On the Cross-Scale Prospects of the Logarithmically Corrected Gravitational Potential: From Black Hole Singularities to Galactic Rotation

This paper proposes an extremely simple logarithmically modified gravitational potential, whose most prominent feature is the cross-scale unity from the black hole "singularity" to galactic dynamics: through the sign reversal of the gravitational potential at the microscopic scale (r<r*≈8.792×10-11m), it dynamically prevents any matter from collapsing into the "singularity". Under this mechanism, it a priori predicts the angular diameter of black hole shadows and the orbital velocities of high-speed stars orbiting them without introducing any free parameters (such as spin), and ultimately extends to explaining galactic rotation dynamics. By analyzing the mathematical asymptotic behavior of all dark matter halo models, we arrive at a core finding: adding a simple logarithmic correction term to the original Newtonian gravitational potential: \( Φ(r)=-\frac{GM}{r}-\frac{kG_h M^2 (ln⁡r+1)}{r} \). enables the possibility of both eliminating singularities and explaining the flattening of galaxy rotation curves within the same theoretical framework (where G_h is defined as the quantum gravitational constant (\( G_h=ℏc^2 G^3/8≈3.5224×10^{-49}kg^{-2}m^3 s^{-2} \)), and its unconventional dimensionality, we believe, can be explained by the compactification of coupled spacetime dimensions). The logarithmic term “ln⁡r" is the key to achieving the cross-scale effect of "repulsion at short distances and attraction at long distances". Through multiple cross-scale verifications—including a priori prediction of black hole shadows (Sgr A*, M87*) that agree with EHT observations without introducing additional free parameters (such as spin); a priori calculation of the perihelion velocities of high-speed stars (S4714, S62) that match observations; and posterior fitting of galactic rotation curve data (Milky Way, Andromeda Galaxy, NGC2974)—spanning nearly 30 orders of magnitude from black hole singularities to galaxies, we initially prove that the framework exhibits high consistency with observations in both strong gravitational fields (black holes) and weak gravitational fields (galaxies). Based on this, we further provide unique quantitative a priori predictions (without adjusting spin α and inclination i) for the angular diameters of shadows of six candidate black holes (such as NGC4261, M84) that have not yet been observed by EHT, and look forward to future verification. Core feature: The logarithmic correction is not introduced to address any single phenomenon. It originates from the universal result of the asymptotic mass distribution ρ(r)~r-3 of dark matter halos, and is consistently reflected in: 1) the regularization of the central gravitational potential; 2) the formation of black hole shadows; 3) the dynamics of high-speed stars; 4) galactic rotation curves. These manifestations form an inseparable whole. This framework not only achieves, for the first time, a unified description of gravity from the microscopic to the macroscopic scale (requiring only ordinary matter mass) but also provides an observable and reproducible empirical framework for quantum gravity theory, potentially freeing it from the long-standing research method of pure mathematical modeling (distant from actual observations) and transitioning to physical verification.