The Meaning of Gravity in Einstein’s General Relativity (GR) and the Theory of Entropicity (ToE)

Core Divergence Between Einstein’s Geometric Interpretation and the Entropic Interpretation of Gravity

Preamble: A Unified Theory of Gravitation (UToG)

Gravity is one of the most fundamental phenomena in nature, yet its conceptual interpretation differs profoundly between Einstein’s General Relativity (GR) and the Theory of Entropicity (ToE), as formulated and developed by John Onimisi Obidi. Both frameworks reproduce the same class of observable gravitational effects, but they do so from radically different ontological foundations. In GR, gravity is understood as a manifestation of the curvature of spacetime, with the geometry of spacetime fully encoding the gravitational interaction. In contrast, ToE interprets gravity as an emergent entropic effect arising from the structure, gradients, and evolution of a fundamental entropic field.

The central objective of this exposition is to clarify this divergence and to show how ToE reframes gravitational interaction within a broader entropic ontology. In this view, the geometric description of gravity in GR is not rejected, but rather reinterpreted as an effective, macroscopic representation of deeper entropic dynamics. This provides a conceptual pathway toward a Unified Theory of Gravitation (UToG), in which geometric and entropic descriptions are seen as different levels of representation of a single underlying entropic substrate.

1. Gravity in General Relativity: Curvature of Spacetime

In Einstein’s General Relativity, gravity is not treated as a force in the Newtonian sense but as a geometric property of spacetime. The presence of mass–energy, encoded in the stress–energy tensor, determines the curvature of spacetime through the Einstein field equations. Free‑falling bodies move along geodesics, which are the “straightest possible paths” in this curved geometry. Phenomena such as the perihelion precession of Mercury, gravitational lensing, gravitational redshift, and gravitational time dilation are interpreted as direct consequences of this curved spacetime structure.

In GR, the statement that “a body follows the shortest distance between two points” is formalized by the notion of a geodesic. In a curved spacetime, a geodesic is not necessarily the shortest path in the Euclidean sense; rather, it is the path that extremizes the spacetime interval. The motion of a free‑falling body is determined entirely by the metric tensor, which encodes the geometry of spacetime. No external force is required to explain the motion; instead, the body is said to be in free fall, following the natural geometry of spacetime. Gravity is therefore fully encoded in the metric and its curvature, and the gravitational interaction is intrinsically geometric.

2. Gravity in the Theory of Entropicity: Entropic Gradients and Maximization

The Theory of Entropicity (ToE) rejects the idea that the curvature of spacetime is the fundamental origin of gravity. Instead, ToE posits that gravity emerges from the structure, gradients, and curvature of a fundamental entropic field. Physical systems evolve toward configurations that maximize entropy, in accordance with the second law of thermodynamics. The entropic field determines which configurations of matter, energy, and information are accessible, and how trajectories in configuration space evolve over time.

Within this framework, gravitational attraction is interpreted as the macroscopic manifestation of entropic optimization. Bodies do not move along geometrically straight paths but along paths that maximize entropic accessibility. What GR interprets as the curvature of spacetime is reinterpreted in ToE as the effective projection of deeper entropic constraints into a geometric language. The observed curvature of trajectories is thus not a primitive geometric fact but a reflection of the underlying entropic structure.

A paradigmatic example is the perihelion shift of Mercury. In GR, this is explained by the curvature of spacetime in the vicinity of the Sun. In ToE, the same effect is modeled as arising from entropy‑driven corrections to an effective potential governing orbital motion. The trajectory’s curvature is then understood as a macroscopic shadow of the entropic field’s configuration. This entropic interpretation aligns with Louis de Broglie’s thermodynamic perspective, in which wave phenomena arise from hidden thermodynamic processes. ToE extends this thermodynamic viewpoint to gravity, treating gravitational behavior as a manifestation of entropic optimization rather than fundamental geometric deformation.

3. The Meaning of “Shortest Distance Between Two Points” in GR and ToE

3.1 Interpretation in General Relativity

In General Relativity, a free‑falling body follows a geodesic, which is the path that extremizes the spacetime interval. This is often colloquially described as the “shortest distance between two points,” but in a curved spacetime this phrase must be understood in a generalized sense. The geodesic is:

the path that requires no external force, the path that is “straight” relative to the curved geometry, and the path determined entirely by the metric tensor.

