Physics:Theory of Entropicity(ToE) Derives Einstein's Mercury Precession

Abstract

The anomalous perihelion precession of Mercury, amounting to 43 arcseconds per century, was historically explained by Einstein’s General Relativity (1915). More recently, alternative approaches based on entropy and emergent gravity have shown that the same result can be derived without invoking spacetime curvature. This article presents a comparative overview of the Newtonian, Einsteinian, and entropic derivations of Mercury’s perihelion precession, with emphasis on the Theory of Entropicity(ToE), first formulated and developed by John Onimisi Obidi.

Introduction

Mercury’s orbit exhibits a small but measurable precession of its perihelion. While Newtonian mechanics accounts for most of this effect through planetary perturbations, an unexplained residual of 43 arcseconds per century remained. Einstein’s General Relativity (GR) provided the first successful theoretical explanation. However, entropy‑based approaches, inspired by the holographic principle, the Unruh effect, and Bekenstein–Hawking entropy, have demonstrated that the same correction can emerge from thermodynamic principles.

Newtonian Prediction

In Newtonian gravity, the potential is: \[ V(r) = -\frac{GMm}{r} \]

The orbital equation in Binet form is: \[ \frac{d^2u}{d\theta^2} + u = \frac{GM}{L^2}, \quad u = \frac{1}{r} \]

The solution yields closed elliptical orbits. Newtonian theory therefore predicts no intrinsic perihelion precession, failing to account for Mercury’s anomaly.

Einstein’s General Relativity

Einstein’s relativistic correction modifies the effective potential: \[ V_{\text{eff}}(r) = -\frac{GMm}{r} + \frac{L^2}{2mr^2} - \frac{GML^2}{c^2mr^3} \]

The additional \(1/r^3\) term arises from spacetime curvature. Perturbative solution of the orbital equation gives the precession per orbit: \[ \Delta \phi = \frac{6\pi GM}{a(1-e^2)c^2} \]

For Mercury, this matches the observed 43 arcseconds per century.

Entropic Gravity Framework

The entropic approach draws on several principles:

• Unruh effect: acceleration ↔ temperature

\[ T = \frac{\hbar a}{2\pi c k_B} \]

• Bekenstein–Hawking entropy:

\[ S = \frac{k_B A}{4\ell_p^2} \]

• Holographic principle: information/entropy scales with area.

• Entropic force law:

\[ F \Delta x = T \Delta S \]

From these, the Newtonian potential acquires an entropy‑driven correction: \[ V(r) = -\frac{GMm}{r} + \alpha \frac{GMm}{r^3} \]

where \(\alpha\) encodes entropic contributions.

Orbital Equation with Entropic Correction

Substituting into the Binet equation: \[ \frac{d^2u}{d\theta^2} + u = \frac{GM}{L^2} + 3\alpha u^2 \]

The extra \(3\alpha u^2\) term plays the same role as Einstein’s relativistic correction. Perturbative solution yields: \[ \Delta \phi = \frac{6\pi GM}{a(1-e^2)c^2} \]

Thus, the entropic framework reproduces Einstein’s result.

Comparative Summary

Aspect	Newton (Classical)	Einstein (GR)	Entropicity (Entropy‑Based)
Core Idea	Force at a distance	Spacetime curvature	Emergent entropic effect
Potential	\(-GMm/r\)	\(-GMm/r + L^2/2mr^2 - GML^2/c^2mr^3\)	\(-GMm/r + \alpha GMm/r^3\)
Orbit	Closed ellipse	Precessing ellipse	Precessing ellipse
Precession	None	\(6\pi GM/[a(1-e^2)c^2]\)	Same formula
Interpretation	Action‑at‑a‑distance	Geometry of spacetime	Thermodynamic/statistical tendency

Interpretation

• Newton: Gravity as a force.
• Einstein: Gravity as geometry.
• Entropicity: Gravity as entropy maximization.

The entropic derivation demonstrates that Mercury’s perihelion precession is not unique to curved spacetime but can also emerge from the statistical thermodynamics of information and entropy.

See Also

• General relativity
• Emergent gravity
• Unruh effect
• Holographic principle
• Bekenstein–Hawking entropy