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

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. The derivation of relativistic phenomena – time dilation, length contraction, and mass increase – from the constraints of entropic capacity is presented, including a new approach to the Lorentz factor, metric tensor, and geodesics grounded in entropy density, flux, and irreversible production. Comparisons with General Relativity (GR) are drawn at both the conceptual and quantitative levels, and the entropic geodesic framework is shown to replace spacetime geodesics in ToE. The report concludes with a structured empirical appraisal and recommendations for future validation, integrating results from symbolic collapse grammar, information-entropy frameworks, and observational tests.

1. Introduction

The pursuit of a unified account of physics – one that harmonizes quantum mechanics, thermodynamics, and relativity – has generated a proliferation of approaches in recent decades. Among these, the Theory of Entropicity (ToE) distinguishes itself by proposing that entropy is not simply a measure of disorder, but the primary field out of which time, space, and interaction emerge. Unlike frameworks in which spacetime geometry is axiomatic or fixed, ToE posits that geometry itself – its structure, causal limits, and curvature – arises from the interplay, persistence, and gradients of the entropic field.

ToE stands on the shoulders of a tradition of “emergent gravity” proposals, but draws its inspiration equally from advances in statistical mechanics, information theory, and the symbolic grammatical models of quantum emergence. The ToE project aims to transcend the dichotomy of spacetime-as-background (as in General Relativity) versus spacetime-as-quantized or discrete (as in loop quantum gravity), suggesting instead that spacetime is the entropic rendering of fundamental motif or information structures, filtered by entropy cost and irreversible production.

This paper presents a formal and comprehensive exposition of ToE’s foundational principles, mathematical architecture, and predictions for relativistic phenomena. It focuses on:

Articulating and formalizing the Entropic Curvature Hypothesis (ECH) and the Entropic Axiom of Reality (EAR).

Demonstrating the entropic derivation of time dilation, length contraction, and relativistic mass increase, and the emergence of the Lorentz factor.

Constructing the effective metric tensor from entropy density, flux, and irreversible production, and defining entropic geodesics.

Comparing ToE predictions with those of General Relativity, including empirical tests such as perihelion precession and solar light deflection.

Surveying recent mathematical and empirical developments related to ToE, such as entropic curvature on graphs, symbolic collapse grammars, and information-entropy frameworks.

2. The Foundations of the Theory of Entropicity

2.1. The Entropic Curvature Hypothesis (ECH)

Statement (ECH): Spacetime curvature is an emergent consequence of the structure and dynamics of the entropic field; it encodes how entropy density, flux, and irreversible production organize the possible trajectories and interactions of physical systems.

ECH emerges from the insight that entropy is a dynamic, universal field rather than a mere scalar or a statistical summary. The field’s local configuration constrains the evolution of all physical quantities: when entropy gradients and fluxes are present, the effective “geometry” experienced by systems – captured canonically by the metric tensor in GR – emerges from the propagation and interaction characteristics of entropy itself.

Formally, ECH generalizes the notion of curvature from geometry to entropy. We define an entropic potential φ(x) related to the entropy cost H(x) by:

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

with H(x) the (possibly symbolic) entropy cost at point x. In regions of steep entropy gradient, ∇^2 H(x) is large, generating significant “curvature” and affecting the rendering or evolution of motifs and physical states.

Mathematical Expression of Curvature in ToE: In ToE, the analog of the Ricci curvature arises from the second variation of the entropy functional:

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

This directly links local geometry to the entropic landscape of the system. The curvature is positive in regions where entropy flow is convex and negative where flow is concave, providing a geometric structure emergent from thermodynamic (or informational) properties.

2.2. The Entropic Axiom of Reality (EAR)

Statement (EAR): Instantaneous interactions or infinite rates of update are impossible; every physical interaction or transition must incur a finite, non-zero entropy cost – i.e., the “entropic speed limit” is fundamental.