The geometry is fundamental, and the motion of bodies is a consequence of that geometry. The geodesic equation, derived from the variational principle applied to the spacetime interval, encodes this idea mathematically. The gravitational interaction is thus fully captured by the geometric structure of spacetime.

3.2 Interpretation in the Theory of Entropicity

In the Theory of Entropicity, a free‑falling body follows the path that maximizes entropic accessibility. This path is not a geometric shortest path but an entropically optimal path. The trajectory is determined by:

the gradients of the entropic field, the entropic curvature of the configuration space, and the system’s intrinsic drive toward maximal entropy.

In this formulation, the entropic field is fundamental, and geometry is emergent. The path of motion is an entropic extremum rather than a purely geometric extremum. Thus, while GR identifies the geodesic as a geometric extremal of the spacetime interval, ToE identifies the physical trajectory as an extremal of an entropy functional. The two descriptions can coincide in their predictions for observable trajectories, particularly in the weak‑field limit, but they differ in their underlying ontological commitments.

4. The Meaning of “Falling” in a Gravitational Field

4.1 Falling in General Relativity

In General Relativity, a body “falls” because spacetime is curved by the presence of mass–energy. The body follows a geodesic in this curved spacetime, and no force acts on it in the local sense; it is in free fall. Gravity is not a force but a geometric inevitability. The apparent acceleration of the body is a manifestation of the curvature of spacetime, not of a force acting within spacetime.

4.2 Falling in the Theory of Entropicity

In the Theory of Entropicity, a body “falls” because the entropic field possesses a gradient. The system evolves toward configurations of higher entropy, and the trajectory followed is the entropically optimal path. Gravity is therefore not a force but an entropic inevitability. In this view, falling is the process of maximizing entropy under the constraints imposed by the entropic field.

This interpretation places gravitational motion in the same conceptual category as diffusion, heat flow, and other processes that are driven by entropy gradients. The difference is that, in ToE, the entropic field is fundamental and universal, and gravitational phenomena are one of its macroscopic manifestations.

5. Comparison Table: Gravity in GR vs Gravity in ToE

6. Synthesis: Gravity as Geometry vs Gravity as Entropy

General Relativity provides a geometric description of gravity that has been extraordinarily successful in explaining a wide range of phenomena. The Theory of Entropicity does not contradict the empirical predictions of GR but instead reinterprets their origin. In this sense, GR describes how gravity behaves, while ToE aims to explain why gravity behaves in that way.

In ToE, the curvature that GR attributes to spacetime is understood as an effective macroscopic representation of a deeper entropic field. The entropic field is the substrate, and geometry is the emergent language through which macroscopic gravitational phenomena appear. This shift from geometry to entropy as the fundamental explanatory principle is the core conceptual divergence between the two theories and forms the basis for a unified entropic interpretation of gravitation.

Appendix: Extra Matter 1 – Explanatory Notes on GR and ToE

A.1 Overview of the Dual Descriptions of Gravity

Einstein’s General Relativity (GR) and the Theory of Entropicity (ToE) both provide highly accurate descriptions of gravitational phenomena, yet they are grounded in fundamentally different ontological assumptions. GR is a geometric theory in which gravity is encoded in the curvature of spacetime, while ToE is an entropic theory in which gravity emerges from the dynamics of an underlying entropic field.

A.2 Einstein: Gravity as Curvature of Spacetime

In General Relativity, gravity is a geometric consequence of mass–energy shaping the curvature of spacetime. The stress–energy tensor determines how spacetime bends, and free‑falling bodies follow geodesics, which can be interpreted as paths of least action in the geometric sense. Classic gravitational phenomena, including the perihelion precession of Mercury, the bending of starlight by massive bodies, gravitational redshift, and time dilation, are all understood as direct manifestations of this curvature. GR is therefore a purely geometric theory: the metric tensor and its curvature encode everything that is conventionally referred to as “gravity.”

