The Obidi Field Equations of Motion (OFEoM): Variational and Conceptual Foundations of the Action Principle of the Theory of Entropicity (ToE)

The Obidi Field Equations of Motion (abbreviated as OFEoM or simply OFE) constitute the fundamental dynamical law of the Theory of Entropicity (ToE). They are obtained from the Obidi Action Principle (OAP), which promotes the entropic field \( S(x) \) to a universal, generative entity. In this formulation, entropy is no longer treated as a statistical descriptor of macroscopic ensembles but as a fundamental scalar field defined on spacetime, encoding the entropic accessibility of each region and governing the evolution of matter, geometry, and motion.

Within this framework, the entropic field \( S(x) \) plays a role analogous to that of the metric in General Relativity (GR) or the wavefunction in Quantum Mechanics. The OFE determine how the underlying entropic substrate flows, organizes, and constrains the universe. They form the core of the Master Entropic Equation (MEE), which serves as the unifying dynamical equation of ToE, in the same way that Einstein’s field equations govern the dynamics of spacetime in GR.

The Obidi Action: Variational Foundation of the Theory of Entropicity

The dynamics of the entropic field are formulated through a variational principle. The starting point is the Obidi Action, defined on a spacetime manifold \( M \) endowed with a metric \( g_{\mu\nu} \). The action functional is written as

where \( \sqrt{-g} \) denotes the square root of the negative determinant of the metric \( g_{\mu\nu} \), \( S(x) \) is the entropic field, and \( T_{\mu\nu} \) is the matter stress–energy tensor. The function \( \mathcal{L} \) is the entropic Lagrangian density, which encodes the kinetic, potential, and interaction structure of the theory.

A general and physically motivated form of the entropic Lagrangian is

where \( A(S) \) is an entropic stiffness function controlling the response of the field to spatial and temporal gradients, \( V(S) \) is an entropic potential, and \( F(S, T_{\mu\nu}) \) encodes the coupling between matter and the entropic field. The constant \( \eta \) is a coupling parameter that sets the strength of this interaction.

The first term, \( A(S) g^{\mu\nu} \nabla_\mu S \nabla_\nu S \), plays the role of a kinetic term for the entropic field, but with a coefficient that can depend on the field itself, allowing for nonlinear self‑interaction. The second term, \( V(S) \), represents an effective entropic potential that can encode preferred configurations or vacuum structures of the entropic substrate. The third term, \( \eta F(S, T_{\mu\nu}) \), provides a mechanism by which matter and energy content influence, and are influenced by, the entropic field.

Crucially, in ToE, the field \( S(x) \) is not interpreted as a conventional matter field. Instead, it is the generative substrate from which matter, geometry, and dynamical laws emerge. The Obidi Action therefore defines a field theory of entropy in which the entropic field is primary and all other physical structures are secondary manifestations.

Derivation of the Obidi Field Equations (OFE)

The Obidi Field Equations are obtained by applying the Euler–Lagrange formalism to the Obidi Action. For a scalar field in curved spacetime, the Euler–Lagrange equation takes the form

The derivation proceeds by computing each contribution explicitly, starting from the Lagrangian density \( \mathcal{L}(S, \nabla S, g_{\mu\nu}, T_{\mu\nu}) = A(S) g^{\mu\nu} \nabla_\mu S \nabla_\nu S + V(S) + \eta F(S, T_{\mu\nu}). \)

3.1 Derivative with respect to the entropic gradient

The dependence of \( \mathcal{L} \) on the derivatives of \( S \) arises solely through the kinetic term. Writing \( \nabla_\mu S = \partial_\mu S \) for a scalar field, one has

This expression represents the canonical momentum density conjugate to the entropic field with respect to spacetime derivatives.

3.2 Divergence term in curved spacetime

The first term in the Euler–Lagrange equation involves the divergence of the quantity \( \sqrt{-g} \, \frac{\partial \mathcal{L}}{\partial (\partial_\mu S)} \). Substituting the expression obtained above yields

Using the identity \( \nabla_\mu V^\mu = \frac{1}{\sqrt{-g}} \partial_\mu (\sqrt{-g} V^\mu) \) for any vector field \( V^\mu \), this can be written as

This term captures the covariant divergence of the entropic flux associated with the field \( S(x) \).

