From Information Geometry to Information Gravity, Information Geometry as the Origin of Einstein’s Gravity: Correspondence of the Obidi Action and the Einstein–Hilbert Action in the Theory of Entropicity (ToE), Theory of Entropicity (ToE) Living Review Letters Series (ToE LRLS) — Letter III
Obidi shows that Lorentzian spacetime geometry emerges from information geometry via a controlled entropy‑gradient disformal transformation, and that the curvature of this emergent metric reproduces the Einstein gravity of General Relativity (GR).
From Information Geometry to Information Gravity: Information Geometry as the Origin of Einstein's Gravity in General Relativity (GR)
https://doi.org/10.13140/RG.2.2.14211.26405
https://doi.org/10.17605/OSF.IO/PT9U8
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John Onimisi Obidi
Research Lab, The Aether
Email: jonimisiobidi@gmail.com
Canonical Archive: https://entropicity.github.io/Theory-of-Entropicity-ToE/
May 31, 2026
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Keywords:
Theory of Entropicity (ToE); Obidi Action; Obidi Transformation; Obidi Metric; Obidi Relativistic Reduction Theorem; Einstein–Hilbert Action; Information Gravity; Information Geometry; Fisher–Rao metric; Fubini–Study metric; Bures metric; Amari–Čencov α-connections; Rényi–Tsallis entropy; Master Entropic Equation; Vuli–Ndlela Integral; Emergent Gravity; Entropic Field; General Relativity; Levi–Civita connection; Hybrid Metric-Affine Space (HMAS); Entropic Cosmological Constant.
John Onimisi Obidi’s framework, titled "From Information Geometry to Information Gravity," outlines the core mathematical and philosophical foundation of his Theory of Entropicity (ToE). The core thesis is that physical spacetime, gravity, and matter are not fundamental, but rather emergent macroscopic properties generated by a deeper, underlying statistical manifold governed by information geometry. [1, 2, 3, 4, 5]Instead of viewing information geometry as a mere mathematical tool for data analysis, Obidi introduces an ontological shift that treats statistical distinguishability as the primary engine of reality. [3, 6]
🧠 The Ontological Shift: Distinguishability to Distance [6]
In classical information geometry, metrics like the Fisher–Rao metric (for classical systems) and the Fubini–Study metric (for quantum states) are used to measure the statistical "distinguishability" between probability distributions. Obidi maps these abstract properties directly onto physical reality: [3, 6, 7]
The Entropic Field $S(x)$: Entropy is redefined as a fundamental, continuous scalar field whose gradients and dynamics generate all physical phenomena. [3]
Distance from Distinguishability: Physical distance in spacetime is a macro-projection of how distinguishable two underlying informational states are. [4, 6]
The Arrow of Time: By isolating the Amari–Čencov $\alpha$-connections (specifically where $\alpha=0$), the theory derives a naturally time-asymmetric, irreversible flow that defines the classical Levi-Civita connection used in General Relativity. [3, 8, 9]
📐 The Obidi Action Principle
To bridge abstract mathematical information and dynamic physical force, the framework introduces the Obidi Action (featuring both Local and Spectral variations). [10, 11]
The Variational Principle: Minimizing or stationarizing the Obidi Action describes the continuous, irreversible rearrangement of underlying informational degrees of freedom. [6, 8]
Haller–Obidi Correspondence (HOC): This correspondence establishes an identity between thermodynamic entropy and classical action ($H \propto \int L \, dt$). Under this rule, the Principle of Least Action becomes a principle of extremal entropy—meaning physical trajectories naturally maximize or extremize entropic flow. [1, 12]
🌌 Information Gravity: Emergent Spacetime
Under the Obidi Equivalence Principle (OEP), spacetime geometry is an isomorphic, coarse-grained projection of the underlying information-geometric manifold. Gravity is stripped of its status as a fundamental attractive force: [4, 12]
Entropic Gradients: Gravity emerges naturally from the tendency of the entropic field to redistribute itself. [12]
Curvature Generation: The curvature coupling term within the Obidi Action ensures that variations and gradients in entropy generate physical spacetime curvature ($R$). [12]
General Relativity as a Limit: When the entropic field is perfectly uniform, the field equations reduce exactly to the standard Einstein–Hilbert action of General Relativity. When $S(x)$ varies, it introduces non-linear corrections that offer testable deviations from classical gravity (e.g., modified gravitational lensing). [12, 13]
🧪 Physical & Quantum Consequences
Rather than relying on holographic boundaries or virtual particles, ToE attempts a top-down unification that impacts several physics anomalies: [11, 14]
Entropic Speed Limit (ESL): The cosmic speed of light ($c$) is reinterpreted as the maximum allowable rate for entropic redistribution.
The No-Rush Theorem: This places a universal bound on quantum reconfigurations, successfully predicting a finite, non-zero duration for the formation of quantum entanglement ($\approx 232$ attoseconds).
Universal Constants: The framework bridges the quantum, thermodynamic, and geometric worlds through a core constraint equation: $\hbar c = k_B T_S \ell_S$. [12, 14, 15]
Would you like to take a deeper look at the mathematical structure of the Obidi Action, explore its specific non-linear corrections to General Relativity, or compare how it differs from Erik Verlinde’s Entropic Gravity?
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John Onimisi Obidi’s Theory of Entropicity (ToE) redefines gravity. Moving from Information Geometry (the geometry of statistical distinguishability) to Information Gravity, Obidi argues that entropy and information form the foundation of physical reality. In this framework, spacetime and gravity are not fundamental; rather, gravity emerges naturally from the gradients of an informational/entropic manifold. [1, 2, 3, 4]
1. Information Geometry as the Substrate
In conventional statistics, information geometry measures how distinguishable two probability distributions are, using tools like the Fisher–Rao metric. Obidi makes an ontological shift: he asserts that this abstract statistical manifold is the underlying, actual substrate of physical reality before spacetime emerges. [1, 2, 3]
2. The Obidi Action and Emergent Spacetime
At the core of the theory lies the Obidi Action, a foundational variational principle. Through a process of coarse-graining, the fundamental information geometry (which contains the Fisher–Rao metric and the Fubini–Study metric for quantum states) undergoes an informational phase transition. This projection forms the macroscopic four-dimensional spacetime we observe, with the Einstein-Hilbert action arising as a natural consequence of this entropic rearrangement. [1, 2, 3, 4, 5]
3. Gravity as an Entropic Gradient
Instead of the classical understanding—where matter curves spacetime—the Theory of Entropicity proposes that entropy curves existence. Key mechanics include: [1]
The Obidi Equivalence Principle (OEP): Spacetime curvature and geodesics are directly isomorphic to the curvature and geodesics of the underlying information-geometric manifold. [1]
Information Gravity: Gravity is not a pulling force, but rather the macro-projection of entropic gradients. Mass is interpreted as an information-curvature density. [1]
No-Rush Theorem & Arrow of Time: The theory relies on the \(\alpha \)-connections (representing irreversibility), placing a finite time constraint on entropic reconfiguration, which naturally dictates the arrow of time and quantum entanglement limits. [1, 2]
You can read more about the mathematical and philosophical frameworks directly on the Medium Publication by Obidi or via the Authorea Pre-Print.
If you want to dive deeper into this, let us know:
Would you like to compare Obidi’s Information Gravity to older frameworks like Erik Verlinde's Entropic Gravity?
Are you interested in the mathematical formulation of the Obidi Action or how it accounts for dark matter? [1, 2]