The Theory of Observational Relativity Serial Report 4: Reexamining Einstein's prediction on Mercury

2.1 From Ptolemy to Copernicus: The doctrine of circular orbits

The concept of The Globe began with Ptolemy’s geocentric theory (see Figure 1 (a)).¹⁰ Ptolemy’s geocentric theory was formed around the 2nd century AD, the core idea of which can be summarized as: (1) the earth is a huge sphere; (2) the earth is the center of the universe; (3) the sun, moon, and stars all revolve around the earth. Naturally, in Ptolemy’s geocentric theory, the orbits of the sun, moon, and stars around the earth are idealized as standard circles centered around the earth.

Figure 1. The Evolution of Celestial Motion Images: From the Great Earth to the Globe, from the circle to the ellipse, from the closed ellipse to the precessing ellipse.

Annotation (a) Ptolemy's geocentric theory. The earth is a ball, that is, the Globe; the earth is the center of the universe, and the sun, moon, and stars revolve around the earth in idealized standard circles.

Annotation (b) Copernicus' heliocentric theory. The sun is the center of the universe, the moon revolves around the earth in an idealized standard circle, and the earth and other planets revolve around the sun in idealized standard circles; the orbits of all planets form the concentric circles centered around the sun.

Annotation (c) Kepler's three laws and Newton's planetary orbit. According Kepler, the orbit of a planet around a star is a closed ellipse with the sun at one of its focal points; Newton's equation of planetary motion supports Kepler's three laws.

Annotation (d) The orbital precession of planets. Astronomical observation shows that the orbit of a planet around a star is not closed ellipse, and the orbit or perihelion of a planet is not fixed, but keeps precessing.

In the 16th century AD, Copernicus created The Heliocentric Theory (see Figure 1(b)),^10,11 moving the center of the universe from the earth to the sun. Copernicus’s heliocentric theory has taken a big step in the right direction of mankind’s cognizing and understanding of the universe: all the planets revolve around the sun; the earth is one of the planets, which rotates itself while it is revolving around the sun; the moon revolves around the earth. Of course, in Copernicus’ heliocentric theory, whether the moon revolves around the earth or planets revolve around the sun, they follow standard circular orbits, and the orbits of all planets form concentric circles with the sun as the center. Naturally, the sun is located at the center of the circle.

Then, the sun became the center of the universe.

2.2 From Kepler to Newton: The doctrine of closed elliptical orbits

In the 17th century AD, Kepler established three significant laws about planetary motion based on the astronomical observation data accumulated by Tycho, that is, Kepler’s three laws (see Figure 1(c))^10,12: the orbital law, the area law, and the harmonic law. Kepler’s orbit law is also known as the ellipse law: the orbit of a planet around a star is an ellipse, and the star is located at one of the foci of the ellipse. Kepler’s orbital law tells that the orbit of a celestial body is not an idealized standard circle.

It is worth noting that Kepler’s elliptical orbits are idealized as standard ellipses, immutable and closed, and do not drift and spiral. In other words, Kepler’s elliptical orbits have no orbital or perihelion precession.

After Kepler’s three laws, Newton’s three laws and law of universal gravitation were successively born.¹³

Based on Newton’s laws, Kepler’s three laws can be theoretically derived. Thus, human being’s physics began to evolve from phenomenological physics to theoretical physics. Newtonian mechanics can describe more general conic orbits: the orbit of a celestial body moving in a gravitational field follows a standard conic curve that can be a circle, ellipse, parabola, or hyperbola. Applying Newton’s law of universal gravitation to the celestial two-body problem, one can establish the theoretical model of a planet orbiting a star, that is, Newton’s equation of planetary motion, which supports Kepler’s orbital law: the orbit of a planet orbiting a star is an idealized standard and closed ellipse without drift or spiral.

However, astronomical observation shows that the orbit of a planet is not an idealized standard and closed ellipse. As depicted by Figure 1(d), the orbit or perihelion of a planet is always precessing. In the solar system, the orbital or perihelion precession of Mercury is the most prominent and is found through astronomical observation to be about 5600.73 arcsec every 100 Earth Years.

All theories or models in physics are idealized. The two-body model (M, m) of a star and a planet is an extremely idealized system: gravity is an action at a distance; (M, m) is an isolated system, where both the star M and the planet m are mass points, M is stationary, and m orbits M. This is the reason why Newton’s planetary orbits keep immutable and standard closed ellipses and cannot predict the orbital or perihelion precession of planets.

So, can Einstein’s theory of general relativity and his equation of planetary motion predict the orbital or perihelion precession of planets?

2.3 Einstein’s planetary orbits: The doctrine of orbital precession

Given the countless objective and non-idealized factors in the objective physical world, it is naturally and inevitable for the actual planetary orbits to be drifting or spiraling. On the contrary, it is difficult for us to imagine that a celestial body would move following the identical closed elliptical orbit eternally. In fact, for Mercury, it only precesses by 13.5″ per revolution around the sun. In this regard, Newton’s equation of planetary motion is already quite precise.

Mercury, as the planet closest to the sun in the solar system, has the most prominent orbital precession: about 5600.73 arcsec per century. Physicists have conducted a corrective calculation for the objective and non-idealized factors of the two-body system of Sun and Mercury. Deducting the effects of the precession of the equinoxes (about 90%) and the perturbations from other planets (especially Venus, Earth, and Jupiter) (about 10%) from the actual observation data of 5600.73, totaling 5557.62 arcsec, finally there is only 43.11″ left that has not found a definite reason.

