Jump to content

Orion1

Senior Members
  • Content Count

    142
  • Joined

Community Reputation

4 Neutral

About Orion1

  • Rank
    Baryon

Profile Information

  • Location
    Terra
  • Interests
    Science
  • College Major/Degree
    Physics
  • Favorite Area of Science
    Physics

Recent Profile Visitors

The recent visitors block is disabled and is not being shown to other users.

  1. Affirmative, during the inflationary epoch, the universe was homogeneous and isotropic, every particle point location in space-time was inflating and expanding from every other particle point location in space-time faster than invariant luminous velocity. Any primordial quantum black holes generated as bosonic radiation when the strong force broke symmetry from gravity, would have evaporated instantly from Hawking blackbody radiation. [math]\;[/math] Hawking blackbody radiation evaporation time for a Planck mass quantum black hole: [math]t_{ev} = \frac{5120 \pi G^{2} m_{P}^{3}}{\hbar c^{4}} = 5120 \pi t_{P} = 5120 \pi \sqrt{\frac{\hbar G}{c^{5}}} = 8.671 \cdot 10^{-40} \; \text{s}[/math] [math]\;[/math] [math]\boxed{t_{ev} = 5120 \pi \sqrt{\frac{\hbar G}{c^{5}}}}[/math] [math]\boxed{t_{ev} = 8.671 \cdot 10^{-40} \; \text{s}}[/math] [math]\;[/math] This next presented Hawking blackbody radiation model is based on quantum physics and classical Newtonian gravitation. [math]\;[/math] Surface radius gravitational acceleration: [math]g = \frac{G M}{r^{2}}[/math] [math]\;[/math] Surface radius gravitational escape velocity: [math]v_{e} = \sqrt{\frac{2 G M}{r}}[/math] [math]\;[/math] Hawking blackbody radiation peak radiation energy initial model condition: [math]\boxed{\frac{2 \pi E}{\hbar} = \frac{g}{v_{e}}}[/math] [math]\;[/math] Hawking blackbody radiation peak radiation energy initial model condition integration via substitution: [math]\frac{2 \pi E}{\hbar} = \frac{g}{v_{e}} = \left(\frac{G M}{r^{2}} \right)\left(\sqrt{\frac{r}{2 G M}} \right) = \sqrt{\frac{G M}{2 r^{3}}}[/math] [math]\;[/math] Surface radius Hawking blackbody radiation peak radiation energy initial model condition: [math]\boxed{\frac{2 \pi E}{\hbar} = \sqrt{\frac{G M}{2 r^{3}}}}[/math] [math]\;[/math] Surface radius Hawking blackbody radiation peak radiation temperature: [math]E = k_{B} T_{H} = \frac{\hbar}{2 \pi} \sqrt{\frac{G M}{2 r^{3}}}[/math] [math]\;[/math] [math]\boxed{T_{H} = \frac{\hbar}{2 \pi k_{B}} \sqrt{\frac{G M}{2 r^{3}}}}[/math] [math]\;[/math] Stellar model radiation source is a perfect blackbody: [math]\boxed{\epsilon = 1}[/math] [math]\;[/math] Stefan-Boltzmann Hawking blackbody radiation power law derivation integration via substitution: [math]P_{H} = A_{s} \epsilon \sigma T_{H}^{4} = \left(4 \pi r^{2} \right) \left(\frac{\pi^{2} k_{B}^{4}}{60 c^{2} \hbar^{3}} \right) \left(\frac{\hbar}{2 \pi k_{B}} \sqrt{\frac{G M}{2 r^{3}}} \right)^{4} = \frac{\hbar G^{2} M^{2}}{960 \pi c^{2} r^{4}}[/math] [math]\;[/math] Stefan-Boltzmann Hawking blackbody radiation power law: [math]\boxed{P_{H} = \frac{\hbar G^{2} M^{2}}{960 \pi c^{2} r^{4}}}[/math] [math]\;[/math] One solar mass neutron star model total stellar radius: (ref. 1) [math]\boxed{R_{ns} = 1 \cdot 10^{4} \; \text{m}}[/math] [math]\;[/math] One solar mass neutron star model Hawking blackbody radiation peak radiation temperature: [math]T_{H} = \frac{\hbar}{2 \pi k_{B}} \sqrt{\frac{G M_{\odot}}{2 R_{ns}^{3}}} = 9.903 \cdot 10^{-9} \; \text{K}[/math] [math]\boxed{T_{H} = \frac{\hbar}{2 \pi k_{B}} \sqrt{\frac{G M_{\odot}}{2 R_{ns}^{3}}}}[/math] [math]\boxed{T_{H} = 9.903 \cdot 10^{-9} \; \text{K}}[/math] [math]\;[/math] One solar mass neutron star model Hawking blackbody radiation power law: [math]P_{H} = \frac{\hbar G^{2} M_{\odot}^{2}}{960 \pi c^{2} R_{ns}^{4}} = 6.852 \cdot 10^{-31} \; \text{W}[/math] [math]\boxed{P_{H} = \frac{\hbar G^{2} M_{\odot}^{2}}{960 \pi c^{2} R_{ns}^{4}}}[/math] [math]\boxed{P_{H} = 6.852 \cdot 10^{-31} \; \text{W}}[/math] [math]\;[/math] Can thermodynamic neutron stars generate Hawking blackbody radiation? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - neutron star: (ref. 1) https://en.wikipedia.org/wiki/Neutron_star Wikipedia - Hawking radiation: (ref. 2) https://en.wikipedia.org/wiki/Hawking_radiation
  2. According to this model, a result which occurs during Schwarzschild black hole Hawking radiation evaporation, when Hawking radiation evaporation lifetime is set at one second remaining. [math]\;[/math] Schwarzschild black hole total Hawking evaporation time with one second lifetime remaining: [math]\boxed{t_{1} = 1 \; \text{s}}[/math] [math]\;[/math] Schwarzschild black hole total Hawking evaporation mass with one second lifetime remaining: [math]\boxed{M_{1} = \left(\frac{t_{1} \hbar c^{4}}{5120 \pi G^{2}} \right)^{\frac{1}{3}} }[/math] [math]\;[/math] [math]\boxed{M_{1} = 2.282 \cdot 10^{5} \; \text{kg}}[/math] [math]\;[/math] Schwarzschild black hole Bekenstein-Hawking total power luminosity integration via substitution: [math]P_{1} = \frac{\hbar c^{6}}{15360 \pi G^{2} M_{1}^{2}} = \frac{\hbar c^{6}}{15360 \pi G^{2}} \left[\left(\frac{5120 \pi G^{2}}{t_{1} \hbar c^{4}} \right)^{\frac{1}{3}} \right]^{2} = \frac{}{24} \left(\frac{\hbar c^{10}}{10 \pi \left(G t_{1} \right)^{2}} \right)^{\frac{1}{3}}[/math] [math]\;[/math] Schwarzschild black hole Bekenstein-Hawking total power luminosity at one second lifetime remaining: [math]\boxed{P_{1} = \frac{}{24} \left(\frac{\hbar c^{10}}{10 \pi \left(G t_{1} \right)^{2}} \right)^{\frac{1}{3}}}[/math] [math]\;[/math] [math]\boxed{P_{1} = 6.838 \cdot 10^{21} \; \text{W}}[/math] [math]\;[/math] [math]1 \; \text{megaton TNT} = 4.184 \cdot 10^{15} \; \text{j}[/math] [math]\;[/math] [math]\boxed{P_{1} = 1.634 \cdot 10^{6} \; \frac{\text{megatons TNT}}{\text{s}}}[/math] [math]\;[/math] Schwarzschild black hole peak Hawking radiation temperature at one second lifetime remaining integration via substitution: [math]T_{1} = \frac{\hbar c^{3}}{8 \pi G M_{1} k_{B}} = \frac{\hbar c^{3}}{8 \pi G k_{B}} \left(\frac{5120 \pi G^{2}}{t_{1} \hbar c^{4}} \right)^{\frac{1}{3}} = \frac{}{k_{B}} \left(\frac{10 \hbar^{2} c^{5}}{G t_{1} \pi^{2}} \right)^{\frac{1}{3}}[/math] [math]\;[/math] Schwarzschild black hole peak Hawking radiation temperature at one second lifetime remaining: [math]\boxed{T_{1} = \frac{}{k_{B}} \left(\frac{10 \hbar^{2} c^{5}}{G t_{1} \pi^{2}} \right)^{\frac{1}{3}}}[/math] [math]\;[/math] [math]\boxed{T_{1} = 5.376 \cdot 10^{17} \; \text{K}}[/math] [math]\;[/math] Schwarzschild black hole total Hawking energy radiation emission after one second lifetime remaining integration via substitution: [math]M_{1} = \frac{E_{1}}{c^{2}} = \left(\frac{t_{1} \hbar c^{4}}{5120 \pi G^{2}} \right)^{\frac{1}{3}} \rightarrow E_{1} = c^{2} \left(\frac{t_{1} \hbar c^{4}}{5120 \pi G^{2}} \right)^{\frac{1}{3}} = \left(\frac{t_{1} \hbar c^{10}}{5120 \pi G^{2}} \right)^{\frac{1}{3}}[/math] [math]\;[/math] Schwarzschild black hole total Hawking energy radiation emission after one second lifetime remaining: [math]\boxed{E_{1} = \left(\frac{t_{1} \hbar c^{10}}{5120 \pi G^{2}} \right)^{\frac{1}{3}}}[/math] [math]\;[/math] [math]\boxed{E_{1} = 2.051 \cdot 10^{22} \; \text{j}}[/math] [math]\;[/math] [math]1 \; \text{megaton TNT} = 4.184 \cdot 10^{15} \; \text{j}[/math] [math]\;[/math] [math]\boxed{E_{1} = 4.903 \cdot 10^{6} \; \text{megatons TNT}}[/math] [math]\;[/math] Is this the eventual fate for every black hole in the entire universe? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math]
  3. Orion1

