Corrections to the Classical Shapiro Time Delay in General Relativity(GR) from the Entropic Force-Field Hypothesis(EFFH)

In this paper, we rigorously derive the Shapiro time delay directly from the Entropic Force- Field Hypothesis (EFFH), reformulating the gravitational time delay by considering the effects of entropy gradients on the metric. Our goal is to demonstrate that the entropic field introduces additional time delay terms distinct from General Relativity (GR), replacing the role of the Kalb-Ramond (KR) field. We first derive the standard Shapiro time delay in GR using the Schwarzschild metric. We then introduce the entropic modification of spacetime metrics by incorporating an entropy-driven correction term. By setting the conditions for light propagation in the new entropy-modified metric, we derive the coordinate time delay and demonstrate the additional entropic terms. Finally, we impose constraints from the Theory of Entropicity (ToE) to determine the functional form of the entropic correction. Our results suggest that the entropy gradients can introduce measurable deviations in time delay from standard GR predictions, providing a potential empirical validation of EFFH. This derivation supports the hypothesis that entropy gradients influence time evolution, offering an alternative explanation for phenomena traditionally attributed to gravitational waves.