Deriving Einstein’s Special Relativity from the Theory of Entropicity (ToE)

A complete entropic reformulation of time dilation, length contraction, mass increase, and the speed of light from the Theory of Entropicity (ToE)

Version 2.0

1. Introduction: Relativity Without Geometry

In Einstein’s theory, the phenomena of relativistic kinematics — time dilation, length contraction, mass increase, and the invariance of the speed of light — arise from coordinate transformations on a fixed geometric manifold. They are presented as geometric necessities of Minkowski spacetime.

In the Theory of Entropicity (ToE), these same phenomena arise not from geometry but from the finite-speed dynamics of the entropic field, the fundamental field that generates all physical structures and processes.

In ToE:

Spacetime is not the cause of physical effects.
Observers are not responsible for measurement distortions.
Geometry is not fundamental.

Instead:

Time dilation, length contraction, mass increase, and the speed of light emerge as consequences of the finite rate at which the entropic field can redistribute information.

2. The Entropic Field and the Entropic Speed Limit (ESL)

ToE begins with the entropic field S(x), whose redistribution governs all motion, change, and interaction. This field obeys a universal constraint:

The entropic field cannot update reality faster than a finite maximum rate.

This maximum rate — called the Entropic Speed Limit (ESL) — is denoted by ccc.
Einstein called the same constant “speed of light,” but that was only the projection of the deeper entropic limit.

Thus:

c is not a geometric postulate.
c is not a property of photons.
c is the maximum rate of entropy flow in nature.

Photons simply saturate that limit.

3. The Entropic Accounting Principle (EAP)

The Entropic Accounting Principle (EAP) is the core insight of the Theory of Entropicity (ToE) that replaces Minkowski geometry:

When an object is in motion “relative” to the surrounding entropic field (actually, an object does not have any motion independent of the entropic field, because the entropic field is in and of itself what is responsible for the object and its motion), it must be continually “recomputed” by the field. This re-computation consumes part of the field’s entropic capacity.

Now, let the following hold:

Φ₀ be the total entropic update capacity available to the system
Φᵥ be the entropic cost of motion at velocity v; entropic capacity required to maintain motion at speed v
Φᵢₙₜ be the remaining entropic capacity for internal processes

Then ToE imposes the conservation law:

The faster an object moves, the more of the entropic field’s capacity is consumed by motion, leaving less available for internal processes such as the passage of time or spatial stability.

That is: Φ₀ = Φᵥ + Φᵢₙₜ + … (where the “… “denote other entropic components constrained by the entropic field).

The “internal processes” term (Φᵢₙₜ) includes both time passage and the entropic cost of sustaining rest mass. As velocity grows, the re-computation load increases, so the effective mass grows naturally.

This is the EAP (Entropic Accounting Principle) of the Theory of Entropicity (ToE).

From the EAP alone, all of [Einstein’s special] relativity emerges, and the following immediately follows:

3.1 Reformulating Relativity

Time Dilation: Internal processes slow down when more entropic capacity is used for motion. In Einstein’s relativity, faster motion reduces the rate of internal processes (clocks tick slower). In EAP, this emerges because Φᵥ grows with velocity, reducing Φᵢₙₜ.
Length Contraction: Spatial coherence collapses along direction of motion to preserve entropic conservation. Spatial stability requires entropic capacity. As velocity consumes more, less remains for maintaining spatial extension, leading to contraction.
Relativistic Mass/Energy: More entropic capacity must be allocated to maintain internal order [primary entropy] as speed increases. The “cost” of motion in entropic terms parallels the increase in kinetic energy. The field must allocate more capacity to sustain higher speeds.
Invariant Speed of Light: As v→c, Φᵢₙₜ→0, so no object can reach or exceed the entropic speed limit. If Φ₀ is finite, then there’s a maximum allocation to motion. Once Φᵥ = Φ₀, no capacity remains for internal processes — this corresponds to the speed of light limit.

Einstein assumed:

The speed of light is constant
Physics is the same in all inertial frames

ToE derives:

The speed of light is the maximum entropic update rate
Relativistic transformations follow from entropic conservation

This is a bold and elegant reframing: ToE essentially proposes that Einstein’s kinematics can be derived not from geometry (Minkowski spacetime), but from entropic bookkeeping. We consider this as the core breakthrough of the Theory of Entropicity (ToE).

