From Information Geometry to Information Gravity — Information Geometry as the Origin of Einstein's Gravity: Correspondence of the Obidi Action and the Einstein–Hilbert Action in the Theory of Entropicity (ToE)
Volume I, Part I (The Explanatory, Non-mathematical Part)
Preamble
The Theory of Entropicity (ToE) proposes that entropy is a fundamental field whose dynamics give rise to what we perceive as space‑time, matter and forces. The central object of the theory is the Obidi action—a functional of an entropic scalar field and the metric of a four‑dimensional manifold. When this action is varied with respect to the entropic field and the metric, it yields the Master Entropic Equation and the entropic Einstein equations, respectively. These equations generalize the Einstein–Hilbert action of general relativity by coupling entropy to curvature and reduce, in appropriate limits, to the Fisher–Rao information metric of classical information geometry and to the Einstein field equations. This letter, the third volume (Letter III) in the Theory of Entropicity (ToE) Living Review Letters Series (ToE LRLS), gives a comprehensive analysis of how information geometry becomes physical geometry and how the Obidi action corresponds to the Einstein–Hilbert action. After reviewing the foundations of information geometry, we introduce the entropic field and its action, derive the entropic field equations and examine how they map onto the Einstein–Hilbert action. We then discuss external approaches to emergent gravity—thermodynamic derivations and entropic gravity—and highlight the Haller–Obidi correspondence that identifies the classical action with entropy. Finally, we outline predictions and implications of ToE and comment on prospects for unification.
1 Foundations of information geometry
1.1 Statistical manifolds and the Fisher information metric
Information geometry studies spaces of probability distributions (statistical manifolds) as differentiable manifolds. Each point on such a manifold corresponds to a probability distribution or, in quantum theory, to a density matrix. To quantify distinguishability between nearby distributions one defines a Riemannian metric. John Baez explains that the covariance matrix of fluctuations around a maximum‑entropy state is positive‑definite and can be viewed as an inner product on the tangent space; the resulting Riemannian metric is called the Fisher information. In intuitive terms, if one considers a family of distributions parameterised by variables theta^a, the metric measures how fast expectation values change when parameters vary.
In the classical case, the Fisher–Rao metric on a family of probability densities p(x; theta) is given by an integral over x of p(x; theta) times the product of the derivatives of log p with respect to the parameters. This metric is invariant under re‑parameterizations and arises naturally when deriving lower bounds on estimation errors [the Cramér–Rao bound (CRB)]. It underlies the method of maximum entropy and plays a central role in inference theory. Baez emphasises that the metric can be visualised by imagining fluctuations forming an ellipsoid around a mean state; the eigenvectors of the covariance matrix define the principal axes of this ellipsoid.
1.2 Algorithmic information and information geometry
Information theory enters physics through both algorithmic complexity and probabilistic information. Kolmogorov complexity measures the minimal description length of a string or state. We find in the ToE literature that algorithmic complexity forms the first rung on a hierarchy: Kolmogorov complexity leads to statistical information and, through Fisher information, to information geometry. Probability distributions are viewed as points on a curved manifold, and the Fisher information metric provides a quantitative notion of curvature.
1.3 Information geometry as physical geometry
The Theory of Entropicity (ToE) reinterprets information geometry as physical geometry. In the ToE paper "Einstein and Bohr Finally Reconciled on Quantum Theory", Obidi emphasizes that ToE “reinterprets geometry itself as the result of entropy flow”. Curvature is not postulated but emerges from gradients in the entropic field; motion, interaction and structure are driven by irreversible entropic constraints rather than by a prior geometric background. Consequently, the speed of light c becomes the maximum rate of entropic redistribution [the entropic speed limit (ESL)], and gravity emerges as an entropic phenomenon. Thus, ToE subsumes information geometry into a dynamical theory of space‑time and matter.
2 The entropic field and the Obidi action
2.1 Entropy as a fundamental field
ToE posits that entropy, usually a measure of disorder or missing information, is a real scalar field S(x) defined on a four‑dimensional manifold. The entropic manifold M_S is a smooth, connected Lorentzian manifold equipped with a metric g_{mu nu}. Space‑time itself is emergent: what we perceive as space‑time is a particular dynamical regime of the entropic manifold. Localised excitations of the entropic field correspond to matter, and entropic gradients manifest as forces.