In 1915, after the establishment of his special relativity⁸ in 1905, Einstein established his general relativity.⁹ Einstein applied his general relativity to the celestial two-body problem and built his equation of planetary motion. Compared with Newton’s equation of planetary motion, Newton’s equation of planetary motion has an additional item: 3GM/c²r² that can be refer to as the item of orbital precession. Thus, by calculation, Einstein made a theoretical prediction: Mercury’s perihelion precesses by 43.03″ per century, which just right coincides with the 43.11″ in the actual observed value of 5600.73 arcsec that have not yet found a home.

Although many physicists still have doubts and propose that it should not be appropriate to draw a final conclusion, the mainstream physics community believes that the 43.03″ supports Einstein’s theory of general relativity.

An early success of Einstein’s theory of general theory was exactly because of the interpretation for the 43.11″ perihelion precession of Mercury per century.

3.1 Observation agents and observational locality

The OR serial reports never tire of talking about the most important concept in the theory of OR: Observation Agents, and discussing their Observational Locality.

Human perception and cognition of the objective world not only depend on observation but is also restricted by observation. The theory of OR has discovered that all theoretical systems or spacetime models in physics, including the Galilean transformation and the Lorentz transformation, as well as, Newtonian mechanics and Einstein relativity theory, must be branded with observation.

Observation is to sense or perceive the objective physical world and obtain the information about observed objects. The information of an observed object must be transmitted from the observed object to the observer through a certain observation medium at a certain speed, so that the observer can perceive and cognize the observed object. This led to the formation of the concept of Observational Agent in the theory of OR.

Observation Agent: The observation agent OA(η) is an observation system (P, M(η),O) employs the observation medium M(η) to transmit the information of the observed object P to the observer O at a certain speed η; the matter wave M(η) transmitting observed information is referred to as the information wave of OA(η), and the matter particles constituting the information wave of OA(η) are referred to as the informons of OA(η).

Železnikar ever employed Informon to refer to an information entity and analogized it with an electron.¹⁴

For the issue of the orbital or perihelion precession of Mercury, the observation data come from the astronomical observations of optical astronomy with the optical observation agent OA(c) (η=c). The observed object P is Mercury, the observation medium M(c) is light, the information wave is light wave, the informons are photons, the information-wave speed η is the speed of light c, and O is the observer on the earth.

In theory, all forms of matter motion, not just light or photons, can serve as observation media to transmit observed information to observers; the speed η at which the observation agent OA(η) transmits observed information through the observation medium M(η) can be any value: 0<η<∞ or η→∞, not just the speed of light c.

The ear is mankind’s acoustic agent OA (v_S) (η=v_S≈ 340 m/s), and the eye is mankind’s optical agent OA(c) (η=c). And OA(η) is the General Observational Agent (0<η<∞;η→∞) of the theory of OR.

According to the principle of locality, the speed η of the information wave of an objectively existing observation agent OA(η) must be finite or limited: η<∞. This is the Observational Locality of OA(η), which is described as a principle in the theory of OR^1–5: The Principle of Observational Locality -- it must take time for observed information to cross space.

In the objective physical world, no speed of a form of matter motion can be infinite. Therefore, an observation agent OA(η) that mankind can employ must have a limited information-wave speed and have the observation locality (η<∞). In other words, human perception of the objective physical world must be restricted by the observational locality, and cannot represent the objective and real physical world, but rather a certain image of the objective physical world presented in mankind’s subjective world.

A theoretical system or spacetime model in human being’s physics must implicitly have the observation agent of its own. Newtonian mechanics is the theory of the idealized observation agent OA_∞, which in a certain sense represents the objectively real physical world; Einstein relativity theory, including the special and the general, is a theory of the optical observation agent OA(c), which presents us with an optical image of the physical world, not the objectively physical reality or really physical existence.

3.2 Gravitational deflection due to different observation agents

Einstein predicted based on his equivalence principle that the trajectory of light would be curved in a gravitational field, that is, Einstein’s famous prediction of the gravitational deflection of light.

In fact, Newton’s theory of universal gravitation can also lead to this conclusion.

Before the formal establishment of general relativity, the gravitational deflection angle of light calculated by Einstein was the same as that calculated with Newtonian mechanics. However, after the formal establishment of general relativity, the gravitational deflection angle δ_E of light calculated by Einstein based on his general relativity is twice the gravitational deflection angle δ_N of light calculated with Newtonian mechanics.

Newton’s gravitational-deflection angle δ_N:

So, is Newton or Einstein right?

Einstein proposed that, by observing solar eclipses, the deflection angle of starlight passing over the surface of the sun could be determined and thereby verify his theoretical prediction of gravitational deflection of light.

In 1919, during a solar eclipse on the coast of West Africa, Eddington and his team measured the deflection angle of starlight, which concluded that the deflection angle was δ=1.61″±0.40″,^15,16 tending to support Einstein’s prediction of gravitational deflection of light. At that time, the British newspaper The Times published a full page news article titled Scientific Revolution -- Newton’s Ideas Overturned. Up to now, almost all observations of total solar eclipses tend to support Einstein’s theoretical prediction,¹⁷ rather than Newton’s.

However, according to the theory of OR, this does not mean that Einstein is more right than Newton. Throughout history, the observation agent employed by mankind to observe solar eclipses has been the optical agent OA(c), and naturally, its observation conclusions tend to support Einstein’s theory of general relativity.

The theory of OR has already clarified^1–5: under different observation agents, the trajectory of light or photons in a gravitational field will present different degrees of deflection.

It should be pointed out that, during the observation of solar eclipses, starlight is both the observed object P and the observation medium M(c) or the information wave of the optical agent OA(c) transmitting the information of starlight at the speed of light c. Therefore, the speed of light c in Einstein’s starlight deflection angle δ_E (Eq. (2)) is not only the speed v of the light or photons as the observed object P but also the speed of the information wave or informons of OA(c) -- the information of starlight is carried and transmitted by starlight itself.

