Newton's Law of Gravity Is Valid Only When Gravitational Lensing Has a Negligible Impact

This paper focuses on the long-standing issue of galactic rotation, i.e. the discrepancy between observed rotation rates and predictions based on Newtonian gravity. While Newton's law holds within the solar system, it fails to explain the higher-than-expected rotation speeds of galaxies, particularly at greater distances from their centers. This study finds that Newton's gravitational formula applies only when gravitational lensing has a negligible impact. We examine the gravitational force on objects farther from the galactic center, where the lensing effect of a supermassive black hole becomes significant. This lensing causes the actual gravitational force to exceed Newtonian predictions, with the discrepancy increasing with distance. This effect offers an alternative explanation for the galactic rotation curve without the need for dark matter. Our findings suggest that Newton's formula requires revision to account for gravitational lensing, a key factor in understanding galactic dynamics and its broader cosmic implications.