2.2 Definition of the Obidi action
The dynamics of the entropic field are encoded in the Obidi action. In Letter IIA the action is introduced as an axiom and written schematically as an integral over the entropic manifold of four terms:
Kinetic term: proportional to (partial_mu S)^2. It controls the propagation and transport of entropy. The constant alpha sets the entropic stiffness and ensures that the master equation is second order.
Entropic potential V(S): a self‑interaction potential that selects preferred entropic phases. Its form is application‑dependent and may drive spontaneous symmetry breaking or phase transitions.
Curvature coupling beta R_ent(S): couples the entropic field to the curvature of the entropic manifold. R_ent is the entropic Ricci scalar, and beta controls the strength of the coupling. Varying the action with respect to the metric shows that this term is the origin of gravity in the Theory of Entropicity (ToE).
Effective matter Lagrangian L_meff: describes localised, solitonic excitations of the entropic field that appear as matter. These excitations are not fundamental but emerge from entropic dynamics.
2.3 Variational principle and field equations
The Obidi action is extremized with respect to both the entropic field and the metric. The variational principle (Axiom III of ToE) states that physical configurations satisfy delta S_O/delta S = 0 and delta S_O/delta g_{mu nu} = 0. The resulting Euler–Lagrange equations form a coupled system:
Master Entropic Equation (MEE): a non‑linear wave equation of the form alpha Box S + V'(S) + f'(S) R = 0, where Box is the covariant wave operator and f'(S) arises from the curvature coupling.
Entropic Einstein equations: f(S) G_{mu nu} + (g_{mu nu} Box − nabla_mu nabla_nu) f(S) = (1/2) T_{mu nu}(S), where G_{mu nu} is the Einstein tensor and T_{mu nu}(S) is the entropic stress–energy tensor.
The first equation determines how entropy evolves under curvature, while the second tells how entropy shapes geometry. They both constitute what is called the Obidi Field Equations (OFE). Together they show that in ToE there is no separate matter sector; entropy acts both as “matter” and as “source” of curvature. In the flat‑space limit (vanishing curvature coupling) the MEE reduces to a wave equation, recovering standard field theories, whereas the entropic Einstein equations reduce to the Einstein equations of general relativity (GR).
2.4 Entropic configuration metric and Fisher–Rao limit
The Theory of Entropicity (ToE) defines a metric on the configuration space of the entropic field. When the field is parameterized by a finite number of parameters theta^a, the induced metric on parameter space is obtained by integrating products of derivatives of S(x; theta) over space–time. In the special case of flat space with no curvature coupling and vanishing potential, and when the entropic field is interpreted as the logarithm of a probability density (S = ln p), this metric reduces to the Fisher–Rao information metric. This result establishes an explicit bridge between ToE and classical information geometry: the entropic configuration metric generalizes the Fisher–Rao metric from probability distributions on flat space to entropic fields on curved manifolds. It also implies a natural ultraviolet cutoff through the Obidi curvature invariant OCI = ln 2. The inverse square root of OCI defines a minimum curvature radius beyond which the classical description breaks down.
2.5 Complexified entropic field and emergent gauge fields
While the entropic field is a real scalar, it possesses an internal structure. Letter IIA decomposes the entropic field into an amplitude rho and a phase Theta, defining a complex field E(x) = rho(x) e^{i Theta(x)}. In this picture the gradient of the phase acts as a connection on a U(1) fibre bundle and the curvature of this connection yields the electromagnetic field. The analogy with superfluid order parameters leads to a derivation of Maxwell’s equations from the Obidi action. Although electromagnetic phenomena are not the focus of this letter, the construction illustrates that gauge fields and interactions emerge from entropic orientation and thus supports the programmatic ambition of ToE to derive all interactions from entropy.
3 Correspondence between the Obidi and Einstein–Hilbert actions
3.1 The Einstein–Hilbert action
In general relativity, the gravitational field is described by the metric g_{mu nu}. Einstein’s field equations are derived from the Einstein–Hilbert action
S = (1 / 2 kappa) ∫ R sqrt(-g) d^4x,
where g = det(g_{mu nu}), R is the Ricci scalar and kappa = 8 pi G c^{-4}. The Einstein–Hilbert action yields the Einstein equations through the principle of stationary action. It is typically supplemented by a matter action L_M whose variation defines the stress–energy tensor.