However, Newton’s observational agent is the idealized agent OA_∞, whose information-wave speed η is idealized as infinity and informon momentum p_η=h_η/λ is idealized as infinitesimal. The speed of light c in Newton’s starlight deflection angle δ_N (Eq. (1)) is only the speed v of the light or photons as the observed object P, but not the speed of the information wave or informons of OA_∞.

The support of the observation of the optical agent OA(c) for Einstein’s prediction of the gravitational deflection of light does not mean that Einstein is more right than Newton.

On the contrary, Newton’s idealized agent OA_∞ has no observational locality or perturbation effects, and therefore could be referred to as God’s eye, representing in a sense the objective reality and physical existence. If mankind has the idealized agent OA_∞, then his observation must tend to support Newton’s prediction of the gravitational deflection of light rather than Einstein’s.

It is thus clear that Newton’s theory is right and true, and Einstein’s theory is only an approximation.

Newton’s gravitational deflection angle δ_N=0.875″ is the objectively real deflection of starlight sweeping over the sun, whereas Einstein’s gravitational deflection angle δ_E=1.75″ of starlight contains the observational effects of the optical agent OA(c), that is the apparent phenomena caused by the observational locality (c<∞) of OA(c): Δδ=δ_E−δ_N=0.875″ is not the really physical existence.

For the detailed statement on the gravitational deflection of light, see Chapter 17 of the 2^st volume GOR in Observational Relativity.^1–4

3.3 Gravitational redshift due to different observation agents

Even before the formal establishment of general relativity, Einstein predicted based on his equivalence principle that the frequency of light would decay in a gravitational field, that is, Einstein’s famous prediction of the gravitational redshift of light:

Einstein’s prediction of the gravitational redshift of light is supported by the observation of the solar spectrum¹⁸ and the experiment based on Mossbauer effect.¹⁹

In fact, based on Newton’s theory of universal gravitation, one can also get the conclusion that the frequency of light in a gravitational field will decay or present redshift.

Everyone was looking forward to another debate on the gravitational redshift of light between Einstein’s theory of general relativity and Newton’s theory of universal gravitation. However, the gravitational redshift Z_PN of light based on Newton’s theory of universal gravitation is almost the same as the gravitational redshift Z_E of light based on Einstein’s theory of general relativity, with only a difference in the second-order small quantity. It is difficult to distinguish or pass judgement on the two by observation or experiment²⁰:

Thus, the observation of solar spectra and Mossbauer experiment seem to be both a support for Einstein and a support for Newton.

It should be pointed out that both the observation of solar spectra and the experiment of Mossbauer gravitational-redshift employed the optical agent OA(c). Therefore, the observational and experimental conclusions should support Einstein’s theory of general relativity as the theory of the optical agent OA(c) and Einstein’s prediction of gravitational redshift. However, it is unreasonable and confusing that the observational and experimental conclusions of the optical agent OA(c) support the theory of Newton’s theory of universal gravitation as the theory of the idealized agent OA_∞ and Newton’s prediction of gravitational redshift.

The mainstream physics community does not fully understand the role played by observation agents in observations and experiments. They still have no convincing answer to the similarity between Newton and Einstein’s predictions on the gravitational redshift of starlight.

The theory of GOR has already clarified that the current so-called Newtonian gravitational-redshift equation of light (Eq. (4)) is pseudo Newtonian,^1–4,6 not purely based on Newtonian mechanics, but a mixture of classical mechanics, relativity theory, and quantum theory. This is precisely the reason why the pseudo Newtonian gravitational-redshift Z_PN (Eq. (4)) approximates Einstein’s gravitational-redshift Z_E (Eq. (3)).

The gravitational-redshift phenomenon of light is the manifestation that the kinetic energy of photons decays, that is, the redshift of kinetic energy. In essence, it is the transformation from the kinetic energy to potential energy of photons in a gravitational field, following the principle of energy conservation. Based on the principle of energy conservation, the theory of OR equivalently transforms the definition of the gravitational redshift of light from that of frequency redshift Z=Δf/f to that of kinetic-energy redshift Z=ΔK/K.

In this way, the theory of OR has deduced the equation of the gravitational redshift of light purely based on Newtonian classical mechanics^1–4:

Now, Newton’s gravitational-redshift equation of light (Eq. (5)) is different from both Einstein’s Eq. (3) and the pseudo Newtonian Eq. (4). This is reasonable: the redshift of light in a gravitational field should naturally be different from the perspectives of different observation agents. Of course, in a gravitational field, light may present not only redshift but also blueshift.

Similar to the case of gravitational deflection of light, the support from the optical observation agent OA(c) for Einstein’s theory of gravitational redshift (Eq. (3)) does not mean that Einstein is more right than Newton. If mankind has the idealized agent OA_∞, then his observation must tend to support the theory of gravitational redshift purely based on Newtonian mechanics (Eq. (5)).

For the detailed statement on the gravitational redshift of light and the theoretical derivation of the Newtonian gravitational-redshift equation (Eq. (5)), see Chapter 18 of the 2^nd volume GOR in Observational Relativity.^1–4

3.4 Orbital precession due to different observation agents

This article aims to re-examine Einstein’s theoretical prediction about the orbital or perihelion precession of Mercury from the perspective of the general observation agent OA(η) (0<η<∞; η→∞) of the theory of OR.

