# Cosmological Dynamics vs Kinematics ## The question Do cosmological dynamics (GM, field equations, spacetime curvature) contain information that is independent from kinematics (orbital period, semi-major axis, eccentricity)? ## Null hypothesis **H₁:** Cosmological dynamics use variables (GM) that are independent from kinematics. Therefore dynamic solutions will diverge from kinematic solutions. **H₀:** Cosmological dynamics variables (GM) are not independent from kinematics. Therefore kinematic solutions will not diverge from dynamic solutions. ## What GM actually is Newton and Einstein both use GM in their equations. The standard interpretation is that G is a fundamental constant and M is a physical mass, making GM an independent dynamical quantity. Kepler's third law (1619): $GM = \frac{4\pi^2 a^3}{T^2}$ This is not a derived relationship. It is the definition. GM is computed from the semi-major axis $a$ and period $T$. There is no independent measurement of GM that does not trace back to an orbit. Cavendish (1798) measured $G$ in a shed with a torsion balance 179 years after Kepler, allowing $M$ to be extracted from the ratio. He did not discover a new physical quantity. He calibrated a unit conversion factor to split a kinematic ratio into two parts. Equivalently: $GM = v^2 a$ where $v = 2\pi a / T$. GM is velocity squared times distance. It is kinematics with a label. ## The precession test The apsidal precession of an orbit has two forms. **Gerber's Kinematic Equation (1898)** $c^2 = \frac{6\pi\mu}{a(1-e^2)\psi}$ where $\mu = 4\pi^2 a^3 / \tau^2$ (Kepler's ratio). Gerber solved for $c$ as the unknown. No $G$, no $M$. **Einstein's Eq. 13 (1915)** $\psi = \frac{6\pi \, GM}{a \, c^2 (1 - e^2)}$ Appears to depend on mass through GM. **These are the same equation.** Einstein's GM is Gerber's $\mu$ is Kepler's $4\pi^2 a^3/T^2$. Substituting: $\psi = \frac{24\pi^3 a^2}{T^2 c^2(1-e^2)} = \frac{6\pi}{1-e^2}\left(\frac{v}{c}\right)^2$ Every dynamic variable cancels. What remains is $a$, $T$, $e$, and $c$. ## The ε₀μ₀ form Since $c = 1/\sqrt{\varepsilon_0 \mu_0}$, squaring gives $1/c^2 = \varepsilon_0 \mu_0$. Substituting: $\psi = \frac{24\pi^3 a^2 \varepsilon_0 \mu_0}{T^2(1-e^2)}$ The precession is a function of orbital geometry and the physical properties of the vacuum medium. No $c$, no $G$, no $M$, no field equations. ## Test results ### All planets, Moon, and S2 | Body | System | e | $\psi$ v/c ("/100yr) | $\psi$ ε₀μ₀ ("/100yr) | Match | |------|--------|---|----------------------|------------------------|-------| | Mercury | Sun | 0.2056 | 42.98 | 42.98 | ✓ | | Venus | Sun | 0.0068 | 8.62 | 8.62 | ✓ | | Earth | Sun | 0.0167 | 3.84 | 3.84 | ✓ | | Mars | Sun | 0.0934 | 1.35 | 1.35 | ✓ | | Jupiter | Sun | 0.0484 | 0.06 | 0.06 | ✓ | | Saturn | Sun | 0.0542 | 0.01 | 0.01 | ✓ | | Uranus | Sun | 0.0472 | ~0 | ~0 | ✓ | | Neptune | Sun | 0.0086 | ~0 | ~0 | ✓ | | Moon | Earth | 0.0549 | 0.06 | 0.06 | ✓ | | S2 | Sgr A* | 0.88466 | 4465 | 4465 | ✓ | Zero divergence. Three systems, ten bodies, eccentricities from 0.0068 to 0.8847. ### GM independence test GM was doubled and the Kepler constraint enforced three ways: | Test | $\psi_k$ | $\psi_d$ | Match | |------|----------|----------|-------| | OG Mercury | 42.98 | 42.98 | ✓ | | 2xGM (solve a) | 68.23 | 68.23 | ✓ | | 2xGM (solve T) | 121.57 | 121.57 | ✓ | | 2xGM (1.5xa) | 44.12 | 44.12 | ✓ | Four tests, three Kepler partitions, zero divergence. Doubling GM changes the precession only because it changes $a$ or $T$, which changes $v$. The "mass" did nothing. ### Derived quantities (all kinematic) Everything GR claims as its predictions, derived from $a$, $T$, $e$, and $\varepsilon_0 \mu_0$: | Quantity | Mercury | S2 (Sgr A*) | Mainstream | Source | |----------|---------|-------------|------------|--------| | Precession | 42.98 "/100yr | 12.01 '/orbit | 42.98, ~12' | Einstein 