A.3 ToE: Gravity as an Entropic Phenomenon

The Theory of Entropicity reinterprets gravity not as curvature of spacetime itself but as a macroscopic expression of entropic configurations. The entropic field possesses gradients, curvature, and constraints, and physical systems evolve in ways that maximize entropy in accordance with the second law of thermodynamics. Gravitational attraction emerges from entropy gradients, trajectories curve because systems follow entropically favorable paths, and what GR calls “spacetime curvature” is understood as the effective shadow of deeper entropic dynamics.

For instance, the perihelion shift of Mercury can be modeled in ToE as arising from entropy‑driven corrections to an effective potential. The observed curvature of the orbit is then a macroscopic reflection of underlying entropic constraints rather than a primitive geometric fact. This entropic interpretation resonates with Louis de Broglie’s thermodynamic view of quantum behavior, in which wave phenomena arise from hidden thermodynamic processes. ToE extends this idea to gravitational behavior, treating it as a manifestation of entropic optimization rather than fundamental geometric deformation.

Appendix: Extra Matter 2 – Mathematical Explanatory Notes

B.1 Gravity and Motion on Earth in the Theory of Entropicity

B.1.1 Core Principle: Motion Follows Entropic Extremization

In the Theory of Entropicity, the motion of a body is governed by an entropic action functional. Denote the local entropic potential, or entropic field value, at spacetime point \( x \) and time \( t \) by \( S(x,t) \). The entropic action is defined as

\( \mathcal{S}_{\text{ent}} = \int S(x,t)\, dt. \)

The physical trajectory is obtained by imposing the condition that this entropic action is extremized:

\( \delta \mathcal{S}_{\text{ent}} = 0. \)

This is the entropic analogue of the Euler–Lagrange principle. In Newtonian mechanics, the extremized quantity is the action

\( \mathcal{S} = \int L\, dt, \)

where \( L \) is the Lagrangian. In General Relativity, the extremized quantity is the spacetime interval, expressed as

\( \delta \int ds = 0. \)

In ToE, by contrast, the extremized quantity is the entropy functional \( \mathcal{S}_{\text{ent}} \). This entropic extremization principle is the fundamental equation of motion in the entropic framework.

B.1.2 The Entropic Force on Earth

In the weak‑field, low‑velocity limit appropriate to motion near the Earth’s surface, the Theory of Entropicity reduces to a simple entropic force law. The entropic force is given by

\( F_{\text{ent}} = T \nabla S, \)

where \( T \) is the effective local thermodynamic temperature of the entropic field and \( \nabla S \) is the entropy gradient. This structure is analogous to entropic forces encountered in polymer physics, black hole thermodynamics, and entropic gravity models.

On Earth, the entropic field is assumed to have a radial gradient of the form

\( \nabla S(r) \propto \frac{1}{r^{2}}, \)

where \( r \) is the radial distance from the Earth’s center. Consequently, the entropic force scales as

\( F_{\text{ent}} \propto \frac{1}{r^{2}}, \)

which reproduces the familiar Newtonian gravitational law

\( F = \frac{G M m}{r^{2}}, \)

where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( m \) is the mass of the test body. In this interpretation, Newton’s law is the macroscopic shadow of an underlying entropic gradient.

B.1.3 Equation of Motion for a Falling Body in ToE

A body of mass \( m \) subject to the entropic force experiences an acceleration determined by

\( m \frac{d^{2}x}{dt^{2}} = T \nabla S. \)

Near the Earth’s surface, one can identify

\( T \nabla S = m g, \)

where \( g \) is the effective gravitational acceleration. This yields the familiar equation of motion

\( \frac{d^{2}x}{dt^{2}} = g. \)

Integrating this equation under standard initial conditions leads to the well‑known kinematic relation

\( x(t) = x_{0} + v_{0} t + \frac{1}{2} g t^{2}, \)

where \( x_{0} \) is the initial position and \( v_{0} \) is the initial velocity. The crucial point is that, while the form of the equation coincides with Newtonian mechanics, the interpretation is different. In Newtonian gravity, a force pulls the body downward. In General Relativity, the body follows a geodesic in curved spacetime. In the Theory of Entropicity, the body follows the path of maximal entropy increase.