3.2 Mapping the Obidi action to the Einstein–Hilbert action
Comparing the Obidi action with the Einstein–Hilbert action reveals a structural correspondence:
The curvature coupling term beta R_ent(S) in the Obidi action plays the role of the Ricci‑scalar term in the Einstein–Hilbert action. When S is constant (so its derivatives vanish) and f(S) is a constant coefficient, variation with respect to the metric reduces the curvature term to one proportional to R, yielding the standard gravitational Lagrangian. Thus the entropic Einstein equations collapse to G_{mu nu} = kappa T_{mu nu} when f(S) is constant.
The kinetic and potential terms of the Obidi action generalise the matter Lagrangian L_M. In particular, when the entropic field is interpreted as matter, the entropic stress–energy tensor reduces to the usual stress–energy tensor in field theory.
Hence, the Obidi action contains the Einstein–Hilbert action as a limiting case and extends it by promoting entropy to a dynamical field coupled to curvature. In this sense ToE is not an ad hoc modification but a natural generalisation of general relativity within an information‑theoretic framework.
4 Information geometry as the origin of gravity
4.1 Thermodynamic derivation of Einstein’s equations
Long before the Theory of Entropicity (ToE), researchers explored connections between thermodynamics and gravity. Ted Jacobson’s 1995 paper showed that the Einstein field equations can be derived by assuming that horizon entropy is proportional to area and that the Clausius relation delta Q = T dS holds for all local Rindler horizons through each space‑time point. Jacobson argues that requiring this relation for every local horizon forces the gravitational field equations to be the Einstein equations; thus the Einstein field equation is an equation of state. This thermodynamic perspective implies that gravity is emergent rather than fundamental and provides an early example of entropic gravity.
4.2 Entropic gravity and emergent gravity
Erik Verlinde (2010—2011) proposed that gravity is an entropic force. In his derivation, space emerges holographically, and gravity is identified with an entropic force arising from changes in the information associated with the positions of material bodies. A relativistic generalisation of his argument leads directly to the Einstein field equations. The entropic gravity program summarises this idea: gravity is described as an entropic force based on string theory, black‑hole physics and quantum information; it is an emergent phenomenon arising from quantum entanglement and obeys the second law of thermodynamics. Verlinde’s work and subsequent developments (e.g., modified Newtonian dynamics) support the view that gravity may be a manifestation of information and entropy.
4.3 Information‑geometric derivation of space‑time
Ariel Caticha extended information geometry to the dynamical origin of space‑time. In his 2019 paper, he models a blurred space, where points have finite resolution, using the method of maximum entropy. Such a blurred space is endowed with a metric from information geometry; the geometry of any embedded space‑like surface is given by its information geometry. Requiring that space evolve in local time so that it sweeps out a four‑dimensional manifold reproduces the Einstein equations for vacuum gravity. Caticha thus derives general relativity from the geometry of inference, strengthening the connection between information geometry and gravity.
4.4 Gravity from Entropy in Bianconi’s Dressed Einstein Equations
Ginestra Bianconi has advanced a distinct information‑theoretic route to gravity by treating entropy as a geometric quantity capable of modifying the Einstein field equations. In her framework, space‑time is not derived from inference or coarse‑graining but is instead “dressed’’ by entropic contributions arising from the statistical structure of quantum states and networks. Central to this approach is the use of Araki’s quantum relative entropy, which quantifies distinguishability between quantum states and serves as a measure of informational curvature.
Bianconi’s formulation distinguishes between the background space‑time metric and a matter‑induced entropic metric generated from quantum relative entropy. In her approach, the classical metric \(g_{\mu\nu}\) describes the underlying geometric scaffold, while the entropic metric arises from variations of Araki’s quantum relative entropy between neighboring quantum states. By comparing these two metrics, she identifies how informational curvature—encoded in the entropic response of quantum matter—modifies the background geometry. The difference between the two metrics produces additional geometric terms that “dress’’ the Einstein tensor, yielding her dressed Einstein field equations. This comparison shows that matter does not merely curve space‑time through stress–energy; it also induces entropic deformations of the metric itself, revealing gravity as partly driven by the informational structure of quantum states.
Bianconi shows that when quantum relative entropy is applied to fields defined on a network‑like discretization of space‑time, it induces entropic corrections to the Einstein tensor. These corrections appear as additional geometric terms in the field equations, producing what she calls the dressed Einstein equations. In this formulation, the classical Einstein tensor \(G_{\mu\nu}\) is supplemented by an entropic tensor derived from variations of quantum relative entropy with respect to the underlying metric. The resulting equations describe a geometry whose curvature is influenced not only by matter and energy but also by the informational structure of quantum states.
