior Schwarzschild metric - Other formulations: (ref. 3) https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric#Other_formulations Austrian Reports on Gravitation - Interior Schwarzschild Solution And Free Fall - Rainer Burghardt: (ref. 4) http://www.arg.or.at/Wpdf/WIff.pdf Wikipedia - Interior Schwarzschild metric: (ref. 5) https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric#Mathematics Wikipedia - Exterior Reissner-Nordstrom metric: (ref. 6) https://en.wikipedia.org/wiki/Reissner–Nordström_metric#The_metric Quantum (dis)charge of black hole interiors - Christiane Klein: (ref. 7) https://arxiv.org/pdf/2103.03714.pdf Wikipedia - Vaidya metric: (ref. 8) https://en.wikipedia.org/wiki/Vaidya_metric
  2. In Einstein's theory of general relativity, the interior metric or interior fluid solution, is an exact solution to the Einstein field equations and Einstein-Maxwell equations that describes the gravitational field and the space-time geometry in the interior of a non-rotating or rotating neutral or charged spherically symmetric body of mass M, which consists of an incompressible fluid and constant density throughout the body and has zero pressure at the surface and that the electric charge and angular momentum of the mass may be zero or non-zero, and the universal cosmological constant is zero. For a non-zero charged mass, the metric takes into account the Einstein-Maxwell field energy of an electromagnetic field within the space-time geometry. The space-time geometry is in Boyer-Lindquist coordinates. [math]\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}[/math] [math]\;[/math] [math]\color{blue}{\text{"You will do well to expand your horizons." - Fortune Cookie}}[/math] [math]\;[/math] Reference: Wikipedia - Four-gradient As a Jacobian matrix for the SR Minkowski metric tensor: (ref. 1) https://en.wikipedia.org/wiki/Four-gradient#As_a_Jacobian_matrix_for_the_SR_Minkowski_metric_tensor Wikipedia - Schwarzschild radius: (ref. 2) https://en.wikipedia.org/wiki/Schwarzschild_radius Wikipedia - Interior Schwarzschild metric - Other formulations: (ref. 3) https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric#Other_formulations Austrian Reports on Gravitation - Interior Schwarzschild Solution And Free Fall - Rainer Burghardt: (ref. 4) http://www.arg.or.at/Wpdf/WIff.pdf Wikipedia - Interior Schwarzschild metric: (ref. 5) https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric#Mathematics Wikipedia - Exterior Reissner-Nordstrom metric: (ref. 6) https://en.wikipedia.org/wiki/Reissner–Nordström_metric#The_metric
  3. In Einstein's theory of general relativity, the exterior metric or exterior fluid solution, is an exact solution to the Einstein field equations and Einstein-Maxwell equations that describes the gravitational field and the space-time geometry in the exterior of a non-rotating or rotating neutral or charged spherically symmetric body of mass M, which consists of an incompressible fluid and constant density throughout the body and has zero pressure at the surface and that the electric charge and angular momentum of the mass may be zero or non-zero, and the universal cosmological constant is zero. For a non-zero charged mass, the metric takes into account the Einstein-Maxwell field energy of an electromagnetic field within the space-time geometry. The space-time geometry is in Boyer-Lindquist coordinates. There is a theoretical discontinuity with a Kerr metric tensor element: (ref. 5, ref. 6, pg. 238, ref. 7, pg. 3, eq. 2.7) [math]g_{tt} \neq -\left(1 - \frac{r_{s} r}{\Sigma} \right)[/math] This Kerr metric tensor element [math]g_{tt}[/math] is missing the [math]\Delta [/math] rotation term [math]a^{2}[/math] in the entire numerator, and a [math]\Sigma[/math] term in the entire denominator. (ref. 8` pg. 135` tbl. 1) [math]\color{blue}{\text{The theoretical Kerr metric tensor element presented here is:}}[/math] [math]\boxed{g_{tt} = -\frac{\Delta}{\Sigma} = -\left(\frac{r^{2} - r_{s} r + a^{2}}{r^{2} + a^{2} \cos^{2} \theta} \right)}[/math] [math]\color{blue}{\text{This theoretical Kerr metric tensor element matches with the exterior Kerr-Newman metric tensor element when } (Q = 0) \text{.}}[/math] [math]\color{blue}{\text{And this Kerr tensor matches with all the other exterior metric tensor elements when } (J = 0) \text{ and or when } (m = 0) \text{.}}[/math] [math]\;[/math] [math]\color{blue}{\text{Table 1. Metric tensor components and symmetry.}}[/math] [math]\begin{array}{l*{5}c} \text{Metric tensor} & \text{theorem} & \text{identity with applied state} & \text{cited state} & \text{Symmetry} & \text{State} \\ \text{} & dt^{2} & dt^{2} & dt^{2} & \text{} \\ \text{Minkowski} & -1 & -1 & -1 & \text{Spherical} & m = 0,J = 0,Q = 0 \\ \text{Schwarzchild} & -\left(\frac{r^{2} - r_{s} r}{r^{2}} \right) & -\left(1 - \frac{r_{s}}{r} \right) & -\left(1 - \frac{r_{s}}{r} \right) & \text{Spherical} & m \neq 0,J = 0,Q = 0 \\ \text{Reissner-Nordstrom} & -\left(\frac{r^{2} - r_{s} r + r_{Q}^{2}}{r^{2}} \right) & -\left(1 - \frac{r_{s}}{r} \right) & -\left(1 - \frac{r_{s}}{r} \right) & \text{Spherical} & m \neq 0,J = 0,Q = 0 \\ \text{Ellipsoid} & -\left(\frac{r^{2} + a^{2}}{r^{2} + a^{2} \cos^{2} \theta} \right) & -\left(\frac{r^{2} + a^{2}}{r^{2} + a^{2} \cos^{2} \theta} \right) & -\left(\frac{r^{2} + a^{2}}{r^{2} + a^{2} \cos^{2} \theta} \right) & \text{Ellipsoid} & m = 0,J \neq 0,Q = 0 \\ \text{Kerr} & -\left({\frac{r^{2} - r_{s}r + a^{2}}{r^{2} + a^{2} \cos^{2} \theta}} \right) & -\left({\frac{r^{2} - r_{s}r + a^{2}}{r^{2} + a^{2} \cos^{2} \theta}} \right) & \underline{-\left(1 - \frac{r_{s} r}{r^{2} + a^{2} \cos^{2} \theta} \right)} & \text{Ellipsoid} & m \neq 0,J \neq 0,Q = 0 \\ \text{Kerr } \left(\text{cited} \right) & \underline{-\left(1 - \frac{r_{s} r}{r^{2} + a^{2} \cos^{2} \theta} \right)} & \underline{-1} & \underline{-1} & \boxed{\text{Incorrect}} & m = 0,J \neq 0,Q = 0 \\ \text{Kerr-Newman} & -\left(\frac{r^{2} - r_{s} r + a^{2} + r_{Q}^{2}}{r^{2} + a^{2} \cos^{2} \theta} \right) & -\left(\frac{r^{2} - r_{s} r + a^{2}}{r^{2} + a^{2} \cos^{2} \theta} \right) & -\left(\frac{r^{2} - r_{s} r + a^{2}}{r^{2} + a^{2} \cos^{2} \theta} \right) & \text{Ellipsoid} & m \neq 0, J \neq 0, Q = 0 \\ \end{array}[/math] [math]\;[/math] [math]\color{blue}{\text{The underlined metric tensors highlight the matrix locations of the theoretical discontinuities.}}[/math] [math]\;[/math] [math]\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}[/math] [math]\;[/math] [math]\color{blue}{\text{"You will do well to expand your horizons." - Fortune Cookie}}[/math] [math]\;[/math] Reference: Wikipedia - Four-gradient As a Jacobian matrix for the SR Minkowski metric tensor: (ref. 1) https://en.wikipedia.org/wiki/Four-gradient#As_a_Jacobian_matrix_for_the_SR_Minkowski_metric_tensor Wikipedia - Schwarzschild radius: (ref. 2) https://en.wikipedia.org/wiki/Schwarzschild_radius Wikipedia - Schwarzschild metric: (ref. 3) https://en.wikipedia.org/wiki/Schwarzschild_metric Wikipedia - Exterior Reissner-Nordstrom metric: (ref. 4) https://en.wikipedia.org/wiki/Reissner–Nordström_metric#The_metric Wikipedia - Exterior Kerr metric: (ref. 5) https://en.wikipedia.org/wiki/Kerr_metric The Racah Institute of Physics - Gravitational field of a spinning mass - Roy Kerr (ref. 6) http://old.phys.huji.ac.il/~barak_kol/Courses/Black-holes/reading-papers/Kerr.pdf Arxiv - The Kerr Metric - Saul A. Teukolsky: (ref. 7) https://arxiv.org/pdf/1410.2130.pdf Academic Journals - A derivation of the Kerr metric by ellipsoid coordinate transformation - Yu-ching Chou: (ref. 8) https://academicjournals.org/journal/IJPS/article-full-text-pdf/AEE776964987 Wikipedia - Exterior Kerr-Newman metric: (ref. 9) https://arxiv.org/pdf/astro-ph/9801252.pdf Wikipedia - Exterior Kerr-Newman metric: (ref. 10) https://en.wikipedia.org/wiki/Kerr-Newman_metric Science Direct - Rotating black hole and Kerr metric - Pierre Binetruy: (ref. 11) https://www.sciencedirect.com/topics/physics-and-astronomy/kerr-metric
