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) — Publication Review of Letter III

 

The paper titled "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)" explores a theoretical framework where gravity is not a fundamental force, but an emergent phenomenon arising from the geometry of information. [1, 2]

Here is a breakdown of the core concepts, mathematical correspondences, and implications of the research work in ToE's Letter III.

Core Concepts of the Paper

• Information Geometry: This mathematical field applies differential geometry to probability theory. It treats probability distributions as points on a smooth manifold (an information manifold), where distances are measured using information metrics like the Fisher Information Metric.
• Emergent Gravity: Aligning with ideas like Erik Verlinde's entropic gravity, this theory posits that spacetime and gravity emerge from the statistical behavior of underlying quantum or informational bits.
• The Theory of Entropicity (ToE): A specific framework developed to unify physics by treating entropy and entropic informational density as the primary drivers of physical laws.
• The Obidi Action: A novel informational action principle introduced in this framework. It quantifies the change or flow of Entropic information across an Entropic information manifold. [3, 4, 5, 6, 7]

The Key Mathematical Correspondence

The central achievement posited in the paper is mapping informational dynamics directly to physical spacetime curvature. It bridges two distinct action principles:

[ Informational Realm ]                   [ Physical Realm ]
  Obidi Action (I_O)     ===========>   Einstein-Hilbert Action (I_EH)
  (Fisher Info Metric)                    (Spacetime Metric tensor g_μν)

1. The Obidi Action ($I_O$): Formulated using the Fisher Information Metric, it measures the volume and curvature of probability spaces.
2. The Einstein–Hilbert Action ($I_{EH}$): The foundational action in General Relativity that yields Einstein's field equations when varied with respect to the spacetime metric ($g_{\mu\nu}$).
3. The Bridge: The paper demonstrates that under specific physical constraints, the thermodynamic and statistical fluctuations of information (the Obidi Action) mathematically reduce to the scalar curvature ($R$) of physical spacetime. Thus, minimizing [extremizing] informational uncertainty is equivalent to the way matter is formed/created and how matter bends spacetime. [8, 9]

Major Implications for Physics

• Resolution of the Spacetime Singularity: If spacetime is made of discrete bits of information, continuous singularities (like the centers of black holes) might just be mathematical illusions caused by looking at information at the wrong scale.
• Quantum Gravity Bridge: General Relativity (gravity) and Quantum Mechanics (which is deeply rooted in probability and information states) are naturally unified if gravity is shown to be purely informational.
• Cosmological Constant Solution: Dark energy and the expansion of the universe could potentially be reinterpreted as the natural thermodynamic expansion of the universe's total information capacity. [10]

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[1] https://zenodo.org

[2] https://www.firstprinciples.org

[3] https://link.springer.com

[4] https://inspirehep.net

[5] https://www.aldebaran.cz

[6] https://www.cambridge.org

[7] https://www.cambridge.org

[8] https://www.youtube.com

[9] https://scoap3-prod-backend.s3.cern.ch

[10] https://arxiv.org

 

In the Theory of Entropicity (ToE), gravity is not a fundamental force, but a macroscopic thermodynamic manifestation of information flow. Information geometry—the foundation of ToE—posits that space and curvature emerge as statistical manifolds. [1, 2, 3]

At the core of this transition is the mathematical correspondence between the Obidi Action and the Einstein–Hilbert Action. [1, 2, 3]

 

1. The Foundation: Information Geometry

Instead of treating space and geometry as predefined, ToE begins with statistical manifolds where "distance" represents the distinguishability between different quantum or thermodynamic states. [1]

• The Information Metric: Using the Fisher-Rao metric or quantum Fubini-Study metric, the theory builds an underlying statistical substrate.
• The Amari-Čencov \(\alpha \)-connection: Statistical manifolds feature a one-parameter family of connections. ToE identifies \(\alpha = 0\) as the physically relevant state. Because it is both torsion-free and metric-compatible, this connection is mathematically equivalent to the Levi-Civita connection of General Relativity. [1, 2, 3]

 

2. The Obidi Action vs. The Einstein–Hilbert Action

General Relativity relies on extremizing the Einstein–Hilbert action, which yields the classical spacetime curvature of Einstein's equations. ToE introduces a statistical analog to this variational principle: [1, 2]

• The Einstein-Hilbert Action: In standard gravity, the action is defined by integrating the Ricci scalar \(R\) over the spacetime volume \(V\), where \(\mathcal{L}_{EH} = \frac{c^4}{16\pi G} R\).
• The Obidi Action: This principle governs the dynamics of the entropic field \(S(x)\). It is constructed directly from the properties of the underlying information geometry rather than assumed spatial metrics. [1, 2, 3]

 

3. The Curvature Transfer & Emergence

Rather than assuming a predefined geometry, ToE uses a Curvature Transfer Theorem (CTT): [1]

1. The dynamics of the information manifold are driven by the Obidi Action, extremizing to form the Master Entropic Equation (MEE) — the Obidi Field Equations (OFE).
2. The MEE/OFE serves as the entropic equivalent to the Einstein field equations.
3. In the macroscopic thermodynamic limit, the Riemann curvature tensor of physical space (\(R_{S}\)) emerges directly as the pushforward of the information Riemann tensor (\(R_{I}\)). [1, 2, 3, 4]

Through this framework, the Obidi Action rigorously maps how a universe optimizing its entropy flow creates the illusion of classical spacetime. Consequently, Einstein's field equations cease to be fundamental laws and instead become emergent statistical identities. [1, 2, 3]

 

To Learn More

Explore the conceptual and mathematical foundations of this framework through the official ToE publication on Authorea or read an overview of how Obidi transforms information geometry into spacetime on Medium. [1]