This entropic dressing has several implications. First, it provides a mechanism by which quantum information contributes directly to gravitational dynamics, offering a bridge between quantum theory and general relativity without invoking holography or string‑theoretic dualities. Second, it suggests that the cosmological constant may partially arise from entropic contributions associated with the statistical complexity of quantum fields. Third, the framework naturally incorporates network geometry, allowing space‑time to be modeled as a complex system whose curvature reflects both physical fields and informational interactions.
Bianconi’s work thus positions entropy not merely as a thermodynamic or statistical descriptor but as a source of geometric structure capable of modifying gravitational dynamics. Her dressed Einstein equations demonstrate how quantum relative entropy can act as an effective gravitational field, reinforcing the broader theme that information‑theoretic quantities may play a foundational role in the emergence and evolution of space‑time.
4.5 ToE’s entropic manifold and emergent curvature
In Obidi's Theory of Entropicity (ToE) the curvature coupling term in the Obidi action ensures that entropy gradients generate curvature. The entropic field enters both the MEE and the entropic Einstein equations; the term f'(S) R in the MEE shows that curvature acts as an effective potential for entropy, while the term involving second derivatives of f(S) in the entropic Einstein equations demonstrates that variations in entropy feed into curvature. Gravity arises not from a fundamental attractive force but from the tendency of entropy to [re-]distribute itself. When the entropic field is constant [uniform], the curvature term reduces to the Einstein–Hilbert term, but when S varies, the resulting corrections encode deviations from general relativity. This ToE viewpoint aligns with other entropic approaches but is unique in deriving both matter and geometry from a single entropic field.
5 Haller–Obidi correspondence and the entropy–action principle
5.1 Entropy–action identity in classical mechanics
John Haller showed that the self‑information (entropy) of a classical particle is proportional to the time integral of its energy minus its Lagrangian. The entropy–action identity reads H = (2 / hbar) ∫ (m c^2 − L) dt. Haller argued that this identity implies that the principle of least action is a consequence of the second law of thermodynamics: extremizing the action is equivalent to extremizing entropy.
5.2 Haller–Obidi correspondence in field theory
Letter IIA in the ToE Living Review Letters Series (ToE LRLS) generalises Haller’s result to field theory. The Haller–Obidi correspondence (HOC) states that the entropy–action identity for a particle of mass m is H = (2 / hbar) ∫ (m c^2 − L) dt and extends this to field theory by defining the entropic Lagrangian L_ent = m c^2 − (hbar / 2) (u^mu partial_mu S); and the corresponding action from this Lagrangian is called the Obidi-Haller Action (OHA). The Haller–Obidi correspondence (HOC) implies that in the Theory of Entropicity (ToE) the classical action is not merely analogous to entropy—it is entropy, up to a constant. The principle of least action thus becomes the ToE principle of extremal entropy: physical trajectories maximise (or extremize) entropy. This audacious and momentous insight unifies dynamics and thermodynamics and underpins the variational principle of the Obidi action of the Theory of Entropicity (ToE).
6 Beyond gravity: emergent gauge fields and unification
While this letter focuses on gravity, the Obidi action also yields gauge interactions. The complexified entropic field introduces an amplitude and a phase; the gradient of the phase defines a gauge connection, and the curvature of this connection gives rise to the electromagnetic field. The derivation of Maxwell’s equations in Letter IIA of ToE (ToE LRLS) shows that standard electromagnetic theory emerges from the phase sector of the entropic field when one imposes the frozen‑amplitude approximation. More generally, the gauge principle arises from symmetries of the Obidi action: continuous symmetries correspond to conserved currents via the entropic Noether theorem (ENT). Thus, the Theory of Entropicity (ToE) promises a unified origin of gravity and gauge interactions from a single entropic field.
7 Predictions and implications of the Theory of Entropicity (ToE)
The Theory of Entropicity (ToE) makes several novel predictions beyond those of general relativity and standard field theory:
Entropic speed limit (ESL): The entropic speed limit c_ent = sqrt(kappa / rho_S) defines the maximum rate at which entropy can propagate. In Letter IIA of ToE (ToE LRLS) the speed of light c is identified with this entropic propagation speed, providing an explanation for the value of c.
