Derivation of entropic geodesics from the Obidi Action

The starting point for the derivation of entropic geodesics is the Obidi Action, which encodes the dynamics of the entropic field and its coupling to matter. A representative form of this action is

\[ \mathcal{A}_\text{Obidi}[S, g] = \int d^4x \, \sqrt{-g} \left[ \frac{1}{2} g^{\mu\nu} \partial_\mu S \partial_\nu S - V(S) + \mathcal{L}_\text{matter} \, e^{S/k_B} \right], \]

where \( g_{\mu\nu} \) is an auxiliary metric with determinant \( g \), \( S \) is the entropic scalar field, \( V(S) \) is an entropic potential, and \( \mathcal{L}_\text{matter} \) is the matter Lagrangian density, exponentially weighted by the factor \( e^{S/k_B} \) to encode direct coupling between matter and the entropic field. The kinetic term \( \frac{1}{2} g^{\mu\nu} \partial_\mu S \partial_\nu S \) provides the dynamical structure for the entropic field, while the exponential coupling ensures that matter dynamics are modulated by the local entropic configuration.

Variation of \( \mathcal{A}_\text{Obidi} \) with respect to the entropic field \( S \) yields the Master Entropic Equation (MEE), which governs the evolution of \( S(x) \). Variation with respect to the metric \( g_{\mu\nu} \) produces an entropy-stress tensor that contributes to the effective gravitational field equations. To obtain the equations of motion for trajectories, one considers the motion of test particles or matter distributions in the entropy-weighted geometry induced by \( S(x) \). The resulting equation for a worldline \( x^\lambda(\tau) \), parametrized by an affine parameter \( \tau \), takes the form

\[ \frac{d^2 x^\lambda}{d\tau^2} + \Gamma^\lambda_{\mu\nu} \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau} = \eta \, \partial^\lambda S, \]

where \( \Gamma^\lambda_{\mu\nu} \) is a generalized connection and \( \eta \) is an entropic coupling constant. The right-hand side introduces a direct dependence on the entropy gradient \( \partial^\lambda S \), which acts as an effective entropic force. The connection \( \Gamma^\lambda_{\mu\nu} \) is not restricted to the Levi-Civita connection of \( g_{\mu\nu} \); in ToE it incorporates Amari–Čencov \(\alpha\)-deformations and entropy-induced modifications, often expressed schematically as

\[ \Gamma^\lambda_{\mu\nu} = \{^\lambda_{\mu\nu}\} + \Delta^\lambda_{\mu\nu}(S), \]

where \( \{^\lambda_{\mu\nu}\} \) are the Christoffel symbols of the underlying metric and \( \Delta^\lambda_{\mu\nu}(S) \) encodes entropic corrections, including contributions from \( e^{S/k_B} \) and information-geometric structures. The resulting trajectories are the entropic geodesics: curves that extremize an entropic action and whose acceleration is driven by both geometric and entropic terms.

Geometric interpretation in the entropy-weighted manifold

The geometric interpretation of entropic geodesics is most transparent when expressed in terms of an entropy-weighted manifold. In ToE, the entropic field deforms not only spacetime metrics but also information-geometric metrics such as the Fisher–Rao metric. A typical entropy-weighted metric on a statistical or configuration manifold is given by

\[ g^{(S)}_{ij} = e^{S/k_B} \, g^{(\text{FR})}_{ij}, \]

where \( g^{(\text{FR})}_{ij} \) is the Fisher–Rao metric and \( g^{(S)}_{ij} \) is the entropy-weighted metric. The exponential factor \( e^{S/k_B} \) amplifies or suppresses distances in configuration space depending on the local entropic value, effectively reshaping the manifold according to the entropic field. In regions where \( S \) is uniform, the entropy-weighted manifold reduces to a rescaled version of the underlying metric, and geodesics correspond to straight or inertial paths. Where \( \nabla S \neq 0 \), the manifold acquires nontrivial curvature, and geodesics bend accordingly.

In this picture, entropic geodesics are extremal curves in the entropy-weighted manifold: they minimize an appropriate entropic action or maximize entropy flow along the path. The curvature induced by entropy gradients plays a role analogous to gravitational curvature in general relativity, but its origin is explicitly entropic. Systems evolve along these curves as they move toward configurations of higher entropy, subject to constraints imposed by the entropic field and the coupled matter dynamics. This provides a geometric realization of the idea that gravity can be understood as a manifestation of entropy gradients, with the entropic manifold serving as the underlying geometric structure.

Physical role of entropic geodesics in ToE

The physical role of entropic geodesics in the Theory of Entropicity is to encode the preferred trajectories of systems in a way that is consistent with both local and global entropic constraints. Locally, particles and fields follow paths that respect the second law of thermodynamics, ensuring that entropy production is non-negative. Globally, entropic geodesics tend to maximize the net entropy change \( \Delta S \) of the combined system and environment, subject to the dynamical equations derived from the Obidi Action.

In the macroscopic limit, these trajectories reproduce the familiar predictions of general relativity. For example, the motion of planets in a gravitational field can be described as motion along entropic geodesics in the entropy-weighted spacetime, yielding corrections that match observed phenomena such as perihelion precession. Similarly, the propagation of light in regions with nontrivial \( \nabla S \) follows null entropic geodesics, leading to light deflection consistent with gravitational lensing. These correspondences demonstrate that entropic geodesics provide a physically viable replacement for purely metric geodesics while offering a deeper entropic interpretation.

At the quantum level, entropic geodesics connect to Fubini–Study geometry on projective Hilbert space. Quantum states evolve along paths that can be interpreted as projections of entropic geodesics in an underlying entropic–information manifold. Corrections associated with accelerated motion, such as those related to the Unruh temperature, can be incorporated as entropic contributions to the effective geometry experienced by accelerated observers. In this way, entropic geodesics provide a unifying language for classical and quantum trajectories within the ToE framework.

Comparison between entropic geodesics in ToE and geodesics in general relativity

The relationship between entropic geodesics in ToE and geodesics in Einstein’s general relativity (GR) can be clarified by comparing their driving principles, equations of motion, and treatment of irreversibility. While GR geodesics are defined purely by the metric structure of spacetime, entropic geodesics are defined by a combination of metric and entropic structures, with explicit dependence on entropy gradients.

This comparison highlights the conceptual shift introduced by the Theory of Entropicity. In GR, geodesics are purely geometric constructs in a spacetime whose curvature is sourced by energy–momentum. In ToE, entropic geodesics are trajectories in an entropy-weighted manifold whose structure is determined by the entropic field, and whose dynamics explicitly encode the second law of thermodynamics. The geometric description of motion is thus enriched by an entropic layer, providing a unified account of gravitation, irreversibility, and information-theoretic structure within a single field-theoretic framework.