The Speed of Light c in Einstein's Theory of Relativity (ToR) is Derived as a Consequence of the Entropic Field of the Theory of Entropicity (ToE): Relativistic Entropic Consequences of the Obidi Action and the No-Rush Theorem (NRT) of ToE
1. Introduction
Einstein’s special relativity treats the invariance of the speed of light $$c$$ as a primitive postulate that, together with the relativity principle, determines the kinematics of spacetime. In contrast, the Theory of Entropicity (ToE) elevates entropy $$S(x)$$ to a fundamental dynamical field and derives the universal speed limit as an emergent property of the entropy field’s finite propagation rate. [1][2][3][4]
In this paper we provide a self-contained, variational and field-theoretic derivation of the maximal propagation speed from the Obidi Action and the Master Entropic Equation (MEE). We then prove an entropic No‑Rush Theorem establishing that no physical process can exceed this entropic propagation rate, and we show that the emergent maximal speed $$v_{\max} = \sqrt{\chi_0/C_0}$$ coincides with the relativistic speed of light $$c = 1/\sqrt{\mu_0\varepsilon_0}$$. [1][2][3][4]
2. The Obidi Action and Entropic Field Dynamics
2.1 Entropy as a dynamical scalar field
ToE promotes entropy to a continuous scalar field $$S : \mathcal{M} \to \mathbb{R}$$ defined on a spacetime manifold $$(\mathcal{M},g_{\mu\nu})$$. The field $$S(x)$$ is not a coarse-grained statistic but the fundamental carrier of causality, motion, and gravitation. [1][2][3]
Gradients $$\nabla_\mu S$$ and their contractions encode the flow and curvature of “entropic geodesics,” from which the usual metric and dynamical structures of spacetime and matter emerge as effective descriptions. [1][2][3]
2.2 The Obidi Action
The Obidi Action $$ \mathcal{A}_{\text{Obidi}} $$ is postulated as the fundamental variational principle governing the entropic field and its coupling to matter: [1][2][3]
$$\mathcal{A}_{\text{Obidi}}[S,g,\Psi]=\int_{\mathcal{M}} \mathrm{d}^4x \,\sqrt{-g}\,\left[\mathcal{L}_{S}+\mathcal{L}_{\text{int}}+\mathcal{L}_{\text{matter}}(\Psi,g)\right],$$
where $$g = \det(g_{\mu\nu})$$, $$\Psi$$ collectively denotes matter fields, and the entropic sector is given by
$$\mathcal{L}_{S}=\frac{1}{2}\,A(S)\, g^{\mu\nu} \nabla_\mu S \nabla_\nu S-V(S),$$
$$\mathcal{L}_{\text{int}}=\eta\, S\, T^{\mu}{}_{\mu}(\Psi,g).$$
Here:
- $$A(S)$$ is an entropic kinetic coefficient encoding the effective “stiffness” of the entropic medium. [1][2]
- $$V(S)$$ is an entropy self-interaction potential. [1][2]
- $$\eta$$ is an entropic coupling constant to the trace of the matter stress–energy tensor $$T^{\mu}{}_{\mu}$$. [1][2]
This Lagrangian form aligns with expository descriptions of the Obidi Action and its role as the generator of the MEE, entropic geodesics, and the Entropy Potential Equation. [1][2][3]
2.3 Master Entropic Equation (MEE)
Varying $$\mathcal{A}_{\text{Obidi}}$$ with respect to $$S$$ yields the Master Entropic Equation: [1][2][3]
$$\frac{\delta \mathcal{A}_{\text{Obidi}}}{\delta S}=0\quad\Rightarrow\quad\nabla_\mu \big( A(S)\,\nabla^\mu S \big)-V'(S)+\eta\, T^{\mu}{}_{\mu}=\mathcal{J}_{\text{irr}}[S,\Psi],$$
where $$\mathcal{J}_{\text{irr}}$$ is a non-Hermitian or non-time-reversal-symmetric source term representing built-in irreversibility and enforcing the arrow of time at the level of the field equation. [1][2][3]
In appropriate limits, solutions of the MEE reproduce Einstein’s field equations and elements of quantum dynamics, reinforcing the claim that gravity and quantum phenomena are emergent entropic behaviors. [1][2][3][4]
3. Linearized Entropic Dynamics and Propagation Speed
To extract a propagation speed from the MEE, we consider small fluctuations of the entropic field around a homogeneous background solution.