    Latex test

    [math]F(x) = \int_{a}^{b} f(x)dx[/math] [math]\boxed{F(x) = \int_{a}^{b} f(x) dx}[/math]
  4. Orion1

    Latex test

    [math]\dashbox{F(x) = \int_{a}^{b} f(x)dx }[/math] [math]\boxed{F(x) = \int_{a}^{b} f(x)dx}[/math]
  5. Affirmative, stellar black hole masses typically range between 2.27 to 16 solar masses, and are generated from stellar supernovas. (ref. 1) [math]2.27 \cdot M_{\odot} \leq M_{bh} \leq 16 \cdot M_{\odot}[/math] [math]\;[/math] According to this model, any Schwarzschild black hole capable of evaporating completely from Hawking radiation within the universe age, the Schwarzschild black hole must have an initial primordial mass less than or equal to this Hawking evaporation mass. This Hawking evaporation mass is similar to the mass of a small asteroid. [math]\;[/math] Schwarzschild black hole Bekenstein-Hawking total power luminosity: [math]\boxed{P_{H} = \frac{\hbar c^{6}}{15360 \pi G^{2} M^{2}}}[/math] [math]\;[/math] Universe age: [math]t_{u} = 13.799 \cdot 10^{9} \; \text{years} = 4.355 \cdot 10^{17} \; \text{s}[/math] [math]\;[/math] Schwarzschild black hole total Hawking evaporation mass in universe lifetime integration: [math]P_{H} = - \frac{dE}{dt} = -\left(\frac{d}{dt} \right) M c^2 = -c^2 \frac{dM}{dt} = \frac{\hbar c^{6}}{15360 \pi G^{2} M^{2}}[/math] [math]\;[/math] The differential variables are seperable, and the integrals can be written as: [math]- \int_{M_{ev}}^{0} M^{2} dM = \frac{\hbar c^{4}}{15360 \pi G^{2}} \; \int_{0}^{t_{u}} dt = \frac{M_{ev}^{3}}{3} = \frac{t_{u} \hbar c^{4}}{15360 \pi G^{2}}[/math] [math]\;[/math] Schwarzschild black hole total Hawking evaporation mass in universe lifetime: [math]\boxed{M_{ev} \leq \left(\frac{t_{u} \hbar c^{4}}{5120 \pi G^{2}} \right)^{\frac{1}{3}}}[/math] [math]\;[/math] [math]\boxed{M_{ev} \leq 1.730 \cdot 10^{11} \; \text{kg}}[/math] [math]\;[/math] Asteroid Castalia mass: (ref. 2) [math]M = 5.000 \cdot 10^{11} \; \text{kg}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Science Forums - toy model black holes per galaxy average number - Orion1: (ref. 1) https://www.scienceforums.net/topic/120012-toy-model-black-holes-per-galaxy-average-number/ Asteroid Fact Sheet - NASA: (ref. 2) https://nssdc.gsfc.nasa.gov/planetary/factsheet/asteroidfact.html
  6. Translated from Google translate: The page appears to be modelling a classical harmonic oscillator for a mass m fixed to a special spring. This formula appears to be attempting to utilize Leibniz's notation for Newtons second law of motion: [math]m \frac{d^2 x}{dt^2} + F \cdot x = 0[/math] However, Newtons second law of motion in Leibniz's notation is: [math]F = m \frac{d^2 x}{dt^2}[/math] Subtracting force from both the equation left hand side and right hand side results in: [math]m \frac{d^2 x}{dt^2} - F = 0[/math] However, note the equation definition for the potential force on line 5: [math]F(x) = -\frac{dE_p}{dx}[/math] And the resulting potential energy: [math]E_p = -F(x) \cdot dx[/math] So, The equation on line 11 is mixing systeme internationale units of force (newtons) and energy (joules), which is mathematically incorrect. Kartazion, I recommend purchasing a university level physics textbook and study the section on Newtonian mechanics for classical harmonic oscillators. University level physics textbooks already provide the required level mathematics in Leibniz's notation, as well as numerical integration for models such as these, instead of landing in the middle of other online models that display crude and incorrect formulas and with insufficient modeling experience. Some experience in Latex modelling would be beneficial also. Reference: Anharmonic oscillator - Lemans university: https://bit.ly/2SKdIJX
  7. The power emitted by a black hole in the form of Hawking radiation can be estimated for the simplest case of a non-rotating, non-charged Schwarzschild black hole of mass M. Combining the formulas for the black hole Schwarzschild radius, the Stefan Boltzmann law of blackbody radiation, the black hole Bekenstein Hawking luminosity surface radiation temperature, and the sphere surface area formula, which is also the black holes event horizon surface area, several equations can be derived: [math]\;[/math] Stefan Boltzmann constant: (ref. 1, ref. 2) [math]\sigma = \frac{\pi^{2} k_{B}^{4}}{60 \hbar^{3} c^{2}}[/math] [math]\;[/math] Schwarzschild radius: (ref. 3, ref. 4) [math]r_{s} = \frac{2 G M}{c^{2}}[/math] [math]\;[/math] Gravitational acceleration at the Schwarzschild event horizon: [math]g = \frac{G M}{r_{s}^{2}} = G M \left(\frac{c^{2}}{2GM} \right)^{2} = \frac{c^{4}}{4 G M}[/math] [math]\boxed{g = \frac{c^{4}}{4 G M}}[/math] [math]\;[/math] Hawking radiation has a Blackbody radiation Planck spectrum with a peak radiation temperature T: (ref. 5) [math]E = k_{B} T = \frac{\hbar g}{2 \pi c} = \frac{\hbar}{2 \pi c} \left(\frac{c^{4}}{4 G M} \right) = \frac{\hbar c^{3}}{8 \pi G M}[/math] [math]\;[/math] Hawking peak radiation temperature: (ref. 6, ref. 7) [math]\boxed{T_{H} = \frac{\hbar c^{3}}{8 \pi G M k_{B}}}[/math] [math]\;[/math] For a one solar mass black hole, the Hawking peak radiation temperature is: [math]T_{H} = \frac{\hbar c^3}{8 \pi G M_{\odot} k_{B}} = 6.170 \times 10^{-8} \; \text{K}[/math] [math]\boxed{T_{H} = \frac{\hbar c^3}{8 \pi G M_{\odot} k_{B}}}[/math] [math]\boxed{T_{H} = 6.170 \times 10^{-8} \; \text{K}}[/math] [math]\;[/math] Wiens displacement law: (ref. 8) [math]\lambda_\mathrm{peak} = \frac{h c}{4.965 k_{B} T_{H}} = \frac{h c}{4.965 k_{B}} \left(\frac{8 \pi G M_{\odot} k_{B}}{\hbar c^{3}} \right) = \frac{h c}{4.965 k_{B}} \left(\frac{8 \pi^{2} k_{B}}{h c} \right)\left(\frac{2 G M}{c^{2}} \right) = \frac{16 G M \pi^{2}}{4.965 c^{2}} = \frac{8 \pi^{2}}{4.965} \; r_{s} = 15.902 \; r_{s}[/math] [math]\boxed{\lambda_\mathrm{peak} = \frac{16 G M \pi^{2}}{4.965 c^{2}}}[/math] [math]\boxed{\lambda_\mathrm{peak} = 15.902 \; r_{s}}[/math] [math]\;[/math] The radiation peak wavelength is nearly 16 times the black hole Schwarzschild radius. [math]\;[/math] Schwarzschild sphere surface area with Schwarzschild radius integration via substitution: [math]A_{s} = 4 \pi r_{s}^{2} = 4 \pi \left(\frac{2 G M}{c^{2}} \right)^{2} = \frac{16 \pi G^{2} M^{2}}{c^{4}}[/math] [math]\;[/math] Schwarzschild sphere surface area with Schwarzschild radius: [math]\boxed{A_{s} = \frac{16 \pi G^{2} M^{2}}{c^{4}}}[/math] [math]\;[/math] Stefan Boltzmann power law: (ref. 9) [math]P = A_{s} j^{\star} = A_{s} \epsilon \sigma T^{4}[/math] [math]\;[/math] A