1915, GRAVITY 2020 | | Central mass | 1.00 M☉ | 4.15×10⁶ M☉ | 1 M☉, 4.154×10⁶ | GRAVITY 2019 | | Schwarzschild radius | 2,953 m | 12.3×10⁶ m | ~3 km | Wikipedia | | Light deflection (r_sun) | 1.75" | N/A | 1.75" | Eddington 1919 | | Grav. redshift (peri) | 9.62 m/s | 104 km/s | ~200 combined | GRAVITY 2020 | ## Orbital decay (gravitational wave emission) The Hulse-Taylor binary pulsar (PSR B1913+16) is cited as the strongest evidence for gravitational waves. | Year | Event | |------|-------| | 1963 | Peters & Mathews derive the quadrupole radiation formula | | 1974 | Hulse & Taylor discover the binary pulsar PSR B1913+16 | | 1975/76 | Wagoner predicts the orbital decay rate using Peters' formula | | 1979 | Taylor et al. first detect the orbital decay from timing data | | 1982, 1989 | Taylor & Weisberg publish increasingly precise measurements | | 1993 | Nobel Prize awarded to Hulse and Taylor | | 2010 | Weisberg, Nice & Taylor publish 30+ year dataset: observed/predicted = 0.997 ± 0.002 | The quadrupole formula gives the radiated power in terms of $G^4 m_1^2 m_2^2(m_1+m_2)/c^5$. This appears dynamic. But the observed quantity is not power. It is $\dot{T}$, the period derivative, which is dimensionless (seconds per second). Dimensionless observables do not need $G$ or $M$. Substituting $GM = 4\pi^2 a^3/T^2$, $v = 2\pi a/T$, and connecting power to period change via orbital energy: $\dot{T} = -\frac{192\pi}{5} \cdot \eta \cdot \frac{(v/c)^5}{(1-e^2)^{7/2}} \cdot f(e)$ where $f(e) = 1 + \frac{73}{24}e^2 + \frac{37}{96}e^4$ and $\eta = m_1m_2/(m_1+m_2)^2$ is the symmetric mass ratio. No $G$. No $M$. Same pattern as precession. Precession is $(v/c)^2$. Orbital decay is $(v/c)^5$. Both kinematic. The mass ratio $\eta$ for a double pulsar is $a_1 a_2/(a_1+a_2)^2$, a ratio of semi-major axes, which is kinematic. In the medium form: $\dot{T} = -\frac{192\pi}{5} \cdot \eta \cdot \frac{v^5(\varepsilon_0\mu_0)^{5/2}}{(1-e^2)^{7/2}} \cdot f(e)$ | Quantity | Formula | PSR B1913+16 | Observed | |----------|---------|-------------|----------| | $\dot{T}$ | $-(192\pi/5)\eta(v/c)^5 f(e)/(1-e^2)^{7/2}$ | $-2.4 \times 10^{-12}$ | $-(2.423 \pm 0.001) \times 10^{-12}$ | The "strongest evidence for gravitational waves" reduces to the fifth power of the orbital velocity relative to the medium's propagation rate, times the mass ratio and a geometric eccentricity factor. The dynamic variables canceled. $\dot{T}$ does not diverge from kinematics. Full derivation: [[1963_Peters_Mathews_Gravitational_Radiation]] ## Comets and non-gravitational forces It is claimed that adding "dynamical corrections" (non-gravitational forces from outgassing) to comet orbit calculations improves perihelion predictions, proving that dynamics beyond kinematics are required. Analysis of the standard model (Marsden, Sekanina & Yeomans 1973) reveals that the non-gravitational parameters A1, A2, A3 are fitted FROM the same orbital observations they claim to correct. The g(r) function was "selected essentially at random" (Marsden et al. 1973, p.211). Eight different functional forms with different assumed physics produce comparable fits (Krolikowska 2004). The parameters change sign over time. "Dark comets" exhibit non-gravitational acceleration without any visible outgassing, contradicting the claimed physical mechanism. Full analysis: [[Comets]] ## Companion: Artemis II trajectory The Artemis II cislunar trajectory (launched 1 April 2026) was designed using NASA's Copernicus software with GRACE/GRAIL gravity models and DDEABM numerical integration. Every input (GM values, spherical harmonics) is derived from satellite orbit tracking — kinematics. Every optimization variable is a position, velocity, angle, or time. The equation of motion $\ddot{r} = -GM\mathbf{r}/r^3$ is kinematic ($m$ cancelled). The operational burn plan uplinked to Orion contains zero masses, forces, or