What are Entropic Geodesics in the Obidi Action of the Theory of Entropicity (ToE)? Derivations, Geometric Interpretations, Physical Roles and Comparison With the Geodesics of Einstein's Relativity 

Entropic Geodesics in the Theory of Entropicity (ToE) represent the fundamental paths that particles and information follow in the entropy field $$ S(x) $$, derived directly from varying the Obidi Action. They generalize general relativity's geodesics by replacing metric curvature with entropy gradients $$ \nabla S $$, enforcing motion as the path of least entropy disruption or maximum irreversible flow.[8][2][1]

Derivation of Entropic Geodesics from the Obidi Action

The Obidi Action $$ \mathcal{A}_\text{Obidi}[S, g] = \int d^4x \sqrt{-g} \left[ \frac{1}{2} g^{\mu\nu} \partial_\mu S \partial_\nu S - V(S) + \mathcal{L}_\text{matter} e^{S/k_B} \right] $$ is extremized with respect to both the entropy field $$ S $$ and the auxiliary metric $$ g_{\mu\nu} $$. Varying yields the Master Entropic Equation (MEE) for field dynamics and the geodesic equation for trajectories:  

$$ \frac{d^2 x^\lambda}{d\tau^2} + \Gamma^\lambda_{\mu\nu} \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau} = \eta \partial^\lambda S, $$  

where $$ \Gamma $$ includes Amari-Čencov connections deformed by $$ e^{S/k_B} $$, and $$ \eta $$ is the entropic coupling.[8][2][4]

Geometric Interpretation of Entropic Geodesics 

These geodesics trace extremal curves in the **entropy-weighted manifold** $$ g^{(S)}_{ij} = e^{S/k_B} g^{(\text{FR})}_{ij} $$, blending Fisher-Rao information metric with exponential entropy boosting. Uniform $$ S $$ gives straight inertial paths; gradients $$ \nabla S $$ curve them, mimicking gravity as systems seek higher-entropy states (e.g., collapse increases local order but total entropy via radiation).[2][3]

Physical Role of Entropic Geodesics 

Particles follow entropic geodesics to conserve the second law locally while maximizing global $$ \Delta S $$, recovering GR limits like light deflection and perihelion advance. Quantum paths emerge via Fubini-Study projections, with Unruh temperature corrections for acceleration.[1][4]

Comparison of ToE's Entropic Geodesics to Einstein's General Relativity (GR) Geodesics

| Aspect              | GR Geodesics                  | Entropic Geodesics [8] |

|---------------------|-------------------------------|-----------------------------|

| Driving Principle  | Spacetime curvature $$ R_{\mu\nu} $$ | Entropy gradient $$ \partial S $$ |

| Path Equation      | Metric Christoffel symbols   | α-deformed + $$ \eta \nabla S $$ |

| Irreversibility    | None (timelike reversible)   | Built-in via $$ S_\text{irr} $$ |

Citations:

[1] A Brief Note on Some of the Beautiful Implications ... https://johnobidi.substack.com/p/a-brief-note-on-some-of-the-beautiful

[2] John Onimisi Obidi 1 1Affiliation not available October 15, 2025 https://d197for5662m48.cloudfront.net/documents/publicationstatus/284761/preprint_pdf/0304242fc1b6f7dfc2e1da6d68e30f89.pdf

[3] John Onimisi Obidi 1 1Affiliation not available October 17, 2025 https://d197for5662m48.cloudfront.net/documents/publicationstatus/285164/preprint_pdf/c7acf1b70b62c5ae001365c123d20350.pdf

[4] 1 Introduction 2 The Entropic Reformulation of the Unified https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/68f6f66c5dd091524f8f362e/original/transformational-unification-through-the-theory-of-entropicity-to-ea-reformulation-of-quantum-gravitational-correspondence-via-the-obidi-action-and-the-vuli-ndlela-integral.pdf

[5] Further Expositions on the Theory of Entropicity (ToE) and ... https://www.cambridge.org/engage/coe/article-details/69513828083c11e4a170b0b2

[6] The Theory of Entropicity (ToE) Derives and Explains Mass ...www.cambridge.org › coe › assets › orp › resource › item › original › the-... https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/6900d89c113cc7cfff94ef3a/original/the-theory-of-entropicity-to-e-derives-and-explains-mass-increase-time-dilation-and-length-contraction-in-einstein-s-theory-of-relativity-to-r-to-e-applies-logical-entropic-concepts-and-principles-to-verify-einstein-s-relativity.pdf

[7] A Simple Explanation of the Unifying Mathematical ... https://www.authorea.com/users/896400/articles/1348176-a-simple-explanation-of-the-unifying-mathematical-architecture-of-the-theory-of-entropicity-toe-crucial-elements-of-toe-as-a-field-theory

[8] A New Theory Says Gravity May Come From Entropy— ... https://www.popularmechanics.com/science/a64069299/gravity-entropy-unified-theory/