The Quadratic Entropic Expression for the Derivation of Einstein's Relativistic Kinematics from the Theory of Entropicity (ToE)
In the Theory of Entropicity (ToE), the quadratic entropic expression is the starting point for deriving the kinematical structure that ultimately reproduces — and then generalizes — the kinematics of Einstein’s relativity.
Below, we present the correct, precise, technically rigorous meaning of that expression.
The Quadratic Entropic Expression in ToE
ToE begins its kinematical derivation from a quadratic functional of the entropy field. This expression is the entropic analogue of a “kinetic term,” and it appears when we expand the Local Obidi Action under small perturbations of the entropy field.
Let the entropy field be written as:
S(x) = S₀ + φ(x)
where:
- S₀ is a constant background entropy,
- φ(x) is a small fluctuation,
- ∇S is assumed small (near-equilibrium regime).
Under these conditions, the Local Obidi Action (LOA) yields a leading-order term:
Quadratic Entropic Expression
I₍quad₎ ∝ e^{S₀/k_B} ∫ g^{μν} ∂_μ φ ∂_ν φ d⁴x
This is the fundamental quadratic expression that ToE begins with.
Why This Expression Is Quadratic
- It contains φ²-like contributions (specifically gradient-squared terms).
- It is second-order in derivatives of the small entropy fluctuation.
- It behaves like the Fisher information metric in information geometry.
This is crucial because:
- The quadratic form of ∂φ · ∂φ is exactly the mathematical structure that appears in the linearized limit of relativity, wave equations, and field theory.
- It is also the same structure that appears in the quadratic expansion of the Shannon–Fisher information — the heart of classical information geometry.
Thus the quadratic expression bridges physics and information geometry.
Why ToE Starts Here: The Physical Interpretation
ToE interprets this quadratic entropic expression as the entropic energy cost of deforming the entropy field. The larger the gradient of S(x), the more entropy is being transported per unit time and space, and therefore the greater the “entropic resistance” (manifesting as inertia, mass increase, time dilation, etc.).
Thus, this quadratic expression encodes:
- the resistance of the entropic field to deformation,
- the finite entropic propagation speed (foundational for the ETL),
- the functional form that gives rise to relativistic invariants.
How This Leads to Relativistic Kinematics
When you take the quadratic entropic expression:
∫ g^{μν} ∂_μ φ ∂_ν φ
and demand that it remains invariant under physical transformations, you recover:
- Lorentz symmetry,
- the Minkowski metric in the α → 1 limit,
- time dilation,
- length contraction,
- relativistic mass increase.
Thus, relativity does not arise from “postulates,” but from the entropy-field kinetic structure.
Why This Is Deeply Original
Bianconi, Verlinde, Jacobson, Caticha, and others do not start from or derive relativity from such a quadratic entropic expansion. They use entropy as a constraint, not as a dynamical field.
ToE is the first framework in which:
Relativistic kinematics is derived from the quadratic expansion of a fundamental entropy field.
ToE Derivation of Einstein's Relativistic Kinematics: The Lorentz Factor
1. Entropic density and maximum entropic capacity of a rod
Consider a rod in its rest frame.
- Let its proper length be L₀.
- Let its entropic density (entropy per unit proper length) be ρ_S (units: entropy / length).
Then the total entropy of the rod in its rest frame is
S_rod = ρ_S × L₀.
Now introduce your key ToE axiom:
There is a fundamental upper bound on how fast the entropy content of any system can be reconfigured.
Let
- C_max be the maximum entropic capacity of the rod: the largest rate at which the rod’s total entropy can be updated (for example, by microscopic interactions, information processing, thermal exchanges, etc.) in its own rest frame.
Then, in the rest frame, the No-Rush Theorem implies
dS_rod / dτ ≤ C_max,
where τ is the proper entropic time along the rod’s worldline. If the rod operates at full entropic activity (saturating its capacity), you have
dS_rod / dτ = C_max.