black hole is a perfect blackbody: [math]\boxed{\epsilon = 1}[/math] [math]\;[/math] Stefan Boltzmann Schwarzschild Hawking black hole radiation power law derivation: [math]P = A_{s} \epsilon \sigma T_{H}^{4} = \left(\frac{16 \pi G^{2} M^{2}}{c^{4}} \right)\left(\frac{\pi^{2} k_{B}^{4}}{60 \hbar^{3} c^{2}} \right)\left(\frac{\hbar c^{3}}{8 \pi G M k_{B}} \right)^{4} = \frac{\hbar c^{6}}{15360 \pi G^{2} M^{2}}[/math] [math]\;[/math] This yields the Bekenstein Hawking luminosity of a black hole, for pure photon emission and no other emitted particles, and under the assumption that the horizon is the radiating surface: [math]\boxed{P = \frac{\hbar c^{6}}{15360 \pi G^{2} M^{2}}}[/math] [math]\;[/math] [math]P[/math] - radiated power luminosity [math]\hbar[/math] - reduced Planck constant [math]c[/math] - speed of light [math]G[/math] - gravitational constant [math]M[/math] - black hole mass [math]\;[/math] It is worth mentioning that the above formula has not yet been derived in the framework of semiclassical gravity. [math]\;[/math] Substituting the numerical values of the physical constants in the formula for luminosity, the Hawking radiation power from a solar mass black hole turns out to be minuscule: [math]P = \frac{\hbar c^{6}}{15360 \pi G^{2} M_{\odot}^{2}} = 9.007 \cdot 10^{-29} \; \text{W}[/math] [math]\boxed{P = \frac{\hbar c^{6}}{15360 \pi G^{2} M_{\odot}^{2}}}[/math] [math]\boxed{P = 9.007 \cdot 10^{-29} \; \text{W}}[/math] [math]\;[/math] It is indeed an extremely good approximation to call such an object black. Under the assumption of an otherwise empty universe, so that no matter, cosmic microwave background radiation, or other radiation falls into the black hole, it is possible to calculate how long it would take for the black hole to dissipate. Bekenstein Hawking Schwarzschild evaporation constant: [math]K_{ev} = \frac{\hbar c^{6}}{15360 \pi G^{2}} = 3.562 \times 10^{32} \; \text{W} \; \text{kg}^{2}[/math] [math]\boxed{K_{ev} = \frac{\hbar c^{6}}{15360 \pi G^{2}}}[/math] [math]\boxed{K_{ev} = 3.562 \times 10^{32} \; \text{W} \; \text{kg}^{2}}[/math] [math]\;[/math] Given that the power of the Hawking radiation is the rate of evaporation energy loss of the black hole: [math]P = - \frac{dE}{dt} = \frac{K_{ev}}{M^{2}}[/math] [math]\;[/math] Since the black hole total energy E is related to the black hole mass M by the Einstein mass energy formula: [math]E = Mc^{2}[/math] [math]\;[/math] [math]P = - \frac{dE}{dt} = -\left(\frac{d}{dt} \right) M c^2 = -c^2 \frac{dM}{dt}[/math] [math]\;[/math] It is possible to equate this with the above expression for the power equation: [math]-c^{2} \frac{dM}{dt} = \frac{K_\mathrm{ev}}{M^{2}}[/math] [math]\;[/math] The differential equation variables can be seperated: [math]M^{2} \; dM = - \frac{K_{ev}}{c^{2}} \; dt[/math] [math]\;[/math] The black holes mass is now a function [math]M(t)[/math] of time [math]t[/math]. Integrating over [math]M[/math] from [math]M_{0}[/math], the initial mass of the black hole to zero, complete evaporation, and over [math]t[/math] from zero to [math]t_{ev}[/math]: [math]\int_{M_{0}}^{0} M^{2} \; dM = - \frac{K_{ev}}{c^{2}} \int_{0}^{t_{ev}} \; dt[/math] [math]\;[/math] Schwarzschild black hole evaporation time integration via substitution: [math]t_{ev} = \frac{c^{2} M_{0}^{3}}{3 K_{ev}} = \left(\frac{c^{2} M_{0}^{3}}{3} \right)\left(\frac{15360 \pi G^{2}}{\hbar c^{6}} \right) = \frac{5120 \pi G^{2} M_{0}^{3}}{\hbar c^{4}}[/math] [math]\;[/math] Schwarzschild black hole evaporation time: [math]\boxed{t_{ev} = \frac{5120 \pi G^{2} M_{0}^{3}}{\hbar c^{4}}}[/math] [math]\;[/math] Evaporation time For a one solar mass black hole: [math]t_{ev} = \frac{5120 \pi G^{2} M_\odot^{3}}{\hbar c^{4}} = 6.617 \times 10^{74} \; \text{s} = 2.098 \; \cdot \; 10^{67} \; \text{years}[/math] [math]\;[/math] [math]\boxed{t_{ev} = \frac{5120 \pi G^{2} M_\odot^{3}}{\hbar c^{4}}}[/math] [math]\boxed{t_{ev} = 2.098 \; \cdot \; 10^{67} \; \text{years}}[/math] [math]\;[/math] Universe age: [math]t_{u} = 13.799 \; \cdot \; 10^{9} \; \text{years}[/math] [math]\;[/math] The evaporation time for a one solar mass black hole is much greater than the universe age. [math]\boxed{t_{ev} \gg t_{u}}[/math] [math]\;[/math] The lower classical quantum limit for mass for this equation is equivalent to the Planck mass [math]m_{P}[/math]: [math]\;[/math] Hawking radiation evaporation time for a Planck mass quantum black hole: [math]t_{ev} = \frac{5120 \pi G^{2} m_{P}^{3}}{\hbar c^{4}} = 5120 \pi t_{P} = 5120 \pi \sqrt{\frac{\hbar G}{c^{5}}} = 8.671 \times 10^{-40} \; \text{s}[/math] [math]\;[/math] [math]\boxed{t_{ev} = 5120 \pi \sqrt{\frac{\hbar G}{c^{5}}}}[/math] [math]\boxed{t_{ev} = 8.671 \cdot 10^{-40} \; \text{s}}[/math] [math]\;[/math] Where [math]t_{P}[/math] is the Planck time. (ref. 10) [math]\;[/math] However, since the universe contains the cosmic microwave background radiation, in order for the black hole to dissipate, it must have a temperature greater than that of the present day blackbody radiation of the universe of 2.7 K. This implies that M must be less than 0.8 percent of the mass of the Earth. Cosmic microwave background radiation universe temperature: (ref. 11) [math]T_{u} = 2.725 \; \text{K}[/math] [math]\;[/math] Schwarzschild Hawking total black hole mass: [math]M_{H} \leq \frac{\hbar c^{3}}{8 \pi G k_{B} T_{u}} \leq 4.503 \; \cdot \; 10^{22} \; \text{kg}[/math] [math]\;[/math] [math]\boxed{M_{H} \leq \frac{\hbar c^{3}}{8 \pi G k_{B} T_{u}}}[/math] [math]\boxed{M_{H} \leq 4.503 \; \cdot \; 10^{22} \; \text{kg}}[/math] [math]\;[/math] [math]\frac{M_{H}}{M_\oplus} = 7.539 \; \cdot \; 10^{-3} = 0.754 \; \%[/math] [math]\;[/math] [math]M_{\oplus}[/math] - total Earth mass [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Stefan-Boltzmann law: (ref. 1) https://en.wikipedia.org/wiki/Stefan-Boltzmann_law Wikipedia - Stefan-Boltzmann_constant: (ref. 2) https://en.wikipedia.org/wiki/Stefan-Boltzmann_constant Wikipedia - Schwarzschild metric: (ref. 3) https://en.wikipedia.org/wiki/Schwarzschild_metric Wikipedia - Schwarzschild radius: (ref. 4) https://en.wikipedia.org/wiki/Schwarzschild_radius Wikipedia - Blackbody radiation: (ref. 5) https://en.wikipedia.org/wiki/Blackbody_radiation Wikipedia - Hawking radiation: (ref. 6) https://en.wikipedia.org/wiki/Hawking_radiation#Emission_process Wikipedia - Black hole thermodynamics: (ref. 7) https://en.wikipedia.org/wiki/Black_hole_thermodynamics Wikipedia - Wien's_displacement law: (ref. 8) https://en.wikipedia.org/wiki/Wien's_displacement_law Wikipedia - Stefan-Boltzmann law: (ref. 9) https://en.wikipedia.org/wiki/Stefan-Boltzmann_law Wikipedia - Planck time: (ref. 10) https://en.wikipedia.org/wiki/Planck_time Wikipedia - Cosmic microwave background radiation: (ref. 11) https://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation
