How Does the Theory of Entropicity (ToE) Derive the Lorentz Factor of Einstein's Relativity from Entropic Invariants?

The Theory of Entropicity (ToE) derives the Lorentz factor $$\gamma_e = \left(1 - \frac{v^2}{c_e^2}\right)^{-1/2}$$ of Einstein's Relativity from the invariance of the entropic cone $$(c_e s_0)^2 - (v s)^2 = \text{const}$$, where $$c_e$$ is the entropic propagation speed and $$s$$ is entropy density.[1]

## Entropic cone invariant

The core postulate of Obidi's Theory of Entropicity (ToE) is the quadratic form $$(c_e s_0)^2 - j^2 = \text{const}$$, a Minkowski-like pseudo-norm on the entropic 2-vector $$(c_e s, j)$$, with $$j = v s$$ as entropy flux and $$s_0$$ the rest-frame density.[1] Boosts preserving this invariant require the standard Lorentz group structure, yielding $$\gamma_e$$ explicitly from the transformation matrix that maps rest-frame $$(c_e s_0, 0)$$ to moving-frame $$(c_e s(v), v s(v))$$.[1]

## Entropy density boost

Solving the invariant gives $$s(v) = \gamma_e s_0$$, where $$\gamma_e$$ emerges as the scaling needed to satisfy the ToE cone equation under motion.[1] This is not assumed but solved for: motion increases density by the factor that keeps total entropy conserved while respecting the finite $$c_e$$.

## Consistency of ToE's Kinematic derivation across effects

The same $$\gamma_e$$ then governs all relativistic kinematics via conservation:

- Mass: $$m(v) = \gamma_e m_0$$ ($$m \propto s$$).

- Length: $$L(v) = L_0 / \gamma_e$$ ($$\Sigma = s L = \text{const}$$).

- Time: $$\tau(v) = \gamma_e \tau_0$$ (fixed $$\Delta S$$ per cycle).[1]

This closes under velocity addition $$\beta_{e,\text{tot}} = (\beta_1 + \beta_2)/(1 + \beta_1 \beta_2)$$, confirming the group structure from entropic principles alone.[1]

Citations:

[1] The Theory of Entropicity (ToE) Derives and Explains Mass ...www.cambridge.org › coe › assets › orp › resource › item › original › the-... https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/6900d89c113cc7cfff94ef3a/original/the-theory-of-entropicity-to-e-derives-and-explains-mass-increase-time-dilation-and-length-contraction-in-einstein-s-theory-of-relativity-to-r-to-e-applies-logical-entropic-concepts-and-principles-to-verify-einstein-s-relativity.pdf

[2] The Theory of Entropicity (ToE) Derives and Explains Mass Increase ... https://client.prod.orp.cambridge.org/engage/coe/article-details/6900d89c113cc7cfff94ef3a

[3] Special Relativity as an Emergent Symmetry of Entropy ... https://sciety-labs.elifesciences.org/articles/by?article_doi=10.20944%2Fpreprints202505.0078.v2

[4] The Relativistic Boltzmann Equation and Two Times https://pmc.ncbi.nlm.nih.gov/articles/PMC7818102/

[5] Derivations of the Lorentz transformations https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations

[6] Physics: Deriving the Lorentz Factor from Scratch https://www.youtube.com/watch?v=45mjNAWJ5iY

[7] 5.6: The Lorentz Transformation https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/05:__Relativity/5.06:_The_Lorentz_Transformation

[8] How is the Lorentz Factor in special relativity derived? https://www.reddit.com/r/askscience/comments/8hfytt/how_is_the_lorentz_factor_in_special_relativity/