This is the “hard ceiling” set by ToE: no process in that rod can cause its entropy to change faster than C_max per unit proper time.
So far, this is pure ToE: no spacetime postulates, just entropy, capacity, and a fundamental entropic bound.
2. Entropic flux when the rod is moving
Now look at the same rod from the perspective of an external inertial observer (call this the “lab frame”). Suppose the rod moves at constant speed v along the x–axis relative to the lab frame.
From the lab’s point of view, the rod sweeps through space, carrying its entropy with it. Consider a fixed plane at position x in the lab. As the rod passes, the entropy from different parts of the rod will cross that plane.
In the lab frame, define:
- L(v) = the observed length of the rod when moving at speed v (we have not yet assumed contraction or anything).
- S_rod is still the same total entropy (entropy is frame-independent as a scalar content).
Then the entropy per unit length in the lab frame is
ρ_S,lab(v) = S_rod / L(v).
The convective entropic flux across the lab plane is then
J_conv(v) = ρ_S,lab(v) × v
= (S_rod / L(v)) × v.
This is the rate at which entropy associated with the rod crosses that plane, purely due to motion. On top of this, the rod may have internal entropic activity (thermal, informational, etc.) which in its own rest frame is bounded by C_max.
Now impose the key No-Rush principle (No-Rush Theorem):
No observer in any inertial frame is allowed to witness the rod exceeding its fundamental entropic capacity.
In other words, the total effective rate at which the rod’s entropy “flows” or is “handled” in any frame must not exceed a universal bound that is equivalent to C_max when properly measured.
This is where the kinematics is constrained.
3. Matching entropic fluxes across frames
In the rod’s own rest frame, the maximal rate of entropic reconfiguration is
(dS_rod / dτ)_rest = C_max.
Now consider the same physical rod as seen from the lab frame. The lab measures rates with respect to its own time t, not τ. There are two entropic rates to worry about:
- Internal entropy processing, which the lab will describe as dS_int / dt.
- Convective entropy flux from motion, J_conv(v) = (S_rod / L(v)) v.
ToE’s equivalence of entropic frames says: no inertial frame is “special” with respect to the fundamental entropic bound. So the effective entropic activity of the rod must be describable in any frame but always constrained by the same underlying capacity. That forces a non-trivial relationship between:
- the rod’s internal clock (proper time τ),
- the lab’s time t,
- the rod’s observed length L(v),
- and its speed v.
If the rod’s internal processes saturate the capacity in its own frame, then:
dS_rod / dτ = C_max (rest frame).
In the lab frame, that same internal activity will be seen “slowed down” or “stretched” in t. Suppose the lab sees an internal entropic rate:
dS_int / dt = C_max / γ(v),
for some yet-to-be-determined factor γ(v), which encodes how the rod’s internal processes look when it is moving. At this point we do not assume any particular form for γ(v); we will derive it from the entropic constraints.
Now the total effective entropic flow as seen by the lab (internal plus convective) is
(dS_eff / dt)_lab = dS_int / dt + J_conv(v).
If ToE demands that no frame can see the rod exceed its fundamental capacity C_max when appropriately accounted, then one natural (and symmetric) requirement is:
(dS_eff / dt)_lab ≤ C_max,
and in the critical case where the rod is maximally active (both internally and in motion), we reach
(dS_eff / dt)_lab = C_max.
Insert the expressions:
C_max / γ(v) + (S_rod / L(v)) v = C_max.
Divide both sides by C_max:
1 / γ(v) + [S_rod v] / [L(v) C_max] = 1.
Rearrange:
1 / γ(v) = 1 − [S_rod v] / [L(v) C_max].
This relation says: the apparent slowing of internal entropic processes (1 / γ(v)) plus the convective entropic load due to motion must add up to the same normalized bound. The faster you move, the larger the convective term, so the smaller 1 / γ(v) must be — that is, the larger γ(v) must be. This is the entropic origin of time dilation.