  4. [math]\color{blue}{\text{Planck force:} \; (\text{ref. 1})}[/math] [math]F_{P} = \frac{c^4}{G}[/math] [math]\;[/math] [math]\color{blue}{\text{Coulomb's law of electrostatic force:} \; (\text{ref. 2})}[/math] [math]F_{C} = \frac{Q^{2}}{4 \pi \varepsilon_{0} r_{Q}^{2}}[/math] [math]\;[/math] [math]\color{blue}{\text{Planck force is equivalent to Coulomb force:}}[/math] [math]\boxed{F_{P} = F_{C}}[/math] [math]\;[/math] [math]\color{blue}{\text{Planck force is equivalent to Coulomb force integration via substitution, solve for } r_{Q} \text{:}}[/math] [math]\frac{c^4}{G} = \frac{Q^{2}}{4 \pi \varepsilon_{0} r_{Q}^{2}}[/math] [math]r_{Q}^{2} = \frac{Q^{2} G}{4 \pi \varepsilon_{0} c^{4}}[/math] [math]\;[/math] [math]\color{blue}{\text{Reissner-Nordstrom black hole metric charge radius:} \; (\text{ref. 3})}[/math] [math]\boxed{r_{Q} = \frac{Q_{bh}}{2 c^{2}} \sqrt{\frac{G}{\pi \varepsilon_{0}}}}[/math] [math]\;[/math] [math]\color{blue}{\text{Schwarzchild radius:} \; (\text{ref. 4})}[/math] [math]r_{s} = \frac{2 G M}{c^{2}}[/math] [math]\;[/math] [math]\color{blue}{\text{Solve for a Reissner-Nordstrom one solar mass black hole event horizon maximum attainable stable charge magnitude } Q_{bh}}[/math] [math]\color{blue}{\text{for a thin spherical shell of accumulated charged quantum particles:}}[/math] [math]\;[/math] [math]\color{blue}{\text{Reissner-Nordstrom black hole metric charge radius is equivalent to Schwarzchild radius:}}[/math] [math]\boxed{r_{Q} = r_{s}}[/math] [math]\;[/math] [math]\color{blue}{\text{Reissner-Nordstrom metric charge radius is equivalent to Schwarzchild radius integration via substitution:}}[/math] [math]\frac{Q_{bh}}{2 c^{2}} \sqrt{\frac{G}{\pi \varepsilon_{0}}} = \frac{2 G M_{\odot}}{c^{2}}[/math] [math]Q_{bh} = 4 M_{\odot} \sqrt{\pi \varepsilon_{0} G} = 3.427 \cdot 10^{20} \; \text{C}[/math] [math]\;[/math] [math]\color{blue}{\text{Reissner-Nordstrom one solar mass black hole event horizon maximum attainable stable charge magnitude } Q_{bh}}[/math] [math]\color{blue}{\text{for a thin spherical shell of accumulated charged quantum particles:}}[/math] [math]\boxed{Q_{bh} = 4 M_{\odot} \sqrt{\pi \varepsilon_{0} G}}[/math] [math]\boxed{Q_{bh} = 3.427 \cdot 10^{20} \; \text{C}}[/math] [math]\;[/math] [math]\color{blue}{\text{Does a Reissner-Nordstrom black hole metric charge result from a thin spherical shell of accumulated charged quantum particles}}[/math] [math]\color{blue}{\text{near the black hole event horizon?}}[/math] [math]\;[/math] [math]\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}[/math] [math]\;[/math] "Take charge of your thoughts. You can do what you will with them." - Plato [math]\;[/math] Reference: Wikipedia - Planck force: (ref. 1) https://en.wikipedia.org/wiki/Planck_units#Derived_units Wikipedia - Coulomb's law: (ref. 2) https://en.wikipedia.org/wiki/Coulomb's_law Wikipedia - Exterior Reissner-Nordstrom metric: (ref. 3) https://en.wikipedia.org/wiki/Reissner–Nordström_metric#The_metric Wikipedia - Schwarzschild radius: (ref. 4) https://en.wikipedia.org/wiki/Schwarzschild_radius
  5. [math]\color{blue}{\text{Neutron stars cosmological composition parameter:} \; (\text{ref. 1, pg. 3})}[/math] [math]\Omega_{ns} = 5 \cdot 10^{-5}[/math] [math]\;[/math] [math]\color{blue}{\text{Black holes cosmological composition parameter:} \; (\text{ref. 1, pg. 3})}[/math] [math]\Omega_{bh} = 7 \cdot 10^{-5}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model Milky Way neutron stars per galaxy average number:}}[/math] [math]\frac{N_{ns}}{N_g} = \frac{\Omega_{ns} M_{mw}}{\Omega_b M_{as}} = 2.138 \cdot 10^{9} \; \frac{\text{neutron stars}}{\text{galaxy}}[/math] [math]\boxed{\frac{N_{ns}}{N_g} = \frac{\Omega_{ns} M_{mw}}{\Omega_b M_{as}}}[/math] [math]\boxed{\frac{N_{ns}}{N_g} = 2.138 \cdot 10^{9} \; \frac{\text{neutron stars}}{\text{galaxy}}}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model Milky Way black holes per galaxy average number:}}[/math] [math]\boxed{\frac{N_{bh}}{N_{g}} = \frac{\Omega_{bh} \Omega_s M_{mw}}{\Omega_{b}^{2} M_{as}}}[/math] [math]\boxed{\frac{N_{bh}}{N_{g}} = 1.489 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model Milky Way total supernovae per galaxy average number:}}[/math] [math]\frac{N_{sn}}{N_g} = \left(\frac{N_{ns}}{N_g} + \frac{N_{bh}}{N_{g}} \right) = 2.287 \cdot 10^{9} \; \frac{\text{supernovae}}{\text{galaxy}}[/math] [math]\boxed{\frac{N_{sn}}{N_g} = 2.287 \cdot 10^{9} \; \frac{\text{supernovae}}{\text{galaxy}}}[/math] [math]\;[/math] [math]\color{blue}{\text{Milky Way galaxy oldest Population II star age - HD 140283:} \; (\text{ref. 2})}[/math] [math]t_{s} = 13.761 \cdot 10^{9} \; \text{years}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model Milky Way galaxy supernovae average rate per year:}}[/math] [math]\Gamma_{sn} \geq \frac{N_{sn}}{N_g t_{s}} \geq \frac{}{t_{s}} \left(\frac{N_{ns}}{N_g} + \frac{N_{bh}}{N_{g}} \right) \geq 0.166 \; \frac{\text{supernovae}}{\text{galaxy year}}[/math] [math]\boxed{\Gamma_{sn} \geq \frac{}{t_{s}} \left(\frac{N_{ns}}{N_g} + \frac{N_{bh}}{N_{g}} \right)}[/math] [math]\boxed{\Gamma_{sn} \geq 0.166 \; \frac{\text{supernovae}}{\text{galaxy year}}}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model Milky Way galaxy years per supernova average rate:}}[/math] [math]\boxed{\Gamma_{sn}^{-1} \leq t_{s} \left(\frac{N_{ns}}{N_g} + \frac{N_{bh}}{N_{g}} \right)^{-1}}[/math] [math]\boxed{\Gamma_{sn}^{-1} \leq 6.017 \; \frac{\text{galaxy years}}{\text{supernova}}}[/math] [math]\;[/math] [math]\color{blue}{\text{What is the peak luminosity for a supernova capable of producing a neutron star within this mass domain?}}[/math] [math]\boxed{0.595 \cdot M_{\odot} \leq M_{ns} \leq 2.27 \cdot M_{\odot}}[/math] [math]\;[/math] [math]\color{blue}{\text{What is the peak luminosity for a supernova capable of producing a black hole within this mass domain?}}[/math] [math]\boxed{11.979 \cdot M_{\odot} \leq M_{bh} \leq 13.783 \cdot M_{\odot}}[/math] [math]\;[/math] [math]\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}[/math] [math]\;[/math] "Stars, too, were time travelers. How many of those ancient points of light were the last echoes of suns now dead?" - Ransom Riggs [math]\;[/math] Reference: The Cosmic Energy Inventory: (ref. 1) http://arxiv.org/pdf/astro-ph/0406095v2.pdf Wikipedia - Oldest Known Star in Milky Way Galaxy - HD 140283: (ref. 2) https://en.wikipedia.org/wiki/HD_140283 Wikipedia - Supernova: (ref. 3) https://en.wikipedia.org/wiki/Supernova
  6. Source code correction... [math]d\tau^{2} = -\left(1 - \frac{r_{s}}{R} + \frac{r_{Q}^{2}}{R^{2}} \right) dt^{2} + \ldots[/math] [math]d\tau \text{ - proper time observer clock at radial distance R from the center}[/math] [math]dt \text{ - time measured by an observer clock at infinity}[/math] I searched for the motivation of Hans Reissner to include charge in general relativity. This scientist wrote in the cited paper: (ref. 1) "After Mr. Einstein, by explaining the perihelion movement of Mercury, the fertility of his new covariant field equations of gravity and thus of the having shown the postulate of the most general relativity and elsewhere given the generally covariant version of the Maxwell-Lorentz equations on the electromagnetic field, it seemed to me the next task to investigate the influence of the self-gravity of the electric field of spherical symmetry in a simple example. I started from the hope of finding a static cohesion of elementary charges through their own gravity without having to leave the ground of Maxwell's theory. Einstein's gravitation admittedly distorts the field of the elementary electric charge in a certain, incidentally, extremely slight way, but by its very nature it cannot cancel out the mutual electrostatic repulsion of the charge elements." - Hans Reissner I was able to locate a paper on electrically charged strange quark stars. The model appears to utilize the exterior Reissner-Nordstrom metric integrated with Gauss' law. (ref. 2) I hope at least this metric solution provides a student or professor studying general relativity the opportunity to examine the effects that general relativity has on metric interiors and charge, as it did with Karl Schwarzschild and Hans Reissner, plus various other college and university students and scientists. In the letter Schwarzschild wrote to Einstein, he concluded by saying, "As you see, the war treated me kindly enough, in spite of the heavy gunfire, to allow me to get away from it all and take this walk in the land of your ideas." - Karl Schwarzschild Any discussions and/or peer reviews about this specific topic thread? Reference: About the self-gravity of the electric field according to Einstein's theory - Hans Reissner: (ref. 1) http://old.phys.huji.ac.il/~barak_kol/Courses/Black-holes/reading-papers/Reissner.pdf https://zenodo.org/record/1447315/files/article.pdf Electrically Charged Strange Quark Stars - Rodrigo Picanco Negreiros: (ref. 2) https://arxiv.org/pdf/0907.5537.pdf
  7. In Einstein's theory of general relativity, the interior Reissner-Nordstrom metric (also interior Reissner-Nordstrom solution or interior Reissner-Nordstrom fluid solution) is an exact static solution for the gravitational field in the interior of a non-rotating charged spherically symmetric body of mass M, which consists of an incompressible fluid and constant density throughout the body and has zero pressure at the surface. This is a static solution that does not change over time. [math]\;[/math] [math]\eta_{\mu \nu} - \text{perturbed non-dynamical background metric}[/math] [math]\color{blue}{\text{General Relativity Minkowski flat spacetime metric:} \; (\text{ref. 1})}[/math] [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] [math]\color{blue}{\text{Schwarzschild radius:} \; (\text{ref. 2})}[/math] [math]r_{s} = \frac{2 G M}{c^{2}}[/math] [math]\;[/math] [math]r_{g} \text{ - radial coordinate at the body's surface}[/math] [math]\;[/math] [math]\color{blue}{\text{Exterior Schwarzschild metric:} \; (\text{ref. 3, ref. 4})}[/math] [math]ds^{2} = c^{2}{d \tau}^{2} = -\left(1 - {\frac{r_{s}}{r}} \right) c^{2} dt^{2} + \left(1 - {\frac{r_{s}}{r}} \right)^{-1} dr^{2} + r^{2} \left(d \theta^{2} + \sin^{2} \theta \; d \varphi^{2} \right)[/math] [math]\;[/math] [math]d\tau^{2} = -\left(1 - \frac{r_{s}}{R} \right) dt^{2} + \ldots[/math] [math]d\tau \text{ - proper time observer clock at radial distance R from the center}[/math] [math]dt \text{ - time measured by an observer clock at infinity}[/math] [math]\;[/math] [math]\color{blue}{\text{Interior Schwarzschild metric radial parameters:} \; (\text{ref. 5})}[/math] [math]\mathcal{R} ^{2} = \frac{r_{g}^{3}}{r_{s}}[/math] [math]\cos \eta_{g} = \sqrt{1 - \frac{r_{g}^{2}}{\mathcal{R}^{2}}} \; \; \; \; \; \; \cos \eta = \sqrt{1 - \frac{r^{2}}{\mathcal{R}^{2}}}[/math] [math]\;[/math] [math]\color{blue}{\text{Interior Schwarzschild metric total redshift factor:} \; (\text{ref. 6, pg. 2, eq. 2.2})}[/math] [math]\alpha_{T} = \frac{3 \cos \eta_{g} - \cos \eta}{2}[/math] [math]\;[/math] [math]\color{blue}{\text{Interior Schwarzschild metric:} \; (\text{ref. 7})}[/math] [math]ds^{2} = c^{2} d\tau^{2} = -\alpha_{T}^{2} c^{2} dt^{2} + \frac{dr^{2}}{\cos^{2} \eta} + r^{2}\left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math] [math]\;[/math] [math]ds^{2} = c^{2} d\tau^{2} = -\left(\frac{3 \cos \eta_{g} - \cos \eta}{2} \right)^{2} c^{2} dt^{2} + \frac{dr^{2}}{\cos^{2} \eta} + r^{2}\left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math] [math]\;[/math] [math]ds^{2} = c^{2} d\tau^{2} = -\frac{1}{4} \left(3 \sqrt{1 - \frac{r_{g}^{2}}{\mathcal{R}^{2}}} - \sqrt{1 - \frac{r^{2}}{\mathcal{R}^{2}}} \right)^{2} c^{2} dt^{2} + \left(1 - \frac{r^{2}}{\mathcal{R}^{2}} \right)^{-1} dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math] [math]\;[/math] [math]ds^{2} = c^{2} d\tau^{2} = -\frac{1}{4} \left(3 \sqrt{1 - \frac{r_{s}}{r_{g}}} - \sqrt{1 - \frac{r_{s} r^{2}}{r_{g}^{3}}} \right)^{2} c^{2} dt^{2} + \left(1 - \frac{r_{s} r^{2}}{r_{g}^{3}} \right)^{-1} dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math] [math]\;[/math] [math]\color{blue}{\text{Reissner-Nordstrom metric charge radius:} \; (\text{ref. 8})}[/math] [math]r_{Q}^{2} = \frac{Q^{2} G}{4 \pi \varepsilon_{0} c^{4}}[/math] [math]\;[/math] [math]\color{blue}{\text{Exterior Reissner-Nordstrom metric:} \; (\text{ref. 8})}[/math] [math]ds^{2} = c^{2} d\tau^{2} = -\left(1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right) c^{2} dt^{2} + \left(1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right)^{-1} \; dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math] [math]\;[/math] [math]d\tau^{2} = -\left(1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right) dt^{2} + \ldots[/math] [math]d\tau \text{ - proper time observer clock at radial distance R from the center}[/math] [math]dt \text{ - time measured by an observer clock at infinity}[/math] [math]\;[/math] [math]\color{blue}{\text{Interior Reissner-Nordstrom metric radial parameters:}}[/math] [math]\mathcal{R} ^{2} = \frac{r_{g}^{3}}{r_{s}}[/math] [math]\boxed{\cos \eta_{g} = \sqrt{1 - \frac{r_{g}^{2}}{\mathcal{R}^{2}} + \frac{r_{Q}^{2}}{\mathcal{R}^{2}}}} \; \; \; \; \; \; \boxed{\cos \eta = \sqrt{1 - \frac{r^{2}}{\mathcal{R}^{2}} + \frac{r_{Q}^{2}}{\mathcal{R}^{2}}}}[/math] [math]\;[/math] [math]\color{blue}{\text{Interior Reissner-Nordstrom metric total redshift factor:}}[/math] [math]\boxed{\alpha_{T} = \frac{3 \cos \eta_{g} - \cos \eta}{2}}[/math] [math]\;[/math] [math]\color{blue}{\text{Interior Reissner-Nordstrom metric:}}[/math] [math]ds^{2} = c^{2} d\tau^{2} = -\alpha_{T}^{2} c^{2} dt^{2} + \frac{dr^{2}}{\cos^{2} \eta} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math] [math]\;[/math] [math]ds^{2} = c^{2} d\tau^{2} = -\left(\frac{3 \cos \eta_{g} - \cos \eta}{2} \right)^{2} c^{2} dt^{2} + \frac{dr^{2}}{\cos^{2} \eta} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math] [math]\;[/math] [math]ds^{2} = c^{2} d\tau^{2} = -\frac{1}{4} \left(3 \sqrt{1 - \frac{r_{g}^{2}}{\mathcal{R}^{2}} + \frac{r_{Q}^{2}}{\mathcal{R}^{2}}} - \sqrt{1 - \frac{r^{2}}{\mathcal{R}^{2}} + \frac{r_{Q}^{2}}{\mathcal{R}^{2}}} \right)^{2} c^{2} dt^{2} + \left(1 - \frac{r^{2}}{\mathcal{R}^{2}} + \frac{r_{Q}^{2}}{\mathcal{R}^{2}} \right)^{-1} dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math] [math]\;[/math] [math]\boxed{ds^{2} = c^{2} d\tau^{2} = -\frac{1}{4} \left(3 \sqrt{1 - \frac{r_{s}}{r_{g}} + \frac{r_{s} r_{Q}^{2}}{r_{g}^{3}}} - \sqrt{1 - \frac{r_{s} r^{2}}{r_{g}^{3}} + \frac{r_{s} r_{Q}^{2}}{r_{g}^{3}}} \right)^{2} c^{2} dt^{2} + \left(1 - \frac{r_{s} r^{2}}{r_{g}^{3}} + \frac{r_{s} r_{Q}^{2}}{r_{g}^{3}} \right)^{-1} dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)}[/math] [math]\;[/math] [math]\color{blue}{\text{Metrics summary:}}[/math] [math]\color{blue}{\text{Exterior Schwarzschild metric:}}[/math] [math]ds^{2} = c^{2}{d \tau}^{2} = -\left(1 - {\frac{r_{s}}{r}} \right) c^{2} dt^{2} + \left(1 - {\frac{r_{s}}{r}} \right)^{-1} dr^{2} + r^{2} \left(d \theta^{2} + \sin^{2} \theta \; d \varphi^{2} \right)[/math] [math]\;[/math] [math]\color{blue}{\text{Exterior Reissner-Nordstrom metric:}}[/math] [math]ds^{2} = c^{2} d\tau^{2} = -\left(1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right) c^{2} dt^{2} + \left(1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right)^{-1} \; dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math] [math]\;[/math] [math]\color{blue}{\text{Interior Schwarzschild metric:}}[/math] [math]ds^{2} = c^{2} d\tau^{2} = -\frac{1}{4} \left(3 \sqrt{1 - \frac{r_{s}}{r_{g}}} - \sqrt{1 - \frac{r_{s} r^{2}}{r_{g}^{3}}} \right)^{2} c^{2} dt^{2} + \left(1 - \frac{r_{s} r^{2}}{r_{g}^{3}} \right)^{-1} dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math] [math]\;[/math] [math]\color{blue}{\text{Interior Reissner-Nordstrom metric:}}[/math] [math]\boxed{ds^{2} = c^{2} d\tau^{2} = -\frac{1}{4} \left(3 \sqrt{1 - \frac{r_{s}}{r_{g}} + \frac{r_{s} r_{Q}^{2}}{r_{g}^{3}}} - \sqrt{1 - \frac{r_{s} r^{2}}{r_{g}^{3}} + \frac{r_{s} r_{Q}^{2}}{r_{g}^{3}}} \right)^{2} c^{2} dt^{2} + \left(1 - \frac{r_{s} r^{2}}{r_{g}^{3}} + \frac{r_{s} r_{Q}^{2}}{r_{g}^{3}} \right)^{-1} dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)}[/math] [math]\;[/math] [math]\color{blue}{\text{Schwarzschild innermost stable circular orbit:} \; (\text{ref. 9})}[/math] [math]r_{1} = 3 r_{s} = \frac{6 G M}{c^{2}}[/math] [math]\;[/math] [math]\color{blue}{\text{Can a stable orbit accretion disk or spherical plasma shell of charged quantum particles accumulate at the Schwarzschild innermost stable circular orbit, generating a charged black hole?}}[/math] [math]\;[/math] [math]\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}[/math] [math]\;[/math] Reference: Wikipedia - Four-gradient As a Jacobian matrix for the SR Minkowski metric tensor: (ref. 1) https://en.wikipedia.org/wiki/Four-gradient#As_a_Jacobian_matrix_for_the_SR_Minkowski_metric_tensor Wikipedia - Schwarzschild radius: (ref. 2) https://en.wikipedia.org/wiki/Schwarzschild_radius Wikipedia - Schwarzschild metric: (ref. 3) https://en.wikipedia.org/wiki/Schwarzschild_metric Wikipedia - Gravitational redshift - Spherically symmetric gravitational field: (ref. 4) https://en.wikipedia.org/wiki/Gravitational_redshift#Spherically_symmetric_gravitational_field Wikipedia - Interior Schwarzschild metric - Other formulations: (ref. 5) https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric#Other_formulations Austrian Reports on Gravitation - Interior Schwarzschild Solution And Free Fall - Rainer Burghardt: (ref. 6) http://www.arg.or.at/Wpdf/WIff.pdf Wikipedia - Interior Schwarzschild metric: (ref. 7) https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric#Mathematics Wikipedia - Exterior Reissner-Nordstrom metric: (ref. 8 ) https://en.wikipedia.org/wiki/Reissner–Nordström_metric#The_metric Wikipedia - Black hole - Innermost stable circular orbit: (ref. 9) https://en.wikipedia.org/wiki/Black_hole#Innermost_stable_circular_orbit_(ISCO)
  8. Broken Reference links repaired. Wikipedia - Observable universe total stellar number: (ref. 7) https://en.wikipedia.org/wiki/Star#Distribution Wikipedia - Milky Way Galaxy: (ref. 8) https://en.wikipedia.org/wiki/Milky_Way
  9. I am attempting to look within the rather large error bars for the estimates of the number of black holes in a galaxy. The calculated solutions are within the parameters of the mainstream conclusion. Revision complete for year 2020 data... [math]\color{blue}{\text{Planck satellite baryonic cosmological composition parameter:} \; (\text{ref. 1, pg. 11, ref. 2, pg. 3})}[/math] [math]\Omega_{b} = 0.0495[/math] [math]\;[/math] [math]\color{blue}{\text{Black holes cosmological composition parameter:} \; (\text{ref. 2, pg. 3})}[/math] [math]\Omega_{bh} = 0.00007[/math] [math]\;[/math] [math]\color{blue}{\text{Solar mass:} \; (\text{ref. 3})}[/math] [math]M_{\odot} = 1.9885 \cdot 10^{30} \; \text{kg}[/math] [math]\;[/math] [math]\color{blue}{\text{Milky Way galaxy mass:} \; (\text{ref. 4, pg. 1})}[/math] [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] [math]\color{blue}{\text{Toy model stellar baryon composition:} \; (\text{ref. 2, pg. 3})}[/math] [math]\Omega_s = \left(\Omega_{ms} + \Omega_{wd} + \Omega_{ns} \right) = 2.460 \cdot 10^{-3}[/math] [math]\boxed{\Omega_s = 2.460 \cdot 10^{-3}}[/math] [math]\Omega_{ms} - \text{main sequence stars cosmological composition parameter}[/math] [math]\Omega_{wd} - \text{white dwarf stars cosmological composition parameter}[/math] [math]\Omega_{ns}- \text{neutron stars cosmological composition parameter}[/math] [math]\;[/math] [math]\color{blue}{\text{Total stellar class number:} \; (\text{ref. 5})}[/math] [math]n_c = 7[/math] [math]\color{blue}{\text{key: 1 O, 2 B, 3 A, 4 F, 5 G, 6 K, 7 M}}[/math] [math]\Omega_f - \text{main sequence stars stellar class fraction}[/math] [math]N_s - \text{total observable stellar number}[/math] [math]M_s - \text{main sequence stellar mass}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model average stellar mass:} \; (\text{ref. 