Physics:Theory of Entropicity(ToE) and the Emergence of Relativistic Geometry

Short description: Theory of Entropicity (ToE) and the emergence of relativistic geometry from an entropic field

Abstract

The Theory of Entropicity(ToE), as first formulated and developed by John Onimisi Obidi, proposes a structural inversion of relativistic ontology: spacetime curvature is not fundamental; rather, it emerges from the organization and dynamics of a universal entropic field. This article formalizes the Entropic Curvature Hypothesis (ECH), states the Entropic Axiom of Reality (EAR), and derives core relativistic kinematics—time dilation, length contraction, and velocity‑dependent inertia (“mass increase”)—from entropic capacity constraints. A Minkowski‑like invariant arises as an expression of capacity conservation, and the standard Lorentz transformations follow from invariance and linearity. We outline how effective spacetime metrics emerge from entropy density, flux, and production, and we propose programmatic tests that distinguish entropic curvature from purely geometric curvature.

Introduction

In General Relativity (GR), mass–energy curves spacetime, and free bodies follow geodesics of that curved manifold. The Theory of Entropicity (ToE) proposes a complementary inversion: the entropic field—defined by entropy density, flux, and local production—has primary structure (curvature), and the familiar spacetime geometry is the emergent bookkeeping of that structure. In ToE, time is the count of entropic events, distance is entropic separation, and geometry is the organization of irreversible entropic flows. This perspective aims to explain relativistic effects (time dilation, length contraction, and velocity‑dependent inertia) as consequences of a finite, conserved entropic capacity that must be shared between internal evolution and transport (motion), under a finite propagation limit of entropic influence.

Entropic curvature hypothesis (ECH)

Statement

The primary curvature in nature is the curvature of the entropic field. The familiar spacetime curvature of GR emerges as an effective description of this underlying entropic structure.

Operational reinterpretations

• Elapsed time ↔ entropic increment: An operational tick corresponds to a finite entropic event; without nonzero entropic increment, “time” is undefined.
• Spatial separation ↔ entropic separation: Distance registers the entropic reconfiguration required to connect states along an entropic geodesic (path of least entropic resistance).
• Dynamics ↔ entropic geodesics: Free motion extremizes an entropic action, appearing as geodesics of an emergent effective metric.

Entropic axiom of reality (EAR)

Overarching principle

Instantaneous interactions are impossible; every interaction must pay an entropy cost. This implies a finite characteristic propagation speed of entropic influence and a conserved update capacity per system.

Consequences

1. No entropy, no time: Without finite entropic increments, there are no ticks; operational time is undefined.
2. No entropic gradients, no geometry: In the absence of gradients in entropy density/flux, the effective metric is trivial/degenerate.
3. With gradients, geometry emerges: Stable patterns of entropic flow induce an effective metric interpreted as spacetime curvature.

Entropic genesis of metric

ToE elevates entropy from statistic to field; the effective metric summarizes how flows organize under constraints and irreversibility. Heuristically:

[math]\displaystyle{ g_{\mu\nu}^{\text{eff}} \;\sim\; \mathcal{F}\!\left[\, \partial_\mu \Lambda \, \partial_\nu \Lambda,\; J_\mu,\; \Pi_{\mu\nu}^{(\mathrm{irr})} \,\right], }[/math]

where [math]\displaystyle{ \Lambda }[/math] is an entropy density functional, [math]\displaystyle{ J_\mu }[/math] an entropy flux, and [math]\displaystyle{ \Pi_{\mu\nu}^{(\mathrm{irr})} }[/math] encodes irreversible production and constraint asymmetry. Observables such as gravitational redshift, lensing, and time dilation are read as consequences of variations in this entropic structure.

Notation and primitives

• [math]\displaystyle{ \sigma \gt 0 }[/math]: invariant total entropic capacity of a system (per unit intrinsic evolution).
• [math]\displaystyle{ R_{\text{int}} }[/math]: internal update rate (per unit lab time).
• [math]\displaystyle{ R_{\text{move}} }[/math]: transport update rate associated with uniform translation (per unit lab time).
• [math]\displaystyle{ u_\ast }[/math]: finite characteristic propagation speed of entropic influence (an experimental property of the entropic field).
• [math]\displaystyle{ t }[/math]: lab frame time; [math]\displaystyle{ \tau }[/math]: system’s proper (intrinsic) time.
• [math]\displaystyle{ \mathbf{v} = d\mathbf{x}/dt }[/math]: velocity in the lab frame; [math]\displaystyle{ v = \|\mathbf{v}\| }[/math].
• Occupancies (normalized channel usage): [math]\displaystyle{ a \equiv R_{\text{int}}/\sigma }[/math], [math]\displaystyle{ b \equiv R_{\text{move}}/\sigma }[/math].

