Entropic foundations and the Einstein limit

The Theory of Entropicity (ToE), formulated and developed by John Onimisi Obidi, provides a derivation of the Einstein Field Equations (EFE) of General Relativity (GR) starting from an entropic action for a fundamental entropy field \(S(x)\). The central idea is that the metric field equations obtained from this entropic action reduce to the standard Einstein equations in an appropriate smooth, near‑equilibrium limit, in which the entropy field approaches a reference configuration \(S_0\). In this regime, the entropic contributions organize themselves into an effective stress–energy tensor and cosmological‑constant‑like terms, so that the resulting metric dynamics coincide with those of GR.

Obidi Action as the starting point

The derivation begins from a variational principle in which the fundamental field is the scalar entropy field \(S(x)\). The corresponding action functional incorporates three essential ingredients: a geometric term built from the curvature scalar, a kinetic term for the entropy field, and a distinguishability potential that measures the deviation of the entropy configuration from a reference state \(S_0\). A representative form, used in the Einstein‑derivation work, is the [trivial form of the] Obidi Action

\[ A_{\text{ToE}}[g, S] = \int \mathrm{d}^4 x \,\sqrt{-g}\, \big[ \alpha^2 R[g] - \beta^2 g^{\mu\nu} \nabla_\mu S \nabla_\nu S - \lambda\, D(S, S_0) \big], \]

where \(g_{\mu\nu}\) is the metric, \(g\) its determinant, \(R[g]\) the Ricci scalar curvature, \(\nabla_\mu S\) the covariant gradient of the entropy field, and \(D(S, S_0)\) a local distinguishability functional of Kullback–Leibler type that quantifies the deviation of \(S\) from a reference configuration \(S_0\). The constants \(\alpha\), \(\beta\), and \(\lambda\) are coupling parameters that set the relative strength of curvature, entropic kinetics, and distinguishability contributions.

In this formulation, the geometric term \(\alpha^2 R[g]\) plays a role analogous to the Einstein–Hilbert term in GR, but it is embedded within a broader entropic field theory. The kinetic term \(\beta^2 g^{\mu\nu} \nabla_\mu S \nabla_\nu S\) endows the entropy field with dynamics, allowing it to propagate and form gradients. The distinguishability potential \(\lambda D(S, S_0)\) encodes a preference for configurations close to the reference state \(S_0\), and its structure is crucial for defining the equilibrium and near‑equilibrium regimes in which the Einstein limit is recovered.

Entropy variation and the Master Entropic Equation

Varying the above Obidi Action with respect to the entropy field \(S(x)\) yields the Master Entropic Equation, which is a nonlinear wave‑type equation governing the dynamics of the entropic field. In the simplified case, the variation leads to an equation of the form

\[ \beta^2 \nabla_\mu \nabla^\mu S = \lambda \, \ln \!\frac{S}{S_0}, \]

where \(\nabla_\mu \nabla^\mu\) denotes the covariant d'Alembertian operator associated with the metric \(g_{\mu\nu}\). The right‑hand side involves the logarithm of the ratio \(S / S_0\), reflecting the Kullback–Leibler‑type structure of the distinguishability functional \(D(S, S_0)\). This equation describes how deviations of the entropy field from its reference configuration propagate and relax under the combined influence of geometry and the distinguishability potential.

To analyze the behavior near equilibrium, one considers a small perturbation around the reference configuration, writing \(S = S_0 + \delta S\) with \(|\delta S| \ll |S_0|\). Linearizing the logarithmic term in this regime yields

\[ \ln \!\frac{S}{S_0} = \ln \!\left( 1 + \frac{\delta S}{S_0} \right) \approx \frac{\delta S}{S_0}, \]

so that the Master Entropic Equation reduces to a Klein–Gordon‑like equation for the perturbation \(\delta S\):

\[ \beta^2 \nabla_\mu \nabla^\mu \delta S = \lambda \, \frac{\delta S}{S_0}. \]

This can be written in the standard Klein–Gordon form with an effective mass term determined by the curvature of the distinguishability potential around \(S_0\). In this interpretation, the entropy field supports propagating “curvature waves” of entropy, whose dynamics are governed by the background geometry and the structure of the entropic potential.

