Abstract

While well-tested at solar system scales, Newtonian gravity exhibits anomalies at larger scales and accelerations below $$10^{-9} m/s^2$$. Although Yukawa-like modifications can reconcile these anomalies with general relativity up to solar system scales, they face challenges at galactic scales. Notably, gravitational anomalies at accelerations $$\le 10^{-9} m/s^2$$ for separations down to $$50\, \mu$$ m remain undetected experimentally. This paper presents a mathematical framework for Yukawa modification of Newtonian gravity in weak acceleration regimes $$(\le 10^{-9} m/s^2)$$ and small separations ( $$\le 30 \, \mu$$ m) using an extended space model with an extra dimension. This model suggests a vacuum-sourced inertia effect from peculiar photons, with implications for the Mach principle. These photons’ entanglement scale ( $$\le 30\, \mu$$ m) hints at the possibility of longer-range photon entanglement via analytic continuation, opening new avenues for research in quantum gravity and cosmology.