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Theory of Entropicity (ToE)

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  1. Foundations of the Theory of Entropicity (ToE): Ambitious and Promising at Once
  2. A Rigorous Derivation of Newton’s Laws from the Obidi Curvature Invariant (OCI = ln 2) of ToE
  3. Power and Significance of ln 2 in the Theory of Entropicity (ToE)
  4. On the Tripartite Foundations of the Theory of Entropicity (ToE): Prolegomena to Physics
  5. Derivation of the ToE Curvature Invariant ln 2 Using Convexity and KL (Araki-Umegaki) Divergence
  6. On the Foundational and Unification Achievements of the Theory of Entropicity (ToE): From GR to QM and Beyond
  7. On the Unification Efforts of the Theory of Entropicity (ToE): Mathematical Expositions and Trajectory
  8. The Theory of Entropicity (ToE) as a New Foundational Edifice of Physics
  9. Monograph Architecture of the Theory of Entropicity (ToE)
  10. Iterative Solutions of the Complex Obidi Field Equations (OFE) of the Theory of Entropicity (ToE)

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Entropic Derivation of Relativistic Length Contraction and Mass Increase in the Theory of Entropicity (ToE)

How the Theory of Entropicity (ToE) Derives Einstein’s Relativistic Length Contraction and Mass Increase

Entropic reinterpretation of relativistic kinematics

The Theory of Entropicity (ToE) provides a unified derivation of relativistic length contraction and relativistic mass increase by treating them as consequences of entropy conservation and entropy redistribution within a fundamental entropic field, rather than as independent kinematic postulates of spacetime geometry. In this framework, physical objects are characterized by an entropy density \(s\) defined with respect to the entropic field, and relativistic effects arise from how this density transforms under motion while preserving appropriate entropic invariants. The usual Lorentz factor appears as an entropic amplification factor, encoding how motion modifies the local entropy density and thereby constrains spatial extension and inertial response.

Shared entropic foundation and the entropic cone relation

Both length contraction and mass increase in ToE are derived from a single underlying relation, the entropic cone relation, which plays a role analogous to the Minkowski norm in special relativity. This relation connects the rest entropy density, the moving entropy density, and the entropic speed limit. It is expressed as

\[ (c_e s_0)^2 - (v s)^2 = \text{const}, \]

where \(c_e\) is the entropic speed limit, empirically coinciding with the observed speed of light \(c\), \(s_0\) is the rest entropy density, and \(s(v)\) is the entropy density of the same system when it moves with velocity \(v\). The constant on the right‑hand side encodes an invariant entropic quantity associated with the object, analogous to an invariant norm in relativistic kinematics.

Solving this relation for the moving entropy density yields

\[ s(v) = \frac{s_0}{\sqrt{1 - \dfrac{v^2}{c_e^2}}} = \gamma_e \, s_0, \]

where

\[ \gamma_e = \left( 1 - \frac{v^2}{c_e^2} \right)^{-1/2} \]

is the entropic Lorentz factor. This factor quantifies how motion amplifies the local entropy density of a system relative to its rest value. The same \(\gamma_e\) will appear in the derivations of length contraction, mass increase, and time dilation, providing a unified entropic origin for all three relativistic effects.

Length contraction as entropic density compensation

Consider a one‑dimensional rod at rest in its own frame, with rest length \(L_0\) and rest entropy density \(s_0\). The total entropy associated with the rod in this frame is then

\[ \Sigma = s_0 \, L_0, \]

where \(\Sigma\) denotes the total entropy of the rod along its length. In the ToE framework, this total entropy is treated as a conserved quantity under changes of inertial frame, provided no internal processes or external interactions alter the entropic content of the system.

When the rod moves with velocity \(v\) relative to a given frame, its entropy density is no longer \(s_0\) but is boosted according to the entropic cone relation:

\[ s(v) = \gamma_e \, s_0. \]

Let \(L(v)\) denote the length of the rod measured in the frame where it moves with velocity \(v\). Conservation of total entropy requires that

\[ \Sigma = s_0 L_0 = s(v) L(v). \]

Substituting the boosted entropy density into this relation gives

\[ s_0 L_0 = \gamma_e s_0 L(v), \]

and, assuming \(s_0 \neq 0\), one obtains

\[ L(v) = \frac{L_0}{\gamma_e}. \]

This is precisely the standard relativistic length contraction formula, but here it is derived as a consequence of entropy conservation in the presence of a motion‑induced increase in entropy density. The rod’s length must contract in the direction of motion to compensate for the increased entropy density and preserve the total entropic content \(\Sigma\). In this interpretation, length contraction is not a primitive geometric postulate but an emergent effect of how the entropic field redistributes and concentrates entropy under motion.

Relativistic mass increase as entropic inertia amplification

In the Theory of Entropicity, mass is not treated as a fundamental, irreducible parameter but is linked to the entropy density of a system through an entropic constitutive relation. This relation expresses the idea that inertia arises from the entropic resistance of the underlying field to changes in the state of the system. At the simplest level, this is captured by a proportionality of the form

\[ m \propto s, \]

where \(m\) is the mass and \(s\) is the entropy density. For a system at rest, one has a rest mass \(m_0\) associated with the rest entropy density \(s_0\). When the system moves with velocity \(v\), the entropy density is boosted to \(s(v) = \gamma_e s_0\), and the same constitutive relation implies that the mass becomes

\[ m(v) = \gamma_e \, m_0. \]

This is the familiar relativistic mass increase formula, but here it is interpreted as a direct consequence of the entropic amplification of the system’s internal resistance to acceleration. As the velocity approaches the entropic speed limit \(c_e\), the entropy density and hence the effective inertia grow without bound, reflecting the increasing difficulty of further accelerating the system. In this picture, the relativistic mass increase is not an arbitrary kinematic rule but a manifestation of the deeper entropic structure of matter and motion.