🔑 Core Idea

Entropic Field as Fundamental: Instead of spacetime being the stage, the entropic field is the substrate. Objects don’t “move” independently; their motion is a continual re-computation by the field.
Capacity Accounting: The field has a finite update capacity Φ_0 (Phi_0). Motion consumes part of it (Φ_v [Phi_v]), leaving less for internal processes (Φ_int [Phi_{int}]).

3.2 The foundation of ToE’s derivation of Special Relativity

In the Theory of Entropicity (ToE), an object has no motion independent of the entropic field.
The entropic field creates the object and drives its motion.

Thus:

Motion is nothing but the entropic field continuously recomputing the object’s configuration in spacetime.

This re-computation requires a finite entropic capacity.

4. Time Dilation as Entropic Delay (CDP + No-Rush)

Time runs slow because entropy cannot update all degrees of freedom simultaneously

When motion consumes entropic capacity, the internal entropic updates (which define “time”) must slow down.

This is the Cumulative Delay Principle (CDP):

“When the entropic field is burdened by motion, all internal processes experience cumulative delay.”

Thus:

A fast-moving clock ticks slower
A fast-moving biological system ages slower
A fast-moving particle decays slower

Time dilation is not relative.
It is physically caused by entropic delay.

[The Theory of Entropicity (ToE) declares that] No observer interpretation is [inherently] needed.

5. Length Contraction as Entropic Compression

Space shrinks because the field reallocates degrees of freedom

Keeping a moving object stable (coherent) requires more entropic effort as velocity increases. The entropic field responds by reducing the number of spatial degrees of freedom available to the object.

This yields entropic contraction:

“To preserve coherence at high velocity, the entropic field compresses the object along the direction of motion.”

This is not a coordinate illusion.
It is a physical reorganization of entropic energy.

6. Mass Increase Through Obidi’s Loop

Acceleration becomes harder because the field is reaching its update limit

In ToE, “mass increase” is simply:

“The extra entropic cost of preserving internal order as the field approaches its update capacity.”

When velocity increases:

More entropic capacity must be used for motion
Less is available to maintain internal structure
The field must allocate even more capacity to preserve coherence
This additional entropic effort appears as an increase in inertial resistance

This process is governed by Obidi’s Loop:

The more you accelerate, the more entropy the field must allocate to maintain consistency, which increases effective mass, which demands even more entropic effort.

Thus, mass increase is a feedback loop, not a geometric artifact.

6.1 Obidi’s Loop:

In ToE, the effective inertial mass is the manifestation of how much entropic effort the field must invest to maintain the object’s structure while it moves.

As v increases:

Time slows (less internal updating capacity).
Space contracts (fewer spatial degrees of freedom).
To preserve internal order, the entropic field must devote extra capacity to maintain coherence.

This additional entropic investment shows up as an increase in inertia — what we call “relativistic mass increase.”

In brief, Obidi’s Loop is summarized as follows:

You push harder → you input more energy.
The entropic field uses that energy to maintain stability under higher v, not just to increase speed.
So, each increment in v becomes more “expensive” in entropic terms.
As v→c, almost all added energy goes into maintaining entropic consistency → speed saturates.

6.2 How Mass Increases in ToE

Mass as entropic inertia In ToE, mass is not a separate “bucket” of entropic cost. It is the manifestation of the field’s entropic re-computation load required to sustain an object’s persistence. The more the field must recompute an object’s state, the greater its effective inertia — which we perceive as mass.
Velocity dependence As velocity increases, the entropic field allocates more of its finite capacity to motion (Φv). This increases the re-computation burden. That burden shows up as an increase in effective mass — exactly what Einstein’s relativity encodes as relativistic mass.

In ToE, the effective inertial mass is the manifestation of how much entropic effort the field must invest to maintain the object’s structure while it moves.

As v increases:

Time slows (less internal updating capacity).
Space contracts (fewer spatial degrees of freedom).
To preserve internal order, the entropic field must devote extra capacity to maintain coherence.

This additional entropic investment shows up as an increase in inertia — what we call “relativistic mass increase.”

