The Meaning of Gravity in Einstein's General Relativity (GR) and the Theory of Entropicity (ToE): Core Divergence between Einstein’s Geometric Interpretation of Gravity and the Theory of Entropicity (ToE)’s Entropic Interpretation of Gravity

Preamble: A Unified Theory of Gravitation (UToG)

Gravity is one of the most fundamental phenomena in nature, yet its interpretation differs profoundly between Einstein’s General Relativity (GR) and the Theory of Entropicity (ToE), as first formulated and further developed by John Onimisi Obidi. Both frameworks reproduce the same observable gravitational effects, but they do so from radically different ontological foundations. GR treats gravity as a geometric deformation of spacetime, while ToE interprets gravity as an emergent entropic effect arising from the structure and evolution of the entropic field. Understanding this divergence is essential for appreciating how ToE reframes gravitational interaction within a broader entropic ontology.

1. Gravity in General Relativity: Curvature of Spacetime

In Einstein’s General Relativity, gravity is not a force but a geometric property of spacetime. Mass–energy determines the curvature of spacetime through the Einstein field equations, and free‑falling bodies follow geodesics, which are the “straightest possible paths” in this curved geometry. The familiar gravitational phenomena—Mercury’s perihelion precession, gravitational lensing, gravitational redshift, and time dilation—are all interpreted as consequences of this curvature.

In GR, the statement “a body follows the shortest distance between two points” means that the body follows a geodesic, which is not necessarily the shortest path in Euclidean terms but the path that extremizes the spacetime interval. The geometry itself dictates the motion; no force acts on the body. Gravity is therefore fully encoded in the metric and its curvature.

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

The Theory of Entropicity rejects the idea that curvature of spacetime is fundamental. Instead, ToE posits that gravity emerges from the structure, gradients, and curvature of the entropic field. Systems evolve toward configurations that maximize entropy, in accordance with the second law of thermodynamics. The entropic field determines which configurations are accessible and how trajectories evolve.

In this view, gravitational attraction is the macroscopic manifestation of entropic optimization. Bodies move along paths that maximize entropic accessibility, not geometric straightness. What GR interprets as curvature of spacetime is reinterpreted in ToE as the effective shadow of deeper entropic constraints.

For example, the perihelion shift of Mercury arises from entropy‑driven corrections to the effective potential governing orbital motion. The curvature of the observed trajectory is not a geometric primitive but a reflection of the entropic field’s structure.

This interpretation aligns with Louis de Broglie’s thermodynamic perspective, in which wave phenomena arise from hidden thermodynamic processes. ToE extends this idea to gravity: the apparent curvature of motion is a thermodynamic consequence of entropic gradients.

3. What Does “Shortest Distance Between Two Points” Mean in GR vs ToE?

In General Relativity

A free‑falling body follows a geodesic, which is the path that extremizes the spacetime interval. This is often described as the “shortest distance between two points,” but in curved spacetime this means:

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

The geometry is fundamental; motion is a consequence.

In the Theory of Entropicity

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

• entropy gradients,
• entropic curvature,
• the system’s drive toward maximal entropy.

The entropic field is fundamental; geometry is emergent.

Thus, GR’s geodesic is a geometric extremum, while ToE’s trajectory is an entropic extremum.

4. What Does It Mean for a Body to “Fall” in a Gravitational Field?

In General Relativity

A body “falls” because:

• spacetime is curved by mass–energy,
• the body follows a geodesic in that curved spacetime,
• no force acts on the body; it is in free fall.

Gravity is not a force but a geometric inevitability.

In the Theory of Entropicity

A body “falls” because:

• the entropic field has a gradient,
• the system evolves toward configurations of higher entropy,
• the trajectory is the entropically optimal path.

Gravity is not a force but an entropic inevitability.

In ToE, falling is the process of maximizing entropy under the constraints of the entropic field.

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. The Theory of Entropicity does not contradict GR’s predictions but reinterprets their origin. GR describes how gravity behaves; ToE explains why it behaves that way.

In ToE, the curvature that GR attributes to spacetime is an effective macroscopic representation of the deeper entropic field. The entropic field is the substrate; geometry is the emergent language through which macroscopic gravitational phenomena appear.

Appendix: Extra Matter - 1 (More Explanatory Notes)

How Einstein’s General Relativity and the Theory of Entropicity (ToE) Describe Gravity

Einstein’s General Relativity (GR) and the Theory of Entropicity (ToE) both explain gravitational phenomena with extraordinary precision, but they do so from fundamentally different ontological foundations.