Unified entropic trade‑offs and the Entropic Resistance Principle

The derivations of length contraction and mass increase in ToE are not isolated constructions; they are part of a broader unifying principle known as the Entropic Resistance Principle (ERP). This principle states that motion redistributes the available entropy flux of a system between different functional roles: internal cycles (such as those defining clocks and structural stability) and external propulsion. As velocity increases, a larger fraction of the entropic capacity is effectively “diverted” toward sustaining motion, leaving less available for internal processes and spatial extension.

Within this framework, time dilation, length contraction, and mass increase are three complementary manifestations of the same underlying entropic trade‑off. Time dilation reflects the slowing of internal cycles as entropic resources are reallocated; length contraction reflects the compression of spatial extension required to maintain a fixed total entropy under increased density; and mass increase reflects the growth of entropic inertia as the system’s state becomes more resistant to further change. The common factor \(\gamma_e\) arises naturally from the entropic cone relation and governs all three effects simultaneously.

This entropic unification contrasts with the traditional geometric derivations in special relativity, which typically rely on separate thought experiments involving light clocks, moving rods, and energy–momentum relations. In ToE, the same entropy field dynamics and conservation laws account for all relativistic kinematic effects in a single conceptual framework, rooted in the finite propagation speed and redistribution properties of the entropic field.

Entropic summary of relativistic effects

Effect Conserved or controlling quantity Consequence of \(s(v) = \gamma_e s_0\)
Length contraction Total entropy \(\Sigma = s L\) of the object along its length, treated as invariant under changes of inertial frame in the absence of internal processes. The boosted entropy density \(s(v) = \gamma_e s_0\) requires the length to adjust as \[ L(v) = \frac{L_0}{\gamma_e}, \] so that \(\Sigma = s_0 L_0 = s(v) L(v)\) remains constant.
Mass increase Entropy–mass mapping via the constitutive relation \(m \propto s\), linking inertia to entropic resistance. The same density boost \(s(v) = \gamma_e s_0\) implies \[ m(v) = \gamma_e m_0, \] so the effective mass increases with velocity as a direct reflection of enhanced entropic inertia.
Time dilation Entropy per cycle of an internal clock, with each “tick” associated with a minimal entropic increment \(\Delta S\). The entropic resistance to rapid reconfiguration leads to \[ \tau(v) = \gamma_e \tau_0, \] where \(\tau_0\) is the proper time interval per cycle and \(\tau(v)\) is the dilated interval in the moving frame.

References

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    Comprehensive encyclopedia‑style entry introducing the conceptual, mathematical, and ontological structure of the Theory of Entropicity (ToE).
    https://grokipedia.com/page/Theory_of_Entropicity
  2. Grokipedia — John Onimisi Obidi
    Scholarly profile of John Onimisi Obidi, originator of the Theory of Entropicity (ToE), including philosophical and historical motivation, background and research contributions.
    https://grokipedia.com/page/John_Onimisi_Obidi
  3. Google Blogger — Live Website on the Theory of Entropicity (ToE)
    Public‑facing platform containing explanatory essays, conceptual introductions, and updates on the Theory of Entropicity (ToE).
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  4. LinkedIn — Theory of Entropicity (ToE)
    Professional organizational page providing institutional updates and academic outreach related to the Theory of Entropicity (ToE).
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  5. Medium — Theory of Entropicity (ToE)
    Collection of essays and conceptual expositions on the Theory of Entropicity (ToE).
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  6. Substack — Theory of Entropicity (ToE)
    Serialized research notes, essays, and public communications on the Theory of Entropicity (ToE).
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    Indexed scholarly profile and research presence for the Theory of Entropicity (ToE) within the SciProfiles ecosystem.
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    Editorially curated scientific encyclopedia entry, documenting the Theory of Entropicity (ToE)'s conceptual, philosophical, and mathematical structures.
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  9. Encyclopedia.pub — Theory of Entropicity (ToE): Path to Unification of Physics and the Laws of Nature
    A formally maintained, technically curated scientific encyclopedia entry, presenting an expansive overview of the Theory of Entropicity (ToE)'s conceptual, philosophical, and mathematical foundations.
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    Academic papers, drafts, and research notes on the Theory of Entropicity (ToE) hosted on Academia.edu .
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    Principal Figshare repository link for research outputs on the Theory of Entropicity (ToE).
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  13. OSF (Open Science Framework)
    Open‑access repository hosting research materials, datasets, and papers related to the Theory of Entropicity (ToE).
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  14. ResearchGate — Publications on the Theory of Entropicity (ToE)
    Indexed research outputs, citations, and academic interactions related to the Theory of Entropicity (ToE).
    https://www.researchgate.net/search.Search.html?query=John+Onimisi+Obidi&type=publication
  15. Social Science Research Network (SSRN)
    Indexed scholarly works and papers on the Theory of Entropicity (ToE) within the SSRN research repository.
    https://papers.ssrn.com/sol3/cf_dev/AbsByAuth.cfm?per_id=7479570
  16. International Journal of Current Science Research and Review (IJCSRR)
    Peer‑reviewed publication relevant to the Theory of Entropicity (ToE).
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  17. Cambridge University — Cambridge Open Engage (COE)
    Early research outputs and working papers hosted on Cambridge University’s open research dissemination platform.
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  18. GitHub Wiki — Theory of Entropicity (ToE)
    Open‑source technical wiki, documenting the canonical structure, equations, and formal development of the Theory of Entropicity (ToE).
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  19. Canonical Archive of the Theory of Entropicity (ToE)
    Authoritative, version‑controlled archive of the full Theory of Entropicity (ToE) monograph, including derivations and formal definitions.
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