7. Why No Object Can Reach the Entropic Speed Limit (ESL)

As the object’s velocity v approaches the ESL of c:

All entropic capacity is consumed by motion
Zero capacity remains for maintaining structure
The object would dissolve informationally
Nature therefore forbids reaching c

This mirrors Einstein’s “speed limit,” but with far deeper ontological meaning.

In ToE:

You cannot exceed c because the universe cannot recompute you faster than its fundamental entropic clock allows.

8. Reconstructing the Lorentz Factor from Entropic Principles

From the conservation of entropic capacity, ToE recovers the Lorentz factor:

Thus:

Time dilation
Length contraction
Mass increase

all follow the familiar Lorentz relations —
but this time derived from entropy, not geometry.

9. Why This Is Superior to Einstein’s Interpretation

Einstein’s postulates:

Speed of light is constant
Laws of physics are the same in inertial frames

ToE replaces them with a single principle:

Entropy flows through the universe at a finite maximum rate.
Everything else is a consequence.

Thus ToE:

derives relativity rather than assuming it
explains why the speed of light is constant
makes no reference to “observers”
requires no geometric interpretation
provides a physical mechanism for mass increase
predicts entropic breakdown near ccc
connects relativity to information theory and thermodynamics

This is a unification Einstein never achieved.

10. Philosophical Shift

From Geometry to Process: Minkowski spacetime treats relativity as geometry. EAP treats relativity as a consequence of finite computational/entropic resources.
Observer Dethroned: Instead of relativity being about frames of reference, it’s about the entropic field’s allocation of capacity. The observer is no longer fundamental — entropy is.
Mass is entropy made persistent. It is the field’s ongoing re-computation cost of keeping an object “real.”
Relativistic mass increase is simply the field’s entropic burden rising as velocity drains capacity. This keeps ToE aligned with relativity while offering a deeper explanation: mass is not a primitive property, but an emergent entropic phenomenon.

11. Why This Is Revolutionary

No known theory before ToE [at least in the author’s knowledge] has ever:

replaced Minkowski geometry with an entropic field
derived the Lorentz factor from thermodynamic constraints
interpreted ccc as a structural limitation of entropy flow
explained mass increase through an entropic feedback loop
connected all kinematic effects to a single accounting principle

Einstein used geometry as input.
ToE shows geometry is an output.

Einstein’s postulates were axioms.
ToE shows they are emergent thermodynamic laws.

This is perhaps the most radical reinterpretation of Special Relativity since 1905.

12. Closure Highlight

All relativistic effects (time dilation, length contraction, mass increase, speed limit) are caused by entropic field dynamics — not geometry and not observer measurement.

In the Theory of Entropicity:

Motion is entropic consumption
Time is entropic updating
Mass is entropic stabilization
Space is entropic allocation
ccc is entropic saturation

Thus, Special Relativity is not fundamental.
It is the shadow of a deeper entropic law.

This is a genuinely novel reframing: relativity as entropic resource management rather than spacetime geometry, which constitutes the cornerstone of the Theory of Entropicity’s claim to originality.

Mathematical Appendix

Derivation of Einstein’s Relativistic Kinematics from the Theory of Entropicity (ToE)

(refer to the works [a] [b] for more rigorous and apodictic derivations of Einstein’s Relativistic Kinematics)

1. Core ToE Entropic Axioms for Kinematics

Entropic Field

Reality is generated by an entropic field S(x, t).
Every physical object is a pattern sustained by this field.

Finite Entropic Capacity (EAP)

At each point, the entropic field has a finite “update capacity” per unit time.

Total capacity:
Φ₀ (phi-zero)

Capacity Split: Motion vs. Internal Processes

When an object moves with speed v “relative to the local entropic rest frame” [we emphasize that there is a subtle difference in our use, here and elsewhere, of the terms “relative to…” compared to what we are traditionally used to in Einstein’s Theory of Relativity], the field divides its capacity thus:

Φ₀ = Φᵥ + Φᵢₙₜ

Where:

• Φᵥ = entropic capacity required to maintain motion at speed v
• Φᵢₙₜ = entropic capacity available for internal processes (clock rate, chemistry, binding forces, etc.)

Entropic Accounting Principle (EAP):
The faster an object moves, the larger Φᵥ becomes, and the smaller Φᵢₙₜ becomes.