field strengths. Full analysis: [[Artemis_II_Trajectory]] Source notes: [[2020_Batcha_Artemis_I_Trajectory_Design]], [[2023_Eckman_Artemis_I_Trajectory_Operations]] ## Companion: Pluto Pluto's discovery (1930), presented as a second triumph of dynamical prediction after Neptune, fails even more completely. The perturbations that motivated the search were not real: they were artifacts of a 0.5% error in Neptune's mass, which vanished after Voyager 2 (Standish 1993). Pluto's actual mass (0.002 Earth masses) is 3,300 times too small to cause any measurable perturbation. Lowell himself admitted only "a general direction alone is predicable." Brown (Yale) and Russell (Princeton) called it accidental within months. The JHU New Horizons team calls it "coincidence and great dedication, rather than mathematical precision." Full analysis: [[Pluto]] ## Companion: Neptune The discovery of Neptune (1846), presented as the greatest triumph of gravitational dynamics, does not survive scrutiny. The predicted orbital elements (distance 20% off, eccentricity 12x off, mass 2x off, period 32% off) were all wrong. Le Verrier himself rejected Neptune as his predicted planet. The positional accuracy came from a kinematic direction constraint and an empirical spacing law (Bode's Law), not from the dynamics. The historical record was fabricated and physically stolen for 150 years. Full analysis: [[Neptune]] ## Companion: the N-body failure This note addresses the two-body case, where dynamics reduce to kinematics (all dynamic variables cancel). A companion note addresses the N-body case, where the dynamic framework fails to produce a solution at all: [[Three_Body_Null]] For two bodies, dynamics are equivalent to kinematics. For three or more bodies, dynamics are insufficient. In neither case do the dynamic variables carry independent information. ## Conclusion The dynamic variables used canceled out and were shown to be ratios of the kinematic parameters themselves, i.e. not independent from kinematics. Not a single kinematic solution diverged from the dynamic. This applies equally to Newton and Einstein. Newton's GM is the same $4\pi^2 a^3/T^2$. The quadrupole formula for gravitational wave emission reduces to $(v/c)^5$ when expressed as the observable $\dot{T}$. The non-gravitational force corrections to comet orbits are empirical curve fitting parameters extracted from the observations, not independent dynamics ([[Comets]]). Therefore H₀ was not refuted. H₁ cannot be true. ## Calculator [Open in Google Sheets](https://docs.google.com/spreadsheets/d/1XXwcbWtMnIpDmL8BZHYLI5Jle3kjyCJxrz9XpBw5eJA/edit?gid=0#gid=0) Full documentation: [[precession_sheet]] ## Sources GRAVITY Collaboration (2020). "Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole." *Astronomy & Astrophysics*, 636, L5. arXiv: [2004.07187](https://arxiv.org/abs/2004.07187) Gerber, P. (1898). "Die räumliche und zeitliche Ausbreitung der Gravitation." *Zeitschrift für Mathematik und Physik*, 43, 93-104. [[1898_Spatial_and_Temporal_Propagation_of_Gravity]] Einstein, A. (1915). "Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie." *Sitzungsberichte der Preussischen Akademie der Wissenschaften*, 831-839. [[1915_Perihelion_Motion_of_Mercury]] Peters, P. C. and Mathews, J. (1963). "Gravitational Radiation from Point Masses in a Keplerian Orbit." *Physical Review*, 131(1), 435-440. [[1963_Peters_Mathews_Gravitational_Radiation]] Marsden, B. G., Sekanina, Z., and Yeomans, D. K. (1973). "Comets and nongravitational forces. V." *The Astronomical Journal*, 78(2), 211-225. [[1973_Marsden_Sekanina_Yeomans_Nongravitational_V]] Sekanina, Z. (2021). "The End of an Era in Cometary Astronomy: The Deceleration of Comet Encke." arXiv:2109.14829. [[2021_Sekanina_Deceleration_Comet_Encke]] Krolikowska, M. (2004). "Long-period comets with non-gravitational effects." *Astronomy & Astrophysics*, 