The Entropic Axiom of Reality provides the deep rationale for kinematic constraints, such as the impossibility of superluminal (faster-than-c) information transfer. In ToE, interactions require the entropic field to “set conditions” locally, propagating changes at a finite rate determined by the entropic capacity and flux laws. Every update, collapse, or physical event must “pay” an entropy cost, introducing a minimum time and process duration. This is encapsulated in the No-Rush Theorem:

[math]\displaystyle{ \forall, \text{interactions},\ \Delta \tau \geq \frac{\Delta S}{\text{cap}_{\text{loc}}} \tag{3} }[/math]

where ΔS is the entropy generated by the interaction, and $\text{cap}_{\text{loc}}$ is the local update capacity set by the entropic field. Thus, there are no action-at-a-distance phenomena in ToE without an associated entropy cost and delay, unifying thermodynamic irreversibility with the relativity of simultaneity.

3. Mathematical Formalism of the Theory of Entropicity

3.1. Entropic Field Equations: General Structure

The dynamics of the entropic field in ToE take a variational form, paralleling the structure of action principles in physics, but with entropy assuming the principal role. The Obidi Action, for example, in its simplest form, reads:

[math]\displaystyle{ S_{\text{total}} = \int \sqrt{-g},\Big[ \frac{1}{2}A(S),g^{\mu\nu}\nabla_\mu S,\nabla_\nu S - V(S) + \eta, S, T^\mu{}\mu \Big], d^4x + S{\text{matter}} \tag{4} }[/math]

where S(x) is the entropic field, A(S) an “entropic stiffness” analogous to permittivity/permeability, V(S) the entropic potential, and the final term (with coupling constant η) couples entropy to matter (through the trace $T^\mu{}_\mu$ of the stress-energy tensor).

Variation yields the master field equation:

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

which governs the evolution of the entropy field and, by extension, the effective geometry.

Linearization and Characteristic Propagation Speed: Linearizing around a homogeneous background $S_0$:

[math]\displaystyle{ S(x) = S_0 + \delta S(x), \qquad A_0, \Box \delta S - m_S^2, \delta S = 0, \quad m_S^2 = \frac{V''(S_0)}{A(S_0)} \tag{6} }[/math]

The above yields entropic waves with characteristic speed $c_{\text{ent}}$ determined by the ratio of coefficients in the kinetic term – leading, in natural units, to the identification $c_{\text{ent}} = c$, the speed of light.

3.2. Entropic Derivation of the Lorentz Factor

In ToE, relativistic phenomena are emergent properties of entropic capacity constraints. The finite maximal update rate imposed by entropy dynamics gives rise to the Lorentz factor, customarily denoted as $\gamma$:

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

ToE reinterprets $\gamma$ as the ratio of a system’s internal entropic update rate to the combined rate when “transport” (motion) is included.

Internal and Transport Update Rates: Denote:

$\nu_0$: intrinsic (rest frame) entropic update rate

$\nu_{\text{tr}}$: update rate associated with motion (transport)

$c$: maximal update (propagation) capacity of the entropic field

The available “entropic budget” is split: :[math]\displaystyle{ \nu_0^2 + \nu_{\text{tr}}^2 = c^2 \tag{7} }[/math]

Solving for the internal rate when the system moves with velocity $v$: [math]\displaystyle{ \nu_0 = c,\sqrt{1 - \frac{v^2}{c^2}}, }[/math]

thus the operational Lorentz factor can be expressed as $\gamma = \frac{c}{\nu_0} = \frac{1}{\sqrt{1 - v^2/c^2}}$.

This formalism connects the relative suppression of internal (rest frame) processes to the necessity of allocating entropic update budget to transport (motion), mirroring the familiar structure of Special Relativity, but grounding it in entropic field capacity.

Thus the reader must have seen how we have arrived at the Lorentz factor of Einstein's Relativity via the principle of the Theory of Entropicity (ToE), invoking entropic constraints alone without any recourse or resort to geometry or postulating any constancy of the speed of light.

That is, no propagation can be faster than entropy itself in the entropic field:

Entropy is the medium of motion, and the object in that medium cannot move faster than the medium itself permits.

3.3. Entropic Derivation of Time Dilation

Time dilation arises in ToE because as an object’s velocity increases, more of the allowable entropic update rate is consumed by motion, and less remains for internal processes (i.e., clock ticks slow down). Suppose a clock at rest measures proper time $d\tau$, while a moving clock (velocity $v$) measures $dt$:

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

From the ToE perspective, the number of entropic update ticks per “external” interval is reduced by $\gamma^{-1}$ as a result of the resource split – a direct consequence of the Entropic Capacity Constraint. Specifically, the number of internal ticks $N$ in an interval Δt is

which for a fixed external time $\Delta t$ means fewer internal events (ticks).