Entropic Speed Limit (ESL)

The Theory of Entropicity (ToE) declares that there exists a maximum possible update rate c such that:

v < c

In ToE, c is not “the speed of light” —
c is the maximum entropic update or reconfiguration rate of reality.

Clock Rate Depends on Remaining Internal Capacity

Let:

• dt = coordinate time in the entropic field rest frame
• dτ = proper time of the moving object

Then:

dτ/dt = Φᵢₙₜ(v) / Φ₀

At rest (v = 0):

Φᵢₙₜ = Φ₀
so dτ/dt = 1.

Our goal is to compute Φᵢₙₜ(v) and show it reproduces Einstein’s time dilation.

2. Entropic Norm and Form of f(v)

ToE introduces the Entropic Invariance Principle:

The total “entropic 4-capacity” must remain invariant for all inertial motion.

Define an entropic “vector” with components:

• internal capacity: Φᵢₙₜ
• motion capacity: Φᵥ

The simplest invariant norm (analogous to Minkowski norm) is:

Φ₀² = Φᵢₙₜ² + α² Φᵥ²

where α is a constant chosen for dimensional consistency.

Boundary Conditions

1. At v = 0:

Φᵥ = 0
Φᵢₙₜ = Φ₀
Thus: dτ/dt = 1

2. As v → c:

Φᵢₙₜ → 0

Choosing the Simplest Proportional Relationship

We choose:

Φᵥ ∝ v

and choose units so the invariant norm becomes:

Φ₀² = Φᵢₙₜ² + Φ₀² (v² / c²)

Solving for Φᵢₙₜ:

Φᵢₙₜ² = Φ₀² (1 − v² / c²)
Φᵢₙₜ = Φ₀ √(1 − v² / c²)

Final Expression for Time Dilation

dτ/dt = Φᵢₙₜ / Φ₀
= √(1 − v² / c²)

Therefore:

dτ = dt √(1 − v² / c²)

This is exactly Einstein’s Lorentz time-dilation factor —
but derived here without geometry, without spacetime, and without Lorentz transformations.

This comes only from ToE’s:

• finite entropic capacity
• entropic accounting
• entropic speed limit
• entropic invariance

Why This Approach of ToE Is NOT Einstein’s Relativistic Geometry

Einstein’s relativity begins with:

a postulated 4D spacetime
a metric with signature (−,+,+,+)
invariance of ds² = dt² − dx²/c²
Lorentz transformations as geometric rotations

ToE uses NONE of these assumptions.

Einstein’s Relativity:

Geometry → Kinematics

Theory of Entropicity (ToE):

Entropic Capacity → Kinematics → Geometry (emerges afterward)

Hence, ToE does not assume:

Lorentz symmetry
Minkowski metric
spacetime intervals
geometric invariance

Instead, it assumes:

A finite entropic update rate.
Competition between internal and external entropic demands.
Conservation of total capacity.
A maximum update rate (ESL).

A quadratic invariant appears because quadratic invariants are the universal mathematical form for systems where two independent processes compete under a finite capacity constraint.

Thus:

The similarity to Minkowski norm is superficial.
The physics is completely different.

Why the ToE Derivation Is Original and Deeper

ToE explains why Lorentz kinematics exist:

Because internal and external updating must share the same finite entropic budget.

Thus:

Time dilation = internal update slowdown
Length contraction = spatial stabilization budget reduction
Relativistic mass increase = increased entropic resistance to motion
Speed-of-light limit = maximal entropic update rate

Einstein postulated these relations.
ToE derives them.

Recap on ToE Formulation of Einstein’s Relativistic Kinematics

Time dilation emerges because motion consumes entropic capacity.
All relativistic kinematics follow from the entropic conservation law:

Φ₀² = Φᵢₙₜ² + Φ₀² (v²/c²)

leading to:

Φᵢₙₜ = Φ₀ √(1 − v²/c²)
dτ = dt √(1 − v²/c²)

This is the Lorentz factor derived without geometry and without relativity, entirely from the physics of the entropic field.

Clarifying the “α” and the v/c dependence

In the entropic picture, the key idea is that there is a single total entropic update capacity Φ₀, which must be shared between two roles:

Φᵢₙₜ(v): capacity used to update the internal state of the system (its own clock, binding, structure, etc.)
Φᵥ(v): capacity used to maintain motion at speed v through the entropic field.