427, 1117-1126. [[2004_Krolikowska_Long_Period_Comets_NGF]] --- # Appendix A: Kinematic derivations of GR predictions Every quantity that General Relativity claims as a prediction of spacetime curvature can be derived from the same kinematic inputs used for precession: $a$, $T$, $e$, and the medium properties $\varepsilon_0 \mu_0$ (or equivalently $c$). No field equations are required at any step. ## Schwarzschild radius GR defines the Schwarzschild radius (the so-called event horizon) as: $R_s = \frac{2GM}{c^2}$ Since $GM = 4\pi^2 a^3/T^2$: $R_s = \frac{8\pi^2 a^3}{T^2 c^2}$ Or using the medium properties: $R_s = 8\pi^2 a^3 \varepsilon_0 \mu_0 / T^2$ No mass. The "event horizon" is a ratio of orbital geometry to the propagation properties of the medium. For the Sun: $R_s = 2{,}953$ m $\approx 3$ km (matches mainstream). For Sgr A* via S2: $R_s = 12{,}262{,}166$ m (matches EHT constraints). ## Light deflection GR predicts that light passing a massive body at distance $r$ is deflected by: $\theta = \frac{4GM}{rc^2}$ Substituting $GM = 4\pi^2 a^3/T^2$ and $1/c^2 = \varepsilon_0 \mu_0$: $\theta = \frac{16\pi^2 a^3 \varepsilon_0 \mu_0}{T^2 r}$ For starlight grazing the Sun's limb ($r = R_\odot = 695{,}700{,}000$ m), using any Sun planet's orbital parameters: $\theta = 1.75''$ This is Einstein's 1919 eclipse prediction, derived without GR. The deflection is a ratio of orbital geometry, medium properties, and the distance at which light passes. ## Gravitational redshift GR predicts that light escaping a gravitational well is redshifted by: $z = \frac{GM}{rc^2}$ For an orbiting body at pericentre ($r_{peri} = a(1-e)$), substituting $GM = 4\pi^2 a^3/T^2$: $z = \frac{4\pi^2 a^2}{T^2 c^2 (1-e)}$ The equivalent velocity shift is $\Delta v = z \times c$. For S2 at pericentre: $z = 3.47 \times 10^{-4}$, giving $\Delta v = 104$ km/s. GRAVITY measured $\$200 \space km/$s total (gravitational redshift + transverse Doppler combined). ## Pericentre velocity The velocity at closest approach for an elliptical orbit: $v_{peri} = \frac{2\pi a}{T} \sqrt{\frac{1+e}{1-e}}$ Pure kinematics. For S2: $v_{peri} = 7{,}671$ km/s. GRAVITY observed $\approx 7{,}700$ km/s. ## Summary | Quantity | Kinematic formula | Mercury | S2 | Mainstream | |----------|-------------------|---------|-----|-----------| | $R_s$ | $8\pi^2 a^3/(T^2 c^2)$ | 2,953 m | 12.3M m | 3 km, ~12M m | | $\theta$ at $R_\odot$ | $16\pi^2 a^3 \varepsilon_0 \mu_0/(T^2 R_\odot)$ | 1.75" | N/A | 1.75" | | $z$ at peri | $4\pi^2 a^2/(T^2 c^2(1-e))$ | ~0 | 3.47E-4 | part of ~200 km/s | | $v_{peri}$ | $v\sqrt{(1+e)/(1-e)}$ | 58,975 m/s | 7,671 km/s | ~7,700 km/s | The pattern is the same in every case. GR writes a formula with GM. GM is $4\pi^2 a^3/T^2$. The formula reduces to kinematics and medium properties. The dynamic variables contribute nothing that survives. --- # Appendix B: Responses to challenges ## @earthlysceptick: S2 precession kills H₀ ![[earthlysceptick_s2_challenge.png]] > "The elliptical orbit of S2 around Sgr A* precesses by 12 arcminutes per orbit. If S2's motion were described by Keplerian kinematics, S2 would trace the same ellipse during each orbit without apsidal precession. But the orbit has been observed to precess in a rosette pattern. That means your H0 is dead. Try again." S2 orbital parameters (GRAVITY 2020, Gillessen 2017): | Parameter | Value | |-----------|-------| | a | 1025.5 AU = 1.534 × 10¹⁴ m | | T | 16.0518 yr = 5.066 × 10⁸ s | | e | 0.88466 | | v | 1903 km/s | | v/c | 6.348 × 10⁻³ | Gerber's Kinematic Equation applied to S2: $\psi = \frac{6\pi}{1 - 0.88466^2} \times (6.348 \times 10^{-3})^2 = 3.494 \times 10^{-3}\;\text{rad/orbit}$ $\psi = 12.01\;\text{arcmin/orbit}$ GRAVITY measured $\delta\phi \approx 12'$ per orbit with $f_{SP} = 1.10 \pm 0.19$. Your position is a conflation of "Keplerian kinematics" with "closed ellipse."The precession formula uses