How Information Geometry is Transformed Into the Physical Geometry of Spacetime in Obidi's Theory of Entropicity (ToE)
In the Theory of Entropicity (ToE), John Onimisi Obidi declares that information geometry becomes physical spacetime geometry through a "strong physical postulate" that identifies the abstract structures of statistical manifolds with the ontological fabric of reality. [1, 2]
His declaration follows a specific logical and mathematical chain:
1. The Ontological Shift
Obidi's primary move is to stop treating entropy and information geometry as secondary descriptors or mathematical tools. Instead, he posits that the entropic/statistical manifold is the "underlying manifold of reality". In this view, entropy is not a measure of disorder but a fundamental scalar field $S(x)$ whose gradients and dynamics generate physical phenomena. [3, 4, 5, 6, 7]
2. Metric Identification
In standard information geometry, the Fisher-Rao metric (and the quantum Fubini-Study metric) measures the "distinguishability" between states. Obidi identifies this informational distinguishability as the precursor to physical distance. He declares that: [4, 8, 9]
- Informational Curvature = Physical Curvature: The curvature of the information-geometric manifold (built from these metrics) is the same curvature that Einstein describes in General Relativity (GR).
- Emergent Spacetime: Physical 4D spacetime is a "coarse-grained" or "macroscopic projection" of this deeper, higher-dimensional entropic manifold. [2, 10, 11, 12]
3. The Role of the $\alpha$-Connection
The most technical part of his declaration involves the Amari-Čencov $\alpha$-connections. In information geometry, there is a family of connections indexed by $\alpha$. Obidi identifies the $\alpha = 0$ connection as the physical one because it is the unique connection that is both torsion-free and metric-compatible—the exact requirements for the Levi-Civita connection used in General Relativity. [1, 3, 13]
4. Dynamic Generation (The Obidi Action)
He introduces the Obidi Action, a universal variational principle for the entropy field. By varying this action, he derives the Master Entropic Equation (MEE), which he declares to be the entropic analogue (and ancestor) of Einstein’s field equations. This makes physical geometry an output of entropic dynamics rather than a pre-existing stage. [5, 10, 14, 15]
Summary of the Identification
Information Geometry Concept [16] | Physical Spacetime Equivalent |
Entropic/Statistical Manifold | Fundamental Reality (Ontological Manifold) |
Fisher-Rao / Fubini-Study Metric | Pre-spacetime Metric |
$\alpha = 0$ Affine Connection | Levi-Civita Connection of GR |
Entropy Gradients/Curvature | Gravity and Spacetime Curvature |
Distinguishability Limits | Speed of Light ($c$) |
Would you like to look closer at the Master Entropic Equation or how Obidi derives the speed of light from entropic constraints?
[10] https://www.ssrn.com
[16] https://medium.com
In the Theory of Entropicity (ToE), John Onimisi Obidi declares that information geometry becomes physical spacetime geometry by identifying entropy as the fundamental, ontological field of reality rather than a mere statistical measure. [1, 2]
Obidi posits that this entropic field \(S(x)\) acts as a scalar potential that inherently generates all information geometry, which then, through coarse-graining, acts as the precursor to physical spacetime. [1, 2]
Here is how Obidi makes this declaration:
1. Ontological Shift: Information is Reality [1]
Obidi rejects the view that information geometry is just a way to classify probability distributions (epistemic). Instead, he asserts that the statistical/entropic manifold is the actual underlying fabric of reality. [1, 2]
- Fundamental Field: Entropy is treated as a foundational field \(S(x)\) on this differentiable manifold.
- Ontological Metric: Information metrics, such as the Fisher–Rao metric (classical) and Fubini–Study metric (quantum), are elevated to represent the actual metric-affine structure of the entropic manifold. [1]
2. The \(\alpha=0\) Connection Identification
Obidi uses the Amari–Čencov \(\alpha \)-connections, which dictate how probability models change. He declares that only the case where \(\alpha=0\) corresponds to physical spacetime. [1, 2]
- Levi-Civita Correspondence: The \(\alpha=0\) connection is mathematically the Levi-Civita connection of the entropic metric.
- Macroscopic Limit: Obidi identifies this \(\alpha=0\) connection as the physical spacetime connection of General Relativity (GR) at the macroscopic limit. [1]
3. Emergence via Coarse-Graining
Physical spacetime is not assumed to exist fundamentally; it emerges. The 4D spacetime metric \(g_{\mu\nu}(x)\) is declared to be the pullback or coarse-grained version of the deeper entropic metric \(g_{IJ}^{\text{entropic}}\). [1, 2, 3]
- Projection: A subset of coordinates is projected to become the standard 4D spacetime coordinates \((x^\mu, \mu=0,1,2,3)\).
- Entropic Gradients: Gravity is reimagined as the result of gradients in the entropic field. [1, 2]
4. Key Mathematical Bridges
- The Obidi Action: A variational principle is introduced to encode entropic dynamics, which leads to a "Master Entropic Equation" (MEE), serving as the entropic analogue to Einstein's field equations.
- Entropy-Dependent Deformation: The Fisher–Rao metric is deformed by an entropy-dependent factor \(e^{S/k_{B}}\) (derived from the Obidi Action), transforming pure information geometry into a dynamic "entropic geometry". [1, 2, 3]
In essence, Obidi's declaration is:
\(\text{Entropy\ Flow}\rightarrow \text{Information\ Manifold}\rightarrow \text{Coarse-Graining}\rightarrow \text{Physical\ Spacetime}\)
To explore this further, would you like to:
- Look closer at the "Obidi Action" and how it derives the Master Entropic Equation?
- Understand how this theory explains gravity as an entropic force rather than a fundamental force?
- Compare this to Verlinde’s Entropic Gravity?
No comments:
Post a Comment