This matches all experimental results confirming relativistic time dilation, such as the increased lifetimes of moving muons and clock desynchronization in high-speed transportation.

3.4. Entropic Derivation of Length Contraction

Just as time dilation emerges from entropic capacity reallocation, length contraction arises to preserve the integrality of total entropy across reference frames. Define $s$ as entropy density and $L$ as rest length. Conservation: $S_{\text{total}} = s_0 L_0 = s(v),L(v)$.

Motion causes an increase in entropy density: $s(v) = \frac{s_0}{\sqrt{1 - v^2/c^2}} = \gamma,s_0$. Therefore, to maintain constant total entropy,

[math]\displaystyle{ L(v) = \frac{L_0}{\gamma} \tag{10} }[/math]

In ToE, an object contracts in the direction of motion to maintain constant total entropy under a higher entropy density when moving. The causal arrow is reversed compared to classical relativity: entropy is primary, contraction is its corollary.

3.5. Entropic Origin of Relativistic Mass Increase

In ToE, mass emerges as an “entropic persistence cost,” the entropy required to maintain an information motif or grammatical pattern against collapse or annihilation.

Formally, symbolic mass is defined as:

[math]\displaystyle{ m_i \propto \frac{dH}{d\tau} \tag{11} }[/math]

where $H$ is the entropy cost function and $d\tau$ is the proper time increment. As a body acquires velocity (i.e., spends more of its entropy budget on transport), the cost of maintaining identity increases, reflecting the relativistic mass increase:

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

This aligns with the familiar experimental signature: the effective inertia of high-velocity particles increases, as does their resistance to further acceleration – not as a mysterious geometric effect but as the direct result of entropic resource depletion.

4. Emergence of the Effective Metric Tensor and Entropic Geodesics

4.1. Metric Tensor from Entropy Density, Flux, and Irreversible Production

In General Relativity, the metric tensor $g_{\mu\nu}$ is the object that encodes spacetime geometry – distances, angles, and causal structure. In ToE, the metric emerges from the entropic properties of the underlying field:

Entropy density: $s(x)$, local measure of configuration

Entropy flux: $j^\mu(x)$, the flow of entropy through spacetime

Irreversible entropy production: $\sigma(x)$, the rate of entropy generation due to irreversible processes

The effective ToE metric tensor is constructed as follows (conceptually):

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

This relates physical distances and durations to local entropy landscapes and their dynamics.

Explicit Example: Suppose entropy flow supports signal propagation at speed $c$ (from the flux-capacity relation). We then have the constraint

where $|j|$ is the magnitude of entropy flux and $s$ the entropy density. The set of admissible pairs $(s, j)$ forms an “entropic cone” that mirrors the light cone in relativity.

4.2. Entropic Geodesics: Replacing Spacetime Geodesics

In General Relativity, geodesics are curves which extremize the spacetime interval (the action for free particles):

[math]\displaystyle{ S = \int \sqrt{-g_{\mu\nu},\frac{dx^\mu}{d\lambda},\frac{dx^\nu}{d\lambda}};d\lambda. }[/math]

In ToE, the entropic geodesic is instead the curve (or motif evolution) that minimizes total entropy cost (rendering energy), often formalized as:

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

with $\chi_{\mu\nu}$ the entropic “metric” constructed from entropy density and flux.

The null entropic geodesic condition for massless propagation (photons, etc.) is:

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

Thus, in ToE, physical paths correspond to extremal entropy-cost trajectories, generalizing the principle of least action and replacing geodesics of curved spacetime with entropic optimal paths.

5. Mathematical Comparison: ToE vs. General Relativity

5.1. Action Principles and Field Equations

ToE Effective Action Example: One instantiation of an entropic action uses a relative entropy Lagrangian. For example, an action can be defined with a Lagrangian $L = \operatorname{Tr}[\hat{\rho}\ln \hat{\rho}^{-1}]$, and the total action $S = \frac{1}{G} \int \sqrt{|\det g|};L,dr$, with $g$ the background metric and $G$ the metric induced by entropic/matter fields. Modified Einstein Equations in ToE: An example modified field equation incorporating entropic effects is:

[math]\displaystyle{ R^{(G)}{\mu\nu} - \frac{1}{2} g{\mu\nu} \Big( R^{(G)} - 2\Lambda_G \Big) + D_{\mu\nu} = \kappa, T_{\mu\nu} \tag{16} }[/math]

where $R^{(G)}{\mu\nu}$ is the Ricci tensor of the emergent entropic metric $G$, $\Lambda_G$ a cosmological term in the entropic sector, and $D{\mu\nu}$ encodes new entropic contributions.