After the establishment of his general relativity, Einstein applied it to the celestial two-body problem: (M, m). Based on his gravitational-field and gravitational-motion equations, Einstein derived his equation of planetary motion, and made his famous prediction about the orbital or perihelion precession of Mercury: Mercury’s perihelion precesses by 43.03″ every 100 Earth Years.

In the theory of OR, the orbital or perihelion precession of Mercury is listed as the big puzzle BP-10. Indeed, Einstein’s theoretical prediction on the orbital or perihelion precession of Mercury have many doubts.

Astronomical observation shows that Mercury’s perihelion precesses by 5600.73 arcsec per century. But why did Einstein’s prediction only have 43.03″, less than 0.8% of the actual observed value?

Both Newton and Einstein’s equations of planetary motion are idealized models of the celestial two-body system of a star and a planet, in which there is no prior knowledge or information for Newton and Einstein to predict the orbital or perihelion precession of the planet orbiting the star. Therefore, it is impossible for both Newton and Einstein to make a theoretical prediction for the real orbital or perihelion precession of Mercury.

So, how can Einstein’s equation of planetary motion predict that Mercury precesses by 43.03 arcsec every 100 Earth Years? And, why cannot Einstein’s equation of planetary motion predict the other 5557.70″ orbital precession of Mercury per century?

Now, readers have had the knowledge of observation agents and their observation locality. So, readers may have their own answers to the doubts of Einstein’s predictions on Mercury’s orbital or perihelion precession.

4.1 Newton’s celestial two-body problem

All theories or models of human being’s physics must contain certain idealized logical premises or hypotheses. Newton’s equation of planetary motion belongs to the celestial two-body problem, and is a theoretical model of the celestial two-body system, which is highly idealized, and can be described as follows.

Newton’s Celestial Two-Body System (M, m): The large celestial body M and the small celestial body m interact through gravitational interaction, where M is stationary, radiates gravity or gravitational force, and generates a spherically-symmetric gravitational field, and m moves in the gravitational field.

The Idealized Conditions for Newton’s (M, m):

The Coordinate System of Newton’s (M, m): Choose Cartesian 3d coordinates (x, y,z) and corresponding spherical coordinates (r, θ,φ) as depicted in Figure 2(a); set the large celestial body M as the coordinate origin O, the small celestial body m moves in the plane of X-Y (θ=π/2) as depicted in Figure 2(b).

Figure 2. The formalized coordinates of celestial two-body problem.

Annotation (a) Central force and spherical coordinates: The celestial two-body system (M, m) is idealized as an isolated system in the space of spherical-coordinate O(r, θ,φ), where the large celestial body M is located at the coordinate origin of O, and the small celestial body m (a planet or comet or satellite or even photon) moves in the gravitational field of M.

Annotation (b) The orbital plane of the small celestial-body m : According to the property of central force, the motion of mass point m is limited to a fixed plane, e.g. X-Y (θ=π/2).

Newton’s Observation Agent: The idealized observation agent OA_∞, whose information-wave speed η is idealized as infinity (η→∞), with no observational locality -- it takes no time for observed information to cross space.

It is thus clear that Newton’s celestial two-body system is extremely idealized.

4.2 Newton’s model of celestial two-body system

In Newton’s idealized celestial two-body system (M, m), M and m interact through universal gravitation. As depicted in Figure 2(a), the universal gravitation F(r) on m is a sort of central force that always points towards the large celestial body M (at the coordinate origin O), which is a function of the radius vector r: F(r)=F_r r /r (r=|r|).

Omitting theoretical derivation, the motion of mass point m in the gravitational field of M in the corresponding spherical coordinate system (r, θ,φ) can be described by the Binet equation^1–4:

Substituting Newton’s law of universal gravitation into Eq. (6), one has the theoretical model of Newton’s celestial two-body system (M, m) in the form of the Binet equation:

This is the motion equation of the small celestial body m (a planet or comet or satellite or even photon) in the gravitational field of M.

4.3 Newton’s celestial-body orbits

By solving differential Eq. (7), one has the solution

This is the standard conic equation, where M is located at a focal point of the conic section and e is the orbital eccentricity of m.

In the celestial two-body system (M, m), the orbital eccentricity e of m depends on the gravitational constant G and the mass of large celestial body M, as well as, the initial mechanical energy E and angular momentum L of the small celestial body m:

Given the conservation of mechanical energy E and angular momentum L of m, according to Eq. (9), in Newton’s model of celestial two-body system (M, m), the orbital eccentricity e of m is a constant.

The eccentricity e determines the form of the orbit of celestial body m:

Let the large celestial body M be a star and the small celestial body m a planet orbiting M. According to Kepler’s law of ellipses, due to being bound to the star M, the planet m must orbit around M along an elliptical path with an eccentricity of e∈(0,1). The eccentricity of Mercury’s orbit around the sun is e=0.2056; the eccentricity of Earth’s orbit around the sun is e=0.0167. Compared to the orbit of Mercury, the orbit of the earth is closer to a circle.

Newton’s equation of planetary motion (Eq. (7)) and its solution (Eq. (8)) prove Kepler’s first law: the orbit of a planet should be an ellipse.

4.4 Newton’s planetary orbit

If Newton’s model (Eq. (7)) of celestial two-body system (M, m) is employed as that of a star and a planet, then the orbital eccentricity e∈(0,1), that is, Newton’s equation of planetary motion, in which the orbit of the planet m is a standard closed ellipse, with no drift and spiral. Therefore, Newton’s equation of planetary motion cannot predict the orbital or perihelion precession of planets.

