How does the Theory of Entropicity (ToE) Derive Gravity from Entropy Gradients?

The Theory of Entropicity (ToE) derives gravity from entropy gradients via the **Obidi Action**, a variational principle for the entropy field $$ S(x) $$, which generates spacetime curvature and geodesics mimicking Einstein's equations. This treats nonuniform entropy flow as the ontological source of geometry, where gradients $$ \nabla S $$ warp "existence" into gravitational attraction.

Obidi Action

The core functional is  

$$ \mathcal{A}_\text{Obidi}[S] = \int d^4\lambda \sqrt{-g} \left[ \frac{1}{2} (\partial_\mu S)(\partial^\mu S) - V(S) + J(\lambda) S \right], $$  

with kinetic term $$ (\nabla S)^2 $$ driving dynamics like a scalar field.[1][2][3]

Varying with respect to $$ S $$ and the metric $$ g_{\mu\nu} $$ yields the **Master Entropic Equation (MEE)** and entropic field equations. The exponential weighting $$ e^{S/k_B} $$ deforms the metric to $$ g^{(S)}_{\mu\nu} = e^{S/k_B} g_{\mu\nu} $$, coupling entropy to geometry.[1]

Entropy-Weighted Geometry

Entropy gradients deform information metrics (Fisher-Rao, Fubini-Study) into entropic curvature:  

$$ g^{(S)}_{ij} = e^{S/k_B} g^{(\text{FR})}_{ij}. $$  

Amari-Čencov α-connections add irreversibility: $$ \Gamma^\lambda_{\mu\nu} = \{\lambda_{\mu\nu}\} + \frac{\alpha}{2} T^\lambda_{\mu\nu} $$, where $$ T $$ captures entropy asymmetry.[1][3]

This produces emergent Ricci curvature $$ R_{\mu\nu}(S) $$ from $$ \nabla S $$, generalizing Einstein:  

$$ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = \eta T^{(S)}_{\mu\nu}, $$  

with $$ T^{(S)}_{\mu\nu} $$ the entropy-stress tensor.[1]

Geodesics and Gravity Emergence

**Entropic Geodesics** follow paths maximizing entropy flow: particles trace minimum-entropy-disruption trajectories, equivalent to null/timelike geodesics in curved spacetime. Gradients $$ \partial S $$ act like gravitational potentials, yielding Newtonian limits and GR tests (e.g., Mercury precession, light deflection) via higher-order corrections from Unruh/Hawking effects.[2][1]

Uniform $$ S $$ implies flat symmetry; gradients induce "curvature," pulling systems toward equilibrium (gravity).[2]

## Key Predictions

| Effect                  | ToE Derivation from $$ \nabla S $$                  | GR Match [1] |

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

| Perihelion Precession  | Entropy-modified Binet equation, 43"/century      | Exact            |

| Light Deflection       | Entropic variational principle                    | 1.75" by Sun     |

| Cosmological Constant  | Quadratic approximation of Obidi Action           | Small positive   |

Citations:

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

[2] John Onimisi Obidi - Independent Researcher https://independent.academia.edu/JOHNOBIDI

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

[4] On the Theory of Entropicity (ToE) and Ginestra Bianconi's Gravity from ... https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/691437a4a10c9f5ca1db32f3/original/on-the-theory-of-entropicity-to-e-and-ginestra-bianconi-s-gravity-from-entropy-a-rigorous-derivation-of-bianconi-s-results-from-the-entropic-obidi-actions-of-the-theory-of-entropicity-to-e.pdf

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

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

[8] (PDF) Collected Works on the Theory of Entropicity (ToE) Volume I 31 ... https://www.academia.edu/145698037/Collected_Works_on_the_Theory_of_Entropicity_ToE_Volume_I_31_December_2025_V9_S

[9] The Theory of Entropicity (ToE) Lays Down ... https://johnobidi.substack.com/p/the-theory-of-entropicity-toe-lays

[10] Priority Assessment of the Theory of Entropicity (ToE) https://handwiki.org/wiki/Physics:The_Revolutionary_Conceptual_Leap_of_the_Theory_of_Entropicity_(ToE)

[11] Entropic gravity - Wikipedia https://en.wikipedia.org/wiki/Entropic_gravity