Kepler's variables. The orbit precesses because $(v/c)^2 \neq 0$, not because spacetime is curved. No G. No M. No field equations. Three measured quantities ($a$, $T$, $e$) and the propagation speed of the medium ($c$, or equivalently $\varepsilon_0 \mu_0$). The precession of S2 does not falsify H₀. --- ## @haxx_maxx: "This is a joke right?" ![[haxx_maxx_joke.png]] > "Do you actual think you are doing science? This is a joke right?" This is an appeal to ridicule. No counter-argument, no math, no identification of an error. The response assumes that the claim must be wrong because it challenges an established framework, without engaging with the content. What an unfortunate position to publicly hold. What is being presented here is not "science" in the sense of proposing a new theory. It is algebra and kinematics. The substitution $GM = 4\pi^2 a^3/T^2$ is Kepler's third law. It is not disputed by any physicist. The reduction of Einstein's Eq. 13 to Gerber's Kinematic Equation is not disputed either. These are algebraic identities. They are objectively true independent of what shape you believe the Earth is, what cosmological model you prefer, or whether you think GR is correct. The question being asked is narrow and specific: does GM carry information that is independent from the orbital kinematics? The answer, demonstrated across ten bodies in three stellar systems, is no. That is not a joke. It is a falsification test with a clear result. If the result is wrong, show which row in the calculator diverges. Name the body where $\psi_k \neq \psi_d$. Produce a single case where the dynamic form gives a value the kinematic form cannot reproduce. That would falsify H₀. Laughing emojis do not. --- ## @ImNux: Satellites prove orbits, therefore H₀ is irrelevant ![[imnux_satellites.png]] > "I can refute is a irrelevant bs that doesnt matter. Civilians verifying satellites are in space orbiting via physical doppler measurements, personal accurate future orbit calcs, etc. Proves globe/space, proves yall are confused about the irrelevant bs like this, its meaningless." This is a non sequitur. H₀ does not claim satellites don't exist, don't orbit, or can't be tracked. H₀ asks whether GM is independent from kinematics. Nux answered a question that was not asked. Look at his own evidence. His TLE calculation derives orbital velocity from Kepler's third law: $a = \left(\frac{\mu T^2}{4\pi^2}\right)^{1/3} \quad \Rightarrow \quad v = \sqrt{\mu/a}$ where $\mu$ = 398,600 km³/s². That $\mu$ is $GM_{Earth}$. And where does it come from? From satellite orbits. $\mu = 4\pi^2 a^3/T^2$. It is the kinematic ratio. The Doppler measurement then confirms the velocity that was derived from the same ratio. The orbital parameters are using the kinematic identity $GM = 4\pi^2 a^3/T^2$ to predict velocity, confirming it with Doppler, and presenting this as evidence against the claim that $GM = 4\pi^2 a^3/T^2$. The evidence presented is the hypothesis. The Doppler shift is also kinematic. It measures $\Delta f/f = v_r/c$, which gives radial velocity. Velocity is kinematics. The entire chain from TLE to predicted orbit to Doppler confirmation is kinematics verifying kinematics. At no point does an independent dynamical mass enter the calculation. Satellite tracking works. Orbital predictions work. Doppler measurements match. I don't dispute any of that. Irrespective of how we conceptualize satellites work or what you think the shape of Earth is, this null hypothesis is independent of that. All of it is built on the same Keplerian ratios that $H_0$ is about. Your evidence refutes $H_1$ instead of $H_0$. --- ## @JefferyParkins2: "lights in the sky to show the earth is flat" ![[jefferyparkins2_spam.png]] > "So Alan is now using lights in the sky to show the earth is flat. It's a cool self debunk." The null hypothesis does not mention the shape of the Earth. It