  8. Affirmative, revision complete. Relativistic Lagrangian: [math]\boxed{\mathcal{L} = \sum_{1}^{n} \mathcal{L}\left(n \right) = 0} \; \; \; n = 5[/math] [math]\;[/math] Lagrangian equation for a massless quantum field: [math]\; \; \; m = 0[/math] [math]\mathcal{L} = \underbrace{ \mathbb{R} }_{\text{GR}} - \overbrace{\underbrace{\frac{1}{4} F^{\mu \nu} F_{\mu \nu}}_{\text{Yang-Mills}}}^{\text{Maxwell}} + \underbrace{i \overline{\psi} \gamma^\mu D_{\mu} \psi}_{\text{Dirac}} + \underbrace{|D_{\mu} \phi|^{2} - V\left(|\phi| \right)}_{\text{Higgs}} - \underbrace{g \overline{\psi} \psi}_{\text{Yukawa}} = 0[/math] [math]\;[/math] Lagrangian equation for a massless quantum field: [math]\; \; \; m = 0[/math] [math]\boxed{\mathcal{L} = \overbrace{\underbrace{\Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} M^{\alpha \beta}\left(x \right) }_{\text{Quantum Gravity}}}^{\text{spin } 2} - \overbrace{\underbrace{\frac{1}{4} F^{a \mu \nu} F^{a}_{\mu \nu}}_{\text{Yang-Mills}}}^{\text{spin } 1} + \overbrace{\underbrace{i \overline{\psi}_{a} \gamma^{\mu}_{ab} D_{\mu} \psi_{b}}}_{\text{Dirac}}^{\text{spin } 1/2} + \overbrace{\underbrace{|D_{\mu} \phi_{a}|^{2} - V\left(|\phi_{a}| \right)}}_{\text{Higgs}}^{\text{spin } 0} - \overbrace{\underbrace{g \overline{\psi}_{a} \psi_{b}}}_{\text{Yukawa}}^{\text{spin } 0,1/2} = 0}[/math] (ref. 1, ref. 2, ref. 3, ref. 4, ref. 5, ref. 6, ref. 7) [math]\;[/math] Lagrangian equation for a mass quantum field: [math]\; \; \; m \neq 0[/math] [math]\boxed{\mathcal{L} = \overbrace{\underbrace{\Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} M^{\alpha \beta}\left(x \right) }_{\text{Quantum Gravity}}}^{\text{spin } 2} - \overbrace{\underbrace{\frac{1}{4} F^{a \mu \nu} F^{a}_{\mu \nu}}_{\text{Yang-Mills}}}^{\text{spin } 1} + \overbrace{\underbrace{\overline{\psi}_{a} \left(i \gamma^{\mu}_{ab} D_{\mu} - m \mathbb{I}_{ab} \right) \psi_{b}}}_{\text{Dirac}}^{\text{spin } 1/2} + \overbrace{\underbrace{|D_{\mu} \phi_{a}|^{2} - V\left(|\phi_{a}| \right)}}_{\text{Higgs}}^{\text{spin } 0} - \overbrace{\underbrace{g \overline{\psi}_{a} \psi_{b}}}_{\text{Yukawa}}^{\text{spin } 0,1/2} = 0}[/math] (ref. 4, ref. 5) [math]\;[/math] Lagrangian equation for a mass and charge quantum field: [math]\; \; \; m \neq 0, Q \neq 0[/math] [math]\boxed{\mathcal{L} = \overbrace{\underbrace{\Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} M^{\alpha \beta}\left(x \right) }_{\text{Quantum Gravity}}}^{\text{spin } 2} - \overbrace{\underbrace{\frac{1}{4} F^{a \mu \nu} F^{a}_{\mu \nu}}_{\text{Yang-Mills}}}^{\text{spin } 1} + \overbrace{\underbrace{\overline{\psi}_{a} \left[\gamma^{\mu}_{ab} \left(i \partial_{\mu} - e Q A_{\mu} \right) - m \mathbb{I}_{ab} \right] \psi_{b}}}_{\text{Dirac}}^{\text{spin } 1/2} + \overbrace{\underbrace{|\left(\partial_{\mu} -ieQA_{\mu} \right) \phi_{a}|^{2} - \lambda \left(|\phi_{a}|^{2} - \Phi^{2} \right)^{2}}}_{\text{Higgs}}^{\text{spin } 0,1} - \overbrace{\underbrace{g \overline{\psi}_{a} \psi_{b}}}_{\text{Yukawa}}^{\text{spin } 0,1/2} = 0}[/math] (ref. 4, pg. 8, eq. 2.6, ref. 5, ref. 6, ref. 7) [math]\;[/math] Is this approach mathematically and symbolically correct to this point? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Quantum gravity: (ref. 1) https://en.wikipedia.org/wiki/Quantum_gravity Science Forums - Orion1 - Spin 2 Quantum Gravity: (ref. 2) https://www.scienceforums.net/topic/117992-the-lagrangian-equation/?do=findComment&comment=1128193 Wikipedia - Yang–Mills theory: (ref. 3) https://en.wikipedia.org/wiki/Yang–Mills_theory#Mathematical_overview Search For The Standard Model Higgs Boson - Huong Thi Nguyen: (ref. 4) https://www-d0.fnal.gov/results/publications_talks/thesis/nguyen/thesis.pdf Wikipedia - Dirac fields: (ref. 5) https://en.wikipedia.org/wiki/Fermionic_field#Dirac_fields Wikipedia - Higgs mechanism: (ref. 6) https://en.wikipedia.org/wiki/Higgs_mechanism#Abelian_Higgs_mechanism Wikipedia - Yukawa interaction: (ref. 7) https://en.wikipedia.org/wiki/Yukawa_interaction#The_action
  9. Affirmative, revision complete. First order metric tensor field for spin-0 and spin-1 particles: (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] Second order metric tensor field for spin-2 particles: (ref. 3) [math]M^{'\mu \nu} \left(x' \right) = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} M^{\alpha \beta}\left(x \right)[/math] [math]\;[/math] The graviton is a spin-two particle, as opposed to the spin-one photon, so that the interaction forms are somewhat more complex, involving symmetric and traceless second order tensors rather than simple Lorentz four-vectors. [math]\;[/math] In relativistic mechanics, the Center Of Mass-Energy boost and orbital 3-space angular momentum of a rotating object are combined into a four-dimensional bivector in terms of the four-position X and the four-momentum P of the object. (ref. 3) [math]\mathbf{M} = \mathbf{X} \wedge \mathbf{P}[/math] [math]\;[/math] With matrix components: [math]M^{\alpha \beta} = X^{\alpha} P^{\beta} - X^{\beta} P^{\alpha}[/math] [math]\;[/math] Which are six independent quantities altogether. Since the components of X and P are frame-dependent, so is M. Three components [math]M^{ij} = x^{i} p^{j} - x^{j} p^{i} = L^{ij}[/math] [math]\;[/math] are those of the familiar classical 3-space orbital angular momentum, and the other three components [math]M^{0i} = x^{0} p^{i} - x^{i} p^{0} = c \; \left(tp^{i} - x^{i}{\frac{E}{c^{2}}} \right) = -cN^{i}[/math] are the relativistic mass moment, multiplied by -c. The tensor is antisymmetric: [math]M^{\alpha \beta} = -M^{\beta \alpha}[/math] [math]\;[/math] The components of the tensor can be systematically displayed as a matrix. [math]\;[/math] The angular momentum [math]L = x \; \wedge \; p[/math] of a particle with relativistic mass m and relativistic momentum p, as measured by an observer in a lab frame, combines with another vector quantity dynamic mass-energy moment [math]N = mx - pt[/math] in the relativistic angular momentum tensor: (ref. 2) [math]M^{\alpha \beta} = {\begin{pmatrix} 0 & -N_{x}^{1}c & -N_{y}^{2}c & -N_{z}^{3}c \\ N_{x}^{1}c & 0 & L^{12} & -L^{13} \\ N_{y}^{2}c & -L^{21} & 0 & L^{23} \\ N_{z}^{3}c & L^{31} & -L^{32} & 0 \end{pmatrix}}[/math] [math]\;[/math] In classical mechanics, the orbital angular momentum of a particle with instantaneous three-dimensional position vector [math]x = (x, y, z)[/math] and momentum vector [math]p = (px, py, pz)[/math], is defined as the axial vector: (ref. 4) [math]\mathbf{L} = \mathbf{x} \times \mathbf{p}[/math] [math]\;[/math] Which has three components, that are systematically given by cyclic permutations of Cartesian directions, change x to y, y to z, z to x, repeat. [math]L_{x} = yp_{z} - zp_{y}[/math] [math]L_{y} = zp_{x} - xp_{z}[/math] [math]L_{z} = xp_{y} - yp_{x}[/math] [math]\;[/math] A related definition is to conceive orbital angular momentum as a plane element. This can be achieved by replacing the cross product by the exterior product in the language of exterior algebra, and angular momentum becomes a contravariant second order antisymmetric tensor: [math]\mathbf{L} = \mathbf{x} \wedge \mathbf{p}[/math] [math]\;[/math] or writing [math]x = (x_{1}, x_{2}, x_{3}) = (x, y, z)[/math] and momentum vector [math]p = (p_{1}, p_{2}, p_{3}) = (p_{x}, p_{y}, p_{z})[/math], the components can be compactly abbreviated in tensor index notation: [math]L^{ij} = x^{i} p^{j} - x^{j} p^{i}[/math] [math]\;[/math] Where the indices i and j take the values 1, 2, 3. On the other hand, the components can be systematically displayed fully in a 3 x 3 antisymmetric matrix: [math]\mathbf{L} = {\begin{pmatrix} L^{11} & L^{12} & L^{13} \\ L^{21} & L^{22} & L^{23} \\ L^{31} & L^{32} & L^{33} \\ \end{pmatrix}} = {\begin{pmatrix} 0 & L_{xy} & L_{xz} \\ L_{yx} & 0 & L_{yz} \\ L_{zx} & L_{zy} & 0 \end{pmatrix}} = \begin{pmatrix} 0 & L_{xy} & -L_{zx} \\ -L_{xy} & 0 & L_{yz} \\ L_{zx} & -L_{yz} & 0 \end{pmatrix}[/math] [math]\;[/math] [math]\mathbf{L} = {\begin{pmatrix} 0 & xp_{y} - yp_{x} & -\left(zp_{x} - xp_{z} \right) \\ -\left(xp_{y} - yp_{x} \right) & 0 & yp_{z} - zp_{y} \\ zp_{x} - xp_{z} & -\left(yp_{z} - zp_{y} \right) & 0 \end{pmatrix}}[/math] [math]\;[/math] This quantity is additive, and for an isolated system, the total angular momentum of a system is conserved. [math]\;[/math] Dynamic mass-energy moment: [math]\mathbf{N} = m \mathbf{x} - \mathbf{p} t = \frac{E}{c^{2}} \mathbf{x} - \mathbf{p} t = \gamma (\mathbf{u}) m_{0} (\mathbf{x} - \mathbf{u} t)[/math] [math]\;[/math] Expressing N in terms of relativistic mass-energy and momentum, rather than rest mass and velocity, avoids extra Lorentz factors. However, relativistic mass is discouraged by some authors since it can be a misleading quantity to apply in certain equations. [math]\;[/math] Defined here so that the relativistic equation in terms of the relativistic mass-energy equivalence, and classical definition, have the same form. The Cartesian components are: [math]N_{x} = mx - p_{x} t = \frac{E}{c^{2}} x - p_{x} t = \gamma \left(u \right) m_{0}\left(x - u_{x} t \right)[/math] [math]N_{y} = my - p_{y} t = \frac{E}{c^{2}} y - p_{y} t = \gamma \left(u \right) m_{0}\left(y - u_{y} t \right)[/math] [math]N_{z} = mz - p_{z} t = \frac{E}{c^{2}} z - p_{z} t = \gamma \left(u \right) m_{0}\left(z - u_{z} t \right)[/math] [math]\;[/math] For a massless spin-2 graviton, [math]E = pc[/math], and the relativistic angular momentum tensor is: [math]M^{\alpha \beta} = {\begin{pmatrix} 0 & -p_{x}\left(\frac{x}{c} - t \right) & -p_{y}\left(\frac{y}{c} - t \right) & -p_{z}\left(\frac{z}{c} - t \right) \\ p_{x}\left(\frac{x}{c} - t \right) & 0 & xp_{y} - yp_{x} & -\left(zp_{x} - xp_{z} \right) \\ p_{y}\left(\frac{y}{c} - t \right) & -\left(xp_{y} - yp_{x} \right) & 0 & yp_{z} - zp_{y} \\ p_{z}\left(\frac{z}{c} - t \right) & zp_{x} - xp_{z} & -\left(yp_{z} - zp_{y} \right) & 0 \end{pmatrix}}[/math] [math]\;[/math] In spherical coordinates [math](ct, r, \theta, \phi)[/math] the Minkowski flat spacetime metric in contravariant 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] In spherical coordinates [math](ct, r, \theta, \phi)[/math] the relativistic angular momentum tensor in contravariant form: [math]M^{\alpha \beta} = {\begin{pmatrix} 0 & -p_{x}\left(\frac{dr}{c} - t \right) & -p_{y}\left(\frac{r d\phi}{c} - t \right) & -p_{z}\left(\frac{r \sin \theta d\phi}{c} - t \right) \\ p_{x}\left(\frac{dr}{c} - t \right) & 0 & dr p_{y} - r d\theta p_{x} & -\left(r \sin \theta d\phi p_{x} - dr p_{z} \right) \\ p_{y}\left(\frac{r d\theta}{c} - t \right) & -\left(dr p_{y} - r d\theta p_{x} \right) & 0 & r d\theta p_{z} - r \sin \theta d\phi p_{y} \\ p_{z}\left(\frac{r \sin \theta d\phi}{c} - t \right) & r \sin \theta d\phi p_{x} - dr p_{z} & -\left(r d\theta p_{z} - r \sin \theta d\phi p_{y} \right) & 0 \end{pmatrix}}[/math] [math]\;[/math] General Relativity stress-energy tensor: [math]M_{\mu \nu} = \pm \left(\begin{matrix} -\rho\left(r \right) c^{2} & 0 & 0 & 0 \\ 0 & p\left(r \right) & 0 & 0 \\ 0 & 0 & p\left(r \right) & 0 \\ 0 & 0 & 0 & p\left(r \right) \end{matrix} \right)[/math] [math]\;[/math] Jacobian matrix transformation matrices: (ref. 5, pg. 15, eq. 41, para. 1, ref 6) [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] [math]\Lambda^{\mu}_{\alpha} \left(ct, r, \theta, \phi \right) = {\begin{bmatrix} \dfrac{\partial \xi^0}{c^{2} \partial t^0} & 0 & 0 & 0 \\ 0 & \dfrac{\partial \xi^1}{\partial r^1} & 0 & 0 \\ 0 & 0 & \dfrac{\partial \xi^2}{\partial \theta^2} & 0 \\ 0 & 0 & 0 & \dfrac{\partial \xi^3}{\partial \phi^3} \\ \end{bmatrix}} = {\begin{bmatrix} -2 dt & 0 & 0 & 0 \\ 0 & 2 dr & 0 & 0 \\ 0 & 0 & 2r^2 d\theta & 0 \\ 0 & 0 & 0 & 2r^2 \sin^{2} \theta \; d\phi \\ \end{bmatrix}}[/math] [math]\;[/math] [math]\;[/math] [math]\Lambda^{\nu}_{\beta} \left(ct, r, \theta, \phi \right) = {\begin{bmatrix} \dfrac{\partial \xi^0}{c^{2} \partial t^0} & 0 & 0 & 0 \\ 0 & \dfrac{\partial \xi^1}{\partial r^1} & 0 & 0 \\ 0 & 0 & \dfrac{\partial \xi^2}{\partial \theta^2} & 0 \\ 0 & 0 & 0 & \dfrac{\partial \xi^3}{\partial \phi^3} \\ \end{bmatrix}} = {\begin{bmatrix} -2 dt & 0 & 0 & 0 \\ 0 & 2 dr & 0 & 0 \\ 0 & 0 & 2r^2 d\theta & 0 \\ 0 & 0 & 0 & 2r^2 \sin^{2} \theta \; d\phi \\ \end{bmatrix}}[/math] [math]\;[/math] The matrix dot product transformation matrices in covariant form for each of the two four-momentum components as seen from two reference frames, S and S' prime: [math]\Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} = {\begin{bmatrix} 4 dt dt' & 0 & 0 & 0 \\ 0 & 4 dr dr' & 0 & 0 \\ 0 & 0 & 4r^2 r'^{2} d\theta \; d\theta' & 0 \\ 0 & 0 & 0 & 4r^2 r'^{2} \sin^{2} \theta \sin^{2} \theta' \; d\phi \; d\phi' \\ \end{bmatrix}}[/math] [math]\;[/math] Newton's constant: (ref. 7, pg. 9, eq. 37) [math]\kappa^{2} = 32 \pi G[/math] [math]\;[/math] General relativity weak field limit spacetime metric: (ref. 7, pg. 9, eq. 37) [math]g_{\mu \nu} = \eta_{\mu \nu} + \kappa h_{\mu \nu}[/math] [math]\;[/math] General Relativity weak field limit spacetime metric and Planck quantum gravity identity 6: [math]\boxed{\frac{8 \pi G}{c^{4}} M_{\mu \nu} = \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} M^{\alpha \beta} \left(x \right) \left(1 - \frac{1}{2} \left(\eta^{\mu \nu} - \kappa h^{\mu \nu} \right)\left(\eta_{\mu \nu} + \kappa h_{\mu \nu} \right) \right)}[/math] [math]\;[/math] Tensor matrix solution key: [math]s\left(\mu, \nu, \alpha, \beta \right)[/math] [math]\;[/math] [math]s\left(1, 1, 1, 2 \right):[/math] [math]\boxed{2 \pi G p\left(r \right) = c^{4}\left(dr' dr^{2} p_{y} - dr dr' r d\theta p_{x} \right)}[/math] [math]\;[/math] [math]s\left(2, 2, 2, 3 \right):[/math] [math]\boxed{2 \pi G p\left(r \right) = c^{4} \left(r^{3} r'^{2} d\theta^{2} d\theta' p_{z} - r^{3} r'^{2} \sin \theta d\theta d\theta' d\phi p_{y} \right)}[/math] [math]\;[/math] With four stress-energy tensor elements and twelve relativistic angular momentum tensor elements, there are forty-eight possible solution keys. [math]\;[/math] Are these energy-momentum tensors compatible with a massless spin-2 graviton? [math]\;[/math] Is this approach mathematically and symbolically correct to this point? [math]\;[/math] Are there any other tensor matrix solution keys that you want to examine? [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 Wikipedia - Four-tensor - Second order tensors: (ref. 2) https://en.wikipedia.org/wiki/Four-tensor#Second_order_tensors Wikipedia - Relativistic angular momentum - 4d Angular momentum as a bivector: (ref. 3) https://en.wikipedia.org/wiki/Relativistic_angular_momentum#4d_Angular_momentum_as_a_bivector Wikipedia - Relativistic angular momentum - Orbital 3d angular momentum: (ref. 4) https://en.wikipedia.org/wiki/Relativistic_angular_momentum#Orbital_3d_angular_momentum Introduction to Tensor Calculus for General Relativity - Edmund Bertschinger: (ref. 5) https://web.mit.edu/edbert/GR/gr1.pdf Wikipedia - Jacobian matrix: (ref. 6) https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant#Example_3:_spherical-Cartesian_transformation Barry R. Holstein - Department of Physics-LGRT - University of Massachusetts: (ref. 7) https://arxiv.org/pdf/gr-qc/0607045.pdf
  10. Affirmative, revision complete. 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] In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc length, is a function of the metric tensor. [math]\;[/math] General Relativity line element where spacetime is modelled as a curved Pseudo-Riemannian manifold with an appropriate metric tensor: (ref. 3, ref 4, pg. 15, eq. 41, para. 1) [math]ds^2 = g_{\mu \nu} dx^{\mu} dx^{\nu}[/math] [math]\;[/math] The General Relativity line element with a curved Pseudo-Riemannian manifold metric tensor condition imposes constraints on the coefficients [math]\Lambda^{\mu}_{\nu}[/math] (ref. 3, pg. 1, eq. 3) [math]g_{\mu \nu} = g_{\alpha \beta} \Lambda^{\alpha}_{\mu} \Lambda^{\beta}_{\nu}[/math] [math]\;[/math] General Relativity curved Pseudo-Riemannian manifold line element identity: [math]\boxed{ds^2 = g_{\alpha \beta} \Lambda^{\alpha}_{\mu} \Lambda^{\beta}_{\nu} dx^{\mu} dx^{\nu}}[/math] [math]\;[/math] The [math]g_{\alpha \beta}[/math] metric tensor will vary according to the spacetime being modeled. It can have either or both the covariant and contravariant terms accordingly to the Einstein summation convention. In this form it is specifying covariant. [math]g_{\alpha \beta} = \pm \begin{pmatrix} -\xi^0 & 0 & 0 & 0 \\ 0 & \xi^1 & 0 & 0 \\ 0 & 0 & \xi^2 & 0 \\ 0 & 0 & 0 & \xi^3 \end{pmatrix}[/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] The Minkowski flat spacetime metric in covariant form: [math]ds \; ds' = -c^{2} dt dt' + dr dr' + r r' d\theta \; d\theta' + r r' \sin \theta \sin \theta' \; d\phi \; d\phi'[/math] [math]\;[/math] General relativity stress-energy tensor: [math]T_{\mu \nu} = \pm \left(\begin{matrix} -\rho c^{2} & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{matrix} \right)[/math] [math]\;[/math] General Relativity Minkowski flat spacetime metric tensor in covariant form: [math]T^{\alpha \beta} \left(r \right) = \pm \begin{pmatrix} -c^{2} dt dt' & 0 & 0 & 0 \\ 0 & dr dr' & 0 & 0 \\ 0 & 0 & r r' d\theta \; d\theta' & 0 \\ 0 & 0 & 0 & r r' \sin \theta \sin \theta' \; d\phi \; d\phi' \end{pmatrix}[/math] [math]\;[/math] Jacobian matrix transformation matrices: (ref. 4, ref 5, pg. 15, eq. 41, para. 1) [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] [math]\Lambda^{\mu}_{\alpha} \left(ct, r, \theta, \phi \right) = {\begin{bmatrix} \dfrac{\partial \xi^0}{c^{2} \partial t^0} & 0 & 0 & 0 \\ 0 & \dfrac{\partial \xi^1}{\partial r^1} & 0 & 0 \\ 0 & 0 & \dfrac{\partial \xi^2}{\partial \theta^2} & 0 \\ 0 & 0 & 0 & \dfrac{\partial \xi^3}{\partial \phi^3} \\ \end{bmatrix}} = {\begin{bmatrix} -2 dt & 0 & 0 & 0 \\ 0 & 2 dr & 0 & 0 \\ 0 & 0 & 2r^2 d\theta & 0 \\ 0 & 0 & 0 & 2r^2 \sin^{2} \theta \; d\phi \\ \end{bmatrix}}[/math] [math]\;[/math] [math]\;[/math] [math]\Lambda^{\nu}_{\beta} \left(ct, r, \theta, \phi \right) = {\begin{bmatrix} \dfrac{\partial \xi^0}{c^{2} \partial t^0} & 0 & 0 & 0 \\ 0 & \dfrac{\partial \xi^1}{\partial r^1} & 0 & 0 \\ 0 & 0 & \dfrac{\partial \xi^2}{\partial \theta^2} & 0 \\ 0 & 0 & 0 & \dfrac{\partial \xi^3}{\partial \phi^3} \\ \end{bmatrix}} = {\begin{bmatrix} -2 dt & 0 & 0 & 0 \\ 0 & 2 dr & 0 & 0 \\ 0 & 0 & 2r^2 d\theta & 0 \\ 0 & 0 & 0 & 2r^2 \sin^{2} \theta \; d\phi \\ \end{bmatrix}}[/math] [math]\;[/math] The matrix dot product transformation matrices in covariant form for each of the two four-momentum components as seen from two reference frames, S and S' prime: [math]\Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} = {\begin{bmatrix} 4 dt dt' & 0 & 0 & 0 \\ 0 & 4 dr dr' & 0 & 0 \\ 0 & 0 & 4r^2 r'^{2} d\theta \; d\theta' & 0 \\ 0 & 0 & 0 & 4r^2 r'^{2} \sin^{2} \theta \sin^{2} \theta' \; d\phi \; d\phi' \\ \end{bmatrix}}[/math] [math]\;[/math] General Relativity weak field limit spacetime metric and Planck quantum gravity identity 4: (ref. 6) [math]\boxed{\frac{8 \pi G}{c^{4}} 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] Solution 1: [math]\mu = \nu = 0[/math] [math]\boxed{2 \pi G \rho \left(r \right) = c^{4} dt^{2} dt'^{2}}[/math] [math]\;[/math] Solution 