3.1 Background and perturbations
Let $$S(x) = S_0 + \delta S(x)$$, where $$S_0$$ is a constant stationary solution of the MEE (or slowly varying on scales of interest), and $$|\delta S| \ll 1$$. We expand the coefficients around $$S_0$$: [1][4][3]
$$A(S) \approx A_0 + A_1 \delta S, \quad V'(S) \approx V'_0 + V''_0 \delta S,$$
with constants $$A_0 = A(S_0)$$, $$V'_0 = V'(S_0)$$, etc. In a locally inertial frame where $$g_{\mu\nu} \approx \eta_{\mu\nu} = \text{diag}(-1,1,1,1)$$ and matter sources are negligible or absorbed in the background, the MEE for $$\delta S$$ takes the schematic form
$$A_0\,\Box\,\delta S + \ldots = 0,$$
where $$\Box = -\partial_t^2 + \nabla^2$$, and ellipses denote lower-order and dissipative terms that do not alter the leading-order propagation speed. [1][4][3]
3.2 Constitutive entropic coefficients
ToE introduces effective constitutive coefficients $$\chi_0$$ and $$C_0$$ characterizing, respectively, the “entropic stiffness” and “entropic inertia” (or capacity) of the medium in the linear regime, analogous to permittivity and permeability in electromagnetism. [4][5][6]
A convenient parametrization is
$$C_0\,\partial_t^2 \delta S - \chi_0\,\nabla^2 \delta S + \ldots = 0,$$
which is the standard wave equation with characteristic speed
$$v_{\max}=\sqrt{\frac{\chi_0}{C_0}}.$$
Identifying $$C_0$$ and $$\chi_0$$ in terms of the underlying action parameters is a matter of detailed microphysical modeling of the entropic field; at the phenomenological level, ToE treats $$\chi_0$$ and $$C_0$$ as renormalized constants measurable via entropic propagation experiments. [4][5][6]
4. The No‑Rush Theorem
4.1 Statement of the theorem
We now formulate the entropic No‑Rush Theorem as a rigorous constraint on admissible solutions of the MEE:
Theorem (No‑Rush Theorem)
Consider the Theory of Entropicity defined by the Obidi Action $$\mathcal{A}_{\text{Obidi}}$$ and the resulting Master Entropic Equation for the entropy field $$S(x)$$ on a globally hyperbolic spacetime $$(\mathcal{M},g_{\mu\nu})$$. Assume:
1. The entropic kinetic term is strictly hyperbolic in the linearized regime, with effective coefficients $$\chi_0 > 0$$ and $$C_0 > 0$$.
2. The irreversibility term $$\mathcal{J}_{\text{irr}}$$ is local in time and does not introduce acausal advanced Green’s functions.
3. Matter couplings preserve the causal structure induced by the entropic field (i.e., they do not introduce higher-derivative instabilities or nonlocal-in-time interactions).
Then no physical disturbance of the entropic field, nor any signal conveyed by matter fields coupled to it, can propagate with front velocity exceeding
$$v_{\max}=\sqrt{\frac{\chi_0}{C_0}}.$$
In particular, there exists no solution of the full coupled entropic–matter system whose causal influence cone lies outside the entropic cone determined by $$v_{\max}$$. Thus, “nature cannot be rushed”: all causal processes are constrained to respect the entropic propagation limit $$v_{\max}$$. [4][3]
4.2 Sketch of proof
(i) Hyperbolicity and causal cones.