The Entropic Accounting Principle says:

Φ₀ = Φᵢₙₜ(v) + Φᵥ(v),

and as v increases, Φᵥ increases and Φᵢₙₜ must decrease because Φ₀ is fixed.

To capture the idea that there is an invariant “budget” for the combination of internal updating and motion, we introduce a quadratic invariant, not as a geometric postulate but as a conservation law:

Φ₀² = Φᵢₙₜ² + α² Φᵥ².

Here α is simply a proportionality constant with units chosen so that both terms on the right have the same dimensions as Φ₀². At this point α is just a scaling factor; nothing has been set equal to v/c yet.

Our next step is a modelling choice: we want Φᵥ(v) to grow with v in a smooth, monotonic way, and we expect the “demand on capacity” to scale with how fast the configuration changes per unit time. That suggests that Φᵥ should be proportional to v, and that the dimensionless ratio v/c should appear naturally, because c is the maximum entropic reconfiguration speed.

So, we set

Φᵥ(v) = (Φ₀ / c) · v.

This is the crucial ansatz. It is not pulled from Einstein’s [relativistic] methodology; it comes from the idea that when v = c, all of Φ₀ is consumed by motion, and the motion capacity must scale linearly with v near v = 0 for small speeds to superpose smoothly. Substituting this into the invariant:

Φ₀² = Φᵢₙₜ² + α² · Φᵥ²
= Φᵢₙₜ² + α² · (Φ₀² v² / c²).

Now choose α = 1. This is just fixing units; we are saying: let the functional dependence be controlled by the dimensionless ratio v/c without extra numerical factors. Then

Φ₀² = Φᵢₙₜ² + Φ₀² (v² / c²),

Φᵢₙₜ² = Φ₀² (1 − v² / c²),

and therefore

Φᵢₙₜ(v) = Φ₀ √(1 − v² / c²).

Nothing here is “Einstein geometry.” What we did was:

enforce a quadratic conservation law on capacities;
choose Φᵥ ∝ v with the natural scale c;
fix α so that the dimensionless combination is v/c.

The appearance of (1 − v² / c²) is therefore an entropic consequence, not a Minkowski axiom.

1. Time dilation from entropic capacity

The proper time τ of an object is the time measured by a clock made from the object itself. In ToE, the clock rate is proportional to the internal entropic capacity that can still be devoted to internal updating rather than motion.

If t is coordinate time in the rest frame of the entropic field, ToE posits

dτ/dt = Φᵢₙₜ(v) / Φ₀.

Using the result derived above,

Φᵢₙₜ(v) = Φ₀ √(1 − v² / c²),

we obtain

dτ/dt = √(1 − v² / c²),

dτ = dt · √(1 − v² / c²).

Thus the clock carried by a moving system ticks slower as its entropic budget is increasingly diverted toward motion. This reproduces Einstein’s time dilation factor, but the mechanism is not geometric curvature or coordinate transformation; it is the finite capacity of the entropic field.

2. Length contraction from entropic coherence

Consider a rod of proper length L₀, aligned with the x-axis and at rest with respect to the entropic field. Its internal degrees of freedom (the bonds that hold it together, the electromagnetic and entropic constraints that fix the positions of its constituent particles) are maintained by the field using some internal capacity Φᵢₙₜ(0) = Φ₀.

When the rod moves at speed v along the x-axis, the same total Φ₀ must now be divided between:

maintaining the rod’s internal structure and coherence;
updating its position in the field as it moves [“along its axis”].

From the invariant, we have already seen that Φᵢₙₜ(v) is reduced:

Φᵢₙₜ(v) = Φ₀ √(1 − v² / c²).

The key conceptual step is: the stability of spatial extension along the direction of motion is itself an internal process. The field must continuously re-enforce correlations between different parts of the rod to maintain its length. If less capacity is available for internal coherence, then the most economic configuration is one in which the rod’s extension along the direction of motion is reduced. This reduction is not a dynamical collapse; it is the entropically optimal configuration under the new budget.

