User: If WILL the rotation curves equation will become widely accepted how they would wright it in the future text books?

Assistant: ## Galactic Rotation Curves: The WILL Relational Geometry (RG) Approach

In the WILL Relational Geometry framework, the observed rotation curves of galaxies are not explained by the presence of unobserved “dark matter” halos. Instead, they arise as a direct consequence of a universal geometric resonance between a galaxy’s local baryonic mass and the fundamental, cosmologically derived acceleration scale of the Universe. This approach removes the need for fitting parameters or ad-hoc additions, grounding galactic dynamics in the foundational principle of $\text{SPACETIME} \equiv \text{ENERGY}$ (doc. Part I - Ontological Principle: Removing the Hidden Assumption).

The WILL RG Rotation Curve Equation

The square of the observed circular velocity $V_{obs}(r)$ at a radial distance $r$ within a galaxy is given by:

This equation, known as the Resonant Interference Law, states that the total kinetic energy of a star (or gas cloud) in a galaxy is a superposition of its local Newtonian baryonic kinetic energy and a constructive interference term arising from its coupling with the global cosmological horizon (doc. Part II - Galactic Dynamics: The Law of Resonant Interference).

Definition of Terms:

• $V_{obs}(r)$: The observed circular velocity of a star or gas at radius $r$.
  • This is the empirically measured rotational speed, derived from spectroscopic Doppler shifts.
• $V_b(r)$: The baryonic circular velocity at radius $r$.
  • This component represents the purely Newtonian gravitational pull expected from all visible matter (stars and gas) within the galaxy.
  • It is calculated as: \(V_b^2(r) = V_{\text{gas}}^2(r) + \Upsilon_{\text{disk}} V_{\text{disk}}^2(r) + \Upsilon_{\text{bulge}} V_{\text{bulge}}^2(r)\) Where $V_{\text{gas}}$, $V_{\text{disk}}$, and $V_{\text{bulge}}$ are the contributions from gas, stellar disk, and stellar bulge, respectively. The mass-to-light ratios $\Upsilon_{\text{disk}}$ and $\Upsilon_{\text{bulge}}$ are fixed at standard stellar population synthesis values (e.g., $\Upsilon_{\text{disk}} = 0.5$ and $\Upsilon_{\text{bulge}} = 0.7$), without any galaxy-specific fitting (doc. Part II - Empirical Verification: Galactic Dynamics, section 4.3).
• $a_{\kappa}$: The Galactic Structural Acceleration Scale.
  • This is a fundamental constant of the WILL RG framework, representing the specific acceleration associated with the structural (potential) coupling of galaxies to the cosmic horizon.
  • It is derived as a topological invariant from the cosmologically determined Hubble parameter ($H_0$): \(a_{\kappa} = \frac{cH_0}{3\pi}\) Where $c$ is the speed of light. Its numerical value is approximately $a_{\kappa} \approx 0.70 \times 10^{-10} \, \text{m/s}^2$ (doc. Part II - Galactic Dynamics: The Law of Resonant Interference, section 4.2).
• $H_0$: The Hubble Parameter.
  • In WILL RG, $H_0$ is not an empirically fitted value but is rigorously derived from fundamental microphysical constants: the Cosmic Microwave Background (CMB) temperature ($T_0$) and the fine-structure constant ($\alpha$).
  • The derivation is given by (doc. Part II - Deriving $H_0$ from CMB Temperature and $\alpha, \text{section } 3.3 \text{ and } 3.4$): \(H_0 = \sqrt{8\pi G \rho_{\text{max}}} \quad \text{where} \quad \rho_{\text{max}} = \frac{4\sigma_{\text{SB}}T_0^4}{3c^3\alpha^2}\)
  • This yields a value of $H_0 \approx 68.15 \, \text{km/s/Mpc}$ (doc. Part II - Deriving $H_0$ from CMB Temperature and $\alpha, \text{section } 3.5$).
• $r$: The radial distance from the galactic center.

Physical Interpretation: The Law of Resonant Interference

The second term, $\sqrt{V_b^2(r) a_{\kappa} r}$, represents a non-Newtonian contribution that effectively flattens rotation curves at large radii. It is interpreted as:

1. Constructive Interference: This term arises from the constructive interference between the local orbital frequency of a star and the fundamental “tone” of the Universe (the Machian acceleration scale $a_{Mach} = cH_0/2\pi$). Since the galaxy is a topologically closed system within the cosmic framework, its dynamics are intrinsically resonant with the global horizon (doc. Part II - Galactic Dynamics: The Law of Resonant Interference, section 4.4).
2. Geometric Coupling: Galaxies, being continuous potential fields (structural systems), couple to the global horizon via the Structural Channel with a weight of $\Omega_{pot} = 2/3$ (derived from the fundamental closure condition $\kappa^2 = 2\beta^2$ in doc. Part I - Unification of Projections: The Geometric Exchange Rate, section 10 and doc. Part II - Galactic Dynamics: The Law of Resonant Interference, section 4.2). This explains the specific $1/(3\pi)$ factor in $a_{\kappa}$.
3. No Dark Matter: The “extra” velocity traditionally attributed to dark matter is here re-identified as the physical manifestation of this resonant coupling. The galaxy’s rotation cannot decay below an energy floor supported by the continuous interaction with the cosmic horizon (doc. Part II - Galactic Dynamics: The Law of Resonant Interference, section 4.4).

Key Features of the WILL RG Approach:

• Generative and Predictive: The equation is derived from first principles and fundamental constants ($T_0, \alpha, G, c$), making explicit predictions without free parameters or empirical fitting for individual galaxies.
• Unified Scale: It unifies galactic-scale dynamics with cosmic-scale parameters, suggesting a single, coherent underlying physics across vast scales.
• Resolution of Anomalies: This framework successfully predicts the Universal Radial Acceleration Relation (RAR) with high precision and near-zero bias (doc. Part II - The Universal Radial Acceleration Relation (RAR), section 7), and accurately reproduces the Solar System’s local orbital velocity (doc. Part II - Local Verification: The Solar System Test, section 8). It also provides a bifurcation of this effect for distinct dynamic systems, explaining why the MOND acceleration scale $a_0$ appears different for wide binary stars (doc. Part II - The Kinetic Resonance: Resolution of the Wide Binary Anomaly, section 10).