Set the initial angle φ₀ of the orbit of the planet m be zero, then the solution (Eq. (8)) to Newton’s equation of planetary motion can be written as

Equation (10) shows that the perihelion of m is fixed. At the point P (as depicted in Figure 2(b)) of the perihelion of m, φ=2kπ (k=0,1,2,…), u=GM(1+e)/h_K², and r=1/u are all constants. Therefore, there is no orbital or perihelion precession of the planet m.

Or, in other words, at the point P (Figure 2(b)) of the perihelion of m, it holds that du/dφ=0. Let the precession angle of m every revolution be Δφ. Let k=1, that is, the planet m orbits around the star M once. As depicted in Figure 1(d), the planet m travels from the perihelion P to the next perihelion P′, sweeping through an angle of φ =2π +Δφ. Substituting it into Eq. (10), it can be concluded that Δφ=0. This also suggests that there is no any information about the orbital or perihelion precession of planets in Newton’s equation of planetary motion.

So, why cannot Newton’s celestial two-body model predict the orbital precession of planets?

The answer is simple: Newton’s two-body model are too idealized.

Whereas, the objective physical world is not an idealized physical world: gravity is not an action at a long distance, a celestial two-body system (M, m) is never an isolated system, M is not stationary, and both M and m are not mass points. So, the orbit of a planet in the universe is neither a standard circle nor an immutable and standard closed ellipse. The orbital or perihelion precession of a planet is natural and inevitable.

As stated in Sec. 4.1, Newton’s model of the celestial two-body system (M, m) has no prior information for predicting the orbital or perihelion precession of a planet. For instance, Newton’s equation of planetary motion cannot predict the orbital or perihelion precession of Mercury’s 5557.62 arcsec per century due to the lack of prior knowledge or information about the precession of the equinoxes and the perturbations from other celestial bodies.

In particular, according to the theory of OR, there exist apparent orbital-precession phenomena of planets in astronomical observations caused by the observational locality (c<∞) of the optical agent OA(c), which is only a sort of observational effects, rather than the objectively real precession of planetary orbits.

Newton’s observation agent is the idealized agent OA_∞ that has no observational locality (η→∞), and so, it also does not present apparent orbital precession of planets.

Revisiting Newton’s equation of planetary motion and analogizing it with Einstein’s equation of planetary motion and the GOR equation of planetary motion help us recognize the essence of Einstein’s prediction of the 43.03″ precession of Mercury’s perihelion.

5.1 Einstein’s celestial two-body problem

Like Newton’s celestial two-body problem, Einstein’s celestial two-body problem is also highly idealized.

Einstein’s Celestial Two-Body System (M, m): According to Einstein’s theory of general relativity, the large celestial body M is stationary, making the spherically-symmetric spacetime around M curved; The small celestial body m moves in the curved spacetime of M.

In fact, as clarified by the theory of OR,^1–4 Einstein’s so-called spacetime curvature is only a geometrized expression of gravitational interaction and does not represent objectively physical existence.

The Idealized Conditions for Einstein’s (M, m):

It should be pointed out that, in fact, like Newton’s theory of universal gravitation, Einstein’s theory of general relativity also implies the hypothesis that gravity is an action at a distance.^1–4,7

The Coordinate System of Einstein’s (M, m): Like Newton, choose Cartesian 3d coordinates (x, y,z) and corresponding spherical coordinates (r, θ,φ) as depicted in Figure 2(a); set the large celestial body M as the coordinate origin O, the small celestial body m moves in the plane of X-Y (θ=π/2) as depicted in Figure 2(b).

Thus, Einstein and Newton’s equations of planetary motion have the formal comparability.

Einstein’s Observation Agent: The optical observation agent OA(c), whose information-wave speed η is the speed of light c, with the observational locality (c<∞) -- it takes time for observed information to cross space.

In fact, the difference of observation agents is the only and fundamental difference between Einstein and Newton’s equations of planetary motion.

5.2 Einstein’s model of celestial two-body system

For a spherically-symmetric gravitational spacetime, Schwarzschild obtained the exact solution of Einstein field equation, that is, the spacetime metric in the spherical coordinate form of gravitational spacetime observed by the optical agent OA(c), denoted as $g_{\mathit{\mu \nu }}(r,c)$ ²¹:

By substituting Eq. (11) into the equation of the line element ds of Einstein general relativity, one has

By substituting Eq. (11) into Einstein’s gravitational-motion equation of Einstein general relativity, one has

Equation (13) has four equations (μ=0,1,2,3): (1) t=t(τ), (2) r=r(τ), (3) θ=θ(τ), and (4) φ=φ(τ). By analyzing these four equations, one can derive the celestial two-body model of Einstein general relativity with the optical agent OA(c) in the form of the Binet equation:

Contrasting with Newton’s equation of planetary motion (Eq. (7)), it can be seen that Einstein’s equation of planetary motion (Eq. (14)) has an additional item at the right end, that is, the so-called orbital precession item: 3GM/c²r². This means that

For the detailed derivation of Einstein’s equation of planetary motion (Eq. (14)), see reference²² or Chapter 16 of the 2^nd volume GOR in Observational Relativity.^1–4

5.3 Einstein’s planetary orbit and mercury’s perihelion precession

Solving Einstein’s equation of planetary motion (Eq. (14)), one can get a tiny orbital or perihelion precession of planets based on the item of orbital precession 3GM/c²r² in Eq. (14).