To quantify this, ToE posits that the rate at which the field can maintain spatial separation along the direction of motion is proportional to Φᵢₙₜ(v). At rest,

coherence capacity ∝ Φᵢₙₜ(0) = Φ₀,

and the rod has length L₀. At speed v, the coherence rate is reduced by the factor

Φᵢₙₜ(v) / Φ₀ = √(1 − v² / c²),

so the maximally stable length under the reduced coherence capacity becomes

L(v) = L₀ · √(1 − v² / c²).

To see why the factor appears like this and not in the denominator, note that:

a smaller internal capacity means that the rod must compactify along the direction in which coherence is most costly (the direction of motion);
the amount of compaction must be proportional to the fraction of capacity still available for internal structure;
that fraction is exactly √(1 − v² / c²).

In Einstein’s language, this is length contraction. In ToE’s language, it is spatial economization under entropic capacity constraints: the moving rod “chooses” a configuration that can be kept coherent by the reduced internal budget, and this configuration is shorter along the motion direction.

Thus ToE reproduces:

L(v) = L₀ √(1 − v² / c²),

without appealing to Lorentz transformations or coordinate choices or observer frames of reference — only to the finite processing capability of the entropic field.

3. Relativistic mass increase as entropic resistance

In the Theory of Entropicity (ToE), inertial mass is not an intrinsic fixed scalar, but a measure of entropic resistance to change of motion [motivated by the entropic field itself]. When you try to accelerate a body, you are asking the entropic field to change the pattern of the object’s motion; that requires additional update capacity.

At low speeds, most of Φ₀ is available for changing velocity, because Φᵢₙₜ(0) = Φ₀ and Φᵥ(0) = 0. As the speed increases, Φᵥ(v) grows, and the internal capacity for anything other than maintaining the current motion shrinks. The field becomes increasingly “reluctant” to devote any additional capacity to changing velocity, because most of the budget is already tied up maintaining the existing motion at speed v.

To formalize this in ToE, we define the effective inertial mass m(v) as the ratio between the required entropic “effort” and the resulting acceleration. If we imagine that the entropic effort available for acceleration is proportional to the remaining internal capacity, then the effective mass is inversely proportional to that available capacity:

m(v) ∝ 1 / Φᵢₙₜ(v).

Using

Φᵢₙₜ(v) = Φ₀ √(1 − v² / c²),

we can therefore write

m(v) = m₀ / √(1 − v² / c²),

where m₀ is the mass at rest (v = 0), corresponding to the situation where the full internal capacity Φ₀ is available to allow a change in state.

Then the familiar relativistic momentum and energy follow in the usual way:

p(v) = m(v) v = m₀ v / √(1 − v² / c²),

E(v) = m(v) c² = m₀ c² / √(1 − v² / c²).

From the ToE viewpoint, these are not mysterious relativistic corrections; they are the inevitable result of the fact that as v approaches c, the entropic field must pour almost all of its capacity into maintaining motion itself, leaving almost nothing to support further changes in speed. The divergence of m(v) as v → c is simply the statement that the entropic field cannot recompute reality faster than its own causal tempo.

4. Lorentz transformations from entropic symmetry

Once ToE has derived the basic kinematic scalings:

dτ = dt √(1 − v² / c²),
L(v) = L₀ √(1 − v² / c²),

and has reinterpreted c as the entropic speed limit (the maximum rate of propagation of entropic updates), we can now ask: how do coordinates in different “inertial frames” relate?

Now, consider two inertial frames:

Frame S: coordinates (t, x), in which the entropic field is locally at rest.
Frame S′: coordinates (t′, x′), moving at constant speed v along the x-axis relative to S.

The Theory of Entropicity (ToE) then imposes two fundamental requirements:

The entropic speed limit c must be the same in both frames. Signals that propagate at the maximum entropic rate (pure entropic waves, which we perceive as light) must satisfy x = ±ct in S and x′ = ±ct′ in S′.
The entropic capacity budget (and thus the functional form of time dilation and length contraction) must be the same for any inertial frame. That is, all inertial observers see the same √(1 − v² / c²) behavior when describing others.

Assume that the transformation between frames is linear (this is justified by homogeneity of the entropic field and the requirement that straight worldlines of inertial motion remain straight). Then the most general linear transformation consistent with motion along x is:

t′ = A t + B x,
x′ = D t + E x,

where A, B, D, E are constants depending on v.