B.1.4 Why a Body Falls in ToE

In the entropic framework, a body falls because the entropic field around the Earth has a downward gradient. A falling body is simply moving toward regions of higher entropic accessibility. This behavior is analogous to diffusion, heat flow, polymer contraction, and black hole entropy increase, where systems evolve spontaneously toward states of higher entropy. Gravity, in this sense, is not an exception but a particular manifestation of a universal entropic tendency.

B.1.5 How ToE Replaces GR’s “Shortest Distance Between Two Points”

In General Relativity, a free‑falling body follows a geodesic that extremizes the spacetime interval:

\( \delta \int ds = 0. \)

This is the “straightest possible path” in curved spacetime. In the Theory of Entropicity, a free‑falling body follows the path that maximizes entropy production:

\( \delta \int S(x,t)\, dt = 0. \)

This is the “most entropically favorable path.” Thus, GR extremizes geometry, while ToE extremizes entropy. The two approaches yield the same trajectories in the weak‑field limit, but ToE provides a deeper thermodynamic origin for the observed motion.

B.2 Comparison Table: GR vs ToE on Motion and Falling

B.3 The Key Insight

The Theory of Entropicity does not contradict the predictions of General Relativity; rather, it seeks to explain them at a deeper level. GR describes how gravity behaves in terms of geometry. ToE explains why gravity behaves that way in terms of entropy. In this unified perspective:

GR: geometry bends, ToE: entropy drives motion, Newton: force pulls.

All three frameworks yield the same trajectories in the appropriate limits, particularly for motion near the Earth’s surface, but ToE provides a thermodynamic origin for gravitational phenomena, embedding gravity within a universal entropic framework.

References

1. John Onimisi Obidi. Theory of Entropicity (ToE) and de Broglie’s Thermodynamics. Encyclopedia. Available online: https://encyclopedia.pub/entry/59520 (accessed on 14 February 2026).
2. Theory of Entropicity (ToE) Provides the Fundamental Origin for the "Arrow of Time". Available online: https://theoryofentropicity.blogspot.com/2026/02/how-theory-of-entropicity-toe-finalizes.html .
3. Grokipedia — Theory of Entropicity (ToE): https://grokipedia.com/page/Theory_of_Entropicity.
4. Grokipedia — John Onimisi Obidi: https://grokipedia.com/page/John_Onimisi_Obidi.
5. Google Blogger — Live Website on the Theory of Entropicity (ToE): https://theoryofentropicity.blogspot.com.
6. GitHub Wiki — Theory of Entropicity (ToE): https://github.com/Entropicity/Theory-of-Entropicity-ToE/wiki .
7. Canonical Archive — Theory of Entropicity (ToE): https://entropicity.github.io/Theory-of-Entropicity-ToE/ .
8. LinkedIn — Theory of Entropicity (ToE): https://www.linkedin.com/company/theory-of-entropicity-toe/about/?viewAsMember=true .
9. Medium — Theory of Entropicity (ToE): https://medium.com/@jonimisiobidi.
10. Substack — Theory of Entropicity (ToE): https://johnobidi.substack.com/.
11. Figshare — John Onimisi Obidi: https://figshare.com/authors/John_Onimisi_Obidi/20850605 .
12. Encyclopedia — SciProfiles — Theory of Entropicity (ToE): https://sciprofiles.com/profile/4143819.
13. HandWiki — Theory of Entropicity (ToE): https://handwiki.org/wiki/User:PHJOB7.
14. John Onimisi Obidi. Gravitation from Einstein’s GR to Theory of Entropicity (ToE). Encyclopedia. Available online: https://encyclopedia.pub/entry/59524 (accessed on 15 February 2026).
15. John Onimisi Obidi. Theory of Entropicity (ToE): Path to Unification of Physics and the Laws of Nature. Available online: https://encyclopedia.pub/entry/59188.