Generally, the item of orbital precession in Einstein’s equation of planetary motion (Eq. (14)) is a small quantity: 3h_K²/c²r²<<1. So, Einstein’s equation of planetary motion (Eq. (14)) can be solved by making use of the progressive approximation method.²²

By substituting u=GM(1+ecosφ)/h_K² (Eq. (10)) into Eq. (14), one has

The progressive approximation solution of Eq. (15) is

At the perihelion P of the planetary orbit as depicted in Figure 2(b), it holds true that du/dφ=0. Therefore, by taking the derivative with respect to φ at the both sides of Eq. (16), one has

If the φcosφ is not included in Eq. (17), then

Thus, sinφ=0. This suggests that if without the φcosφ, Einstein’s planetary orbits, like Newton’s planetary orbits, were a closed ellipse without precession.

If considering the orbital precession angle Δφ (<<1) of a planet around a star as depicted in Figure 1(d) with k=1, then the angle that the planet sweeps over should be φ=2π +Δφ. Substituting it into Eq. (17), ignoring high-order small quantities, then

Mercury’s eccentricity e≈0.206 is the largest in the solar system. According to the recommendations of the International Standards Organization:

Then, with Eq. (19), one can calculate the orbital or perihelion precession of Mercury during one revolution:

Mercury’s orbital period is T_M=87.961 day; Earth’s orbital period is T_E=365.24219 day. So, Mercury orbit precesses φ=100×Δφ×T_E/T_M=42.73 arcsec per century.

The orbital or perihelion precession angle φ of Mercury calculated by Einstein at that time was 43.03 arcsec, which was extremely consistent with the 43.11″ that has not yet found out a definite reason up to now. In a letter to a friend, Einstein said: “The equation gives the correct numbers for Mercury’s perihelion. You could imagine how happy I am. I couldn’t help but be happy for several days.”

However, astronomical observation shows that the orbital or perihelion precession of Mercury is 5600.73 arcsec per 100 Earth Years, which is far greater than the 43.03″ predicted by Einstein.

Einstein’s 43.03″ is far from the actual observed orbital precession of Mercury, and so it is not enough to confirm that Einstein’s theory of general relativity and equation of planetary motion correctly predicted the orbital or perihelion precession of Mercury.

6.1 GOR celestial two-body problem

Like Einstein’s celestial two-body problem, the GOR celestial two-body problem is also highly idealized.

GOR Celestial Two-Body System (M, m): The large celestial body M and the small celestial body m interact through gravitational interaction, where M is stationary, radiates gravity or gravitational force, and generates a spherically-symmetric gravitational field, and m moves in the gravitational field.

This is consistent with the statement of Newton’s celestial two-body system (M, m).

The Idealized Conditions for Einstein’s (M, m):

Actually, like Newton’s theory of universal gravitation and Einstein’s theory of general relativity, the theory of GOR also implies the hypothesis that gravity is an action at a distance.^1–4,7

The Coordinate System of GOR’s (M, m): Like Newton, choose Cartesian 3d coordinates (x, y,z) and corresponding spherical coordinates (r, θ,φ) as depicted in Figure 2(a); set the large celestial body M as the coordinate origin O, the small celestial body m moves in the plane of X-Y (θ=π/2) as depicted in Figure 2(b).

Then, the GOR, Einstein and Newton’s equations of planetary motion has the formal comparability.

The GOR Observation Agent: The general observation agent OA(η) (0<η<∞; η→∞), whose information-wave speed η can be any speed value, not just the speed of light c, and can even be idealized as infinity.

It is thus clear that the only and fundamental difference between the GOR celestial two-body problem and Newton and Einstein’s celestial body two body problem lies only in their observation agents.

In fact, both Newton’s idealized agent OA_∞ and Einstein’s optical agent OA(c) are just two special cases of the general observational agent OA(η) in the theory of OR.

6.2 The principle of general correspondence

The principle of general correspondence is that proposed by the theory of OR for deducing the gravitational theory of the general observation agent OA(η) (0<η <∞; η→∞), that is, the theory of GOR, particularly for deriving the GOR gravitational-field equation and gravitational-motion equation.^1–4

Actually, the principle of general correspondence is the generalization and unification of Bohr’s principle of correspondence and Galileo’s principle of relativity.

Bohr’s principle of correspondence pertains to the corresponding relationship between quantum physical models and classical physical models, as well as, the corresponding relationship between the optical agent OA(c) and the idealized agent OA_∞, embodying the idea of “All Observation Agents are Equal!”.

Galileo’s principle of relativity pertains to the corresponding relationship of the space-time transformations between different reference frames or different observers, embodying the idea of “All Observers are Equal!”. The Inertial OR (i.e., the theory of IOR) and the general Lorentz transformation of OR further clarified that^1–4 all observers, regardless of their reference frame or their observation agents, are equal or have the equal right, so that their physical models must be the same in form or have the isomorphic and consistent corresponding relationship.

So, the principle of correspondence and the principle of relativity can be unified into the principle of general correspondence under the concept of Equal Rights for All Observers or All Observation Agents.

The Principle of General Correspondence (PGC): The universe or spacetime is symmetric, therefore, all observers in the universe or spacetime, regardless of their reference frame or observation agents, are equal or have the equal right, and so, their physical laws or physical models must be the same in form, or in other words, have the isomorphic and consistent corresponding relationship.

Following the PGC principle, one can perform the isomorphically-consistent transformation of a physical model, law, or even theory between different observers O and O′ or different observation agents OA(η₁) and OA(η₂).

The application of the PGC principle can follow two distinct logical ways.

PGC Logical Way 1:

Based on the PGC principle, directly replacing η₁ with η₂, one can transform the physical quantity U(η₁) observed with OA(η₁) into the physical quantity U(η₂) observed with OA(η₂), and transform the physical models of OA(η₁) into the physical models of OA(η₂) isomorphically and uniformly.