3.3 Derivative with respect to the entropic field

The derivative of the Lagrangian density with respect to the field \( S \) itself receives contributions from all three terms:

where \( A'(S) = \frac{dA}{dS} \) and \( V'(S) = \frac{dV}{dS} \). The first term represents the feedback of the entropic field on its own kinetic structure, the second term encodes the local response of the entropic potential, and the third term captures the variation of the matter–entropy coupling.

3.4 Final form of the Obidi Field Equations

Substituting the expressions for the divergence term and the field derivative into the Euler–Lagrange equation yields

This is the Obidi Field Equation (OFE) in its general form [without the spectral component of the Spectral Obidi Action (SOA), for the sake of simplicity]. It is a nonlinear, self‑coupled, matter‑sourced partial differential equation governing the evolution of the entropic field \( S(x) \). The nonlinearity arises both from the dependence of \( A(S) \) on the field and from the explicit quadratic term in the gradient of \( S \).

Conceptual Structure and Interpretation of the Obidi Field Equations

The Obidi Field Equations (OFE) differ in a fundamental way from conventional field equations in classical and quantum field theory. In standard formulations, fields such as the electromagnetic potential or scalar fields typically propagate on a fixed geometric background, and their self‑interaction is limited to specific nonlinear terms. In contrast, the entropic field \( S(x) \) in ToE is both dynamical and generative: it shapes the effective geometry and constrains the admissible dynamics of matter.

The term \( 2 \nabla_\mu \left( A(S) \nabla^\mu S \right) \) represents a generalized entropic diffusion or propagation term, where the effective “diffusivity” is modulated by the field‑dependent stiffness \( A(S) \). The contribution \( -A'(S) (\nabla S)^2 \) introduces a strong nonlinear feedback: regions with large entropic gradients can significantly alter the local propagation properties of the field. The potential term \( -V'(S) \) encodes preferred configurations or vacuum structures of the entropic substrate, while the matter–entropy coupling term \( -\eta \frac{\partial F}{\partial S} \) allows matter content to act as a source or sink for entropic structure.

Conceptually, the OFE implement a self‑referential dynamics in which the entropic field governs its own evolution and, simultaneously, the evolution of the effective geometry and matter distribution. The metric \( g_{\mu\nu} \) is not an independent primitive but an emergent structure determined by the entropic field and its couplings. This leads naturally to a picture in which geometry and entropy co‑evolve, and in which the familiar gravitational phenomena of General Relativity arise as an effective description of deeper entropic dynamics.

From a control‑theoretic and informational perspective, the OFE can be viewed as a continuous, field‑theoretic analogue of a Hamilton–Jacobi–Bellman type equation, where the entropic field encodes a global optimization landscape. The universe evolves along trajectories that are compatible with the entropic constraints encoded in \( S(x) \), and the OFE express the condition that this evolution extremizes an underlying entropic action.

Entropic Geodesics and the Motion of Test Bodies: Method of the Entropic Cost Functional (ECF)

To describe the motion of test bodies in the entropic background defined by \( S(x) \), the Theory of Entropicity introduces an entropic cost functional (ECF) associated with a worldline \( \gamma \). Let \( x^\mu(\lambda) \) denote a parametrized trajectory with tangent vector \( u^\mu = \frac{dx^\mu}{d\lambda} \). The entropic cost functional is defined as

where \( m \) is the rest mass of the test body and \( \alpha \) is a coupling constant that measures the strength of the interaction between the trajectory and the entropic field. The first term is the usual kinetic contribution associated with motion in a metric background, while the second term represents the entropic contribution to the cost of the trajectory.