Under assumptions (1)–(2), the linearized MEE defines a second-order hyperbolic operator with principal symbol
$$P(k_\mu) = C_0\, (k_0)^2 - \chi_0\,\vec{k}^2,$$
whose characteristic surfaces satisfy $$P(k_\mu)=0$$, i.e.,
$$C_0\,\omega^2 - \chi_0\,\vec{k}^{\,2} = 0\quad\Rightarrow\quad\omega^2 = v_{\max}^2\,\vec{k}^{\,2},\quad v_{\max}^2 = \frac{\chi_0}{C_0}.$$
The associated characteristic cone in spacetime is given by $$|\mathbf{x}| = v_{\max} t$$ (in local inertial coordinates). This cone defines the maximal group and front velocities allowed by the entropic field equations. [4][3]
(ii) Green’s functions and support.
The retarded Green’s function $$G_{\text{ret}}(x-x')$$ of the linearized operator vanishes outside this characteristic cone, by standard results for strictly hyperbolic operators with local coefficients. Thus, any localized perturbation at $$x'$$ can only influence points $$x$$ within $$|\mathbf{x}-\mathbf{x}'|\le v_{\max}(t-t')$$; there is no support outside the entropic cone. [4][3]
(iii) Nonlinear completion and matter coupling.
Nonlinear terms in the full MEE and local couplings to matter fields $$\Psi$$ leave the principal part of the operator unchanged, and therefore cannot enlarge the characteristic cone. Under assumption (3), the coupled system remains hyperbolic with the same set of characteristic cones. Hence, all dynamical fields share the same causal structure set by the entropic field. [4][3]
(iv) Absence of super‑entropic solutions.
Any hypothetical solution exhibiting front velocity $$v > v_{\max}$$ would require characteristics outside the entropic cone, contradicting hyperbolicity and the support properties of $$G_{\text{ret}}$$. Such solutions are therefore excluded from the physical solution space of the theory. This completes the proof of the No‑Rush Theorem. [4][3]
5. Identification of $$v_{\max}$$ with the Speed of Light
5.1 Electrodynamics and the Maxwell wave speed
In vacuum electrodynamics on Minkowski spacetime, Maxwell’s equations yield the wave equation for electromagnetic fields with characteristic speed
$$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}},$$
where $$\varepsilon_0$$ is the vacuum permittivity and $$\mu_0$$ is the vacuum permeability. This $$c$$ is empirically equal to the speed of light and the universal relativistic speed limit in Einstein’s theory. [7]
5.2 Entropic reinterpretation of $$\mu_0$$ and $$\varepsilon_0$$
ToE reinterprets the electromagnetic sector as an emergent effective field theory living on and constrained by the entropic substrate. In this view: [4][5][6]
- The vacuum permittivity $$\varepsilon_0$$ is understood as a measure of the entropic “susceptibility” of the medium to electric field configurations.
- The vacuum permeability $$\mu_0$$ is analogously related to the medium’s entropic response to magnetic configurations or, more abstractly, to the entropic cost of storing field momentum and vorticity.
The effective entropic coefficients $$\chi_0$$ and $$C_0$$ can therefore be related to $$\varepsilon_0$$ and $$\mu_0$$ through identification of the respective energy densities and action functionals, producing
$$\chi_0 = \frac{1}{\mu_0},\qquad C_0 = \varepsilon_0.$$
This mapping is consistent with the interpretation of $$\chi_0$$ as an entropic analogue of stiffness (inverse permeability) and $$C_0$$ as an entropic analogue of capacity (permittivity). [4][5][6]
5.3 Derivation of $$v_{\max} = c$$
Combining the entropic propagation speed with the electromagnetic identifications, we obtain
$$v_{\max}=\sqrt{\frac{\chi_0}{C_0}}=\sqrt{\frac{1/\mu_0}{\varepsilon_0}}=\frac{1}{\sqrt{\mu_0\varepsilon_0}}=c.$$
Thus, the maximal entropic propagation speed $$v_{\max}$$ derived from the Obidi Action and the MEE is numerically and structurally identical to the relativistic speed of light. In ToE, $$c$$ is therefore not a primitive constant but a derived quantity: the maximum rate at which the entropic field can rearrange and thereby transmit causal influence. [4][5][6]
6. Discussion and Outlook
The chain of reasoning established here may be summarized as follows: [1][2][3][4]
1. The Obidi Action defines a dynamical entropy field $$S(x)$$ whose evolution is governed by the Master Entropic Equation.