Metric variation and Einstein‑like tensor equations

Varying the same Obidi Action with respect to the metric \(g_{\mu\nu}\) produces a set of tensor field equations in which geometric curvature is sourced by the entropy field. The variation of the curvature term yields the Einstein tensor \(G_{\mu\nu}\), while the variation of the entropic kinetic and potential terms produces an effective entropic stress–energy tensor. A representative form of the resulting equation is

\[ G_{\mu\nu} = \frac{1}{\alpha^2} \Big[ \beta^2 \big( \nabla_\mu S \nabla_\nu S - \tfrac{1}{2} g_{\mu\nu} (\nabla S)^2 \big) + \lambda g_{\mu\nu} D(S, S_0) \Big], \]

where \(G_{\mu\nu}\) is the Einstein tensor constructed from \(g_{\mu\nu}\), and \((\nabla S)^2 = g^{\rho\sigma} \nabla_\rho S \nabla_\sigma S\). The first term on the right‑hand side has the structure of the stress–energy tensor of a scalar field, built entirely from the gradients of the entropy field. The second term, proportional to \(g_{\mu\nu} D(S, S_0)\), behaves like a potential energy density associated with the distinguishability functional.

This equation has the same formal structure as the Einstein field equations with a specific effective stress–energy tensor:

\[ G_{\mu\nu} = 8\pi G \, T_{\mu\nu}^{\text{(eff)}}, \]

where the effective tensor \(T_{\mu\nu}^{\text{(eff)}}\) is constructed entirely from the entropy field and its potential. In the ToE framework, curvature is therefore not taken as primitive but is induced by the entropic field dynamics, with the metric responding to entropic gradients and distinguishability in a manner analogous to how it responds to matter and energy in GR.

Einstein limit: smooth configurations and weak entropic gradients

The recovery of the standard Einstein Field Equations proceeds by considering a regime in which the entropy field is close to its reference configuration and its gradients are relatively weak and smooth. In this near‑equilibrium limit, one takes \(S \to S_0\) with small deviations and slowly varying gradients. Under these conditions, the distinguishability functional \(D(S, S_0)\) behaves effectively like a cosmological‑constant‑type term, since its value approaches a constant minimum at \(S = S_0\). The contribution \(\lambda g_{\mu\nu} D(S, S_0)\) in the metric equation then plays the role of an effective cosmological constant. This parallels the small positive cosmological constant that Ginestra Bianconi derives in her elegant Gravity from Entropy (GfE) framework, where gravitational behavior and large‑scale curvature emerge from relative entropic principles applied to network manifolds.

At the same time, the entropy‑gradient contribution to the right‑hand side of the metric equation takes the form of a scalar‑field‑like stress–energy tensor or, in suitable approximations, a fluid‑like source with a fixed equation of state. In the smooth limit, these contributions can be reorganized and identified with a conventional stress–energy tensor \(T_{\mu\nu}\) describing matter and energy in GR.

By matching coefficients, one identifies the combination of entropic couplings with the gravitational constant, setting

\[ \frac{1}{\alpha^2} \beta^2 \;\longrightarrow\; 8\pi G, \]

and interpreting the entropic contributions as defining an effective \(T_{\mu\nu}\). In this regime, the metric equation reduces to the familiar Einstein form

\[ G_{\mu\nu} = 8\pi G \, T_{\mu\nu}, \]

so that Einstein’s equations appear as the reversible, near‑equilibrium limit of the more general entropic dynamics. The ToE thus interprets GR as an emergent, coarse‑grained description of an underlying entropic field theory, valid when entropic gradients are weak and the system is close to an equilibrium configuration.

The role of \(\ln 2\) and the Obidi Curvature Invariant

Within this entropic derivation, a distinguished role is played by the quantity \(\ln 2\), which defines the Obidi Curvature Invariant (OCI). This invariant arises as the minimal non‑zero contrast in the distinguishability potential \(D(S, S_0)\), corresponding to the smallest entropic curvature “fold” that yields one bit of physical distinguishability. In other words, \(\ln 2\) quantifies the minimal entropic curvature required for two configurations to be recognized as distinct at the level of the entropic field.

In the GR limit, this invariant does not appear explicitly in the Einstein equations, because it is effectively absorbed into the normalization of the action and into cosmological‑constant‑like terms. Nevertheless, it conceptually anchors the entropic origin of curvature that standard Einstein gravity treats as primitive. The presence of the Obidi Curvature Invariant indicates that curvature, in the ToE framework, is ultimately a manifestation of discrete entropic distinguishability at the level of the fundamental field, even though in the smooth, macroscopic limit it is described by the continuous geometry of GR.

Thus, while the Einstein Field Equations emerge as an effective description in the appropriate limit, the underlying entropic theory retains a deeper structure in which quantities such as \(\ln 2\) encode the minimal units of curvature and information. This reinforces the interpretation of GR as a coarse‑grained entropic theory of geometry, rather than as a fundamental description of spacetime.

References

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2. Grokipedia — John Onimisi Obidi
   Scholarly profile of John Onimisi Obidi, originator of the Theory of Entropicity (ToE), including philosophical and historical motivation, background and research contributions.
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16. International Journal of Current Science Research and Review (IJCSRR)
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