asks whether $GM = 4\pi^2 a^3/T^2$, which is Kepler's third law, an equation that appears in every undergraduate astronomy textbook regardless of cosmological model. The substitution of $GM$ into Einstein's Eq. 13 is taught in standard GR courses. The algebraic reduction to $6\pi(v/c)^2/(1-e^2)$ is verified by every physics department that assigns the problem. None of this is in dispute within mainstream physics. The only claim being made is that the dynamic variables cancel and do not contribute independent information. That claim is testable and has been tested across ten bodies in three systems. Straw man arguments do not constitute a refutation. Twitter flagged this as probable spam. That assessment appears correct. --- ## Gary A Brown / Matt Butler: Wayne (2015) paper ![[gary_brown_wayne_paper.png]] > Gary A Brown: "This paper helps to lay out the detailed analysis for perihelion motion, specifically as it relates to Mercury, as it pertains to both Newtonian mechanics and Einsteinium relativity... I believe you'll find your answers here..." > > Matt Butler: "Really thought you had something there, huh?" I read the paper. It supports H₀ rather than falsifying it. Wayne (2015), a Cornell professor, rewrites Einstein's GR correction as a velocity dependent perturbation to Newton's gravitational potential. His Eqs. 24 and 25 express the precession in terms of gravitational potential energy (PE) and kinetic energy (KE): $\frac{PE \cdot v_\theta^2}{KE_\theta \cdot c^2} = \frac{2GM}{ac^2} \qquad\qquad \psi = \frac{3\pi v_\theta^2}{(PE/KE_\theta) \cdot c^2(1-e^2)}$ Wayne links the precession to an energy caused by an unknown mechanism, then proposes adding a velocity correction to the closed Newtonian formula $F_g = GMm/r^2$: $F_g = \frac{GMm}{r^2}\left(1 + \frac{v_\theta^2}{c^2}\right)$ $F_g = GMm/r^2$ is a closed formula. Without the $(1 + v_\theta^2/c^2)$ term, both masses cancel and you get a standard closed Keplerian ellipse with zero precession. The equation cannot solve the precession without injecting energy into the system. That is the very problem GR was invoked to solve. Wayne admits he cannot identify the source: "I cannot identify the tangential velocity-dependent force with certainty" (p.189). Adding an unidentified force to a closed formula violates the conservation laws the formula was built on. The solution he arrives at does not diverge from the kinematic one. ### PE and KE are kinematic ![[precession_calculator_v3_PE_KE.png]] The PE and KE that Wayne frames as dynamic quantities are derived entirely from the orbital parameters: $KE/m = \frac{1}{2}v^2 = \frac{1}{2}\left(\frac{2\pi a}{T}\right)^2 = \frac{2\pi^2 a^2}{T^2}$ $PE/m = -\frac{GM}{r} = -\frac{4\pi^2 a^3}{T^2 r} = -\frac{4\pi^2 a^2}{T^2} = -v^2 \quad\text{(at average distance } r = a\text{)}$ $PE/KE = \frac{-v^2}{\frac{1}{2}v^2} = -2$ The ratio is $-2$ for every Keplerian orbit. This is the virial theorem. It does not depend on mass, distance, or system. Mercury, S2, Moon, anything. PE and KE are identities of the kinematic components of the orbital parameters, based solely on empirical observation and ratios of $\pi$. ### Wayne's result is Gerber's equation Wayne's Eq. 23: $\psi = \frac{24\pi^3 a^2}{P^2 c^2(1-e^2)}$ Gerber's Kinematic Equation (1898): $c^2 = \frac{6\pi\mu}{a(1-e^2)\psi}$ where $\mu = 4\pi^2 a^3/\tau^2$. Rearranging for $\psi$: $\psi = \frac{6\pi}{1-e^2}\left(\frac{v}{c}\right)^2$ Same equation. Gerber derived it 117 years earlier from a retarded potential propagating through a physical medium at speed $c$, where the $v^2/c^2$ correction arises naturally from the finite propagation time. ### The medium form Substituting $1/c^2 = \varepsilon_0\mu_0$: $\psi = \frac{24\pi^3 a^2 \varepsilon_0 \mu_0}{T^2(1-e^2)}$ No $c$, no $G$, no $M$, no $m$. If Wayne wants to claim friction as the mechanism for his velocity correction, he needs a