2. Linearization around a homogeneous background yields a hyperbolic wave equation with characteristic speed $$v_{\max} = \sqrt{\chi_0/C_0}$$.
3. The No‑Rush Theorem (NGT) proves that no physical disturbance, classical or quantum, can propagate faster than $$v_{\max}$$.
4. Matching the entropic constitutive coefficients to electromagnetic vacuum parameters implies $$v_{\max} = 1/\sqrt{\mu_0\varepsilon_0} = c$$.
ToE doesn't just accept Einstein's second postulate as given—it reveals [c] as the universe's hardwired "maximum entropic processing rate," a dynamical limit baked into the substrate of reality itself.Where Einstein said "light speed is invariant, period," ToE explains why: the entropy field can't reconfigure any faster.
Every causal link—EM waves, gravitational influence, even quantum correlations—must wait for the entropic medium to catch up. Relativity's light cones aren't geometric abstractions; they're the literal shape of allowed entropic flow.
This recasts the second postulate from mystery axiom to derived necessity. No tachyons, no superluminal shortcuts, no acausal loopholes—because none fit the bandwidth of the entropic architecture. Nature literally cannot be rushed beyond $$[v_{\max} = \sqrt{\chi_0/C_0} = c]$$.
In this way, the constancy and universality of $$c$$ emerge as necessary consequences of finite-rate entropic dynamics, rather than axiomatic assumptions. Relativistic kinematics, gravitational geometry, and quantum constraints become manifestations of the same entropic causal structure. [1][2][3][4][5][6]
Citations:
[1] Physics:Implications of the Obidi Action and the Theory of Entropicity (ToE) https://handwiki.org/wiki/Physics:Implications_of_the_Obidi_Action_and_the_Theory_of_Entropicity_(ToE)
[2] A Brief Note on Some of the Beautiful Implications of Obidi's Theory ... https://johnobidi.substack.com/p/a-brief-note-on-some-of-the-beautiful
[3] On the Conceptual and Mathematical Foundations of ... https://client.prod.orp.cambridge.org/engage/coe/article-details/68ea8b61bc2ac3a0e07a6f2c
[4] The Theory of Entropicity (ToE) Derives Einstein's Relativistic Speed ... https://www.cambridge.org/engage/coe/article-details/6908aca0113cc7cfffd949e3
[5] The Theory of Entropicity (ToE) Derives Einstein's Relativistic Speed ... https://www.academia.edu/144796856/The_Theory_of_Entropicity_ToE_Derives_Einsteins_Relativistic_Speed_of_Light_c_as_a_Function_of_the_Entropic_Field_ToE_Applies_Logical_Entropic_Concepts_and_Principles_to_Derive_Einsteins_Second_Postulate_Version_2_0
[6] The Theory of Entropicity (ToE) Derives Einstein's Relativistic Speed ... https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/690a7684ef936fb4a2577e84/original/the-theory-of-entropicity-to-e-derives-einstein-s-relativistic-speed-of-light-c-as-a-function-of-the-entropic-field-to-e-applies-logical-entropic-concepts-and-principles-to-derive-einstein-s-second-postulate.pdf
[7] Entropic gravity - Wikipedia https://en.wikipedia.org/wiki/Entropic_gravity
[8] The Theory of Entropicity (ToE) Derives and Explains Mass Increase ... https://client.prod.orp.cambridge.org/engage/coe/article-details/6900d89c113cc7cfff94ef3a
[9] An introduction to the maximum entropy approach and its ... - PMC https://pmc.ncbi.nlm.nih.gov/articles/PMC5968179/
[10] Exploring the Origin of Maximum Entropy States Relevant ... https://pmc.ncbi.nlm.nih.gov/articles/PMC8870825/
[11] Relativistic Roots of κ-Entropy - PMC https://pmc.ncbi.nlm.nih.gov/articles/PMC11119737/
