Physics:On Key Concepts in Obidi's Emerging Theory of Entropicity(ToE)

John Onimisi Obidi and the Theory of Entropicity (ToE)

Abstract

John Onimisi Obidi's Theory of Entropicity (ToE) presents a unified framework in which entropy is elevated from a statistical measure to a fundamental force-field that governs all interactions. Through the introduction of the Obidi Action (or Master Entropic Equation), Obidi derives entropic field equations that recover classical physics (General Relativity and Quantum Mechanics) as emergent phenomena of entropy-driven dynamics. Central to ToE are novel constructs such as the Entropic Force-Field Hypothesis (EFFH), the Entropic Lorentz Group, and the Entropic Time Limit (ETL) This comprehensive article elucidates the core principles, mathematical formalism, physical implications, and experimental predictions of ToE, situating it within the landscape of contemporary theoretical physics.

Table of Contents

1. Background and Motivation
2. Entropic Force-Field Hypothesis (EFFH)
3. Obidi Action and Master Entropic Equation
4. Derivation of Entropic Field Equations
5. Linearization: Entropic Wave Analysis
6. Entropic Lorentz Group and Invariance
7. Entropic Time Limit (ETL)
8. Unification of General Relativity and Quantum Mechanics
9. Black Hole Physics and Information
10. Experimental Tests: Attosecond Entanglement and Beyond
11. Extensions Beyond the Standard Model
12. Thermodynamic Uncertainty and CPT
13. Philosophical Implications and Future Directions
14. References

1. Background and Motivation

Modern physics is split between General Relativity (the physics of the very large) and Quantum Mechanics (the physics of the very small). Many attempts to unify them, including string theory and loop quantum gravity, posit additional dimensions or quantized spacetime structures. Obidi’s Theory of Entropicity (ToE) takes a radically different stance: entropy itself is the root of all interactions, and gravity and quantum dynamics emerge from an underlying entropic field.

Entropy traditionally quantifies the number of microstates compatible with a macrostate, via Boltzmann’s formula: [math]\displaystyle{ S = k_B \ln \Omega. }[/math] Obidi’s insight is that this measure reflects a physical field permeating spacetime whose dynamics impose universal rate limits, interaction strengths, and time evolution. In this view, entropy gradients drive forces and entropy fluctuations underlie quantum uncertainty.

2. Entropic Force-Field Hypothesis (EFFH)

Entropic Force-Field Hypothesis (EFFH): Entropy is a fundamental field whose variations generate what we perceive as forces. This hypothesis elevates entropy to the same ontological status as the electromagnetic or gravitational field.

Coupling of Fundamental Fields in the Theory of Entropicity (ToE)

The following table outlines the nature of coupling between fundamental fields and their respective sources across three major domains of physics: Electromagnetism, Gravity, and Entropy (as proposed in the Theory of Entropicity). This structure follows the logic of modern field theory and highlights how ToE reinterprets entropy as a dynamical, fundamental field.

Interpretation in ToE Framework

Electromagnetism

In classical and quantum electrodynamics, the electromagnetic field couples to the charge current via the 4-potential . The minimal coupling term in the action is: [math]\displaystyle{ S_{\text{EM}} = \int d^4x, J^\mu A_\mu }[/math]

This defines the interaction of EM fields with charged particles.

Gravity

In general relativity, gravity couples to the energy–momentum tensor through the metric tensor . The coupling appears in the Einstein–Hilbert action: [math]\displaystyle{ S_{\text{grav}} = \int d^4x, \sqrt{-g} \left( \frac{1}{2\kappa} R + T^{\mu\nu} g_{\mu\nu} \right) }[/math]

This framework describes how spacetime curvature interacts with matter and energy.

Entropy (ToE)

The Theory of Entropicity elevates entropy to a dynamical and fundamental field . Unlike conventional thermodynamics, entropy here couples not just to matter but also to vacuum fluctuations and wave structures. Its action includes thermodynamic and field-theoretic coupling terms: [math]\displaystyle{ S_{\text{ent}} = \int d^4x, \left( TS + \beta S + \frac{\mathcal{S}G}{k_B} + \frac{\mathcal{S}{\text{irr}}}{\hbar_{\text{eff}}} \right) }[/math]

Where:

is temperature,

is the inverse temperature,

is the gravitational entropy term,

is the irreversibility correction,

is an entropy-modified Planck constant.