Einstein: Gravity as Curvature of Spacetime

In General Relativity, gravity is not a force but 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—paths of least action—within this curved geometry. Classic phenomena such as:

• Mercury’s perihelion precession
• the bending of starlight
• gravitational redshift and time dilation

are interpreted as direct manifestations of this curvature. GR is therefore a purely geometric theory: the metric and its curvature encode everything we call “gravity.”

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. In ToE, 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.

In this framework:

• gravitational attraction emerges from entropy gradients,
• trajectories curve because systems follow entropically favorable paths,
• and what GR calls “spacetime curvature” is the effective shadow of deeper entropic dynamics.

For example, the perihelion shift of Mercury arises from entropy‑driven corrections to the effective potential governing orbital motion. The observed curvature of trajectories is not a geometric primitive but a macroscopic reflection of underlying entropic constraints.

Connection to Louis de Broglie

This entropic interpretation resonates with Louis de Broglie’s thermodynamic view of quantum behavior, where wave phenomena arise from hidden thermodynamic processes. ToE extends this idea: gravitational behavior is likewise a manifestation of entropic optimization, not geometric deformation.

Appendix: Extra Matter - 2 (Mathematical Explanatory Notes)

Gravity and Motion on Earth in the Theory of Entropicity (ToE)

1. The Core Principle: Motion Follows Entropic Extremization

In ToE, the motion of a body is governed by the entropic action functional:

Sent=∫S(x,t) dt,

where S(x,t) is the local entropic potential (the entropic field value at spacetime point x).

The physical trajectory is the one that maximizes the rate of entropy increase:

δSent=0.

This is the entropic analogue of the Euler–Lagrange principle.

In Newtonian mechanics, the extremized quantity is the action

S=∫L dt.

In GR, the extremized quantity is the spacetime interval

δ∫ds=0.

In ToE, the extremized quantity is the entropy functional.

This is the fundamental equation of motion.

2. The Entropic Force on Earth

In the weak‑field, low‑velocity limit (Earth’s surface), ToE reduces to a simple entropic force:

Fent=T∇S,

where:

• T is the effective local thermodynamic temperature of the entropic field
• ∇S is the entropy gradient

This is the same structure as the entropic force in polymer physics, black hole thermodynamics, and Verlinde’s entropic gravity.

On Earth, the entropic field has a radial gradient:

∇S(r)∝1/r^2.

Thus:

Fent∝1/r^2,

which reproduces Newton’s law:

F=GMm/r^2.

Newton’s law is the macroscopic shadow of an entropic gradient.

3. Equation of Motion for a Falling Body in ToE

A body of mass m experiences an entropic acceleration:

mdx^2/dt^2=T∇S.

On Earth:

T∇S=mg,

so:

dx^2/dt^2=g.

Thus, ToE reproduces the familiar Newtonian equation:

x(t)=x_0+v_0.t+(1/2).g.t^2

But the interpretation is different:

• Newton: a force pulls the body downward
• GR: the body follows a geodesic in curved spacetime
• ToE: the body follows the path of maximal entropy increase

4. Why Does a Body Fall in ToE?

Because the entropic field around Earth has a downward gradient.

A falling body is simply moving toward higher entropic accessibility.

This is analogous to:

• diffusion
• heat flow
• polymer contraction
• black hole entropy increase

In all cases, systems evolve toward maximal entropy.

Gravity is no exception.

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

In GR

A free‑falling body follows a geodesic, which extremizes the spacetime interval:

δ∫ds=0.

This is the “straightest possible path” in curved spacetime.

In ToE

A free‑falling body follows the path that maximizes entropy production:

δ∫S(x,t) dt=0.

This is the “most entropically favorable path.”

Thus:

• GR extremizes geometry
• ToE extremizes entropy

The two give the same trajectories in the weak‑field limit, but ToE provides a deeper thermodynamic origin.

6. Comparison Table: GR vs ToE on Motion and Falling

7. The Key Insight

ToE does not contradict GR’s predictions. It explains them.

GR describes how gravity behaves. ToE explains why gravity behaves that way.

• GR: geometry bends
• ToE: entropy drives motion
• Newton: force pulls

All three give the same trajectories on Earth, but ToE provides the thermodynamic origin.

References

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