5})}[/math] [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] [math]\color{blue}{\text{Toy model average stellar mass upper bound limit:}}[/math] [math]\boxed{M_{as} = 1.184 \cdot 10^{30} \; \text{kg}}[/math] [math]\color{blue}{\text{Observabe universe average stellar mass:} \; (\text{ref. 6, pg. 20})}[/math] [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] [math]\color{blue}{\text{Toy model Milky Way stars per galaxy average number:}}[/math] [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] [math]\color{blue}{\text{Observable universe stars per galaxy average number:} \; (\text{ref. 7, ref. 8})}[/math] [math]\frac{N_s}{N_g} = 1.500 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math] [math]\;[/math] [math]\color{blue}{\text{Observable universe Milky Way galaxy total stellar number:} \; (\text{ref. 8})}[/math] [math]\frac{N_s}{N_g} = 2.500 \cdot 10^{11} \pm 1.500 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}} \; \; \; \; \; \frac{N_s}{N_g} = \left(1.000 \cdot 10^{11} \rightarrow 4.000 \cdot 10^{11} \right) \; \frac{\text{stars}}{\text{galaxy}}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model Milky Way black holes per galaxy average number integration via substitution:}}[/math] [math]\frac{N_{bh}}{N_{g}} = \frac{\Omega_{bh}}{\Omega_{b}} \left(\frac{N_{s}}{N_{g}} \right) = \frac{\Omega_{bh}}{\Omega_{b}} \left(\frac{\Omega_s M_{mw}}{\Omega_b M_{as}} \right) = \frac{\Omega_{bh} \Omega_s M_{mw}}{\Omega_{b}^{2} M_{as}} = 1.489 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model Milky Way black holes per galaxy average number:}}[/math] [math]\boxed{\frac{N_{bh}}{N_{g}} = \frac{\Omega_{bh} \Omega_s M_{mw}}{\Omega_{b}^{2} M_{as}}}[/math] [math]\boxed{\frac{N_{bh}}{N_{g}} = 1.489 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}}[/math] [math]\;[/math] [math]\color{blue}{\text{Synthetic catalog black holes per galaxy average number:} \; (\text{ref. 9, pg. 1})}[/math] [math]\frac{N_{bh}}{N_g} = 1.293 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}[/math] [math]\;[/math] [math]\color{blue}{\text{Observable universe Milky Way black holes per galaxy average number:}}[/math] [math]\boxed{\frac{N_{bh}}{N_{g}} = \frac{\Omega_{bh}}{\Omega_{b}} \left(\frac{N_{s}}{N_{g}} \right)}[/math] [math]\boxed{\frac{N_{bh}}{N_g} = \left(1.414 \cdot 10^{8} \rightarrow 5.657 \cdot 10^{8} \right) \; \frac{\text{black holes}}{\text{galaxy}}}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model Milky Way galaxy average black hole mass integration via substitution:}}[/math] [math]M_{bh} = \frac{\Omega_{bh} M_{mw}}{\Omega_{b}} \left(\frac{N_{g}}{N_{bh}} \right) = \frac{\Omega_{bh} M_{mw}}{\Omega_{b}} \left[\frac{\Omega_{b}^{2} M_{as}}{\Omega_{bh} \Omega_s M_{mw}} \right] = \left(\frac{\Omega_{b}}{\Omega_{s}} \right) M_{as} = 2.379 \cdot 10^{31} \; \text{kg}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model Milky Way galaxy average black hole mass:}}[/math] [math]\boxed{M_{bh} = \left(\frac{\Omega_{b}}{\Omega_{s}} \right) M_{as}}[/math] [math]\boxed{M_{bh} = 2.382 \cdot 10^{31} \; \text{kg}}[/math] [math]\boxed{M_{bh} = 11.979\cdot M_{\odot}}[/math] [math]\;[/math] [math]\color{blue}{\text{Calculated synthetic catalog Milky Way galaxy average black hole mass:}}[/math] [math]\boxed{M_{bh} = \frac{\Omega_{bh} M_{mw}}{\Omega_{b}} \left(\frac{N_{g}}{N_{bh}} \right)}[/math] [math]\boxed{M_{bh} = 2.741 \cdot 10^{31} \; \text{kg}}[/math] [math]\boxed{M_{bh} = 13.783 \cdot M_{\odot}}[/math] [math]\;[/math] [math]\color{blue}{\text{Synthetic catalog Milky Way galaxy average black hole mass:} \; (\text{ref. 9, pg. 1})}[/math] [math]M_{bh} = 2.784 \cdot 10^{31} \; \text{kg}[/math] [math]M_{bh} = 14.000 \cdot M_{\odot}[/math] [math]\;[/math] [math]\color{blue}{\text{Observable universe Milky Way galaxy average black hole mass integration via substitution:}}[/math] [math]M_{bh} = \frac{\Omega_{bh} M_{mw}}{\Omega_{b}} \left(\frac{N_{g}}{N_{bh}} \right) = \frac{\Omega_{bh} M_{mw}}{\Omega_{b}} \left[\frac{\Omega_{b}}{\Omega_{bh}} \left(\frac{N_{g}}{N_{s}} \right) \right] = M_{mw} \left(\frac{N_{g}}{N_{s}} \right) = 6.265 \cdot 10^{30} \; \text{kg} \rightarrow 2.506 \cdot 10^{31} \; \text{kg}[/math] [math]\;[/math] [math]\color{blue}{\text{Observable universe Milky Way galaxy average black hole mass:}}[/math] [math]\boxed{M_{bh} = M_{mw} \left(\frac{N_{g}}{N_{s}} \right)}[/math] [math]\boxed{M_{bh} = \left(6.265 \cdot 10^{30} \; \text{kg} \rightarrow 2.506 \cdot 10^{31} \; \text{kg} \right)}[/math] [math]\boxed{M_{bh} = \left(3.151 \rightarrow 12.603 \right) \cdot M_{\odot}}[/math] [math]\;[/math] [math]\color{blue}{\text{Black hole mass spectrum domain:} \; (\text{ref. 10, ref. 11})}[/math] [math]\boxed{2.27 \cdot M_{\odot} \leq M_{bh} \leq 36 \cdot M_{\odot}}[/math] [math]\;[/math] [math]\color{blue}{\text{Calculation results summary:}}[/math] [math]\color{blue}{\text{Toy model Milky Way black holes per galaxy average number:}}[/math] [math]\boxed{\frac{N_{bh}}{N_{g}} = 1.489 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}}[/math] [math]\color{blue}{\text{Synthetic catalog black holes per galaxy average number:}}[/math] [math]\frac{N_{bh}}{N_g} = 1.293 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}[/math] [math]\color{blue}{\text{Observable universe Milky Way black holes per galaxy average number:}}[/math] [math]\boxed{\frac{N_{bh}}{N_g} = \left(1.414 \cdot 10^{8} \rightarrow 5.657 \cdot 10^{8} \right) \; \frac{\text{black holes}}{\text{galaxy}}}[/math] [math]\color{blue}{\text{Toy model Milky Way galaxy average black hole mass:}}[/math] [math]\boxed{M_{bh} = 11.979 \cdot M_{\odot}}[/math] [math]\color{blue}{\text{Calculated synthetic catalog Milky Way galaxy average black hole mass:}}[/math] [math]\boxed{M_{bh} = 13.783 \cdot M_{\odot}}[/math] [math]\color{blue}{\text{Synthetic catalog Milky Way galaxy average black hole mass:}}[/math] [math]M_{bh} = 14.000 \cdot M_{\odot}[/math] [math]\color{blue}{\text{Observable universe Milky Way galaxy average black hole mass:}}[/math] [math]\boxed{M_{bh} = \left(3.151 \rightarrow 12.603 \right) \cdot M_{\odot}}[/math] [math]\color{blue}{\text{Toy model average stellar mass:}}[/math] [math]\boxed{M_{as} = \left(0.219 \rightarrow 0.595 \right) \cdot M_{\odot}}[/math] [math]\;[/math] [math]\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}[/math] [math]\;[/math] "This above all: to thine own self be true, And it must follow, as the night the day, Thou canst not then be false to any man." - Polonius [math]\;[/math] Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) https://arxiv.org/pdf/1303.5076.pdf The Cosmic Energy Inventory: (ref. 2) http://arxiv.org/pdf/astro-ph/0406095v2.pdf Wikipedia - Sun Sol: (ref. 3) https://en.wikipedia.org/wiki/Sun Mass models of the Milky Way: (ref. 4) http://arxiv.org/pdf/1102.4340v1 Wikipedia - Stellar classification - Harvard spectral classification: (ref. 5) https://en.wikipedia.org/wiki/Stellar_classification#Harvard_spectral_classification On The Mass Distribution Of Stars...: (ref. 6) http://www.doiserbia.nb.rs/img/doi/1450-698X/2006/1450-698X0672017N.pdf Wikipedia - Observable universe total stellar number: (ref. 7) https://en.wikipedia.org/wiki/Star Distribution Wikipedia - Milky Way Galaxy: (ref. 8) https://en.wikipedia.org/wiki/Milky Way Synthetic catalog of black holes in the Milky Way (2020): (ref. 9) https://arxiv.org/pdf/1908.08775.pdf Wikipedia - Tolman-Oppenheimer-Volkoff limit: (ref. 10) https://en.wikipedia.org/wiki/Tolman-Oppenheimer-Volkoff_limit Wikipedia - Black hole: (ref. 11) https://en.wikipedia.org/wiki/Black_hole#Detection_of_gravitational_waves_from_merging_black_holes
  10. Affirmative, that OP post 1, reference 1 link has since been broken after it was posted on September 4, 2019. The updated reference link is still available on post 3, reference 1, and in Reference here. "Once we accept our limits, we go beyond them." - Albert Einstein Reference: Planck 2013 results. XVI. Cosmological parameters: https://arxiv.org/pdf/1303.5076.pdf Toy model black holes per galaxy average number - post 3 - Orion1: https://www.scienceforums.net/topic/120012-toy-model-black-holes-per-galaxy-average-number/?do=findComment&comment=1192794
  11. we do have others that even dispute their existence. There is no misunderstanding, I was merely responding that there is in fact a long standing dispute if black holes exist or not. Any observational evidence for their existence was not available until Cygnus X-1 was discovered in 1964. For any scientific claim, the burden of proof is on the individual making such a claim. Therefore, absent any observational evidence, the rational default position is that they are merely potentially falsifiable "theoretical artifacts" until experimental observational evidence becomes available. Even Albert Einstein doubted the existence of black holes. In a paper written in 1939, Albert Einstein attempted to reject the notion of black holes that his theory of general relativity and gravity, published more than two decades earlier, seemed to predict. Einstein denied several times that black holes could form. In 1939 he published a paper that argues that a star collapsing would spin faster and faster, spinning at the speed of light with infinite energy well before the point where it is about to collapse into a Schwarzchild singularity, or black hole. "The essential result of this investigation is a clear understanding as to why the "Schwarzschild singularities" do not exist in physical reality. Although the theory given here treats only clusters whose particles move along circular paths it does not seem to be subject to reasonable doubt that mote general cases will have analogous results. The "Schwarzschild singularity" does not appear for the reason that matter cannot be concentrated arbitrarily. And this is due to the fact that otherwise the constituting particles would reach the velocity of light." - Albert Einstein Reference: Wikipedia - Cygnus X-1: https://en.wikipedia.org/wiki/Cygnus_X-1 Wikipedia - List of black_holes: https://en.wikipedia.org/wiki/List_of_black_holes The Racah Institute of Physics - On A Stationary System With Spherical Symmetry Consisting Of Many Gravitating Masses - Albert Einstein - 1939: http://old.phys.huji.ac.il/~barak_kol/Courses/Black-holes/reading-papers/Einstein1939.pdf