References

1. Grokipedia — Theory of Entropicity (ToE)
   Comprehensive encyclopedia‑style entry introducing the conceptual, mathematical, and ontological structure of the Theory of Entropicity (ToE).
   https://grokipedia.com/page/Theory_of_Entropicity
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.
   https://grokipedia.com/page/John_Onimisi_Obidi
3. Google Blogger — Live Website on the Theory of Entropicity (ToE)
   Public‑facing platform containing explanatory essays, conceptual introductions, and updates on the Theory of Entropicity (ToE).
   https://theoryofentropicity.blogspot.com
4. LinkedIn — Theory of Entropicity (ToE)
   Professional organizational page providing institutional updates and academic outreach related to the Theory of Entropicity (ToE).
   https://www.linkedin.com/company/theory-of-entropicity-toe/about/?viewAsMember=true
5. Medium — Theory of Entropicity (ToE)
   Collection of essays and conceptual expositions on the Theory of Entropicity (ToE).
   https://medium.com/@jonimisiobidi
6. Substack — Theory of Entropicity (ToE)
   Serialized research notes, essays, and public communications on the Theory of Entropicity (ToE).
   https://johnobidi.substack.com/
7. SciProfiles — Theory of Entropicity (ToE)
   Indexed scholarly profile and research presence for the Theory of Entropicity (ToE) within the SciProfiles ecosystem.
   https://sciprofiles.com/profile/4143819
8. HandWiki — Theory of Entropicity (ToE)
   Editorially curated scientific encyclopedia entry, documenting the Theory of Entropicity (ToE)'s conceptual, philosophical, and mathematical structures.
   https://handwiki.org/wiki/User:PHJOB7
9. Encyclopedia.pub — Theory of Entropicity (ToE): Path to Unification of Physics and the Laws of Nature
   A formally maintained, technically curated scientific encyclopedia entry, presenting an expansive overview of the Theory of Entropicity (ToE)'s conceptual, philosophical, and mathematical foundations.
   https://encyclopedia.pub/entry/59188
10. Authorea — Research Profile of John Onimisi Obidi
    Research manuscripts, papers, and scientific documents on the Theory of Entropicity (ToE).
    https://www.authorea.com/users/896400-john-onimisi-obidi
11. Academia.edu — Research Papers
    Academic papers, drafts, and research notes on the Theory of Entropicity (ToE) hosted on Academia.edu .
    https://independent.academia.edu/JOHNOBIDI
12. Figshare — Research Archive
    Principal Figshare repository link for research outputs on the Theory of Entropicity (ToE).
    https://figshare.com/authors/John_Onimisi_Obidi/20850605
13. OSF (Open Science Framework)
    Open‑access repository hosting research materials, datasets, and papers related to the Theory of Entropicity (ToE).
    https://osf.io/5crh3/
14. ResearchGate — Publications on the Theory of Entropicity (ToE)
    Indexed research outputs, citations, and academic interactions related to the Theory of Entropicity (ToE).
    https://www.researchgate.net/search.Search.html?query=John+Onimisi+Obidi&type=publication
15. Social Science Research Network (SSRN)
    Indexed scholarly works and papers on the Theory of Entropicity (ToE) within the SSRN research repository.
    https://papers.ssrn.com/sol3/cf_dev/AbsByAuth.cfm?per_id=7479570
16. International Journal of Current Science Research and Review (IJCSRR)
    Peer‑reviewed publication relevant to the Theory of Entropicity (ToE).
    https://doi.org/10.47191/ijcsrr/V8-i11%E2%80%9321
17. Cambridge University — Cambridge Open Engage (COE)
    Early research outputs and working papers hosted on Cambridge University’s open research dissemination platform.
    https://www.cambridge.org/core/services/open-research/cambridge-open-engage
18. GitHub Wiki — Theory of Entropicity (ToE)
    Open‑source technical wiki, documenting the canonical structure, equations, and formal development of the Theory of Entropicity (ToE).
    https://github.com/Entropicity/Theory-of-Entropicity-ToE/wiki
19. Canonical Archive of the Theory of Entropicity (ToE)
    Authoritative, version‑controlled archive of the full Theory of Entropicity (ToE) monograph, including derivations and formal definitions.
    https://entropicity.github.io/Theory-of-Entropicity-ToE/