Natural UV cutoff: The Obidi curvature invariant (OCI) sets a minimum curvature radius l_{OCI} = 1 / sqrt(ln 2). Below this scale the continuum description breaks down.
Corrections to Maxwell’s equations: In curved entropic manifolds the curvature coupling term leads to birefringence, dispersion and nonlinear amplitude–phase coupling at very high frequencies, predicting deviations from classical electrodynamics.
Entropy‑driven wave function collapse: Letter I of the ToE Living Review Letters Series (LRLS) argues that quantum measurement collapse is an entropy‑driven phase transition. Thus, [quantum wave function] collapse occurs when entropic flux exceeds a critical threshold, providing a deterministic but irreversible mechanism for wave‑function collapse.
Matter as solitonic entropic excitations: ToE treats [elementary] particles as persistent configurations of the entropic field. Mass emerges from the entropic density, and inertial properties are determined by entropic gradients.
Unified action principle: All known physics—including gravity, gauge fields and quantum dynamics—should arise from limiting regimes of the Obidi action. This is the Obidi Correspondence Principle (OCP). The Vuli‑Ndlela integral (VNI), a path integral weighted by entropic action, underpins quantum mechanics in ToE.
These predictions make ToE falsifiable. Observational tests could include looking for curvature‑dependent birefringence in astrophysical light propagation, deviations from inverse‑square gravity at cosmological scales and entropy‑related corrections to quantum mechanics. The natural UV cutoff suggests modifications to high‑energy phenomena, potentially providing an alternative to Planck‑scale quantum gravity.
8 Discussion and conclusion
This letter (Letter III in the ToE Living Review Letters Series) has shown how information geometry becomes physical geometry in the Theory of Entropicity (ToE). Starting from statistical manifolds endowed with the Fisher–Rao metric, ToE elevates entropy to a dynamic field on a four‑dimensional manifold. The Obidi action, comprising kinetic, potential, curvature and matter terms, generates two coupled field equations—the MEE and the entropic Einstein equations [together referred to as the Obidi Field Equations (OFE)]—that jointly describe the evolution of entropy and geometry. By analysing the structure of the action we have demonstrated that it contains the Einstein–Hilbert action as a special case. When the entropic field is constant (that is, relatively uniform), the curvature coupling reduces to the Ricci scalar term and the entropic Einstein equations reduce to Einstein’s field equations. When the entropic field varies, gravity becomes an information‑theoretic phenomenon: curvature is driven by entropy gradients, and space‑time emerges from an entropic manifold.
The Theory of Entropicity (ToE) thus unifies information geometry, thermodynamics and general relativity. It builds on previous entropic and thermodynamic derivations of gravity, such as Jacobson’s and Verlinde’s, but the Theory of Entropicity (ToE) goes further by embedding matter, gauge fields and quantum mechanics into a single entropic framework. The Haller–Obidi Correspondence (HOC) identifies action with entropy, grounding dynamics in the second law of thermodynamics. The recovery of the Fisher–Rao metric in a suitable limit connects the Theory of Entropicity (ToE) back to classical information geometry, ensuring that the theory (ToE) is consistent with established statistical inference.
Doubtless, much work remains to be undertaken in subsequent Letters in the ToE Living Review Letters Series (ToE LRLS). A detailed derivation of the entropic Einstein equations and their solutions, the quantum theory of entropic fluctuations, and the embedding of the Standard Model gauge group into the entropic framework are necessary to fully assess the full scale viability of Obidi's Theory of Entropicity (ToE). Experimental tests of the predicted modifications to electromagnetism and gravity will determine whether entropy truly underlies the fabric of reality.
But regardless of the outcome, Obidi's Theory of Entropicity (ToE) undoubtedly already provides a powerful conceptual lens: it invites us to view the universe not as a stage with pre‑existing geometric laws but as an information‑theoretic process where space, time, matter and forces emerge from the flow [and gradients] of entropy.
This paper has explored the Theory of Entropicity (ToE) in some depth, bridging information geometry and information gravity. It has examined the Obidi action, its correspondence with the Einstein–Hilbert action, and how entropic field dynamics give rise to gravitational phenomena. The paper has also surveyed historical and contemporary literature, including Jacobson’s thermodynamic derivation of general relativity, Caticha’s information-geometric approach, Verlinde’s entropic gravity, and Haller’s entropy–action principle, integrating these perspectives into a cohesive narrative under one governing Entropic Substrate (ES).