PGC Logical Way 2:

For the detailed statement of the PGC principle, see Chapter 11 of the 2^nd volume GOR in Observational Relativity.^1–4

6.3 GOR model of celestial two-body system

Now, the PGC principle can be applied to the GOR celestial two-body problem to solve the GOR gravitational-field equation and to derive the GOR celestial two-body model or the GOR equation of planetary motion from the GOR gravitational-motion equation.

Einstein’s celestial two-body model adopts the optical agent OA(c), the information wave of which is light or electromagnetic wave and transmits observed information at the speed of light c. The GOR celestial two-body model adopts the general observation agent OA(η), the information-wave speed η of which can be any speed: 0<η<∞ and even η→∞ in theory.

Based on the PGC principle and following PGC logical way 1, by analogizing with Schwarzschild exact solution,²¹ one can obtain the exact solution of GOR gravitational-field equation for a spherically-symmetric gravitational spacetime, that is, the spacetime metric in the spherical coordinate form of gravitational spacetime observed by the general observation agent OA(η), denoted as $g_{\mathit{\mu \nu }}(r,\eta )$ ^1–4:

By substituting Eq. (21) into the equation of the line element ds of the theory of GOR, one has

By substituting Eq. (21) into the GOR gravitational-motion equation of the theory of GOR, one has

Equation (23) has four equations (μ=0,1,2,3): (1) t=t(τ), (2) r=r(τ), (3) θ=θ(τ), and (4) φ=φ(τ). By analyzing these four equations, based on the PGC principle and following PGC logical way 2, analogizing with the logic of deducing Einstein’s celestial two-body model stated in Sec. 5.2,²² one can derive the GOR celestial two-body model of the theory of GOR with the general observation agent OA(η) in the form of the Binet equation:

It is obvious that the GOR celestial two-body model (Eq. (24)) and Einstein’s celestial two-body model (Eq. (14)) has the isomorphically-consistent corresponding relationship.

Like Einstein’s equation of planetary motion (Eq. (14)), the GOR equation of planetary motion (Eq. (24)) also has an item of orbital precession: 3GM/η²r². This suggests that, if η<∞, then the GOR equation of planetary motion is also a nonlinear differential equation, in which the planetary orbit is also drifting and spiraling constantly, not an immutable and standard closed ellipse.

For the detailed derivation of GOR equation of planetary motion (Eq. (24)), see Chapter 16 of the 2^nd volume GOR in Observational Relativity.^1–4

6.4 The unified theory of celestial motion

The theory of GOR has clarified^1–4 that the motion of celestial bodies lies in the action of universal gravitation, rather than spacetime curvature. There are two major theoretical systems for gravity in physics: one is Newton’s theory of universal gravitation¹³; the other is Einstein’s theory of general relativity. So, the theory of celestial motion can also be divided into two major schools: Newton’s theory of celestial motion; Einstein’s theory of celestial motion. Undoubtedly, the unification of the two major schools is of great significance.

Like all the relations in the theory of OR, the GOR celestial two-body model (Eq. (24)), as the general observation agent OA(η), has a high degree of generalization and unification, which has generalized and unified Newton’s celestial two-body model (Eq. (7)) with the idealized agent OA_∞ and Einstein’s celestial two-body model (Eq. (14)) with the optical agent OA(c).

As η→∞, Newton’s celestial two-body model or planetary-motion equation (Eq. (7)) strictly converges to the GOR celestial two-body model or the GOR planetary-motion equation (Eq. (24)):

As η→c, Einstein’s celestial two-body model or planetary-motion equation (Eq. (14)) strictly converges to the GOR celestial two-body model or the GOR planetary-motion equation (Eq. (24)):

So, both Newton’s equation of planetary motion and Einstein’s equation of planetary motion are special cases of GOR equation of planetary motion, serving their respective specific observation agents.

The celestial two-body problem is the most fundamental and representative problem in the theory of celestial motion. The unification of Newton’s equation of planetary motion and Einstein’s equation of planetary motion within the theory of GOR means the unification of Newton’s theory of celestial motion and Einstein’s theory of celestial motion: Newton’s theory of celestial motion serves the idealized agent OA_∞; Einstein’s theory of celestial motion serves as the optical agent OA(c).

In particular, this suggests that the GOR celestial two-body model is logically consistent with both the Newton’s celestial two-body model and the Einstein celestial two-body model. This, from one aspect, confirms the logical self-consistence and theoretical validity of the GOR planetary-motion equation, the GOR celestial two-body model, and even the theory of GOR.

7.1 GOR prediction on planet’s orbital precession

Now, like Einstein’s equation of planetary motion (Eq. (14)), the GOR equation of planetary motion (Eq. (24)) also has an orbital precession item: 3GM/η²r², which seems to be able to predict the orbital or perihelion precession of planets too, e.g. that of Mercury.

Based on the PGC principle, following PGC logical way 1 or PGC logical way 2, by following or analogizing the logic of solving Einstein’s equation of planetary motion (Eq. (14)) stated in Sec. 5.3, one can obtain the progressive approximation solution to GOR equation of planetary motion (Eq. (24)):

At the perihelion P of the planetary orbit as depicted in Figure 2(b), it holds true that du/dφ=0. Therefore, by taking the derivative with respect to φ at the both ends of Eq. (27), one has

If considering the orbital precession angle Δφ (<<1) of a planet around a star as depicted in Figure 1(d) with k=1, then the angle the planet sweeps over should be φ=2π +Δφ. Substituting it into Eq. (28), ignoring high-order small quantities, then one has the GOR precession-angle equation as below:

It is worth noting that the precession angles of both Newton’s planetary orbit and Einstein’s planetary orbit can be calculated with the GOR precession-angle equation (Eq. (29)). According to Eq. (29), under Newton’s idealized agent OA_∞, η→∞, Δφ=0, and so, Newton’s planetary orbit has no precession, which is consistent with the conclusion of Newton’s equation of planetary motion (Eq. (7)); under the optical agent OA(c), η→c, Δφ=0.1029″ for every revolution of Mercury, that is, 42.73″ per century, which is consistent with the conclusion of Einstein’s equation of planetary motion (Eq. (14)).

