From ln 2 to Gravity: The Emergence of Spacetime from Entropic Curvature of the Theory of Entropicity (ToE)
Within the framework of the Theory of Entropicity (ToE), gravity is not a fundamental force transmitted by a field or particle, but the statistical geometry of informational curvature within the universal entropic field .
Wherever this field bends, folds, or differentiates, energy gradients appear — and those gradients are perceived, in conventional physics, as gravitational attraction.
The key to this reinterpretation lies in the Obidi Curvature Invariant (OCI), the universal entropic curvature quantum . This invariant represents the smallest stable fold of the entropic field — the minimal distinguishable curvature between two informational configurations. ToE posits that this elementary fold defines the unit of geometric distinction from which all spacetime structure arises.
1. The Entropic Gradient as Gravitational Potential
In standard thermodynamics, a force appears wherever there is a gradient in entropy or temperature.
In ToE, the same principle applies, but entropy itself is now a field quantity rather than a macroscopic measure. The spatial derivative of the entropic field gives a natural vector quantity:
\vec{F}_S = -\, T_S\, \nabla S(x),
where is the entropic force density.
This force is not postulated but emerges whenever the informational temperature and entropic gradient coexist.
The minus sign expresses that the entropic field seeks to restore uniformity — curvature relaxes toward flatness.
Now, if is locally proportional to energy density, and energy density curves the field through the Master Entropic Equation (MEE) (the ToE analogue of Einstein’s field equations), then the macroscopic result of such relaxation is what we call gravitational acceleration.
2. From the ln 2 Fold to the Gravitational Field
Consider two infinitesimal regions of the entropic field, separated by the minimum stable curvature ratio 2:1, corresponding to an entropic difference .
Let the local temperature be .
The energy difference between these two regions is then:
\Delta E = k_B T_S \ln 2.
If a particle or quantum of matter moves across this gradient, it experiences a net change in informational energy per unit distance.
The spatial derivative of this energy defines the entropic force:
F = -\,\frac{dE}{dx} = -\,k_B \ln 2\, \frac{dT_S}{dx}.
ToE interprets as the local curvature gradient — the rate of change of informational temperature across the manifold. Thus, gravitational acceleration is proportional to the spatial derivative of the informational temperature, or equivalently, to the curvature gradient in the entropic field.
In regions of spacetime where curvature increases — near masses, horizons, or regions of informational compression — is positive, and matter experiences an inward acceleration. This aligns precisely with Einstein’s insight that matter tells spacetime how to curve, and curvature tells matter how to move, but ToE extends it:
Entropy tells energy how to curve, and curvature tells energy how to flow.
3. The Holographic Surface as an Entropic Boundary
The connection between entropic curvature and gravity becomes clearer when considering the holographic principle.
In ToE, every surface enclosing a region of space acts as a local “entropic horizon,” carrying a finite number of informational folds, each corresponding to an area element proportional to .
If each such fold stores one bit of information, then the total entropy on a surface of area is:
S = \frac{k_B A}{4\, L_P^2} \ln 2,
where is the Planck length.
This formula resembles the Bekenstein–Hawking relation but now includes the ln 2 curvature explicitly as the entropic quantization unit of area.
In this view, the smallest possible change in the area of a horizon corresponds to the creation or annihilation of a single ln 2 fold — a minimal geometric event.
Hence, the black hole horizon and every holographic surface are physical manifestations of discrete informational curvature quanta. The “pixels” of spacetime are ln 2-curved folds in the entropic field.
4. Recovering Newton’s Law from Entropic Curvature
ToE allows one to derive Newton’s gravitational law without assuming it.
Consider a holographic screen enclosing a mass at temperature . The energy on the screen is distributed in bits, each carrying . The total number of bits is proportional to the surface area .
Thus:
E = \frac{A}{L_P^2}\, k_B T_S \ln 2.
Using the energy–mass relation and assuming equilibrium at the screen, ToE gives:
M c^2 = 4\pi r^2 \frac{k_B T_S \ln 2}{L_P^2}.
Differentiating with respect to and applying the equipartition condition between matter and informational degrees of freedom yields:
F = -\,\frac{G M m}{r^2},
where arises naturally from the proportionality constants in the entropic field equations.
Thus, Newton’s law of universal gravitation is recovered not from an external postulate, but as a low-curvature limit of the informational field dynamics governed by the ln 2 curvature invariant.
5. Geometric–Thermal Equivalence of Gravitation
This derivation reveals that gravity is not a fundamental force but the macroscopic perception of curvature gradients in the entropic field.
Regions of high curvature (where informational temperature varies sharply) act as gravitational wells.
The acceleration we attribute to gravitational attraction is simply the entropic field’s attempt to restore uniformity — a relaxation of geometric information.
In this light, spacetime itself is an emergent thermogeometric fabric:
- curvature measures informational difference,
- temperature measures reconfiguration rate, and
- gravity measures the resulting entropic flow.
At the smallest scale, each ln 2 fold defines the elementary unit of this structure — the first possible difference in the universe’s informational geometry.
6. Conceptual Consequences
The Obidi Curvature Invariant thereby anchors the unity of physics at its deepest level.
It connects:
- Thermodynamics (through ),
- Information Theory (through the distinguishability threshold of ln 2),
- Geometry (as the minimal curvature fold of ), and
- Gravity (as large-scale entropic relaxation of curvature).
The simplicity of ln 2 conceals a profound universality.
It defines not merely the conversion between bit and entropy, but the quantum of curvature that underlies spacetime itself.
Wherever curvature exists, there is temperature; wherever temperature changes, there is force; and at the threshold between indistinguishable and distinguishable curvature — there lies ln 2, the first spark of structure.
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