Derivations: capacity, dilation, and Lorentz symmetry from the Theory of Entropicity(ToE)

Capacity budget (independence and conservation)

ToE posits orthogonal (independent) channels and a conserved total capacity:

[math]\displaystyle{ \sigma^2 \;=\; R_{\text{int}}^2 \;+\; R_{\text{move}}^2. \tag{1} }[/math]

Normalize by [math]\displaystyle{ \sigma^2 }[/math] to obtain the unit‑circle constraint in 2D capacity space:

[math]\displaystyle{ a^2 \;+\; b^2 \;=\; 1. \tag{2} }[/math]

Transport occupancy and finite propagation

By extensivity (uniform motion scales demand linearly) and saturation at the entropic propagation limit,

[math]\displaystyle{ b \;=\; \frac{v}{u_\ast},\qquad 0 \le v \lt u_\ast. \tag{3} }[/math]

Internal occupancy and proper time

By definition of intrinsic evolution, for any internal state variable [math]\displaystyle{ S }[/math] advancing uniformly with proper time,

[math]\displaystyle{ \frac{dS}{d\tau} \;=\; \sigma, \quad \frac{dS}{dt} \;=\; R_{\text{int}} \;\Rightarrow\; R_{\text{int}} \;=\; \sigma\,\frac{d\tau}{dt}. \tag{4} }[/math]

Hence

[math]\displaystyle{ a \;=\; \frac{R_{\text{int}}}{\sigma} \;=\; \frac{d\tau}{dt}. \tag{5} }[/math]

Time dilation and invariant interval

Combine (2), (3), and (5):

[math]\displaystyle{ \left(\frac{d\tau}{dt}\right)^2 \;+\; \left(\frac{v}{u_\ast}\right)^2 \;=\; 1 \;\;\Rightarrow\;\; \frac{d\tau}{dt} \;=\; \sqrt{1 - \frac{v^2}{u_\ast^2}},\quad \gamma_{\mathrm{e}} \;\equiv\; \frac{dt}{d\tau} \;=\; \frac{1}{\sqrt{1 - v^2/u_\ast^2}}. \tag{6} }[/math]

Multiplying by [math]\displaystyle{ u_\ast^2 dt^2 }[/math] and using [math]\displaystyle{ v^2 dt^2 = d\mathbf{x}^2 }[/math] yields the ToE‑native Minkowski-like invariant:

[math]\displaystyle{ u_\ast^2\,d\tau^2 \;=\; u_\ast^2\,dt^2 \;-\; d\mathbf{x}^2. \tag{7} }[/math]

Lorentz transformations from invariance imposed by the Theory of Entropicity(ToE)

Assume linearity between inertial frames [math]\displaystyle{ S }[/math] and [math]\displaystyle{ S' }[/math] with relative speed [math]\displaystyle{ v }[/math] along [math]\displaystyle{ x }[/math]:

[math]\displaystyle{ x' \;=\; A x + B t,\qquad t' \;=\; D x + E t,\qquad y'=y,\; z'=z. \tag{8} }[/math]

Origin worldline of [math]\displaystyle{ S' }[/math] satisfies [math]\displaystyle{ x' = 0 }[/math] with [math]\displaystyle{ x = vt }[/math]:

[math]\displaystyle{ 0 = A (vt) + B t \;\Rightarrow\; B = -A v. \tag{9} }[/math]

Impose invariance of (7):

[math]\displaystyle{ u_\ast^2 t'^2 - x'^2 \;=\; u_\ast^2 t^2 - x^2. \tag{10} }[/math]

Substitute (8)–(9) and equate coefficients of [math]\displaystyle{ x^2 }[/math], [math]\displaystyle{ xt }[/math], [math]\displaystyle{ t^2 }[/math], giving

[math]\displaystyle{ u_\ast^2 D^2 - A^2 = -1,\qquad u_\ast^2 D E = -A^2 v,\qquad u_\ast^2 E^2 - A^2 v^2 = u_\ast^2. \tag{11} }[/math]

A solution consistent with reciprocity and continuity at [math]\displaystyle{ v=0 }[/math] is

[math]\displaystyle{ D = -\frac{A v}{u_\ast^2},\qquad E = A,\qquad A = \gamma_{\mathrm{e}} = \frac{1}{\sqrt{1 - v^2/u_\ast^2}}. \tag{12} }[/math]

Thus the Lorentz–ToE transformation is given as:

[math]\displaystyle{ \begin{aligned} x' &= \gamma_{\mathrm{e}}\,\big(x - v t\big),\\ t' &= \gamma_{\mathrm{e}}\,\left(t - \frac{v}{u_\ast^2}\,x\right),\\ y' &= y,\;\; z' = z. \end{aligned} \tag{13} }[/math]

Relativistic kinematics (operational)

Time dilation (operational)

A clock at rest in [math]\displaystyle{ S' }[/math] has [math]\displaystyle{ dx'=0 }[/math]. Using (13) and [math]\displaystyle{ dx = v\,dt }[/math] on that worldline:

[math]\displaystyle{ dt' \;=\; \gamma_{\mathrm{e}}\!\left(1 - \frac{v^2}{u_\ast^2}\right) dt \;=\; \frac{dt}{\gamma_{\mathrm{e}}} \;\Rightarrow\; \frac{dt}{d\tau} \;=\; \gamma_{\mathrm{e}}. \tag{14} }[/math]