  12. Orion1 wants his work to be analyzed and shot at.... unlike many. Peer review discussions have proven to be more valuable in improving my research, equations, calculations and solutions. Confucius says: Man with watch always knows the time. Man with two watches is never sure. And two men each with a watch, and time becomes relative. I am not qualified to review your estimates nor the methodology you are using, its just that on many occasions, we do have others that even dispute their existence. The ability to multiply and divide math is the only qualification to review my estimates. I disagree that my equations are incomprehensible. There is observational evidence for the existence of black holes. On 10 April 2019 an image was released of a black hole, which is seen in magnified fashion because the light paths near the event horizon are highly bent. The dark shadow in the middle results from light paths absorbed by the black hole. The image is in false color, as the detected light halo in this image is not in the visible spectrum, but radio waves. The Event Horizon Telescope (EHT), is an active program that directly observes the immediate environment of the event horizon of black holes, such as the black hole at the centre of the Milky Way. In April 2017, The Event Horizon Telescope (EHT) began observation of the black hole in the center of Messier 87. Prior to this, in 2015, the The Event Horizon Telescope (EHT) detected magnetic fields just outside the event horizon of Sagittarius A*. On 14 September 2015 the LIGO gravitational wave observatory made the first-ever successful direct observation of gravitational waves. The signal was consistent with theoretical predictions for the gravitational waves produced by the merger of two black holes: one with about 36 solar masses, and the other around 29 solar masses. This observation provides the most concrete evidence for the existence of black holes to date. if your estimates are correct and there are far more BH's then thought, could that constitute the Dark Matter problem? (or part thereof) MACHO's come to mind. Negative, in my opinion black holes do not constitute dark matter. Although the population densities and masses seem high, their overall compositional parameter is low compared to the dark matter compositional parameter: Black holes cosmological composition parameter: [math]\Omega_{bh} = 0.00007[/math] Dark matter cosmological composition parameter: [math]\Omega_{dm} = 0.268[/math] This means that 0.007% of everything is composed of black holes, while 26.8% of everything is composed of dark matter. Dark energy composition plus dark matter composition plus baryonic matter composition equals one. [math]\Omega_{\Lambda} + \Omega_{dm} + \Omega_{b} = 1[/math] 68.25% + 26.8% + 4.95% = 100% In my opinion, the "Bullet Cluster" rules out all candidate theories for dark matter, except for, (and possibly non-baryonic), extremely small mass quantum particles. Several groups have searched for MACHOs by searching for the microlensing amplification of light. These groups have ruled out dark matter being explained by MACHOs with mass in the range 1×10−8 solar masses (0.3 lunar masses) to 100 solar masses. These searches have ruled out the possibility that these objects make up a significant fraction of dark matter in our galaxy. Observations using the Hubble Space Telescope's NICMOS instrument showed that less than one percent of the halo mass is composed of red dwarfs. This corresponds to a negligible fraction of the dark matter halo mass. Therefore, the missing mass problem is not solved by MACHOs. How many BH's out there may not have accretion disks? (having been around long enough to consume all matter/energy within its region) is another consideration I think. This is a dynamic question that depends mostly on the environment that the black hole was generated in, its age, and the availability of baryonic matter in that environment as accretion fuel. I would expect newly generated black holes to have a bright accretion disk and polar jets, and as time passes, the disk and jets fade away from fuel exhaustion, with the remaining residual baryonic matter being driven away by the accretion and polar jet radiation. However, black holes in binary systems could receive an unstable continuous supply of fuel from their companion star's solar wind, or if within proximity, accreted directly from the companion star's surface, possibly producing intermittent high energy x-ray and gamma radiation bursts. According to the "Synthetic catalog of black holes in the Milky Way", there are 9.3 million black holes in binary systems: Black Holes in binary systems: [math]N_{bhb} = 9.3 \cdot 10^{6} \; \text{black holes}[/math] "Only when it is dark enough can you see the stars." - Martin Luther King, Jr. Reference: Wikipedia - Black hole - Observational evidence/Accretion of matter: https://en.wikipedia.org/wiki/Black_hole#Observational_evidence https://en.wikipedia.org/wiki/Black_hole#Accretion_of_matter Wikipedia - Dark matter https://en.wikipedia.org/wiki/Dark_matter Wikipedia - Bullet Cluster: https://en.wikipedia.org/wiki/Bullet_Cluster Wikipedia - Massive compact halo object https://en.wikipedia.org/wiki/Massive_compact_halo_object Synthetic catalog of black holes in the Milky Way: https://arxiv.org/pdf/1908.08775.pdf
  13. Peer review discussions have proven to be more valuable in improving my research, equations, calculations and solutions than a blog. Time: The indefinite continued progress of existence and events in the past, present, and future regarded as a whole. "travel through space and time" The definition seems simple enough. I am attempting to look within the rather large error bars for the estimates of the number of black holes in a galaxy. The calculated solutions are within the parameters of the mainstream conclusion. The difference is that the prior required only a single equation, while the latter required a sophisticated computer algorithm. Does not oppose but validates the mainstream picture, however, this offers an alternative solution to mainstream physics. In terms of stellar astrophysics, the estimate numbers are equivalent to an arrow hitting a target bulls-eye. Time is the indefinite continued progress of existence and events in the past, present, and future regarded as a whole. I am attempting to look within the rather large error bars for the estimates of the number of black holes in a galaxy. There is no discussion requirement to follow external sources, those external sources exist because the burden of proof is mine. However, anyone interested in astrophysics would be depleting themselves of interesting information by not researching available resource materials. "I would rather have questions that can't be answered than answers that can't be questioned." - Richard Feynman
  14. [math]\color{blue}{\text{Planck satellite baryonic cosmological composition parameter:} \; (\text{ref. 1, pg. 11, ref. 2, pg. 3})}[/math] [math]\Omega_{b} = 0.0495[/math] [math]\;[/math] [math]\color{blue}{\text{Black holes cosmological composition parameter:} \; (\text{ref. 2, pg. 3})}[/math] [math]\Omega_{bh} = 0.00007[/math] [math]\;[/math] [math]\color{blue}{\text{Solar mass:} \; (\text{ref. 3})}[/math] [math]M_{\odot} = 1.9885 \cdot 10^{30} \; \text{kg}[/math] [math]\;[/math] [math]\color{blue}{\text{Milky Way galaxy mass:} \; (\text{ref. 4, pg. 1})}[/math] [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] [math]\color{blue}{\text{Toy model stellar baryon composition:} \; (\text{ref. 2, pg. 3})}[/math] [math]\Omega_s = \left(\Omega_{ms} + \Omega_{wd} + \Omega_{ns} \right) = 2.460 \cdot 10^{-3}[/math] [math]\boxed{\Omega_s = 2.460 \cdot 10^{-3}}[/math] [math]\;[/math] [math]\color{blue}{\text{Total stellar class number:} \; (\text{ref. 5})}[/math] [math]n_c = 7[/math] [math]\color{blue}{\text{key: 1 O, 2 B, 3 A, 4 F, 5 G, 6 K, 7 M}}[/math] [math]\Omega_f - \text{main sequence stars stellar class fraction}[/math] [math]N_s - \text{total observable stellar number}[/math] [math]M_s - \text{main sequence stellar mass}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model average stellar mass:} \; (\text{ref. 5})}[/math] [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] [math]\color{blue}{\text{Toy model average stellar mass upper bound limit:}}[/math] [math]\boxed{M_{as} = 1.184 \cdot 