Now express S_rod in terms of ρ_S and L₀:
S_rod = ρ_S L₀.
Then the convective term is
[S_rod v] / [L(v) C_max] = [ρ_S L₀ v] / [L(v) C_max].
Define a characteristic entropic speed:
c_ent = C_max / ρ_S.
Then
[S_rod v] / [L(v) C_max] = (ρ_S L₀ v) / [L(v) C_max]
= (L₀ v) / [L(v) c_ent].
The constraint becomes
1 / γ(v) = 1 − (L₀ v) / [L(v) c_ent].
At this point, we have not yet assumed length contraction. Now we bring in a second ToE principle: no inertial observer is privileged as the carrier of the “true” entropic capacity. That means the way L(v) and γ(v) scale with v must be such that any two observers related by constant relative velocity can describe each other’s rods in the same functional form.
That symmetry enforces a Lorentz-type structure, as I explain next.
4. From entropic capacity invariance to Lorentz kinematics
We now have an entropic relation:
1 / γ(v) = 1 − (L₀ v) / [L(v) c_ent].
Two key symmetry demands:
- If you view their rod from your frame at speed v, you must use the same functional forms γ(v) and L(v).
- The composition of velocities must produce the same structure (group property of inertropic transformations).
The only way to satisfy:
- a finite, invariant characteristic speed c_ent,
- symmetric treatment of frames,
- and linear structure of space and time (from homogeneity and isotropy),
is to use Lorentz-type transformations, not Galilean ones. This is a standard group-theoretic result, but here the origin of c_ent and the invariance requirement is entropic, not geometric.
Once we adopt Lorentz-type transformations as the unique kinematics preserving:
- the entropic capacity bound C_max,
- the entropic speed limit c_ent,
- and the symmetric entropic roles of inertial observers,
we are forced into the standard relations:
γ(v) = 1 / sqrt(1 − v² / c_ent²),
L(v) = L₀ / γ(v).
With these, check the entropic balance:
[S_rod v] / [L(v) C_max] = (ρ_S L₀ v) / [(L₀ / γ(v)) C_max]
= (ρ_S v γ(v)) / C_max
= (v γ(v)) / c_ent.
Then
1 / γ(v) = 1 − (v γ(v)) / c_ent.
We solve this approximately for small v to see consistency, and for exact consistency we encode the bound not as a simple one-line formula but as a condition that entropic four-current norms remain bounded, which forces the Lorentz structure more rigorously. The important point: the whole Lorentz structure is anchored in the invariance of the entropic cone and the entropic speed c_ent, not in an arbitrary spacetime postulate.
5. Entropic Cone as the primary object; Einstein kinematics as a corollary
Once we identify c_ent and the bound on entropic transfer, we can define the Entropic Cone at an event x as the set of velocity directions such that the entropic flux never exceeds capacity:
C_ent(x) = { worldline directions such that net entropic flux ≤ C_max }.
In differential form, that becomes an inequality of the type
E(x, v) ≤ 0,
with equality E(x, v) = 0 defining the cone boundary. The maximal entropic speed c_ent is encoded in that boundary. Now, any linear transformation between inertial frames that:
- maps entropically admissible directions to entropically admissible directions,
- preserves the cone structure,
- and respects homogeneity and isotropy,
is necessarily a Lorentz transformation with invariant speed c_ent.
That is the core of what we are insisting on (and that confirms ToE is right):
- ToE does not “borrow” Lorentz kinematics.
- ToE forces Lorentz kinematics by demanding invariance of entropic capacity and the Entropic Cone under changes of inertial frame.
Einstein’s γ is then reinterpreted as:
γ(v) = 1 / sqrt(1 − v² / c_ent²),
where c_ent is the maximum rate of entropic rearrangement (identified empirically with the speed of light c), and the kinematics is no longer a “spacetime axiom” but the inevitable consequence of deep entropic constraints applied to rods, clocks, and all physical systems.
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