Length contraction

A rigid rod at rest in [math]\displaystyle{ S' }[/math] has proper length [math]\displaystyle{ L_0 = \Delta x' }[/math] measured at equal [math]\displaystyle{ t' }[/math]. In [math]\displaystyle{ S }[/math], measure at equal [math]\displaystyle{ t }[/math] ([math]\displaystyle{ \Delta t = 0 }[/math]). From (13):

[math]\displaystyle{ \Delta x' \;=\; \gamma_{\mathrm{e}}\,(\Delta x - v\,\Delta t) \;=\; \gamma_{\mathrm{e}}\,\Delta x \;\Rightarrow\; L \;=\; \Delta x \;=\; \frac{L_0}{\gamma_{\mathrm{e}}} \;=\; L_0 \sqrt{1 - \frac{v^2}{u_\ast^2}}. \tag{15} }[/math]

Dynamics and velocity‑dependent inertia

Action and Lagrangian from intrinsic evolution

For free motion, the action accumulates with intrinsic evolution (capacity consumption):

[math]\displaystyle{ S \;\propto\; \int d\tau \;=\; \int \sqrt{1 - \frac{v^2}{u_\ast^2}}\,dt. \tag{16} }[/math]

Choose

[math]\displaystyle{ L(v) \;=\; - m_0\,u_\ast^2\,\sqrt{1 - \frac{v^2}{u_\ast^2}}, \tag{17} }[/math]

so that the Newtonian limit fixes [math]\displaystyle{ m_0 }[/math] as the rest mass.

Momentum and energy

Canonical momentum:

[math]\displaystyle{ \mathbf{p} \;=\; \frac{\partial L}{\partial \mathbf{v}} \;=\; \frac{m_0\,\mathbf{v}}{\sqrt{1 - v^2/u_\ast^2}} \;=\; \gamma_{\mathrm{e}}\,m_0\,\mathbf{v}. \tag{18} }[/math]

Energy via Legendre transform:

[math]\displaystyle{ E \;=\; \mathbf{v}\!\cdot\!\mathbf{p} - L \;=\; \gamma_{\mathrm{e}} m_0 u_\ast^2. \tag{19} }[/math]

Low‑speed expansion of (17):

[math]\displaystyle{ L \;=\; -m_0 u_\ast^2 + \tfrac{1}{2} m_0 v^2 + O(v^4), \tag{20} }[/math]

identifying [math]\displaystyle{ m_0 }[/math] as Newtonian mass and [math]\displaystyle{ E_0 = m_0 u_\ast^2 }[/math] as rest energy (offset).

Effective “mass increase”

Define velocity‑dependent inertia by [math]\displaystyle{ \mathbf{p} = m_{\text{eff}}(v)\,\mathbf{v} }[/math]:

[math]\displaystyle{ m_{\text{eff}}(v) \;=\; \gamma_{\mathrm{e}}\,m_0 \;=\; \frac{m_0}{\sqrt{1 - v^2/u_\ast^2}}, \qquad E \;=\; m_{\text{eff}}(v)\,u_\ast^2. \tag{21} }[/math]

Matter, curvature, and entropy: who sources what?

• GR view: Mass–energy (stress–energy [math]\displaystyle{ T_{\mu\nu} }[/math]) sources curvature; curvature guides motion along spacetime geodesics.
• ToE view: Constraint asymmetry and irreversible production source the entropic field; entropic flows organize trajectories; the observed metric is the effective ledger of these flows.

Framework	Source term	Governing law	Physical path	Observables
General Relativity	Stress–energy [math]\displaystyle{ T_{\mu\nu} }[/math]	Einstein field equations	Spacetime geodesics	Time dilation, gravitational lensing
Theory of Entropicity	Constraint asymmetry + entropy production	Obidi Action / Vuli Ndlela Integral (entropic field equations)	Entropic geodesics (least entropic resistance)	Tick rates (ETL), entropic lensing, emergent metric

Programmatic tests and predictions

• Clock response to engineered entropic environments: Predict and measure ETL shifts (entropic tick‑length) under controlled constraint asymmetries.
• Entropic lensing: Quantify trajectory deflection from calibrated entropy gradients; compare with GR predictions in matched regimes.
• Geodesic selection under dissipation control: Track path changes as irreversibility parameters are tuned.
• Metric emergence from entropy maps: Reconstruct [math]\displaystyle{ g_{\mu\nu}^{\text{eff}} }[/math] from measured [math]\displaystyle{ \Lambda }[/math], [math]\displaystyle{ J_\mu }[/math], and [math]\displaystyle{ \Pi_{\mu\nu}^{(\mathrm{irr})} }[/math].
• Tomography of the entropic field: Use networks of clocks and transport probes to infer curvature of the entropic field.

See also

• Relativity
• Thermodynamics
• Information geometry
• Geodesic (general relativity)
• Statistical field theory

External links

• Official resources and updates on the Theory of Entropicity (ToE): https://handwiki.org/wiki/Category:Theory_of_Entropicity