10^{30} \; \text{kg}}[/math] [math]\color{blue}{\text{Observabe universe average stellar mass:} \; (\text{ref. 6, pg. 20})}[/math] [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] [math]\color{blue}{\text{Toy model Milky Way stars per galaxy average number:}}[/math] [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] [math]\color{blue}{\text{Observable universe stars per galaxy average number:} \; (\text{ref. 7, ref. 8})}[/math] [math]\frac{N_s}{N_g} = 1.500 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}}[/math] [math]\;[/math] [math]\color{blue}{\text{Observable universe Milky Way galaxy total stellar number:} \; (\text{ref. 8})}[/math] [math]\frac{N_s}{N_g} = 2.500 \cdot 10^{11} \pm 1.500 \cdot 10^{11} \; \frac{\text{stars}}{\text{galaxy}} \; \; \; \; \; \frac{N_s}{N_g} = \left(1.000 \cdot 10^{11} \rightarrow 4.000 \cdot 10^{11} \right) \; \frac{\text{stars}}{\text{galaxy}}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model Milky Way black holes per galaxy average number integration via substitution:}}[/math] [math]\frac{N_{bh}}{N_{g}} = \frac{\Omega_{bh}}{\Omega_{b}} \left(\frac{N_{s}}{N_{g}} \right) = \frac{\Omega_{bh}}{\Omega_{b}} \left(\frac{\Omega_s M_{mw}}{\Omega_b M_{as}} \right) = \frac{\Omega_{bh} \Omega_s M_{mw}}{\Omega_{b}^{2} M_{as}} = 1.489 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model Milky Way black holes per galaxy average number:}}[/math] [math]\boxed{\frac{N_{bh}}{N_{g}} = \frac{\Omega_{bh} \Omega_s M_{mw}}{\Omega_{b}^{2} M_{as}}}[/math] [math]\boxed{\frac{N_{bh}}{N_{g}} = 1.489 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}}[/math] [math]\;[/math] [math]\color{blue}{\text{Synthetic catalog black holes per galaxy average number:} \; (\text{ref. 9, pg. 1})}[/math] [math]\frac{N_{bh}}{N_g} = 1.693 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}[/math] [math]\;[/math] [math]\color{blue}{\text{Observable universe Milky Way black holes per galaxy average number:}}[/math] [math]\boxed{\frac{N_{bh}}{N_{g}} = \frac{\Omega_{bh}}{\Omega_{b}} \left(\frac{N_{s}}{N_{g}} \right)}[/math] [math]\boxed{\frac{N_{bh}}{N_g} = \left(1.414 \cdot 10^{8} \rightarrow 5.657 \cdot 10^{8} \right) \; \frac{\text{black holes}}{\text{galaxy}}}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model Milky Way galaxy average black hole mass integration via substitution:}}[/math] [math]M_{bh} = \frac{\Omega_{bh} M_{mw}}{\Omega_{b}} \left(\frac{N_{g}}{N_{bh}} \right) = \frac{\Omega_{bh} M_{mw}}{\Omega_{b}} \left[\frac{\Omega_{b}^{2} M_{as}}{\Omega_{bh} \Omega_s M_{mw}} \right] = \left(\frac{\Omega_{b}}{\Omega_{s}} \right) M_{as} = 2.379 \cdot 10^{31} \; \text{kg}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model Milky Way galaxy average black hole mass:}}[/math] [math]\boxed{M_{bh} = \left(\frac{\Omega_{b}}{\Omega_{s}} \right) M_{as}}[/math] [math]\boxed{M_{bh} = 2.382 \cdot 10^{31} \; \text{kg}}[/math] [math]\boxed{M_{bh} = 11.979\cdot M_{\odot}}[/math] [math]\;[/math] [math]\color{blue}{\text{Synthetic catalog Milky Way galaxy average black hole mass:}}[/math] [math]\boxed{M_{bh} = \frac{\Omega_{bh} M_{mw}}{\Omega_{b}} \left(\frac{N_{g}}{N_{bh}} \right)}[/math] [math]\boxed{M_{bh} = 2.093 \cdot 10^{31} \; \text{kg}}[/math] [math]\boxed{M_{bh} = 10.527 \cdot M_{\odot}}[/math] [math]\;[/math] [math]\color{blue}{\text{Observable universe Milky Way galaxy average black hole mass integration via substitution:}}[/math] [math]M_{bh} = \frac{\Omega_{bh} M_{mw}}{\Omega_{b}} \left(\frac{N_{g}}{N_{bh}} \right) = \frac{\Omega_{bh} M_{mw}}{\Omega_{b}} \left[\frac{\Omega_{b}}{\Omega_{bh}} \left(\frac{N_{g}}{N_{s}} \right) \right] = M_{mw} \left(\frac{N_{g}}{N_{s}} \right) = 6.265 \cdot 10^{30} \; \text{kg} \rightarrow 2.506 \cdot 10^{31} \; \text{kg}[/math] [math]\;[/math] [math]\color{blue}{\text{Observable universe Milky Way galaxy average black hole mass:}}[/math] [math]\boxed{M_{bh} = M_{mw} \left(\frac{N_{g}}{N_{s}} \right)}[/math] [math]\boxed{M_{bh} = \left(6.265 \cdot 10^{30} \; \text{kg} \rightarrow 2.506 \cdot 10^{31} \; \text{kg} \right)}[/math] [math]\boxed{M_{bh} = \left(3.151 \rightarrow 12.603 \right) \cdot M_{\odot}}[/math] [math]\;[/math] [math]\color{blue}{\text{Black hole mass spectrum domain:} \; (\text{ref. 10, ref. 11})}[/math] [math]\boxed{2.27 \cdot M_{\odot} \leq M_{bh} \leq 36 \cdot M_{\odot}}[/math] [math]\;[/math] [math]\color{blue}{\text{Calculation results summary:}}[/math] [math]\color{blue}{\text{Toy model Milky Way black holes per galaxy average number:}}[/math] [math]\boxed{\frac{N_{bh}}{N_{g}} = 1.489 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}}[/math] [math]\color{blue}{\text{Synthetic catalog black holes per galaxy average number:}}[/math] [math]\frac{N_{bh}}{N_g} = 1.693 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}[/math] [math]\color{blue}{\text{Observable universe Milky Way black holes per galaxy average number:}}[/math] [math]\boxed{\frac{N_{bh}}{N_g} = \left(1.414 \cdot 10^{8} \rightarrow 5.657 \cdot 10^{8} \right) \; \frac{\text{black holes}}{\text{galaxy}}}[/math] [math]\color{blue}{\text{Toy model Milky Way galaxy average black hole mass:}}[/math] [math]\boxed{M_{bh} = 11.979 \cdot M_{\odot}}[/math] [math]\color{blue}{\text{Synthetic catalog Milky Way galaxy average black hole mass:}}[/math] [math]\boxed{M_{bh} = 10.527 \cdot M_{\odot}}[/math] [math]\color{blue}{\text{Observable universe Milky Way galaxy average black hole mass:}}[/math] [math]\boxed{M_{bh} = \left(3.151 \rightarrow 12.603 \right) \cdot M_{\odot}}[/math] [math]\color{blue}{\text{Toy model average stellar mass:}}[/math] [math]\boxed{M_{as} = \left(0.219 \rightarrow 0.595 \right) \cdot M_{\odot}}[/math] [math]\;[/math] [math]\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}[/math] [math]\;[/math] Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) https://arxiv.org/pdf/1303.5076.pdf The Cosmic Energy Inventory: (ref. 2) http://arxiv.org/pdf/astro-ph/0406095v2.pdf Wikipedia - Sun Sol: (ref. 3) https://en.wikipedia.org/wiki/Sun Mass models of the Milky Way: (ref. 4) http://arxiv.org/pdf/1102.4340v1 Wikipedia - Stellar classification - Harvard spectral classification: (ref. 5) https://en.wikipedia.org/wiki/Stellar_classification#Harvard_spectral_classification On The Mass Distribution Of Stars...: (ref. 6) http://www.doiserbia.nb.rs/img/doi/1450-698X/2006/1450-698X0672017N.pdf Wikipedia - Observable universe total stellar number: (ref. 7) https://en.wikipedia.org/wiki/Star#Distribution Wikipedia - Milky Way Galaxy: (ref. 8) https://en.wikipedia.org/wiki/Milky_Way Synthetic catalog of black holes in the Milky Way: (ref. 9) https://arxiv.org/pdf/1908.08775.pdf Wikipedia - Tolman-Oppenheimer-Volkoff limit: (ref. 10) https://en.wikipedia.org/wiki/Tolman-Oppenheimer-Volkoff_limit Wikipedia - Black hole: (ref. 11) https://en.wikipedia.org/wiki/Black_hole#Detection_of_gravitational_waves_from_merging_black_holes
  15. [math]\color{blue}{\text{Total stellar class number: (ref. 1)}}[/math] [math]n_c = 7[/math] [math]\;[/math] [math]\color{blue}{\text{key: 1 O, 2 B, 3 A, 4 F, 5 G, 6 K, 7 M}}[/math] [math]\Omega_f - \text{main sequence stars stellar class fraction}[/math] [math]N_s - \text{total observable stellar number}[/math] [math]M_s - \text{main sequence stellar mass}[/math] [math]\;[/math] [math]\color{blue}{\text{Toy model average stellar mass: (ref. 1)}}[/math] [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] [math]\;[/math] [math]\color{blue}{\text{Toy model average stellar mass upper bound limit:}}[/math] [math]\boxed{M_{as} = 1.184 \cdot 10^{30} \; \text{kg}}[/math] [math]\;[/math] [math]\color{blue}{\text{Observabe universe average stellar mass: (ref. 2, pg. 20)}}[/math] [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] [math]\color{blue}{\text{Synthetic catalog black holes per galaxy average number: (ref. 3, pg. 1)}}[/math] [math]\frac{N_{bh}}{N_g} = 1.693 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}[/math] [math]\;[/math] [math]\color{blue}{\text{Synthetic catalog Milky Way galaxy average black hole mass:}}[/math] [math]\boxed{M_{bh} = \frac{\Omega_{bh} M_{mw}}{\Omega_{b}} \left(\frac{N_{g}}{N_{bh}} \right)}[/math] [math]\boxed{M_{bh} = 2.093 \cdot 10^{31} \; \text{kg}}[/math] [math]\boxed{M_{bh} = 10.527 \cdot M_{\odot}}[/math] [math]\;[/math] [math]\color{blue}{\text{Black hole mass spectrum domain: (ref. 4), (ref. 5)}}[/math] [math]\boxed{2.27 \cdot M_{\odot} \leq M_{bh} \leq 36 \cdot M_{\odot}}[/math] [math]\color{blue}{\text{Synthetic catalog Milky Way galaxy average black hole mass:}}[/math] [math]\boxed{M_{bh} = 10.527 \cdot M_{\odot}}[/math] [math]\color{blue}{\text{Toy model average stellar mass:}}[/math] [math]\boxed{M_{as} = \left(0.219 \rightarrow 0.595 \right) \cdot M_{\odot}}[/math] [math]\;[/math] [math]\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}[/math] [math]\;[/math] Reference: Wikipedia - Stellar classification - Harvard spectral classification: (ref. 1) https://en.wikipedia.org/wiki/Stellar_classification#Harvard_spectral_classification On The Mass Distribution Of Stars...: (ref. 2) http://www.doiserbia.nb.rs/img/doi/1450-698X/2006/1450-698X0672017N.pdf Synthetic catalog of black holes in the Milky Way: (ref. 3) https://arxiv.org/pdf/1908.08775.pdf Wikipedia - Tolman-Oppenheimer-Volkoff limit: (ref. 4) https://en.wikipedia.org/wiki/Tolman-Oppenheimer-Volkoff_limit Wikipedia - Black hole - Detection of gravitation waves from merging black holes: (ref. 5) https://en.wikipedia.org/wiki/Black_hole#Detection_of_gravitational_waves_from_merging_black_holes