Varying \( \mathcal{R}[\gamma] \) with respect to the path \( x^\mu(\lambda) \), while keeping the endpoints fixed, yields the corresponding Euler–Lagrange equations. The variation of the kinetic term produces the usual geodesic term \( m \frac{D u^\mu}{D\lambda} \), where \( \frac{D}{D\lambda} \) denotes the covariant derivative along the curve. The variation of the entropic term contributes a force proportional to the gradient of \( S(x) \). The resulting equation of motion is

Dividing by \( m \) and defining \( \kappa = \frac{\alpha}{m} \), one obtains

These trajectories are called entropic geodesics. They generalize the notion of metric geodesics by incorporating the influence of the entropic field. In the absence of other forces, a test body follows a path that extremizes the entropic cost functional, and its acceleration is determined by the gradient of the entropic field. This provides a direct link between the entropic structure of spacetime and the observed motion of bodies.

Newtonian Gravity as the Weak–Field Limit of the Entropic Field

The entropic geodesic equation admits a clear non‑relativistic limit in which it reproduces the familiar form of Newtonian gravity. Consider a regime in which spacetime is approximately flat, velocities are small compared to the speed of light, and the parameter \( \lambda \) can be identified with coordinate time \( t \). In this limit, the spatial components of the entropic geodesic equation reduce to

where \( \mathbf{x}(t) \) denotes the spatial position of the test body and \( \nabla \) is the spatial gradient. Define an effective gravitational potential \( \Phi(\mathbf{x}) \) by

Substituting this definition into the equation of motion yields

which is precisely the Newtonian equation of motion in a gravitational potential \( \Phi \). Thus, in the weak‑field, low‑velocity limit, the entropic field \( S(\mathbf{x}) \) reproduces the familiar Newtonian gravitational dynamics.

To connect this with the standard inverse‑square law, consider a spherically symmetric configuration in which the entropic field depends only on the radial coordinate \( r \). Suppose that outside a localized source, the field takes the form

where \( S_0 \) and \( B \) are constants. The radial gradient is then

with \( \hat{r} \) the radial unit vector. Substituting into the equation of motion gives

To match the Newtonian gravitational acceleration \( \mathbf{a} = -\frac{G M}{r^2} \hat{r} \), one chooses the constants such that

where \( M \) is the mass of the central source and \( G \) is Newton’s gravitational constant. The acceleration then becomes

which is exactly the Newtonian inverse‑square law. This demonstrates that Newtonian gravity emerges naturally as the weak‑field limit of the entropic field dynamics encoded in the OFE and the entropic geodesic principle.

General Relativity as an Emergent Geometric Limit

In General Relativity, the gravitational interaction is encoded in the curvature of spacetime, and the motion of test bodies is described by geodesics of the metric \( g_{\mu\nu} \). In the weak‑field, static limit around a mass distribution, the time–time component of the metric can be written as

where \( \Phi \) is the Newtonian gravitational potential and \( c \) is the speed of light. The geodesic equation in this metric reproduces the Newtonian acceleration \( \mathbf{a} = -\nabla \Phi \) in the appropriate limit.

In the Theory of Entropicity, the effective gravitational potential is not fundamental but arises from the entropic field \( S(x) \). One may therefore define an effective potential \( \Phi_{\text{eff}}(x) \) as a function of the entropic field:

for some monotonic function \( f \). The corresponding effective metric component is then

In the weak‑field regime, the entropic geodesics derived from the entropic cost functional coincide with the metric geodesics of this effective metric. The dynamics of the entropic field, governed by the OFE, determine the function \( S(x) \), which in turn determines \( f(S) \) and hence the effective metric. With appropriate choices of the functions \( A(S) \), \( V(S) \), and the coupling \( F(S, T_{\mu\nu}) \), the resulting effective metric can be arranged to satisfy the Einstein Field Equations in the relevant limit.

In this sense, General Relativity appears as an emergent geometric encoding of a deeper entropic dynamics. The curvature of spacetime is not fundamental but is instead a macroscopic manifestation of the structure and evolution of the entropic field. The OFE thus provide a more primitive description, from which the Einstein equations can be recovered as a limiting case.

Summary and Structural Role of the Obidi Field Equations