These terms appear explicitly in the path integral formulation known as the Vuli-Ndlela Integral used in ToE: [math]\displaystyle{ Z_{\text{ToE}} = \int_{\mathbb{S}} \mathcal{D}[\phi] , e^{\frac{i}{\hbar} S[\phi]}+e^{-\frac{\mathcal{S}G[\phi]}{k_B}}+ e^{-\frac{\mathcal{S}{\text{irr}}[\phi]}{\hbar_{\text{eff}}}} }[/math]

Summary

The above structure emphasizes that in the Theory of Entropicity:

Entropy is not just a statistical descriptor, but a dynamic and universal field.

It couples fundamentally to all matter, radiation, and the vacuum structure.

Its coupling terms resemble those of EM and gravity in that they appear in the action and determine system evolution.

See Also

Electromagnetic field theory

General relativity

Thermodynamic entropy

Theory of Entropicity

Obidi Action

Vuli Ndlela Integral

2.1 Ontological Status

In ToE, the entropic field is:

1. A Lorentz scalar under coordinate transformations.
2. Dynamically coupled to matter and curvature.
3. Possessing its own kinetic and potential terms.

2.2 Comparison ToE's Entropic Field to Other Fields

Field	Fundamental?	Source	Coupling Term
EM	Yes	Charge density	Electromagnetic 4-potential Aₘ (couples via JᵐAₘ)
Gravity	Yes	Energy–momentum tensor	Metric tensor gₘₙ (couples via Tᵐⁿgₘₙ)
Entropy	Yes (ToE)	Matter, waves & vacuum	Microstates, Temperature T (couples via TS or βS, where β=1/k_B T)

3. Obidi Action and Master Entropic Equation

The Obidi Action is the variational starting point for ToE dynamics: [math]\displaystyle{ \mathcal{S}{\text{ToE}} = \int d^4x,\sqrt{-g}\left[ -\frac{1}{2}A(S)g^{\mu\nu}\nabla\mu S \nabla_\nu S - V(S) + \eta S T^\mu{}{\mu} \right] + \mathcal{S}{\text{matter}}[\Phi,g_{\mu\nu}], }[/math] where:

encodes entropic stiffness, analogous to permeability/permittivity.

is the entropic potential controlling self-interactions.

couples entropy to the trace of the stress-energy tensor, ensuring entropy sources matter and vice versa.

3.1 Action Variation and Field Equation

Varying w.r.t. yields: [math]\displaystyle{ A(S),\Box S + \frac{1}{2}A'(S)(\nabla S)^2 - V'(S) + \eta T^\mu{}_{\mu} = 0. }[/math] This is the Master Entropic Equation, governing how entropy evolves in curved spacetime.

4. Derivation of Entropic Field Equations

Field equations for coupled matter–entropy dynamics follow from Euler–Lagrange: [math]\displaystyle{ \frac{1}{\sqrt{-g}}\partial_\mu\left(\sqrt{-g} A(S)g^{\mu\nu}\partial_\nu S\right) - \frac{1}{2}A'(S)(\nabla S)^2 + V'(S) = \eta T^\mu{}{\mu}. }[/math] Similarly, variation w.r.t. g{\mu\nu} gives modified Einstein equations with entropic stress.

5. Linearization: Entropic Wave Analysis

To study small perturbations, set . Linearizing yields: [math]\displaystyle{ A_0\Box \sigma - m_S^2 \sigma = 0, }[/math] with . This admits wave solutions, entropic waves, with characteristic speed: [math]\displaystyle{ c_{\text{ent}}^2 = \frac{\text{coef}(\nabla^2)}{\text{coef}(\partial_t^2)} = 1 \quad(\text{in natural units}). }[/math] Restoring dimensions through factors shows , offering an entropic basis for light speed invariance.

