Deriving the Einstein Field Equations of General Relativity (GR) and Schrodinger Wave Equation of Quantum Mechanics from the Spectral Obidi Action (SOA) of the Theory of Entropicity (ToE)

1. What makes an “action” an action

In any field theory an action has to integrate a Lagrangian density that depends on the field itself and, crucially, on its spacetime variations. The prototype is

$$\mathcal A[\phi]=\int \mathcal L(\phi,\partial_\mu\phi)\,d^4x .$$

Varying $\mathcal A$ with respect to $\phi$ produces equations of motion.
An information–geometric quantity such as the Araki relative entropy,

$$S(\rho\Vert\sigma)=\operatorname{Tr}[\rho(\ln\rho-\ln\sigma)],$$

has no such derivatives—it compares two fixed operators. On its own it cannot generate dynamics.

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2. Making distinguishability a potential term

If we treat an entropy field $S(x)$ as the fundamental variable, the natural way to incorporate “relative entropy” information is to let it weight how far the present configuration $S(x)$ is from a local reference configuration $S_0(x)$. A simple covariant scalar capturing that idea is

$$D(x)=S(x)\ln\!\frac{S(x)}{S_0(x)}-S(x)+S_0(x),$$

the continuum analogue of Kullback–Leibler divergence.
$D(x)$ is positive, vanishes when $S=S_0$, and grows with distinguishability. This plays the role of a potential energy density.

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3. The Spectral Obidi Action

The full action must also contain a geometric or “kinetic” term that governs how the field varies through spacetime.
A minimal generally–covariant form is

$$\boxed{{\displaystyle {\mathcal{A}}_{\text{ToE}}[S]=\int d^{4}x\sqrt{-g}[\frac{\alpha }{2}R[g]-\frac{\beta }{2}{g}^{\mu \nu }{\mathrm{\nabla }}_{\mu }S{\mathrm{\nabla }}_{\nu }S-\lambda D(S,S_0)]\mathrm{.}}}$$

Here

• $R[g]$ – curvature scalar of the metric $g_{\mu\nu}$ induced by the entropic field,

• $\alpha,\beta,\lambda$ – coupling constants,

• $D(S,S_0)$ – the distinguishability potential introduced above.

The first term gives geometric dynamics (like Einstein–Hilbert),
the second term supplies the “kinetic” energy of entropy variations,
and the third term penalizes deviation from equilibrium curvature $S_0$.

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4. Field equation: the Master Entropic Equation

Varying this action with respect to $S$ gives

$$\beta\,\nabla_\mu\nabla^\mu S =\lambda\,\frac{\partial D}{\partial S} =\lambda\,\ln\!\frac{S}{S_0}.$$

This is a nonlinear wave equation whose equilibrium points satisfy $S=S_0$.
Small oscillations around equilibrium obey

$${\mathrm{\nabla }}_{\mu }{\mathrm{\nabla }}^{\mu }\delta S=\frac{\lambda }{\beta S_0}\delta S,$$

showing that departures from equilibrium propagate as curvature waves with an effective mass term $m_S^2=\lambda/(\beta S_0)$.

If we now couple $S(x)$back to the metric through the first term, the resulting Einstein–like equations read

$$G_{\mu \nu }=\frac{1}{\alpha }[\beta ({\mathrm{\nabla }}_{\mu }S{\mathrm{\nabla }}_{\nu }S-\frac{1}{2}{g}_{\mu \nu }(\mathrm{\nabla }S{)}^2)+\lambda g_{\mu \nu }D(S,S_0)],$$

which reduce to Einstein’s equations in the smooth limit where $S\to S_0$ and $D\to0$. The ln 2 unit of distinguishability enters through the normalization of $D(S,S_0)$: when $S/S_0=2$, $D=k_B\ln2$.

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5. Physical meaning

• The Araki–type term $D(S,S_0)$ is no longer a static comparison; it is a potential driving the field toward equilibrium.

• The Spectral Obidi Action unifies geometry ($R[g]$), dynamics ($\mathrm{\nabla }\mathrm{S),\; and\; information\; (}$$D$).

• Extremizing this action yields both the Master Entropic Equation for $S(x)$ and Einstein–like equations for $g_{\mu \mathrm{\nu .}}$

In this construction the constant $\ln \mathrm{2 }$appears as the first non-zero minimum of the potential $D$, identifying it with the smallest distinguishable curvature fold—the Obidi Curvature Invariant.