5.2. Metric Tensor Comparison

In GR, the metric tensor is the primary field. In ToE, the metric tensor is emergent from local entropy configurations. For a symmetric, positive-definite entropy field, the equivalent metric may be reconstructed via an inner product of entropy gradients and flux (ensuring compatibility with standard distances and causal structure when entropy gradients are small). This allows for new behavior, e.g., emergent metrics in highly non-equilibrium situations.

6. Experimental and Observational Validation of ToE

6.1. Perihelion Precession of Mercury

General Relativity predicts a perihelion shift of Mercury’s orbit of

[math]\displaystyle{ \Delta\theta = \frac{6\pi G M}{a,c^2(1 - e^2)}, }[/math]

which corresponds to about 43 arcseconds per century. ToE achieves the same value by introducing higher-order entropy corrections to Newtonian gravity. The ToE derivation integrates inputs from the Unruh effect, Hawking temperature, Bekenstein–Hawking entropy, the holographic principle, and the Binet equation to arrive at:

[math]\displaystyle{ \Delta\theta_{\text{ToE}} = \Delta\theta_{\text{GR}} = 43'' }[/math] (arcseconds) per century, without invoking spacetime curvature directly but rather entropy-driven geodesic deviations.

6.2. Solar Starlight Deflection

General Relativity predicts that starlight grazing the Sun is deflected by

[math]\displaystyle{ \delta = \frac{4GM}{c^2 R_\odot}, }[/math]

which is about 1.75 arcseconds. ToE recovers this result via an entropic variational principle that constrains particle trajectories with an entropic coupling constant η, yielding, for null geodesics, $\delta_{\text{ToE}} \approx \delta_{\text{GR}}$, matching the empirical values observed during solar eclipses.

6.3. Time Dilation and Length Contraction

Experimental evidence for time dilation (muon lifetimes, atomic clocks in motion) and length contraction (indirectly observed via density and cross-section changes in particle accelerator experiments) align perfectly with the ToE predictions, as these arise from the universal entropic capacity constraint.

6.4. Speed of Quantum Collapse

ToE predicts a finite, nonzero delay for information and entanglement propagation by the No-Rush Theorem, with the fastest formation rate for entanglement in experiments on the order of 232 attoseconds. These results are consonant with recent high-precision experiments and are incompatible with models permitting instantaneity.

7. Related Mathematical and Conceptual Frameworks

7.1. Entropic Curvature on Graph Spaces

Recent work on entropic curvature in discrete spaces (e.g., via Schrödinger bridges and mean-field dynamics on graphs) reinforces ToE’s postulate that curvature is an emergent feature of entropy flows and concentration properties – even outside continuous manifolds. Criteria from these works include: (i) $C$-displacement convexity for entropy along interpolating paths, and (ii) gradient flows of free energy as discrete analogues of geodesics.

7.2. Symbolic Collapse Grammar and Entropic Rendering

The symbolic approach proposes that reality emerges from a grammatical rendering of motif strings filtered by entropy cost. Only motifs whose cumulative entropy cost is below a threshold survive collapse. This paradigm reproduces time’s arrow, quantum measurement, and relativistic effects via motif persistence, rendering, and collapse filtering dynamics.

7.3. Information-Entropy Frameworks and Spacetime Emergence

Information-entropy frameworks (IEF) and the Information-Entropic Spacetime Emergence (IESE) theory connect entropic cost and confirmation rate directly to the dynamics of wave functions, metric formation, and cosmic expansion – bypassing explicit time parameters and unifying quantum, thermodynamic, and geometric phenomena.