It is thus clear that Newton’s equation of planetary motion (Eq. (7)) and Einstein’s equation of planetary motion (Eq. (14)) are indeed only two special cases of the GOR equation of planetary motion (Eq. (24)).

7.2 Einstein’s prediction of the 43.03″: An apparent precession

It should be pointed out that, although the GOR equation of planetary motion can also predict the orbital or perihelion precession of planets and conclude that Mercury’s orbital or perihelion precesses by 42.73″ every 100 Earth Years (with current data), it does not mean that the 42.73″ is a part of the objectively existing precession of Mercury.

According to the GOR equation of planetary motion (Eq. (24)) with the GOR precession item 3GM/η²r² and the GOR precession angle Δφ=6πG²M²/η²h_K² (Eq. (29)), the planetary precession presented in the GOR celestial two-body model depends on the observation agent OA(η), or more accurately, depends on the speed η of the information wave of OA(η): for the same planet, its orbital or perihelion precession observed or presented by different observation agents must be different.

This fact indicates that the so-called perihelion precession presented in the GOR equation of planetary motion, including Einstein’s equation of planetary motion as a special case, is not the objectively existing planetary-orbit precession. In essence, it is just a sort of observational effect or an apparent phenomenon caused by the observational locality (η<∞) of the observation agent OA(η).

The theory of OR has been clarified^1–5 that all relativistic effects, including the special (inertial) relativistic and the general (gravitational) relativistic, are observational effects and apparent phenomena, the root and essence of which lie in the observational locality (η<∞) of mankind’s observation agent OA(η). All relativistic effects in Einstein’s theory of relativity are observational effects and apparent phenomena caused by the observation locality (c<∞) of the optical agent OA(c): the speed of light is not really invariant; spacetime is not really curved.

Einstein’s equation of planetary motion, as a model of the optical agent OA(c), is a special case of GOR equation of planetary motion. So, according to Eq. (29) and Eqs. (19-20), Einstein’s theoretical prediction that Mercury’s perihelion precesses by 43.03″ per century is only a relativistic apparent phenomenon caused by the observational locality (c<∞) of the optical agent OA(c).

Mirage does not really exist. However, you may have indeed seen the beauty of a mirage. Seeing may not necessarily be real, however, the history data recorded by optical astronomical observation may indeed record and contain the 43.03″ of Mercury precession predicted by Einstein, although it is only an apparent phenomenon caused by the observation locality (c<∞) of the optical agent OA(c), and not the objectively and really existing precession of Mercury’s perihelion.

Thank God for bestowing light upon mankind. Thanks to light with such a huge speed of 3×10⁸ m⋅s⁻¹, our perception or observation of the physical world can be so close to the objective and real physical reality. Taking the observation of Mercury in optical astronomy as an example, the authenticity is greater than 99% and the non-authenticity is less than 1%.

According to the theory of OR,^1–4,7 the speed of gravitational wave κ must be much greater than the speed of light c. If mankind can truly develop gravitational wave astronomy²³ and the speed κ of gravitational wave is truly as calculated by Laplace: κ>7×10⁶c, ²⁴ or as calculated by Flandern: κ=2×10¹⁰c,²⁵ then the observational locality of the gravitational-wave observation agent OA(κ) will be very small. The apparent precession angle per century of Mercury orbit observed with OA(κ) will only be 43.27×(c²/κ²)≈0 arcsec that is close to that of Newton’s idealized observation agent OA_∞.

7.3 The support for OR from astronomical observation

The theory of OR does not doubt that the orbit or perihelion of a planet in the real universe is precessing. Actually, due to various objective and non-idealized factors, the orbits of all celestial bodies in the universe must not be a standard and closed circle or ellipse. And a planet cannot permanently orbit a star in a fixed plane.

In the objective and real physical world, the orbital or perihelion precession of planets is natural and inevitable, which is not the so-called Abnormal Precession.

However, whether it is Newton’s theory of universal gravitation or Einstein’s theory of general relativity, or even the theory of GOR, the idealized celestial two-body model has no prior knowledge or information about the orbital or perihelion precession of planets. Therefore, it is impossible for Newton and Einstein and even GOR to make theoretical predictions about the objective and real orbital precession of planets.

As is well known, astronomical observation shows that Mercury’s perihelion precesses by 5600.73 arcsec per century. After deducting 5557.62 arcsec caused by equinoxes and perturbations, there are still 43.11 arcsec left that has not yet found the right reason up to now.

There are two possibilities for this:

Neither the first nor the second implies that Einstein’s general theory of relativity or his equation of planetary motion can make theoretical prediction about the real orbital or perihelion precession of Mercury.

If it is the second, then the left 43.11″ has had the right reason: it is not the objective and real orbital or perihelion precession of Mercury, but the observational effect and apparent phenomenon presented by the optical agent OA(c). So, the left 43.11″ of Mercury is not a support for Einstein, but the support for the theory of OR: an observation agent OA(η), such as the optical agent OA(c) (η=c), does indeed present observational effects and apparent phenomena caused by the observational locality (η<∞) of OA(η). In other words, the history astronomical-observation data of Mercury provides another empirical evidence for the theory of OR, including the theory of GOR and the GOR equation of planetary motion.