  16. \[\color{blue}{\text{Solar surface photon radiation temperature:} \; (\text{ref. 1})}\] \[T_{\odot} = 5772 \; \text{K}\] \[\color{blue}{\text{Bose-Einstein solar surface photon radiation spectral radiance peak frequency:}}\] \[\boxed{f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\odot}}{h}}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\boxed{f_{\gamma} = 3.39332 \cdot 10^{14} \; \text{Hz}}\] \[\boxed{f_{\gamma} = 339.332 \; \text{THz}}\] \[\color{blue}{\text{Calculated solar surface photon radiation spectral radiance peak frequency:} \; (\text{ref. 2})}\] \[f_{\gamma} = \frac{\left[W_{0}\left(-3e^{-3} \right) + 3 \right] k_B T_{\odot}}{h}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[T_{1} = 6000 \; \text{K} \; \; \; f_{\gamma} = 352.735 \; \text{THz}\] \[\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}\] Reference: Wikipedia - Sun Sol: (ref. 1) https://en.wikipedia.org/wiki/Sun Wikipedia - Wien's displacement law: (ref. 2) https://en.wikipedia.org/wiki/Wien's_displacement_law#Parameterization_by_frequency Wikipedia - Lambert W function: (ref. 3) https://en.wikipedia.org/wiki/Lambert_W_function
  17. \[\color{blue}{\text{Toy model quantum particle properties chart:}}\] \[\begin{array}{l*{9}{c}r} & \text{identity} & \text{helicity state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} & \text{spectral radiance peak frequency} \\ b & \text{quinton} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda & 2.058 \; \text{THz} \\ b & \text{scalaron} & 0 & 0 & 1 & 1 & 1 & 18.658 \; \text{eV} & \phi & 114.366 \; \text{GHz} \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & 8.167 \; \text{eV} & \nu_{s} & 126.915 \; \text{GHz} \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & 0.038 \; \text{eV} & \nu & 126.915 \; \text{GHz} \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma & 160.229 \; \text{GHz} \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} & 114.366 \; \text{GHz} \\ \end{array}\] \[\color{blue}{\text{Planck's law:}} \; (\color{blue}{\text{ref. 1}})\] \[B\left(f_{\gamma},T_{\gamma} \right) = \frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} \pm 1 \right)}\] \[\color{blue}{\text{A plus sign in the denominator is a Fermi-Dirac distribution, a minus sign in the denominator is a Bose-Einstein distribution.}}\] \[\color{blue}{\text{+ sign - Fermi-Dirac distribution}}\] \[\color{blue}{\text{- sign - Bose-Einstein distribution}}\] \[\color{blue}{\text{Planck's law energy distribution frequency plot.}} \; (\color{blue}{\text{attached graph 1}})\] \[114.366 \; \text{GHz} \; \; \; 126.915 \; \text{GHz} \; \; \; 160.229 \; \text{GHz}\] \[\color{blue}{\text{Planck's law energy distribution frequency plot.}} \; (\color{blue}{\text{attached graph 2}})\] \[2.058 \; \text{THz}\] \[\color{blue}{\text{In this toy model, some quantum particle radiation distributions are embedded within the photon radiation distribution.}}\] \[\color{blue}{\text{Would a quantum particle radiation distribution embedded within the photon radiation distribution induce cosine anisotropy in the photon radiation?}} \; (\color{blue}{\text{ref. 2}})\] \[\color{blue}{\text{If cosine anisotropy is a detectable amount, is it possible to map cosine anisotropy amounts verses frequency within the photon radiation distribution?}} \; (\color{blue}{\text{ref. 2}})\] \[\color{blue}{\text{Based upon the frequency multi-distribution plot, what frequency domain in the photon radiation distribution would you expect to detect cosine anisotropy?}}\] \[\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}\] Reference: Wikipedia - Planck's law: (ref. 1) https://en.wikipedia.org/wiki/Planck's_law Wikipedia - Anisotropy - Physics: (ref. 2) https://en.wikipedia.org/wiki/Anisotropy#Physics
  18. \[\color{blue}{\text{Symbolic identity key:}}\] \[\begin{array}{lcl} \text{s} \text{ - spin quantum number} \\ n_{s} \text{ - spin states total integer helicity number} \\ N_{s} \text{ - species total integer number} \\ N_{n} \text{ - total effective degeneracy number} \\ \end{array}\] \[n_{s} = 2 s + 1 \; \; \; \; \; \; s < 2 \; \; \; \; \; \; m = 0\] \[n_{s} = 2 \; \; \; \; \; \; s \geq 2 \; \; \; \; \; \; m = 0\] \[\text{if } n_{s} \geq N_{s} \text{ then } N_{n} = n_{s}\] \[\text{if } n_{s} \leq N_{s} \text{ then } N_{n} = N_{s}\] \[\color{blue}{\text{Toy model quantum particle properties chart:}}\] \[\begin{array}{l*{9}{c}r} & \text{identity} & \text{helicity state} & \text{spin} & n_{s} & N_{s} & N_{n} & \text{mass} & \text{type} & \text{spectral radiance peak frequency} \\ b & \text{quinton} & 0 & 0 & 1 & 1 & 1 & 0 & \Lambda & 2.058 \; \text{THz} \\ b & \text{scalaron} & 0 & 0 & 1 & 1 & 1 & 18.658 \; \text{eV} & \phi & 114.366 \; \text{GHz} \\ f & \text{sterile neutrino} & +,- & 1/2 & 2 & 3 & 3 & 8.167 \; \text{eV} & \nu_{s} & 126.915 \; \text{GHz} \\ f & \text{neutrino} & +,- & 1/2 & 2 & 3 & 3 & 0.038 \; \text{eV} & \nu & 126.915 \; \text{GHz} \\ b & \text{photon} & +,- & 1 & 2 & 1 & 2 & 0 & \gamma & 160.229 \; \text{GHz} \\ b & \text{graviton} & +,- & 2 & 2 & 1 & 2 & 0 & \text{G} & 114.366 \; \text{GHz} \\ \end{array}\] \[\color{blue}{\text{For massless quantum particles, the transverse modes cannot exist due to Lorentz invariance.}}\] \[\color{blue}{\text{Only positive and negative helicity states remain. For massless scalar particles, only zero helicity states remain.}}\] \[\color{blue}{\text{The spin 1 photon is also restricted to its positive and negative helicity states, and has a total effective degeneracy number of 2.}}\] \[\color{blue}{\text{A massless graviton has only 2 helicity states, and has a total effective degeneracy number of 2.}}\] \[\color{blue}{\text{Dark energy quinton total effective degeneracy number:}}\] \[\boxed{N_{\Lambda} = 1}\] \[\color{blue}{\text{Planck satellite dark energy cosmological composition parameter:} \; (\text{ref. 1, pg. 11})}\] \[\Omega_{\Lambda} = 0.6825\] \[\color{blue}{\text{Bose-Einstein dark energy cosmic quinton background radiation temperature:} \; (\text{ref. 2})}\] \[\boxed{T_{\Lambda} = \frac{}{k_{B}} \left(\frac{45 \Omega_{\Lambda} H_0^2 \hbar^3 c^5}{4 G N_{\Lambda} \pi^3} \right)^{1/4}}\] \[\boxed{T_{\Lambda} = 35.013 \; \text{K}}\] \[\color{blue}{\text{Bose-Einstein dark energy cosmic quinton background radiation spectral radiance peak frequency:}}\] \[\boxed{f_{\Lambda} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\Lambda}}{h}}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\boxed{f_{\Lambda} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right]}{2 \pi} \left(\frac{45 \Omega_{\Lambda} H_0^2 c^5}{4 \hbar G N_{\Lambda} \pi^3} \right)^{1/4}}\] \[\boxed{f_{\Lambda} = 2.05837 \cdot 10^{12} \; \text{Hz}}\] \[\boxed{f_{\Lambda} = 2.058 \; \text{THz}}\] \[\color{blue}{\text{Cosmic photon background radiation temperature at present time:} \; (\text{ref. 4})}\] \[T_{\gamma} = 2.72548 \; \text{K}\] \[\color{blue}{\text{Cosmic neutrino and sterile neutrino background radiation temperature at present time:} \; (\text{ref. 5})}\] \[T_{\nu} = \left(\frac{4}{11} \right)^{\frac{1}{3}} T_{\gamma} = 1.945 \; \text{K}\] \[\boxed{T_{\nu} = 1.945 \; \text{K}}\] \[\color{blue}{\text{Dark matter scalaron and sterile neutrino radiation temperature is equivalent to cosmic neutrino background radiation temperature:}}\] \[\boxed{T_{\phi} = T_{s \nu} = T_{\nu} = 1.945 \; \text{K}}\] \[\color{blue}{\text{Bose-Einstein dark matter cosmic scalaron background radiation spectral radiance peak frequency:}}\] \[\boxed{f_{\phi} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\phi}}{h}}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\boxed{f_{\phi} = 1.14366 \cdot 10^{11} \; \text{Hz}}\] \[\boxed{f_{\phi} = 114.366 \; \text{GHz}}\] \[\color{blue}{\text{Fermi-Dirac dark matter sterile neutrino background radiation spectral radiance peak frequency:}}\] \[\boxed{f_{s \nu} = \frac{\left[W_{0}\left(\frac{3}{e^3} \right) + 3 \right] k_B T_{\nu}}{h}}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\boxed{f_{s \nu} = 1.26915 \cdot 10^{11} \; \text{Hz}}\] \[\boxed{f_{s \nu} = 126.915 \; \text{GHz}}\] \[\color{blue}{\text{Solve for cosmic neutrino background radiation spectral radiance peak frequency:}}\] \[\color{blue}{\text{Neutrino species total effective degeneracy number:}}\] \[N_{\nu} = 3.046\] \[\color{blue}{\text{Neutrino radiation energy density Fermi-Dirac distribution:}}\] \[\epsilon_{\nu} = \frac{4 \pi N_{\nu} \left(k_B T_{\nu} \right)^4}{\left(2 \pi \hbar c \right)^3} \int_{0}^c \frac{E_t \left(v \right)^3}{e^{\frac{E_t \left(v \right)}{E_1 \left(T_{\nu} \right)}} + 1} dv\] \[\color{blue}{\text{Solve Fermi-Dirac distribution first derivative x-axis zero intercept with respect to frequency:}}\] \[\frac{d \epsilon_{\nu}}{df_{\nu}} = \frac{d}{df_{\nu}} \left(\frac{4 \pi N_{\nu} \left(k_B T_{\nu} \right)^4 E_t\left(\omega \right)^3}{\left(2 \pi \hbar c \right)^3 \left(e^{\frac{E_t \left(\omega \right)}{E_1 \left(T_{\nu} \right)}} + 1 \right)} \right) = 0\] \[\frac{d}{df_{\nu}} \left(\frac{4 \pi N_{\nu} \left(k_B T_{\nu} \right)^4 \left(h f_{\nu} \right)^3 }{\left(h c \right)^3 \left(e^{\frac{h f}{k_{B} T_{\nu}}} + 1 \right)} \right) = \frac{12 \pi N_{\nu} f_{\nu}^2 \left(k_B