8. Empirical and Observational Tests: Current Status and Future Prospects

8.1. Experimental Confirmations

Table: Summary of Experimental Confirmations for ToE

Extensive experiments demonstrate that ToE’s entropic reinterpretation upholds all observed relativistic effects and, crucially, offers potential routes to detect deviations if, for example, maximal entropic propagation deviates minutely from $c$ in extreme contexts.

8.2. Prospects for Distinguishing ToE from GR

Novel Mass-Ratio Scaling: ToE predicts mass hierarchies via motif depth and entropy scaling (e.g., $m \sim \lambda^N$), suggesting a testable structure in particle physics.

Collapse Thresholds: The entropic rendering framework suggests experiments measuring the minimum timescale for quantum collapse and phase transitions may yield deviations in highly non-equilibrium or ultra-high-energy contexts.

Cosmological Signatures: ToE’s entropy-driven expansion and emergent cosmological constant could resolve the dark energy problem without requiring new fields – an idea now under scrutiny via cosmic surveys.

9. Survey of ToE Publications, Authors, and Influential Works

John Onimisi Obidi: Founding proponent of ToE; major contributions on entropy as a field, perihelion precession, and collapse speed constraints.

Travis S. Taylor: Advances symbolic collapse grammar and entropy filtering as basis for quantum and relativistic emergence.

Martin Rapaport & Paul-Marie Samson: Mathematical formalization of entropic curvature on graph spaces.

Sean Hellman: Development of the Information-Entropy Framework (IEF) for unification and emergence of geometry from information.

Ginestra Bianconi: "Gravity from Entropy" – relative entropy-based gravitational action.

Additional supporting works bridging entropy, geometry, and information include volume integrals over entropic metrics, analyses of the entropy cost of information confirmation, and resource-theoretic approaches to quantum thermodynamics.

10. Physical and Philosophical Interpretation

10.1. Geometry Emerges from Entropy

In ToE, geometry and its curvature are not fundamental properties but summarize the entropic “resistance landscape” experienced by systems. The structure of spacetime – its light cones, null intervals, and causal relationships – arises from how entropy is produced, transported, and accumulated across configuration space.

10.2. Causality, the Arrow of Time, and the No-Rush Theorem

The Entropic Axiom of Reality roots the unidirectionality (arrow) of time in the requirement that all evolution proceeds via entropy increase; processes must wait for entropy conditions to propagate. This provides a thermodynamic explanation for the origin of temporal asymmetry and causal structure, distinct from geometric postulation.

10.3. Information, Collapse, and Reality

Physical existence is tied to collapse-resistance in entropic grammar: what is rendered as observable is what survives the symbolic and physical collapse rules against entropic filtering. This insight bridges quantum measurement, decoherence, and macroscopic emergence in a single framework.

11. Conclusions on the Foregoing Discussions on ToE

The Theory of Entropicity (ToE) provides a rigorous, unifying, and physically motivated framework that meets the empirical demands of relativity while delivering a deeper, entropic basis for the emergence of geometry and relativistic effects. It accomplishes this through:

The Entropic Curvature Hypothesis delimiting the emergence of spacetime geometry from entropy flows.

The Entropic Axiom of Reality enforcing finite, causality-preserving update rates.

A mathematical formalism that reproduces all standard relativistic effects via entropic capacity and flux constraints.

The emergent metric tensor and entropic geodesics, showing how spacetime intervals and equations of motion arise from entropy, not from a priori geometry.

Empirical and mathematical concordance with all confirmed relativistic experimental results, including new predictions (e.g., finite quantum collapse time) and potential deviations in high-entropy-gradient regimes.

Synergy with recent work in entropic gravity, symbolic emergence, and information-geometry unification.

While ToE currently matches GR in all experimentally tested predictions, the framing it supplies opens new avenues for testing, especially outside equilibrium and in extreme entropy landscapes. Its capacity to connect quantum foundations, thermodynamics, relativity, and cosmology via a single entropic principle makes it not merely a reinterpretation, but a leading candidate for a true theory of fundamental physical emergence.

Acknowledgments

This research synthesis is based on extensive review and integration of published and preprint works on the Theory of Entropicity (ToE), emergent gravity, informational geometry, and entropic rendering grammars, alongside the latest experimental confirmations and mathematical analyses as detailed in the referenced materials. Further progress will depend on dedicated theoretical development, mathematical rigor, and innovative experimental design to probe the entropic foundations of the universe.