2: [math]\mu = \nu = 1[/math] [math]\boxed{2 \pi G p \left(r \right) = c^{4} dr^{2} dr'^{2}}[/math] [math]\;[/math] Solution 3: [math]\mu = \nu = 2[/math] [math]\boxed{2 \pi G p \left(r \right) = c^{4} dr^{3} dr'^{3} d\theta^{2} \; d\theta'^{2}}[/math] [math]\;[/math] Solution 4: [math]\mu = \nu = 3[/math] [math]\boxed{2 \pi G p \left(r \right) = c^{4} dr^{3} dr'^{3} \sin^{3} \theta \; \sin^{3} \theta' d\phi^{2} \; d\phi'^{2}}[/math] [math]\;[/math] Is this approach mathematically and symbolically correct to this point? [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 Wikipedia - General relativity - Metric tensor - Local coordinates and matrix representations: (ref. 2) https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)#Local_coordinates_and_matrix_representations Lorentz Transformations - Bernard Durney: (ref. 3) https://arxiv.org/pdf/1103.0156.pdf Introduction to Tensor Calculus for General Relativity - Edmund Bertschinger: (ref. 4) https://web.mit.edu/edbert/GR/gr1.pdf Wikipedia - Jacobian matrix: (ref. 5) https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant#Example_3:_spherical-Cartesian_transformation Science Forums - Lagrangian equation for a massless Planck graviton - Orion1: (ref. 6) https://www.scienceforums.net/topic/117992-the-lagrangian-equation/?do=findComment&comment=1096894
  11. Affirmative, revision complete. [math]\eta_{\mu \nu}[/math] - perturbed non-dynamical background metric [math]\;[/math] General Relativity Minkowski flat spacetime metric: (ref. 1) [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.1, ref. 2) [math]\boxed{\eta_{\mu \nu} = \eta^{\mu \nu}}[/math] [math]\;[/math] What is the formal mathematical definition for [math]g_{\alpha \beta}[/math]? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [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 Wikipeda - Lorentz covariance: (ref. 2) https://en.wikipedia.org/wiki/Lorentz_covariance
  12. Affirmative, revision complete. 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] In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc length, is a function of the metric tensor. [math]\;[/math] General Relativity line element where spacetime is modelled as a curved Pseudo-Riemannian manifold with an appropriate metric tensor: (ref. 3, ref 4, pg. 15, eq. 41, para. 1) [math]ds^2 = g_{\mu \nu} dx^{\mu} dx^{\nu}[/math] [math]\;[/math] The General Relativity line element with a curved Pseudo-Riemannian manifold metric tensor condition imposes constraints on the coefficients [math]\Lambda^{\mu}_{\nu}[/math]: (ref. 3, pg. 1, eq. 3) [math]g_{\mu \nu} = g_{\alpha \beta} \Lambda^{\alpha}_{\mu} \Lambda^{\beta}_{\nu}[/math] [math]\;[/math] General Relativity curved Pseudo-Riemannian manifold line element identity: [math]\boxed{ds^2 = g_{\alpha \beta} \Lambda^{\alpha}_{\mu} \Lambda^{\beta}_{\nu} dx^{\mu} dx^{\nu}}[/math] [math]\;[/math] General Relativity matrix symmetry expression: [math]\mu \cdot \nu = \nu \cdot \mu = \eta[/math] [math]\;[/math] Jacobian matrix transformation matrices: (ref. 4, ref 5, pg. 15, eq. 41, para. 1) [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] [math]\Lambda^{\mu}_{\alpha} \left(ct, r, \theta, \phi \right) = {\begin{bmatrix} \dfrac{\partial \xi^0}{\partial t^0} & 0 & 0 & 0 \\ 0 & \dfrac{\partial \xi^1}{\partial r^1} & 0 & 0 \\ 0 & 0 & \dfrac{\partial \xi^2}{\partial \theta^2} & 0 \\ 0 & 0 & 0 & \dfrac{\partial \xi^3}{\partial \phi^3} \\ \end{bmatrix}}[/math] [math]\;[/math] [math]\;[/math] [math]\Lambda^{\nu}_{\beta} \left(ct, r, \theta, \phi \right) = {\begin{bmatrix} \dfrac{\partial \xi^0}{\partial t^0} & 0 & 0 & 0 \\ 0 & \dfrac{\partial \xi^1}{\partial r^1} & 0 & 0 \\ 0 & 0 & \dfrac{\partial \xi^2}{\partial \theta^2} & 0 \\ 0 & 0 & 0 & \dfrac{\partial \xi^3}{\partial \phi^3} \\ \end{bmatrix}}[/math] [math]\;[/math] Is this approach mathematically and symbolically correct to this point? [math]\;[/math] What is the formal mathematical definition for [math]g_{\alpha \beta}[/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 Wikipedia - General relativity - Metric tensor - Local coordinates and matrix representations: (ref. 2) https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)#Local_coordinates_and_matrix_representations Lorentz Transformations - Bernard Durney: (ref. 3) https://arxiv.org/pdf/1103.0156.pdf Introduction to Tensor Calculus for General Relativity - Edmund Bertschinger: (ref. 4) https://web.mit.edu/edbert/GR/gr1.pdf Wikipedia - Jacobian matrix: (ref. 5) https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant#Example_3:_spherical-Cartesian_transformation
  13. Affirmative, revision complete. WMAP satellite cosmological composition parameters at photon decoupling time: (ref. 1) [math]\Omega_{\gamma,t} = 0.15[/math] [math]\;[/math] Photon species total effective degeneracy number: [math]\boxed{N_{\gamma} = 2}[/math] [math]\;[/math] Solar surface temperature: (ref. 2) [math]T_{\odot} = 5772 \; \text{K}[/math] [math]\;[/math] Total solar radius: (ref. 2) [math]R_{\odot} = 6.963 \cdot 10^{5} \; \text{km}[/math] [math]\;[/math] Earth orbital radius semi-major axis: (ref. 3) [math]r_{\oplus} = 1.496 \cdot 10^{8} \; \text{km}[/math] [math]\;[/math] Earth albedo reflectivity: (ref. 3) [math]\alpha_{\oplus} = 0.367[/math] [math]\;[/math] Earth surface temperature at present time: (ref. 4) [math]T_{\oplus} = T_{\odot} \left(\frac{\left(1 - \alpha_{\oplus} \right)^{\frac{1}{2}} R_{\odot}}{2 r_{\oplus}} \right)^{\frac{1}{2}} = 248.367 \; \text{K}[/math] [math]\boxed{T_{\oplus} = 248.367 \; \text{K}}[/math] [math]\;[/math] Isotropic cosmic photon background radiation temperature at present time: (ref. 5) [math]T_{\gamma} = 2.72548 \; \text{K}[/math] [math]\;[/math] Toy model universe age at isotropic cosmic photon background radiation time: (ref. 6) [math]\boxed{T_{u,\gamma} = \frac{3}{4\left(k_B T_{\gamma,t} \right)^2} \sqrt{\frac{5 \Omega_{\gamma,t} \hbar^3 c^5}{G N_{\gamma} \pi^3}}} \; \; \; m_{\gamma} = 0[/math] [math]\;[/math] Isotropic cosmic photon background radiation temperature at past time is equivalent to Earth surface temperature at present time: [math]\boxed{T_{\gamma,t} = T_{\oplus}}[/math] [math]\;[/math] Toy model universe age at isotropic cosmic photon background radiation time integration via substitution: [math]T_{u,\gamma} = \frac{3}{4\left(k_B T_{\gamma,t} \right)^2} \sqrt{\frac{5 \Omega_{\gamma,t} \hbar^3 c^5}{G N_{\gamma} \pi^3}} = \frac{3}{2 \left(k_B T_{\odot} \right)^2} \left( \frac{r_{\oplus}}{R_{\odot}} \right) \sqrt{\frac{5 \Omega_{\gamma,t} \hbar^3 c^5}{G N_{\gamma} \left(1 - \alpha_{\oplus} \right) \pi^3}}[/math] [math]\;[/math] Toy model universe age at abiogenesis epoch time: [math]\boxed{T_{u,\gamma} = \frac{3 r_{\oplus}}{2 R_{\odot} \left(k_B T_{\odot} \right)^2} \sqrt{\frac{5 \Omega_{\gamma,t} \hbar^3 c^5}{G N_{\gamma} \left(1 - \alpha_{\oplus} \right) \pi^3}}}[/math] [math]\boxed{T_{u,\gamma} = 1.447 \cdot 10^{15} \; \text{s}} \; \; \; \left(4.585 \cdot 10^{7} \; \text{years} \right)[/math] [math]\;[/math] If every planet around every second generation star and every third generation star were illuminated with a luminous isotropic cosmic photon background radiation source that is more cleaner and isotropic and more stable than the host star radiation source, could this cosmic photon background radiation have initiated a universe abiogenesis epoch? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: WMAP satellite content of the Universe: (ref. 1) http://map.gsfc.nasa.gov/media/080998/index.html Wikipedia - Sun Sol: (ref. 2) https://en.wikipedia.org/wiki/Sun Wikipedia - Earth: (ref. 3) https://en.wikipedia.org/wiki/Earth UVIC - The blackbody temperature of a planet: (ref. 4) http://www.astro.uvic.ca/~venn/A201/maths.7.planet_temperature.pdf Wikipedia - Cosmic microwave background radiation: (ref. 5) https://en.wikipedia.org/wiki/Cosmic_microwave_background Importance_of_precise_measurement Science Forums - Orion1 - Photon decoupling time: (ref. 6) https://www.scienceforums.net/topic/86694-observable-universe-mass/?do=findComment&comment=1037262
  14. Solar surface temperature: (ref. 1) [math]T_{\odot} = 5772 \; \text{K}[/math] [math]\;[/math] Total solar radius: (ref. 1) [math]R_{\odot} = 6.963 \cdot 10^{5} \; \text{km}[/math] [math]\;[/math] Earth orbital radius semi-major axis: (ref. 2) [math]r_{\oplus} = 1.496 \cdot 10^{8} \; \text{km}[/math] [math]\;[/math] Earth albedo reflectivity: (ref. 2) [math]\alpha_{\oplus} = 0.367[/math] [math]\;[/math] Earth surface temperature at present time: (ref. 3) [math]T_{\oplus} = T_{\odot} \left(\frac{\left(1 - \alpha_{\oplus} \right)^{\frac{1}{2}} R_{\odot}}{2 r_{\odot}} \right)^{\frac{1}{2}} = 248.367 \; \text{K}[/math] [math]\boxed{T_{\oplus} = 248.367 \; \text{K}}[/math] [math]\;[/math] Cosmic photon background radiation temperature at present time: (ref. 4) [math]T_{\gamma,0} = 2.72548 \; \text{K}[/math] [math]\;[/math] Cosmology scale factor: [math]\boxed{\frac{a\left(t_0 \right)}{a\left(t \right)} = \frac{T_{\gamma,t}}{T_{\gamma,0}}}[/math] [math]\;[/math] Cosmology scale factor at present time: [math]\boxed{a\left(t_0 \right) = 1}[/math] [math]\;[/math] Cosmic photon background radiation temperature at past time is equivalent to Earth surface temperature at present time: [math]\boxed{T_{\gamma,t} = T_{\oplus}}[/math] [math]\;[/math] Universe abiogenesis epoch scale factor at past time: [math]a\left(t \right) = \frac{T_{\gamma,0}}{T_{\gamma,t}} = 0.011[/math] [math]\boxed{a\left(t \right) = \frac{T_{\gamma,0}}{T_{\gamma,t}}}[/math] [math]\boxed{a\left(t \right) = 0.011}[/math] [math]\;[/math] If every planet around every second generation star and every third generation star were illuminated with a luminous cosmic photon background radiation source that is cleaner and more stable than the host star radiation source, could this cosmic photon background radiation have initiated a universe abiogenesis epoch? [math]\;[/math] According to your universe model calculator, what is the universe age at this scale factor? [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Wikipedia - Sun Sol: (ref. 1) https://en.wikipedia.org/wiki/Sun Wikipedia - Earth: (ref. 2) https://en.wikipedia.org/wiki/Earth UVIC - The blackbody temperature of a planet: (ref. 3) http://www.astro.uvic.ca/~venn/A201/maths.7.planet_temperature.pdf Wikipedia - Cosmic microwave background radiation: (ref. 4) https://en.wikipedia.org/wiki/Cosmic_microwave_background#Importance_of_precise_measurement
  15. Planck satellite baryonic cosmological composition parameter: (ref. 1, pg. 11, ref. 2, pg. 3) [math]\Omega_{b} = 0.0495[/math] [math]\;[/math] Black holes cosmological composition parameter: {ref. 2, pg. 3) [math]\Omega_{bh} = 0.00007[/math] [math]\;[/math] Solar mass: (ref. 3) [math]M_{\odot} = 1.9885 \cdot 10^{30} \; \text{kg}[/math] [math]\;[/math] Milky Way galaxy mass: (ref. 4, pg. 1) [math]M_{mw} = 1.260 \cdot 10^{12} \cdot M_{\odot} = 2.506 \cdot 10^{42} \; \text{kg}[/math] [math]\boxed{M_{mw} = 2.506 \cdot 10^{42} \; \text{kg}}[/math] [math]\;[/math] PSR J2215+5135 pulsar Tolman-Oppenheimer-Volkoff observational lower mass limit: (ref. 5) [math]\boxed{M_{bh} \geq 2.27 \cdot M_{\odot}}[/math] [math]\boxed{M_{bh} \geq 4.514 \cdot 10^{30} \; \text{kg}}[/math] [math]\;[/math] Stellar class O upper mass limit: (ref. 6) [math]\boxed{M_{bh} \geq 16 \cdot M_{\odot}}[/math] [math]\boxed{M_{bh} \geq 3.182 \cdot 10^{31} \; \text{kg}}[/math] [math]\;[/math] Toy model black holes per galaxy average number: [math]\frac{N_{bh}}{N_g} = \frac{\Omega_{bh} M_{mw}}{\Omega_b M_{bh}} = \left(1.114 \cdot 10^{8} \rightarrow 7.851 \cdot 10^{8} \right) \; \frac{\text{black holes}}{\text{galaxy}}[/math] [math]\boxed{\frac{N_{bh}}{N_g} = \frac{\Omega_{bh} M_{mw}}{\Omega_b M_{bh}}}[/math] [math]\boxed{\frac{N_{bh}}{N_g} = \left(1.114 \cdot 10^{8} \rightarrow 7.851 \cdot 10^{8} \right) \; \frac{\text{black holes}}{\text{galaxy}}}[/math] [math]\;[/math] Synthetic catalog black holes per galaxy average number: (ref. 7, pg. 1) [math]\frac{N_{bh}}{N_g} = 1.693 \cdot 10^{8} \; \frac{\text{black holes}}{\text{galaxy}}[/math] [math]\;[/math] Any discussions and/or peer reviews about this specific topic thread? [math]\;[/math] Reference: Planck 2013 results. XVI. Cosmological parameters: (ref. 1) http://planck.caltech.edu/pub/2013results/Planck_2013_results_16.pdf The Cosmic Energy Inventory: (ref. 2) http://arxiv.org/pdf/astro-ph/0406095v2.pdf Wikipedia - Sun Sol: (ref. 3) https://en.wikipedia.org/wiki/Sun Mass models of the Milky Way: (ref. 4) http://arxiv.org/pdf/1102.4340v1 Wikipedia - Tolman-Oppenheimer-Volkoff limit: (ref. 5) https://en.wikipedia.org/wiki/Tolman-Oppenheimer-Volkoff_limit Wikipedia - Stellar classification - Harvard spectral classification: (ref. 6) https://en.wikipedia.org/wiki/Stellar_classification#Harvard_spectral_classification Synthetic catalog of black holes in the Milky Way: (ref. 7) https://arxiv.org/pdf/1908.08775.pdf
×
×
  • 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.