Entropic foundations of the Lorentz factor

The Theory of Entropicity (ToE) derives the Lorentz factor \(\gamma_e = \left(1 - \dfrac{v^2}{c_e^2}\right)^{-1/2}\) of Einstein’s Relativity from the invariance of a fundamental entropic cone (EC), rather than from postulated properties of spacetime geometry. In this formulation, the basic kinematic structure of relativity emerges from the requirement that a specific quadratic entropic form remain invariant under changes of inertial frame. The quantity \(c_e\) is interpreted as the entropic propagation speed, and the field variable of interest is the entropy density \(s\), together with its associated entropy flux. The standard relativistic factor \(\gamma_e\) appears as the unique scaling factor that preserves this entropic invariant under motion.

Entropic cone (EC) invariant and Minkowski-like structure

At the core of Obidi’s Theory of Entropicity is the postulate of an entropic cone (EC) invariant, expressed as a quadratic form

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

where \(c_e\) denotes the entropic propagation speed, \(s_0\) is the rest-frame entropy density, and \(j\) is the entropy flux. The flux is defined by

\[ j = v s, \]

with \(v\) the relative velocity and \(s\) the entropy density in the moving frame. This structure can be viewed as a Minkowski-like pseudo-norm on the entropic two-vector \((c_e s, j)\), analogous to the norm on spacetime four-vectors in special relativity. The invariant quadratic form defines an entropic cone (EC) that constrains admissible transformations between inertial frames.

In the rest frame of the system, the entropic two-vector takes the form

\[ (c_e s_0, 0), \]

since the entropy flux vanishes when the system is at rest. In a frame where the system moves with velocity \(v\), the corresponding two-vector becomes

\[ (c_e s(v), v s(v)). \]

The requirement that the entropic cone invariant (ECI) be preserved under such transformations imposes a specific group structure on the allowed mappings between rest and moving frames. This structure is mathematically equivalent to the Lorentz group, and the associated transformation matrices yield the Lorentz factor \(\gamma_e\) as the unique scaling factor that maintains the invariance of the entropic quadratic form.

Entropy density boost and emergence of \(\gamma_e\)

To obtain the explicit form of the Lorentz factor, one starts from the entropic cone relation (ECR)

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

Evaluating this expression in the rest frame, where \(v = 0\) and \(s = s_0\), gives

\[ (c_e s_0)^2 = \text{const}. \]

In a frame where the system moves with velocity \(v\) and has entropy density \(s(v)\), the same invariant must hold:

\[ (c_e s_0)^2 = (c_e s(v))^2 - (v s(v))^2. \]

Factoring out \(s(v)^2\) on the right-hand side yields

\[ (c_e s_0)^2 = s(v)^2 \left( c_e^2 - v^2 \right). \]

Solving for the moving entropy density \(s(v)\) gives

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

and hence

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

where

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

is the entropic Lorentz factor. In this derivation, \(\gamma_e\) is not assumed a priori; it emerges as the unique scaling factor required to preserve the entropic cone invariant (ECI) under motion. The increase of entropy density with velocity is thus a direct consequence of the invariant structure of the entropic two-vector.

Consistency of the entropic Lorentz factor across relativistic effects

Once the entropy density boost \(s(v) = \gamma_e s_0\) is established, the same factor \(\gamma_e\) propagates consistently through all relativistic kinematic effects in the Theory of Entropicity. This consistency arises from the way entropy conservation and entropic constitutive relations constrain physical quantities such as mass, length, and proper time.

For mass, the theory posits a mapping between mass and entropy density, typically expressed as a proportionality \(m \propto s\). If \(m_0\) is the rest mass associated with \(s_0\), then the boosted density \(s(v) = \gamma_e s_0\) implies

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

reproducing the standard relativistic mass increase as an entropic effect.

For length contraction, consider a one-dimensional object with rest length \(L_0\) and rest entropy density \(s_0\), so that the total entropy is \(\Sigma = s_0 L_0\). In a moving frame, the entropy density becomes \(s(v) = \gamma_e s_0\), and conservation of total entropy requires

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

Substituting the boosted density yields

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

and hence

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

which is the standard relativistic length contraction formula, now derived from entropic invariance.

For time dilation, the theory associates each “tick” of a clock with a fixed entropic increment \(\Delta S\) per cycle. If \(\tau_0\) is the proper time interval corresponding to this increment in the rest frame, then in a moving frame the entropic resistance to rapid reconfiguration implies that the time interval becomes

\[ \tau(v) = \gamma_e \tau_0. \]

Thus, the same \(\gamma_e\) governs mass increase, length contraction, and time dilation, providing a unified entropic origin for all relativistic kinematic effects.

Velocity addition and entropic group structure

The entropic derivation of the Lorentz factor is further reinforced by the consistency of the velocity addition law within the ToE framework. Defining the dimensionless entropic velocity parameter

\[ \beta_e = \frac{v}{c_e}, \]

the composition of two velocities \(v_1\) and \(v_2\) along the same line yields a total effective velocity characterized by

\[ \beta_{e,\text{tot}} = \frac{\beta_1 + \beta_2}{1 + \beta_1 \beta_2}, \]

where \(\beta_1 = v_1 / c_e\) and \(\beta_2 = v_2 / c_e\). This is the standard relativistic velocity addition formula, here obtained as a consequence of preserving the entropic cone invariant (ECI) under successive transformations. The set of transformations that preserve the entropic quadratic form thus forms a group isomorphic to the Lorentz group.

This group structure confirms that the kinematics derived from entropic principles alone are fully consistent with the symmetry structure of special relativity. The Lorentz factor \(\gamma_e\) is therefore not an arbitrary insertion but a necessary consequence of demanding that the entropic two-vector \((c_e s, j)\) transform in a way that preserves the fundamental entropic invariant.

Conceptual significance of the entropic derivation

The entropic derivation of the Lorentz factor in the Theory of Entropicity shows that the core mathematical structure of Einstein’s Relativity can be recovered from a deeper entropic ontology. Instead of postulating the invariance of the spacetime interval and the speed of light, ToE postulates the invariance of an entropic cone (EC) built from entropy density and entropy flux, with \(c_e\) emerging as the maximum entropic propagation speed. The familiar relativistic factor \(\gamma_e\) then arises as the unique scaling required to preserve this entropic invariant under motion.

This perspective integrates relativistic kinematics into a broader framework in which mass, length, and time are all understood as emergent manifestations of how the entropy field redistributes and resists change. The Lorentz factor becomes a measure of entropic deformation induced by motion, unifying relativistic effects under a single entropic principle rather than treating them as separate geometric postulates.

In conclusion, the entropic‑cone (EC) formalism developed within the Theory of Entropicity (ToE) furnishes a remarkably clear and structurally elegant derivation of the Lorentz factor, demonstrating that relativistic time dilation, mass increase, and length contraction arise naturally from the intrinsic geometry of entropic flow rather than from externally imposed kinematical [geomtric] postulates.

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