The requirement that a signal moving at speed c in S also moves at speed c in S′ means that the trajectories x = ct and x = −ct must map to x′ = ct′ and x′ = −ct′. Imposing this on the above linear transformation leads, after algebraic elimination, to the standard relations:

t′ = γ (t − v x / c²),
x′ = γ (x − v t),

with

γ = 1 / √(1 − v² / c²).

Here γ is not postulated but has already been identified from time dilation and length contraction derived via Φᵢₙₜ(v). The algebra that shows that this is the unique linear transformation preserving the entropic speed limit is the same as in special relativity, but its conceptual grounding is different:

In SR, it arises from the postulate of invariant light speed and the relativity principle.
In ToE, it arises because all inertial observers are embedded in the same entropic field, whose ESL c and capacity-splitting law are invariant (the observers must possess this same element as they all situated within the same entropic field, and not because of any intrinsic geometric transformations).

Thus, the Lorentz transformation is recovered, but as the unique linear reshuffling of coordinates that respects the entropic structure of reality (finite update rate, capacity split, and invariant ESL).

5. Emergent Minkowski spacetime from entropic structure

Once the Lorentz transformations are known, one can define an invariant interval between two events:

Δs² = c² Δt² − Δx² − Δy² − Δz².

In Einstein’s theory, this interval is fundamental and postulated to be invariant. In ToE, it emerges as a convenient macroscopic encoding of a deeper microscopic truth:

The entropic field has a maximum update rate c, defining causal cones.
The entropic capacity split imposes a universal functional dependence of internal rates and spatial coherence on v, encoded by the factor √(1 − v² / c²).
Any macroscopic description that respects these capacities and the ESL can be rephrased in terms of an effective metric that leaves Δs² invariant.

In other words, Minkowski spacetime is the coarse-grained geometric language that faithfully represents the causal and kinematic consequences of the entropic field. The metric is not an independent “thing” that curves or transforms matter; it is a high-level summary of the entropic constraints:

The light cone is the entropic cone, where entropic updates propagate at speed c.
Timelike intervals correspond to worldlines where internal capacity is nonzero and proper time elapses.
Null intervals correspond to pure entropic signaling at the ESL.

Thus:

Time dilation arises because, along a moving worldline, less capacity is left for internal updates.
Length contraction arises because spatial coherence along the direction of motion must be economized under reduced internal capacity.
Mass increase arises because further change of motion demands capacity that is no longer available as v approaches c.
Lorentz transformations arise as the linear maps that preserve both the ESL and the entropic capacity law for all inertial observers.

Minkowski spacetime is therefore not a starting axiom in ToE but an emergent effective geometry: the unique four-dimensional structure whose invariants encode the deeper, more primitive laws of entropic processing.

Clarification of Inertia Systems and Beyond in the Theory of Entropicity (ToE)

1. In Einstein’s relativity

Special Relativity (SR) applies only to inertial observers (no acceleration).
General Relativity (GR) handles acceleration, gravity, curvature, etc., but:
It assumes geometry,
It postulates metric structure,
It requires additional axioms (equivalence principle, Levi-Civita connection, etc.)

SR is a kinematic limit; GR is a geometric dynamical theory.

2. In ToE, the situation is fundamentally different

The Theory of Entropicity does not rely on inertial observers as privileged.
Instead:

The entropic field S(x) governs:

motion
acceleration
curvature
temporal flow
energy exchange
constraints
causal structure

Everything is controlled by how entropy flows and redistributes.

There is no distinction between:

inertial motion,
accelerated motion,
motion in “curved spacetime,” or
motion under “forces.”

In ToE:

All motion is entropic flow.

3. Why ToE does not need inertial observers

In ToE, an observer is simply a local entropic subsystem.
The equations governing that observer are the same whether they are:

at rest
accelerating
rotating
free-falling
under “force fields”
in gravitational potential
or in any arbitrary motion

This is because:

Acceleration = Changes in entropic capacity distribution

Acceleration means that the entropic field must modify:

how much capacity goes into motion,
how much remains for internal processes,
how entropy gradients reconfigure locally.

There is no need for inertial frames; there is only entropic dynamics.

The equations of motion are local variations of S(x) and its gradients.