T_{\nu} \right)^4}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)} - \frac{4 \pi h N_{\nu} f_{\nu}^3 \left(k_B T_{\nu} \right)^3 e^{\frac{h f_{\nu}}{k_B T_{\nu}}}}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)^2} = 0\] \[\frac{12 \pi N_{\nu} f_{\nu}^2 \left(k_B T_{\nu} \right)^4}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)} = \frac{4 \pi h N_{\nu} f_{\nu}^3 \left(k_B T_{\nu} \right)^3 e^{\frac{h f_{\nu}}{k_B T_{\nu}}}}{c^3 \left(e^{\frac{h f_{\nu}}{k_B T_{\nu}}} + 1 \right)^2}\] \[\frac{h f_{\nu}}{k_B T_{\nu}} = 3 \left(1 + \frac{1}{e^{\frac{h f_{\nu}}{k_B T_{\nu}}}} \right) \; \; \; \; \; \; \frac{h f_{\nu}}{k_B T_{\nu}} = 3 + \frac{3}{e^{\frac{h f_{\nu}}{k_B T_{\nu}}}} \; \; \; \; \; \; \frac{h f_{\nu}}{k_B T_{\nu}} = \frac{3}{e^{\frac{h f_{\nu}}{k_B T_{\nu}}}} + 3\] \[\pm x = \left(y \pm a \right) e^{y} \; \; \; \; \; \; y = \frac{x}{e^{y}} + a \; \; \; \; \; \; y = W_{k}\left(\frac{x}{e^{a}} \right) + a \; \; \; \; \; \; x = 3 \; \; \; a = 3 \; \; \; k = 0\] \[f_{\nu} = \frac{\left[W_{0}\left(\frac{3}{e^3} \right) + 3 \right] k_B T_{\nu}}{h} = 1.26915 \cdot 10^{11} \; \text{Hz} = 126.915 \; \text{GHz}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\color{blue}{\text{Fermi-Dirac cosmic neutrino background radiation spectral radiance peak frequency:}}\] \[\boxed{f_{\nu} = \frac{\left[W_{0}\left(\frac{3}{e^3} \right) + 3 \right] k_B T_{\nu}}{h}}\] \[\boxed{f_{\nu} = 1.26915 \cdot 10^{11} \; \text{Hz}}\] \[\boxed{f_{\nu} = 126.915 \; \text{GHz}}\] \[\color{blue}{\text{Solve for cosmic photon background radiation spectral radiance peak frequency:}}\] \[\color{blue}{\text{Photon species total effective degeneracy number:}}\] \[\boxed{N_{\gamma} = 2}\] \[\color{blue}{\text{Cosmic photon background radiation temperature at present time:} \; (\text{ref. 4})}\] \[T_{\gamma} = 2.72548 \; \text{K}\] \[\color{blue}{\text{Planck's law:} \; (\text{ref. 6})}\] \[B\left(f_{\gamma},T_{\gamma} \right) = \frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} - 1 \right)}\] \[\color{blue}{\text{Solve Bose-Einstein Planck's law first derivative x-axis zero intercept with respect to frequency:}}\] \[\frac{dB\left(f_{\gamma},T_{\gamma} \right)}{df_{\gamma}} = \frac{d}{df_{\gamma}} \left[\frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} - 1 \right)} \right] = 0\] \[\color{blue}{\text{Planck's law first derivative with respect to frequency:}}\] \[\frac{dB\left(f_{\gamma},T_{\gamma} \right)}{df_{\gamma}} = \frac{d}{df_{\gamma}} \left[\frac{N_{\gamma} h f_{\gamma}^{3}}{c^{2} \left(e^{\frac{h f_{\gamma}}{k_{B} T_{\gamma}}} - 1 \right)} \right] = \frac{3 N_{\gamma} h f_{\gamma}^2}{c^2 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} - \frac{N_{\gamma} h^2 f_{\gamma}^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^2 k_B T_{\gamma} \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2} = 0\] \[\frac{3 N_{\gamma} h f_{\gamma}^2}{c^2 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} = \frac{N_{\gamma} h^2 f_{\gamma}^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^2 k_B T_{\gamma} \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2}\] \[\frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 \left(1 - \frac{1}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \right) \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 - \frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = -\frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} + 3\] \[\pm x = \left(y \pm a \right) e^{y} \; \; \; \; \; \; y = -\frac{x}{e^{y}} + a \; \; \; \; \; \; y = W_{k}\left(-\frac{x}{e^{a}} \right) + a \; \; \; \; \; \; x = -3 \; \; \; a = 3 \; \; \; k = 0\] \[f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h} = 1.60229 \cdot 10^{11} \; \text{Hz} = 160.229 \; \text{GHz}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\color{blue}{\text{Bose-Einstein cosmic photon background radiation spectral radiance peak frequency:}}\] \[\boxed{f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h}}\] \[\boxed{f_{\gamma} = 1.60229 \cdot 10^{11} \; \text{Hz}}\] \[\boxed{f_{\gamma} = 160.229 \; \text{GHz}}\] \[\color{blue}{\text{Observed cosmic photon background radiation spectral radiance peak frequency:} \; (\text{ref. 4})}\] \[f_{\gamma} = 160.23 \; \text{GHz}\] \[\color{blue}{\text{Solve for cosmic photon background radiation spectral radiance peak frequency:}}\] \[\color{blue}{\text{Photon radiation energy density Bose-Einstein distribution:}}\] \[\epsilon_{\gamma} = \frac{4 \pi N_{\gamma} \left(k_B T_{\gamma} \right)^4}{\left( 2 \pi \hbar c \right)^3} \int_{0}^\infty \frac{E_t \left(\omega \right)^3}{e^{\frac{E_t \left(\omega \right)}{E_1 \left(T_{\gamma} \right)}} - 1} d \omega\] \[\color{blue}{\text{Solve Bose-Einstein distribution first derivative x-axis zero intercept with respect to frequency:}}\] \[\frac{d \epsilon_{\gamma}}{df_{\gamma}} = \frac{d}{df_{\gamma}} \left(\frac{4 \pi N_{\gamma} \left(k_B T_{\gamma} \right)^4 E_t\left(\omega \right)^3}{\left(2 \pi \hbar c \right)^3 \left(e^{\frac{E_t \left(\omega \right)}{E_1 \left(T_{\gamma} \right)}} - 1 \right)} \right) = 0\] \[\frac{d}{df_{\gamma}} \left(\frac{4 \pi N_{\gamma} \left(k_B T_{\gamma} \right)^4 \left(h f_{\gamma} \right)^3 }{\left(h c \right)^3 \left(e^{\frac{h f}{k_{B} T_{\gamma}}} - 1 \right)} \right) = \frac{12 \pi N_{\gamma} f_{\gamma}^2 \left(k_B T_{\gamma} \right)^4}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} - \frac{4 \pi h N_{\gamma} f_{\gamma}^3 \left(k_B T_{\gamma} \right)^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2} = 0\] \[\frac{12 \pi N_{\gamma} f_{\gamma}^2 \left(k_B T_{\gamma} \right)^4}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)} = \frac{4 \pi h N_{\gamma} f_{\gamma}^3 \left(k_B T_{\gamma} \right)^3 e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}}{c^3 \left(e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}} - 1 \right)^2}\] \[\frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 \left(1 - \frac{1}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \right) \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = 3 - \frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} \; \; \; \; \; \; \frac{h f_{\gamma}}{k_B T_{\gamma}} = -\frac{3}{e^{\frac{h f_{\gamma}}{k_B T_{\gamma}}}} + 3\] \[\pm x = \left(y \pm a \right) e^{y} \; \; \; \; \; \; y = -\frac{x}{e^{y}} + a \; \; \; \; \; \; y = W_{k}\left(-\frac{x}{e^{a}} \right) + a \; \; \; \; \; \; x = -3 \; \; \; a = 3 \; \; \; k = 0\] \[f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h} = 1.60229 \cdot 10^{11} \; \text{Hz} = 160.229 \; \text{GHz}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\color{blue}{\text{Bose-Einstein cosmic photon background radiation spectral radiance peak frequency:}}\] \[\boxed{f_{\gamma} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{\gamma}}{h}}\] \[\boxed{f_{\gamma} = 1.60229 \cdot 10^{11} \; \text{Hz}}\] \[\boxed{f_{\gamma} = 160.229 \; \text{GHz}}\] \[\color{blue}{\text{Observed cosmic photon background radiation spectral radiance peak frequency:} \; (\text{ref. 4})}\] \[f_{\gamma} = 160.23 \; \text{GHz}\] \[\color{blue}{\text{Cosmic graviton background radiation temperature is equivalent to cosmic neutrino background radiation temperature:}}\] \[\boxed{T_{G} = T_{\nu} = 1.945 \; \text{K}}\] \[\color{blue}{\text{Bose-Einstein cosmic graviton background radiation spectral radiance peak frequency:}}\] \[\boxed{f_{G} = \frac{\left[W_{0}\left(-\frac{3}{e^3} \right) + 3 \right] k_B T_{G}}{h}}\] \[W_{0}\left(z \right) - \text{Lambert W function} \; \left(\text{ref. 3} \right)\] \[\boxed{f_{G} = 1.14366 \cdot 10^{11} \; \text{Hz}}\] \[\boxed{f_{G} = 114.366 \; \text{GHz}}\] \[\color{blue}{\text{Are Fermilab quantum particle detectors capable of detecting cold dark matter quantum particles?} \; (\text{ref. 7})}\] \[\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}\] Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://www.ymambrini.com/My_World/Articles_files/Planck_2013_results_16.pdf Science Forums - Dark energy quinton radiation temperature - Orion1: (ref. 2) https://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=931225 Wikipedia - Lambert W function: (ref. 3) https://en.wikipedia.org/wiki/Lambert_W_function Wikipedia - Cosmic microwave background - Importance of precise measurement: (ref. 4) https://en.wikipedia.org/wiki/Cosmic_microwave_background#Importance_of_precise_measurement Wikipedia - Cosmic neutrino background - Derivation of the CvB temperature: (ref. 5) https://en.wikipedia.org/wiki/Cosmic_neutrino_background#Derivation_of_the_CνB_temperature Wikipedia - Planck's law: (ref. 6) https://en.wikipedia.org/wiki/Planck's_law Fermilab - How scientists at Fermilab search for dark matter particles: (ref. 7) https://bit.ly/3AvGTo9 Science Forums - Toy model calculation versus observation comparison summary - Orion1: (ref. 8 ) https://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=1112539
  19. y=∫f(x)dx Thanks joigus! \[ \color{blue}{\text{test}} \]
  20. @joigus, how did you center your equations? I cannot get \begin{center} command to work.
  21. [math]\begin{align} \text{text} \end{align}[/math] [math]\color{blue}{\text{text}}[/math] text [math]F = ma[/math]
×
×
  • Create New...

Important Information

We have placed cookies on your device to help make this website better. You can adjust your cookie settings, otherwise we'll assume you're okay to continue.