6. Entropic Lorentz Group and Invariance

Define an entropic line element: [math]\displaystyle{ d\sigma^2 = \alpha(S) dt^2 - \beta(S)d\mathbf{x}^2. }[/math] The Entropic Lorentz Group are transformations preserving null paths of . Under ToE constraints (homogeneity, isotropy, finite entropic speed), this reduces to the standard Lorentz group, explaining invariant as an entropic corollary.

7. Entropic Time Limit (ETL)

The Entropic Time Limit sets a fundamental minimum time for any interaction: [math]\displaystyle{ \Delta t_{\min} = \frac{\delta S}{\dot{S}} \sim \frac{\hbar}{k_B , \Delta S}, }[/math] distinct from Planck time. It implies a maximal information transfer rate and influences quantum uncertainty.

8. Unification of General Relativity and Quantum Mechanics

In ToE, both gravitational curvature and quantum uncertainty are consequences of entropic field dynamics:

Gravity: emerges from in the modified Einstein equations.

Quantum: emerges from field fluctuations in , leading to entropic uncertainty relations.

8.1 Entropic Quantization

Quantization of the entropic field yields commutation: [math]\displaystyle{ [S(x), \Pi_S(y)] = i\hbar \delta^3(x-y), }[/math] where is the conjugate momentum, linking entropy to quantum structure.

9. Black Hole Physics and Information

ToE provides a new perspective on the Black Hole Information Paradox: the horizon entropy is naturally the entropic field integrated over the horizon area: [math]\displaystyle{ S_{\text{horizon}} = \int_{\mathcal{H}} S(x), dA, }[/math] and the evaporation process is an entropic relaxation, preserving information in the entropy field.

10. Experimental Tests: Attosecond Entanglement and Beyond

ToE predicts unique signatures in:

Attosecond-timescale entanglement formation.

Deviations in high-precision tests of speed-of-light invariance in extreme entropy gradients.

Modified decoherence rates in mesoscopic systems at cryogenic temperatures.

11. Extensions Beyond the Standard Model

ToE introduces:

Entropic Probability: distribution of microstate pathways weighted by entropic action.

Entropic CPT: combined Charge–Parity–Time reversal as an entropic symmetry.

Thermodynamic Uncertainty Relation: [math]\displaystyle{ \Delta S \Delta t \ge k_B, }[/math] setting bounds on entropy fluctuations over time intervals.

12. Philosophical Implications and Future Directions

ToE reframes existence, causality, and information through an entropic lens:

Existence requires nonzero .

Time’s arrow is entropic flow.

Consciousness may be emergent from high entropic complexity.

Future research aims to:

Formalize entropic gauge theory.

Explore entropic models for quantum computing.

Integrate ToE with observational cosmology.

Summary

John Onimisi Obidi and the Theory of Entropicity (ToE),” covering the following:

The Entropic Force-Field Hypothesis (EFFH)

The Obidi Action / Master Entropic Equation

Derivation of Entropic Field Equations

Linearization and entropic wave analyses

The Entropic Lorentz Group

The Entropic Time Limit (ETL)

Unification of General Relativity and Quantum Mechanics

Black hole information resolution

Experimental tests including attosecond entanglement

Extensions beyond the Standard Model (Entropic Probability, CPT, Thermodynamic Uncertainty)

Philosophical implications

It includes formal theorems.

References

J. O. Obidi, "Theory of Entropicity (ToE) and Obidi Action," (2025).

Bekenstein, J. D. (1973). "Black holes and entropy." Physical Review D, 7(8).

Hawking, S. W. (1975). "Particle creation by black holes." Communications in Mathematical Physics, 43.

Verlinde, E. (2011). "On the origin of gravity and the laws of Newton." JHEP, 2011(4):29.

Landau, L. D., Lifshitz, E. M. (1980). "Statistical Physics, Part 1." (Vol. 5).

Obidi, J.O. (2025). Theory of Entropicity: Entropy as a Fundamental Field. Cambridge Open Engage. DOI:[1]

Verlinde, E. (2011). "On the Origin of Gravity and the Laws of Newton." JHEP.

Jacobson, T. (1995). "Thermodynamics of Spacetime: The Einstein Equation of State." PRL.