physical medium for that friction to act through. The medium is described by $\varepsilon_0$ and $\mu_0$. It was there all along. Wayne's conclusion: "the relativity of space and time is sufficient but not necessary." His own paper confirms that no dynamic variable survived the derivation. H₀ stands. Full analysis: [[2015_Wayne_Perihelion_Velocity_Correction]] --- ## @JosefChose-e6c: PSR B1913+16 orbital decay ![[josefchose_psr_challenge.png]] > "Alan, before commenting on topics you don't fully understand, it would be wise to first engage with the actual data. How do you explain the observed decay of the orbital period in PSR B1913+16, which quantitatively matches the predictions of general relativity for gravitational wave emission?" I engaged with the actual data. Here is the timeline: | Year | Event | |------|-------| | 1963 | Peters & Mathews derive the quadrupole radiation formula | | 1974 | Hulse & Taylor discover binary pulsar PSR B1913+16 | | 1976 | Wagoner predicts orbital decay rate using Peters' formula | | 1979 | Taylor et al. first detect orbital decay from timing data | | 2010 | Weisberg, Nice & Taylor publish 30 year dataset: observed/predicted = 0.997 ± 0.002 | Peters' quadrupole formula (1963, Eq. 16) gives the radiated power as: $\langle P \rangle = \frac{32 G^4 m_1^2 m_2^2 (m_1+m_2)}{5 c^5 a^5 (1-e^2)^{7/2}} f(e)$ This appears to depend on $G$, $m_1$, $m_2$, and $c$. But the observable is not power. The observable is $\dot{T}$, the rate of change of orbital period, which is dimensionless (seconds per second). Connecting power to period change via orbital energy and substituting $GM = 4\pi^2 a^3/T^2$ and $v = 2\pi a/T$, every dynamic variable cancels: $\dot{T} = -\frac{192\pi}{5} \cdot \eta \cdot \frac{(v/c)^5}{(1-e^2)^{7/2}} \cdot f(e)$ where $f(e) = 1 + \frac{73}{24}e^2 + \frac{37}{96}e^4$ and $\eta = m_1m_2/(m_1+m_2)^2$. No $G$. No $M$. The coefficient $192\pi/5 = (32/5) \times 3 \times 2\pi$, all geometric (quadrupole orbital average, spherical radiation integral, Kepler energy-period conversion, velocity definition). The mass ratio $\eta$ for a double pulsar is $a_1 a_2/(a_1+a_2)^2$, a ratio of semi-major axes measured from Doppler. For neutron star binaries, $\eta \approx 0.25$ (equal mass) is a nuclear physics constraint, not a GR derivation. Hulse-Taylor has $\eta = 0.2499$. Using PSR B1913+16 orbital parameters ($a = 1.95 \times 10^9$ m, $T = 27906.98$ s, $e = 0.6171$, $q = 1.039$): | Form | $\dot{T}$ calculated | Observed | |------|----------------------|----------| | $(v/c)^5$ | $-2.41 \times 10^{-12}$ | $-(2.423 \pm 0.001) \times 10^{-12}$ | | $v^5(\varepsilon_0\mu_0)^{5/2}$ | $-2.41 \times 10^{-12}$ | same | The "strongest evidence for gravitational waves" is the fifth power of the orbital velocity relative to the medium's propagation rate, times the mass ratio and a geometric eccentricity factor. No field equations were required to produce this number. It was computed on a spreadsheet using the same kinematic inputs ($a$, $T$, $e$) and the same medium properties ($\varepsilon_0$, $\mu_0$) that reproduce every other GR prediction in this document. The precession of this same system is also computed kinematically: $\psi = 6\pi(v/c)^2/(1-e^2) = 4.23°$/yr, matching the observed 4.2266°/yr. Precession is $(v/c)^2$. Orbital decay is $(v/c)^5$. Same inputs. Same spreadsheet. Same kinematics. The dynamic variables used canceled out. Not a single kinematic solution diverged from the dynamic. H₀ was not refuted. Full derivation: [[1963_Peters_Mathews_Gravitational_Radiation]] Observational data: [[2010_Weisberg_Taylor_PSR_B1913+16]] ### Follow-up: "derive the decay rate and identify the mechanism" ![[josefchose_psr_followup.png]] ![[josefchose_psr_reply.png]] > "Your sheet does not calculate a time evolution of the orbital period, does not derive a decay rate, and does not identify any physical