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6. Summary

The essential move is that ToE does not turn Araki relative entropy itself into an action; rather, it uses its functional form as the potential term within an action that already contains dynamical curvature and derivative terms.
That is what makes the resulting Spectral Obidi Action (SOA) a genuine variational principle capable of producing equations of motion and, in the appropriate limit, the familiar dynamics of general relativity and thermodynamics.

Deriving the Schrodinger Wave Equation from ToE

We must now turn our attention to deriving the Schrodinger wave equation of quantum mechanics from the Theory of Entropicity (ToE).

1. Linearising the entropic field

We start with the background $S_0(x)$ that satisfies the equilibrium (or “classical”) Master Entropic Equation,

$$\beta\,\nabla_\mu\nabla^\mu S_0=\lambda\,\ln\!\frac{S_0}{S_\text{ref}} .$$

Now write the full field as a small fluctuation about that background,

$$S(x)=S_0(x)+\delta S(x),\qquad |\delta S|\ll S_0 .$$

Expanding the distinguishability potential $D(S,S_0)$ to second order gives

$$D(S,S_0)\approx \frac{1}{2S_0}(\delta S)^2 ,$$

so in the small-fluctuation limit the Lagrangian density in the Spectral Obidi Action becomes, up to an overall $\sqrt{-g}$

$$\mathcal L_{\text{quad}} = -\frac{\beta}{2}g^{\mu\nu}\nabla_\mu\delta S\,\nabla_\nu\delta S -\frac{\lambda}{2S_0}(\delta S)^2 .$$

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2. The field equation for small oscillations

Varying this quadratic Lagrangian gives the wave equation

$$\beta\,\nabla_\mu\nabla^\mu \delta S +\frac{\lambda}{S_0}\,\delta S=0 .$$

In flat spacetime, with metric signature $(+,-,-,-)$, this reduces to

$$\partial_t^2\delta S -c_S^2\nabla^2\delta S +\omega_S^2\,\delta S=0 ,$$

where

$$c_S^2=1, \qquad \omega_S^2=\frac{\lambda}{\beta S_0}.$$

This is a Klein–Gordon-type equation describing harmonic oscillations of the entropic field with an effective “mass”

$$m_S=\hbar\,\sqrt{\frac{\lambda}{\beta S_0}}/c^2 .$$

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3. Canonical quantisation

To quantise, we promote $\delta S(x)$ and its conjugate momentum

$$\Pi(x)=\frac{\partial\mathcal L}{\partial(\partial_t\delta S)} =\beta\,\partial_t\delta S$$

to operators satisfying the equal-time commutation relation

$$[\delta S(\mathbf x),\Pi(\mathbf y)]=i\hbar\,\delta^{(3)}(\mathbf x-\mathbf y).$$

Expanding $\delta S(x)$ in plane-wave modes gives

$$\delta S(x)=\!\!\int\!\!\frac{d^3k}{(2\pi)^3\sqrt{2\omega_k}} \Big( a_{\mathbf k}\,e^{-i\omega_k t+i\mathbf k\cdot\mathbf x} +a_{\mathbf k}^\dagger\,e^{i\omega_k t-i\mathbf k\cdot\mathbf x} \Big),$$

with dispersion relation

$$\omega_k^2=c_S^2k^2+\omega_S^2.$$

The operators $a_{\mathbf k}$ and $a_{\mathbf k}^\dagger$ create and annihilate quanta of entropic curvature—small ripples in the $S(x)$field. These excitations are sometimes called entropions in the ToE language: the elementary carriers of entropic disturbance.

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4. Relation to quantum wave equations

When spatial variations are slow and curvature coupling is weak ($\omega_S$ small), each mode satisfies an approximate Schrödinger-type equation for its amplitude $\psi(\mathbf x,t)$,

$$i\hbar\,\partial_t\psi =-\frac{\hbar^2}{2m_S}\nabla^2\psi +V_{\text{eff}}(\mathbf x)\psi ,$$

where $V_{\text{eff}}$ arises from the background $S_0(x)$.
Thus, quantum mechanics appears as the low-energy, single-quantum limit of the entropic field dynamics: wave-like behaviour of small curvature disturbances in an otherwise classical entropic background.

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5. Conceptual picture

• The classical ToE equations describe how the mean entropic curvature $S_0(x)$ shapes geometry and gravitation.

• The quantum limit describes small fluctuations of that curvature—the entropic analogue of matter waves.

• Planck’s constant $\hbar$ sets the scale of quantisation, while the invariant $\ln 2$ fixes the smallest distinguishable amplitude of a curvature fold.

In this view, the Schrödinger and Dirac equations are not independent postulates but linearised, quantised limits of the broader entropic field theory embodied by the Spectral Obidi Action (SOA).