Physics:Einstein's Relativity Results from the Theory of Entropicity(ToE)

Binet–Obidi Equation and Relativistic Orbital Effects from the Theory of Entropicity (ToE)

Here, we derive Einstein's beautiful and classical results of his marvelous General Theory of Relativity (GToR), the perihelion precession of Mercury and the deflection of starlight, from the Theory of Entropicity(ToE) without any resort to spacetime curvature.

Introduction

The Binet–Obidi Equation is a generalization of the classical Binet equation for central-force motion, incorporating the entropic correction term predicted by the Theory of Entropicity(ToE), as first formulated and developed by John Onimisi Obidi. This correction arises from the dynamics of the entropic field and yields the same quantitative predictions for Mercury's perihelion precession and the deflection of starlight as General Relativity (GR), but from a fundamentally different physical origin.

1. Classical Binet Equation

For a particle of mass [math]\displaystyle{ m }[/math] in a central force [math]\displaystyle{ F(r) }[/math], the Binet equation is:

[math]\displaystyle{ \frac{d^2 u}{d\theta^2} + u = -\frac{F(1/u)}{m h^2 u^2} }[/math]

where:

• [math]\displaystyle{ u = 1/r }[/math]
• [math]\displaystyle{ h = r^2 \dot{\theta} }[/math] is the specific angular momentum.

For Newtonian gravity: [math]\displaystyle{ F(r) = -\frac{GMm}{r^2} \quad\Rightarrow\quad \frac{d^2 u}{d\theta^2} + u = \frac{GM}{h^2} }[/math]

which yields closed Keplerian ellipses with no precession.

2. Entropicity Correction

From Obidi's Theory of Entropicity(ToE), the gravitational force is modified by an entropy-gradient term:

[math]\displaystyle{ F_{\text{ToE}}(r) = -\frac{GMm}{r^2} - \frac{\beta\,GMm}{c^2\,r^3} }[/math]

where:

• The [math]\displaystyle{ 1/r^3 }[/math] term is the Obidi entropic correction.
• [math]\displaystyle{ \beta }[/math] is a dimensionless constant determined by observation.

3. Binet–Obidi Equation

Substituting [math]\displaystyle{ F_{\text{ToE}} }[/math] into the above Binet equation, we obtain the following equation:

[math]\displaystyle{ \frac{d^2 u}{d\theta^2} + u = \frac{GM}{h^2} + \frac{\beta\,GM}{c^2\,h^2} u }[/math]

Rewriting the above, we obtain:

[math]\displaystyle{ \frac{d^2 u}{d\theta^2} + \left(1 - \frac{\beta\,GM}{c^2\,h^2}\right) u = \frac{GM}{h^2} }[/math]

This is the Binet–Obidi Equation.

4. Solution and Perihelion Precession

The general solution is:

[math]\displaystyle{ u(\theta) = \frac{GM}{h^2} \left[ 1 + e \cos\left( \sqrt{1 - \frac{\beta\,GM}{c^2\,h^2}}\,\theta \right) \right] }[/math]

For small [math]\displaystyle{ \epsilon = \frac{\beta\,GM}{2c^2\,h^2} \ll 1 }[/math], the angular frequency is: [math]\displaystyle{ \omega \approx 1 - \epsilon }[/math]

After one revolution we obtain:

([math]\displaystyle{ 2\pi }[/math] in [math]\displaystyle{ \theta }[/math]),

so the perihelion advances by:

[math]\displaystyle{ \Delta\theta \approx \frac{\pi\beta\,GM}{c^2 a (1 - e^2)} }[/math]

Using Mercury's observed precession:

([math]\displaystyle{ 43'' }[/math] per century) fixes [math]\displaystyle{ \beta = 3 }[/math],

thus giving:

[math]\displaystyle{ \Delta\theta_{\text{ToE}} = \frac{6\pi GM}{c^2 a (1 - e^2)} }[/math],

which is clinically and classically identical to the GR result.

5. Light Deflection

For light, [math]\displaystyle{ m \to 0 }[/math] but [math]\displaystyle{ h \approx b c }[/math] (impact parameter [math]\displaystyle{ b }[/math]), we therefore have:

[math]\displaystyle{ \frac{d^2 u}{d\theta^2} + u = \frac{\beta\,GM}{c^2 b^2} u }[/math]

Solving perturbatively for [math]\displaystyle{ \beta = 3 }[/math] yields:

[math]\displaystyle{ \delta \phi \approx \frac{4GM}{c^2 b}, }[/math]

thus, once again, matching the GR prediction for starlight grazing the Sun.

6. Significance of Obidi's Contribution in his Theory of Entropicity(ToE)

While GR attributes the [math]\displaystyle{ 1/r^3 }[/math] term to spacetime curvature, the ToE derives it from entropic field dynamics. The Binet–Obidi Equation thus provides a unifying mathematical form for relativistic orbital effects, but with a thermodynamic–informational physical origin.

See Also

• Binet equation
• Perihelion precession of Mercury
• Deflection of light by gravity
• Theory of Entropicity(ToE)
• General Relativity
• Albert Einstein