mechanism that would carry energy away from the system." $-2.41 \times 10^{-12}$ s/s IS the decay rate. That is the rate at which the orbital period decreases per second. Multiply by $3.156 \times 10^7$ and you get $-76$ $\mu$s/yr, which is the annual period decrease that accumulates into the parabolic curve in Weisberg's Fig. 2 ([[2010_Weisberg_Taylor_PSR_B1913+16]]). The sheet derives this number. It matches observation. > "General relativity makes a quantitative prediction based on its masses, eccentricity, and orbital period." The sheet uses eccentricity and orbital period. The "masses" in the GR formula are $GM_1$ and $GM_2$. $GM = 4\pi^2a^3/T^2$. That is $a$ and $T$. The masses are orbital parameters wearing a unit conversion. The prediction matches to 0.2% on the sheet using the same inputs minus the notation. > "You need to identify the physical mechanism responsible for the energy loss." GR's mechanism is gravitational wave energy calculated via the pseudotensor. Peters & Mathews (1963) cite Infeld & Plebanski (1960) for whether this energy "has any physical meaning" and then bypass the question ([[1960_Infeld_Plebanski_Motion_and_Relativity]]). Infeld (Einstein's own collaborator) showed: - Gravitational radiation can be annihilated by coordinate choice (p.166) - "Its existence or non-existence will depend upon the choice of arbitrary harmonic functions" (p.201) - Einstein himself, quoted by Infeld: "We do not have any satisfactory classical theory of radiation" (p.201) The pseudotensor is not invariant under coordinate transformation. A quantity that can be zeroed by changing coordinates is not physical. Neither side has a confirmed mechanism. The difference: the kinematic derivation does not claim one. GR claims one that its own foundational sources declare coordinate dependent, mathematically inadmissible, and unsatisfactory. The orbital decay is $(v/c)^5 \times$ geometry. The only physical parameters that make up the value of $c$ are the vacuum properties $\varepsilon_0$ and $\mu_0$. The equations are ratios of velocity with respect to a medium described by those properties: $v^5(\varepsilon_0\mu_0)^{5/2}$. The null hypothesis asks one question: can GM produce a prediction that diverges from $4\pi^2a^3/T^2$? If it can, H₁ is true and dynamics are independent from kinematics. If it cannot, H₀ stands. Across precession, orbital decay, deflection, redshift, and Shapiro delay, GM has never produced a value that $4\pi^2a^3/T^2$ could not reproduce. Not once. Not for Mercury, not for S2, not for PSR B1913+16. The field equations, the Schwarzschild metric, and the full apparatus of absolute differential calculus were mobilized to arrive at numbers that Kepler's ratio and the medium properties already contain. Until GM makes a prediction that diverges from kinematics, H₀ is not falsified. H₁ cannot be true. --- ## See Also - [[00_Null_Hypothesis_Index]] — Master null hypothesis index - [[Perihelion_Precession_Null]] — Eq. 14 mutual exclusivity with relativity - [[Pseudotensor_Conservation_Null]] — Conservation laws violated by the pseudotensor - [[Gravitational_Waves_Null]] — Do gravitational waves exist? - [[First_Principles_Dynamics]] — Bouw's first principles derivation: dynamics = kinematics times (m/m) - [[2013_Bouw_Geocentricity_Forces_and_First_Principles]] — Source note on Bouw Ch. 27 and Appendix E - [[Artemis_II_Trajectory]] — NASA's Artemis II mission trajectory contains zero independent dynamical variables - [[Textbook_Orbital_Dynamics]] — Standard four slide patched conic textbook derivation (Earth to Moon) reduced to kinematics - [[2017_Zhang_Theory_Design_Special_Space_Orbits]] — Springer engineering textbook on hovering, cruising, and CR3BP. Active control is also kinematic; mass enters only as a unit conversion at the end. Page 44 admits thrust "reduces a part of earth's mass" (sub-keplerian orbit)