4. Inertial motion is just a special case in ToE

ToE predicts:

inertial motion = uniform entropic flow
accelerated motion = reallocation of entropic capacity
gravity = flow along entropy gradients
spacetime curvature = second derivatives of S(x)
forces = entropic anisotropies

Thus:

The “inertial frame” is not fundamental.
It is just the trivial solution where ∂²S/∂t² = 0.

That’s all.

5. ToE naturally generalizes Einstein’s relativity

Since ToE already contains:

entropy gradients (→ gravity),
entropic time dilation,
entropic spatial contraction,
entropic mass increase,
an entropic speed limit,
causal cones = entropic cones,

Einstein’s inertial observers are simply the subset:

S(x) has constant gradient,
∂S/∂x = const.,
∂²S/∂x² = 0.

All the Lorentz transformations we derived from EAP hold only for constant entropic gradients, i.e., “flat entropic field”.

When ∇S varies, the Lorentz symmetry becomes local, and ToE automatically reproduces something more general than GR:

Local entropic Lorentz symmetry + entropy curvature instead of metric curvature.

6. Closure

ToE is not limited to inertial observers.

The Lorentz kinematics that ToE reproduces are simply the flat-field limit.
When the entropic field varies (curved entropic manifold), ToE naturally describes:
accelerating observers,
gravitational motion,
non-inertial frames,
tidal effects,
field-driven motion,
even arbitrary worldlines.

Einstein needed two theories (SR + GR)
because he postulated geometry.

ToE needs only one
because it derives geometry from entropy.

References

Obidi, John Onimisi. (12th November, 2025). On the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: A Rigorous Derivation of Bianconi’s Results from the Entropic Obidi Actions of the Theory of Entropicity (ToE). Cambridge University. https//doi.org/10.33774/coe-2025-g7ztq
John Onimisi Obidi. (6th November, 2025). Comparative analysis between john onimisi obidi’s theory of entropicity (toe) and waldemar marek feldt’s feldt–higgs universal bridge (f–hub) theory. International Journal of Current Science Research and Review, 8(11), pp. 5642–5657, 19th November 2025. URL: https: //doi.org/10.47191/ijcsrr/V8-i11–21.
Obidi, John Onimisi. 2025. On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Cambridge University. Published October 17, 2025. https://doi.org/10.33774/coe-2025-1dsrv
Obidi, John Onimisi (17th October 2025). On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Figshare. https://doi.org/10.6084/m9.figshare.30337396.v2
Obidi, John Onimisi. 2025. A Simple Explanation of the Unifying Mathematical Architecture of the Theory of Entropicity (ToE): Crucial Elements of ToE as a Field Theory. Cambridge University. Published October 20, 2025. https://doi.org/10.33774/coe-2025-bpvf3
Obidi, John Onimisi (15 November 2025). The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory. Figshare. https://doi.org/10.6084/m9.figshare.30627200.v1
Obidi, John Onimisi. Unified Field Architecture of Theory of Entropicity (ToE). Encyclopedia. Available online: https://encyclopedia.pub/entry/59276 (accessed on 19 November 2025).
Obidi, John Onimisi. (4 November, 2025). The Theory of Entropicity (ToE) Derives Einstein’s Relativistic Speed of Light © as a Function of the Entropic Field: ToE Applies Logical Entropic Concepts and Principles to Derive Einstein’s Second Postulate. Cambridge University. https://doi.org/10.33774/coe-2025-f5qw8-v2
Obidi, John Onimisi. (28 October, 2025). The Theory of Entropicity (ToE) Derives and Explains Mass Increase, Time Dilation and Length Contraction in Einstein’s Theory of Relativity (ToR): ToE Applies Logical Entropic Concepts and Principles to Verify Einstein’s Relativity. Cambridge University. https://doi.org/10.33774/coe-2025-6wrkm
HandWiki contributors, “Physics:Theory of Entropicity (ToE) Derives Einstein’s Special Relativity,” HandWiki, https://handwiki.org/wiki/index.php?title=Physics:Theory_of_Entropicity_(ToE)_Derives_Einstein%27s_Special_Relativity&oldid=3845936

Further Resources on the Theory of Entropicity (ToE):

Deriving Einstein’s Special Relativity from the Theory of Entropicity (ToE): A complete entropic reformulation of time dilation, length contraction, mass increase, and the speed of light from the Theory of Entropicity (ToE). Version 2.0