Physics:Einstein's Relativity and Obidi's Theory of Entropicity(ToE)

Short description: Entropic Field Curvature and the Emergence of Relativistic Geometry

Abstract

This research paper rigorously formalizes the Theory of Entropicity (ToE) — first formulated and developed by John Onimisi Obidi —advancing it as a foundational framework for the emergence of relativistic geometry. The analysis pivots on the Entropic Curvature Hypothesis (ECH)—which posits that spacetime curvature is not fundamental but emerges from the structure and dynamics of the entropic field—and the Entropic Axiom of Reality (EAR)—which asserts the impossibility of instantaneous interactions by mandating an entropy cost for every interaction.

Author's Note: Much of the content in this submission can be found in their expanded forms in the following citations[ ].

1. Introduction

The Theory of Entropicity (ToE) proposes that entropy is not merely a measure of disorder, but the primary field from which time, space, and interaction emerge. Unlike frameworks where spacetime geometry is axiomatic, ToE suggests that geometry arises from gradients and dynamics of the entropic field.

2. The Foundations of the Theory of Entropicity

2.1 Entropic Curvature Hypothesis (ECH)

[math]\displaystyle{ \phi(x) = \nabla^2 H(x) \tag{1} }[/math]

[math]\displaystyle{ \mathcal{C}(x) \sim \frac{\delta^2}{\delta x^2} H(x) \tag{2} }[/math]

2.2 Entropic Axiom of Reality (EAR)

[math]\displaystyle{ \forall\ \text{events},\ \Delta \tau \geq \frac{\Delta S}{\text{Capacity}_{\text{local}}} \tag{3} }[/math]

3. Mathematical Formalism Using the Obidi Action

3.1 Entropic Field Equations

The Obidi Action in its simplest form is given as:

[math]\displaystyle{ S_{\text{Obidi}} = \int \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] d^4x + S_{\text{matter}} \tag{4} }[/math]

Then, after a series of algebraic transformations, the resultant equation of motion in the Theory of Entropicity(ToE) becomes:

[math]\displaystyle{ A(S)\Box S + \frac{1}{2}A'(S)(\nabla S)^2 - V'(S) + \eta T^\mu_\mu = 0 \tag{5} }[/math]

3.2 Lorentz Factor from Entropic Capacity - Great Insight from the Theory of Entropicity(ToE)

[math]\displaystyle{ \nu_0^2 + \nu_{\text{move}}^2 = c^2 \tag{7} }[/math]

[math]\displaystyle{ \gamma = \frac{1}{\sqrt{1 - v^2 / c^2}} \tag{8} }[/math]

3.3 Time Dilation

[math]\displaystyle{ dt = \gamma d\tau = \frac{d\tau}{\sqrt{1 - v^2 / c^2}} \tag{9} }[/math]

3.4 Length Contraction

[math]\displaystyle{ L(v) = \frac{L_0}{\gamma} = L_0 \sqrt{1 - \frac{v^2}{c^2}} \tag{10} }[/math]

3.5 Mass Increase

[math]\displaystyle{ m(v) = \gamma m_0 = \frac{m_0}{\sqrt{1 - v^2 / c^2}} \tag{12} }[/math]

4. Emergence of Metric and Geodesics

4.1 Entropic Metric Tensor

[math]\displaystyle{ g^{(\text{eff})}_{\mu\nu} = f[s(x), j^\mu(x), \sigma(x)] \tag{13} }[/math]

4.2 Entropic Geodesics

[math]\displaystyle{ S_{\text{entropic}} = \int d\lambda\, \sqrt{\chi_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}} \tag{14} }[/math]

[math]\displaystyle{ \chi_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda} = 0 \tag{15} }[/math]

5. Comparison: ToE vs General Relativity and the Laws of Newton

Feature	ToE (Theory of Entropicity)	General Relativity
Action	Entropic field action	Einstein-Hilbert action
Metric	Emergent from entropy	Fundamental geometric field
Geodesics	Least entropic resistance	Least spacetime interval

6. What do We Mean by the Path of Least Resistance in [Classical] Physics? A Very Brief Introduction to the Conceptual Foundations of the Theory of Entropicity(ToE)

In [classical] mechanics, or physics in general, the path of least resistance is often invoked to describe how particles or systems evolve between two states. The idea suggests that a physical process follows a trajectory that minimizes opposition—often formalized as the Principle of least action, where the action integral has minimal variation at the endpoints. From Fermat’s principle in optics to Newton’s laws of motion, this notion pervades much of traditional physics.

