Exact solution of the perihelion motion of Mercury from Newtonian mechanics

 ABSTRACT.

The 43 arcsec/100-year error of Mercury's perihelion movement can be precisely solved from Newton's force balance equation. The centrifugal force and universal gravitational balance equation is GM☉ mG／r²＝mI v²／r. The approximate solution of "absolute rest reference frame + Newtonian mechanics" is v²＝GM☉／r. The exact solution of "light's observational reference frame + Newtonian mechanics" is v²=GM☉／(γ r).

By Mpfiz - Own work, Public Domain, Link

 

MATERIALS AND METHODS.

From the balance formula of centrifugal force and universal gravitational force,

GM☉ mG／r²＝mI v²／r　　(1).

In the Post-Newton approximation,

In other words, when viewed from the coordinate system, the gravity balances the inertial force, and it appears that the gravity has disappeared. It means that gravity and acceleration can be canceled out in a small area. However, in order for Eq. (1) to hold, the gravitational mass mG and the inertial mass mI must be equal. In general relativity, the motion in the gravitational field is given by the geodesic curve, and the geodesic equation does not depend on the mass. A geodesic is a metric that is not Euclidean, that is, the shortest path in space-time.
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The post-Newton approximation is an approximation of the Einstein equation in the weak gravitational field. This is a method of approximating the ratio of the speed v of the celestial body to the speed of light c (v²／c²) as an expansion parameter without solving all Einstein equations.

However, the mass equivalence (mG=mI) is a slow approximation for (v≪c), and the moving body has an exact solution of (mG≠mI).

In 1901, Kaufmann first discovered as an experimental fact that the mass of an electron increases as it becomes faster.

The scale factor of the light equivalence principle that distinguishes the change in inertial mass due to scale and the change in gravitational mass due to increase or decrease in energy is

γ＝mI／mG＝c／w＝c／√(c²±v²)　　(2).

Eliminating the mass of Mercury from (Equation 1) with the reference frame of the light velocity (c) of (Equation 2) point at infinity,

γ v²＝GM☉／r    　(3).

 

According to Kepler's third law,

4 π² γ＝GM☉ P²／r³　　(4).

The error of one revolution revolving from the scale factor on the left side is

⊿φ＝4 π²(γ－1)　　(5).

This coefficient 4π is derived from the solid angle around the circumference, and the amount of electricity charged by the point charge is equal to the electric flux around the circumference in the rational system and equal to the flux per solid angle in the non-rational system. If the angle is dimensionless, it does not affect the dimension of electromagnetic quantity.

In arc seconds,

"x＝⊿φ(180／π)＊3600　　(6).

By Francisco Esquembre-Author's own work, CC view-inheritance 3.0, by link.

Mercury orbits the Sun in 88 days, which means that it has rotated about 415 revolutions in 100 years, so the arc of 415 orbits is

"y＝4 π(γ－1)＊180＊3600＊415　　(7).

For those who are particular about eccentricity, we have obtained (v²) from Newtonian mechanics,

v²＝GM☉／r＝GM☉(1－e²)／l　　(8),

It is a good idea to calculate by including such as.

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RESULTS AND DISCUSSION.

The correction from the approximate solution of "Absolute rest reference system + Newtonian mechanics" by general relativity using Schwart-Schild solution and geodesic equation is not accurate.

If we can be obtained the average orbital velocity (v)  from them, the exact solution of the orbital motion of Mercury can be obtained from the ratio (γ) of the speed of light (c) and the average wave speed (w). 

CONCLUSION.

"Light velocity (c) = wave speed (w), gravitational mass (mG) = inertial mass (mI)" only occurs at relative velocity (v = 0). This is the viewpoint seen from the local inertial system that general relativity says. However, the experience value obtained from Newtonian mechanics is another viewpoint assuming point at infinity. This approximation, which adds a second term without converting from the observer's speed of light to the wave speed of the object, causes a delta function problem at full scale of micro and macro.

They pointed out that most treatments of the problem used “delta functions” as a way to approximate the bodies in the system as point masses. As a consequence, the “self-field,” the gravitational field of the body evaluated at its own location, becomes infinite. 

In the principle of light equivalence, after converting to “light's observational reference frame”,

w²＝c²±v².

Since it is dilated to the wave speed (w) of the moving object (v) seen from the observer (c) and treated as a single term, the delta function problem like general relativity does not occur.

E＝c |p|＝h f＝mG c²＝γ mI w².

Light momentum equivalence principle.

Since the mass of a body and the speed of light are closely related to the momentum of light, there are incorrect to immutable them by the classical principle of relativity.

 

Acknowledgment。

Thanks to everyone who parasitized on 5ch and argued, the prospect of this paper improved.

 

Appendix。

1. General relativity effect by relativity of atomic clock mounted on GPS satellite + special relativistic effect of post-Newton approximation,

On the GPS satellites, the progress of time is different from the surface of the earth due to the relativistic effect, so the frequency of the satellite clock is offset (-4.45e-10) in advance and synchronized with UTC (Coordinated Universal Time).

The exact solution of the ratio (wave speed) between the atomic clock mounted on the GPS satellite and the clock on the surface according to this paper is

Surface speed of light (c): 299,792,458 m/s.

Geocentric gravity constant (GM): 3.986e+14 m³/s².

Earth radius (r): 6,378,000 m.

Hyperbolic infinite wave speed, w∞＝√(c²＋2GM／r).

GPS satellite altitude (h): 20,200,000 m.

Orbital velocity of GPS satellites (v): 3,874 m/s.

Wave speed of GPS satellites, wG＝√(w∞²－2GM／(r＋h)－v²).

Difference in how the clock progresses, wG／c＝1＋4.45e-10.

2. Below is an animation of the sun revolving under the influence of another planet.

Everything in the Solar System orbits the center of mass (it's rarely in the center of the Sun!)