In the Theory of Entropicity(ToE), however, we take a different and radical approach to the trajectory of motion or interaction, offering a profound re-conceptualization of this principle. In ToE, the path of least resistance is not merely a trajectory of minimal mechanical effort or minimized action. So, in ToE, we say that the path of nature is always that path that offers the least resistance to the manifestation of entropy itself. This is because entropy is universal, and nature always goes through entropy by default; so that naturally, if a particle must move at all, nature will permit it to move only through the path which does not obstruct Nature's own tendency towards [increasing] entropy. Naturally, therefore, Nature will always favour the path of increasing entropy compared to that path which would reduce entropy. That is why there is attraction or repulsion between objects and why an object would follow a straight line (or curve or geodesic). Any path that would reduce entropy is therefore naturally suppressed or discounted, unless another entropy is imposed on it to act otherwise. This is indeed the foundation of Newton's Laws of Motion, and any form of motion in general. In short, Newton’s laws of motion, and indeed all forms of motion, can be reinterpreted as corollaries of the universal entropic tendency.

In other words, what we do claim in the Theory of Entropicity(ToE) is that Nature does not arbitrarily choose trajectories; rather, it selects those that are most consistent with the fundamental tendency toward entropy increase. Thus, whenever a particle or system undergoes motion, the permitted path is constrained by entropy flow: it must not obstruct the universal drive toward entropic realization.

The Entropic Path Principle

In ToE, entropy is not just a thermodynamic scalar but a dynamical field, denoted as Lambda, that governs the evolution of all physical systems. The entropic field possesses structure, flux, and curvature, and serves as the true substrate out of which space, time, and interaction emerge. Nature does not randomly select paths; rather, it filters out those trajectories that obstruct or delay the increase of entropy.

The Principle of Least Action Redefined by the Theory of Entropicity(ToE):

The trajectory of any system or particle is the one that minimally obstructs the increase of entropy along that path.

This principle is formalized in the Entropic Path Integral and entropic geodesic equation derived from the Obidi Action and the Vuli-Ndlela Integral. It replaces the classical least-action approach with a least-entropy-resistance principle; where the “least resistance” in ToE means:

the path that allows entropy to increase most efficiently, with the least obstruction to its natural flow.

Illustration: Entropic Filtering of Motion:

Let us consider a practical case. Suppose an agent (a particle or observer) attempts to move from point A to point B. According to ToE, such motion is only permitted if the integrated entropy along the path increases:

[math]\displaystyle{ \int_{\text{Path}_{A \to B}} dS(x, t) \gt 0 }[/math]

If no such increase is possible—perhaps due to entropic obstruction, stagnation, or conflicting constraints—nature suppresses that path. The system must either wait (by invoking the No-Rush Theorem) or choose a new trajectory consistent with entropy’s progression. This selection mechanism is not probabilistic but governed by field-determined entropic capacity and flux propagation rates.

Importantly, this interpretation includes non-local and irreversible entropic processes, not just canonical thermodynamic ensembles. Time, in this framework, is an emergent parameter—merely the result of entropic progression along permitted trajectories.

Key Implications in the ToE Framework

• Paths that maximize entropy generation are spontaneously favored. They correspond to entropic geodesics—paths of minimal entropic cost per unit configuration change.

• Paths that suppress entropy are dynamically suppressed unless overruled by external entropic fields or constraints.

• The observed effects of attraction, repulsion, and inertia are not simply due to forces or geometric curvature, but due to entropic field dynamics optimizing itself.

• Straight-line motion or classical geodesics are only special cases where the entropy gradient is uniform or symmetric. These are not simply geometric necessities, but are consequences of the entropic field optimizing itself.

In general, therefore, the true path of motion of any body or particle [or system] is defined by the entropic curvature tensor.

7. Experimental Validation

7.1 Perihelion Precession

[math]\displaystyle{ \Delta\theta = \frac{6\pi GM}{ac^2(1 - e^2)} \tag{GR} }[/math]

7.2 Light Deflection

[math]\displaystyle{ \delta = \frac{4GM}{c^2 R_\odot} \approx 1.75'' \tag{GR} }[/math]

References

John Onimisi Obidi