Precession of planets' orbits between Newtonian gravity and Einstein's general relativity

Published

June 22, 2024

Title

Precession of planets' orbits between Newtonian gravity and Einstein's general relativity

Author

Christian Corda

DOI

10.62684/TJWF4948

Christian Corda

SUNY Polytechnic Institute, 13502 Utica, New York, USA; Istituto Livi, 59100 Prato, Prato, Italy; International Institute for Applicable Mathematics and Information Sciences, B. M. Birla Science Centre, Adarsh Nagar, Hyderabad 500063, India. E-mails: cordac.galilei@gmail.com; christian.corda@liviprato.edu.it

Abstract

The recent result that, contrary to a longstanding conviction older than 160 years, the precession of Mercury perihelion can be achieved in Newtonian gravity with a very high precision by correctly analyzing the situation without neglecting Mercury's mass is confirmed. Orbit's precession does not occur in an inertial (in Newtonian sense) frame of reference, but it occurs in the non-inertial frame of reference of the Sun because in Newtonian theory, the distance which is travelled by a body depends on the frame of reference in which the motion of the body is analyzed. After reviewing a previous solution, the precession is directly obtained from the orbit equation in the non-inertial frame of reference of the Sun. The Newtonian formula of the precession of planets' perihelion breaks down for the other planets because the predicted Newtonian result is indeed too large for Venus and Earth. In a previous paper, it was shown that corrections due to gravitational and rotational time dilation, in an intermediate framework (a particular post-Newtonian approximation of general relativity) which analyzes gravity between Newton and Einstein, solve the problem in accordance with some other results in the literature. By adding such corrections, a result consistent with the one of general relativity was indeed obtained. Those corrections also are re-analyzed in this paper by stressing the difference between general relativity and Newtonian physics by evoking the Einstein Equivalence Principle, and the third postulate of relativity. This framework of gravity between Newton and Einstein seems consistent with global covariance. Finally, it is important to stress that a better understanding of gravitational effects in an intermediate framework between Newtonian theory and general relativity, which is one of the goals of this paper, could, in principle, be crucial for a better understanding of the famous Dark Matter and Dark Energy problems.

Introduction

In 1609, Kepler's astronomical observations permitted us to understand that planets' orbits around the Sun are ellipses, the Sun being one of their foci. This result can be obtained via Newtonian universal gravitation. By considering a particular planet, as the other planets in the solar system also gravitationally interact with the planet in question, their presence generates a complication. By repeating the computation, if one takes into account such a complication, the result will be that the attraction due the other planets generates a precession, orbit after orbit, of the perihelion of the planet. A similar effect arises from the precession of the rotation axis of the Earth. In the case of Mercury, its perihelion advances of 5,600 arcseconds per century, in the direction of the planet's rotation around the Sun. In any case, if one removes the contribution due to the precession of the Earth, which is 5,025 arcseconds, and the contribution of the attraction of the other planets, being this latter computed via Newton's law of gravity, one does not correctly matches the observations, which show that there are 43 arcseconds which are missed. Centennial computations, realized in more than 160 years, generated the universally accepted conclusion that this anomalous rate of advance of the perihelion of the orbit of Mercury cannot be explained within a Newtonian framework. The French Astronomer Urbain Le Verrier in 1859 verbatim stated that "this is an important astronomical problem". In fact, Le Verrier started in 1843 to reanalyze various observations of the perihelion of the orbit of Mercury from 1697 to 1848, by ending that the advance of Mercury's orbit was not consistent with Newtonian gravity. The original discrepancy found by le Le Verrier was of 38′′ arcseconds per tropical century. Simon Newcomb, a Canadian-American astronomer, corrected to 43′′ arcseconds per tropical century the value of the discrepancy in 1882. Scientists proposed a lot of ad hoc and unsuccessful solutions which resulted only in introducing more additional problems. In the 19th century, scientists attempted to achieve the cited discrepancy by hypothesizing the presence of a perturbing planet, called Vulcan. It was assumed that Vulcan was a small planet and closer to the Sun than Mercury. In any case, Vulcan was never observed. Historically, in 1916 Einstein solved the problem via the general relativity theory. The MESSENGER data and the Cassini mission recently fixed the general relativistic value to 42, 98′′ per tropical century. By expressing the perihelion advance in radians per revolution (hereafter we will use polar coordinates), the general relativistic value reads

[math]\displaystyle{ \Delta_\varphi\simeq\frac{24\pi^3a^2}{T^2_0c^2(1-e^2)} = \frac{3\pi r_g}{a(1-e^2)}, }[/math]

(1)

being [math]\displaystyle{ a }[/math] the orbit's semi-major axis, [math]\displaystyle{ T_0 }[/math] the orbital period of Mercury in Newtonian absolute time, [math]\displaystyle{ c }[/math] the speed of light, [math]\displaystyle{ r_g }[/math] the Sun's gravitational radius and [math]\displaystyle{ e }[/math] the eccentricity of Mercury's orbit. The corresponding total angle which is swept by Mercury per revolution is

[math]\displaystyle{ \varphi\simeq\varphi_0\left(1+\frac{12\pi^2a^2}{T^2_0c^2(1-e^2)}\right), }[/math]

(2)

being [math]\displaystyle{ \varphi_0 = 2\pi }[/math] the unperturbed (that means in absence of advance) total angle which is swept by Mercury during its complete revolution around the Sun. If one inserts the numerical values in Eq. (1) like in, the well known value [math]\displaystyle{ \Delta_\varphi\simeq5.02 ∗ 10^7 }[/math] radians per revolution of general relativity is obtained. This value corresponds to about 0,1 arcseconds.

In, we calculated the advance of the perihelion of the orbit of Mercury by using Newtonian gravity. We considered 3 different approaches and the result has been that Mercury's orbit is consistent with Newton's theory with a very high precision by correctly analyzing the situation without neglecting Mercury's mass in the correct frame of reference of the Sun which is non-inertial. Einstein's gravity remains more precise than Newtonian gravity, but, actually, Newton's theory is more powerful than scientists were thinking till now, at least for the case of Mercury. On the contrary, the Newtonian formula of the precession of the perihelion of the other planets breaks down. In fact, the Newtonian value which has been found in resulted too large for Venus and Earth. The problem has been solved by introducing corrections due to gravitational and rotational time dilation. Such corrections permitted indeed to re-obtain the same value of the general relativity theory.

Thus, the two important results in are:

i) That Newtonian gravity cannot predict the anomalous rate of the advance of the perihelion of the orbits of the planets is an erroneous statement. The real problem with Newtonian theory is that the precession's value is too large;

ii) The advance of the planet's perihelion can be obtained with the same precision of the general theory of relativity if one extends Newtonian theory by including the effect of gravitational and rotational time dilation. Two recent interesting works are consistent with the latter results but, with respect to the importance of rotational time dilation surfaced in.

In this new paper the situation is re-analyzed in Newtonian physics. It will be indeed shown that, despite the orbit's precession does not occur in an inertial (in Newtonian sense) frame of reference, it occurs, instead, in the non-inertial frame of reference of the Sun and it is due to the well known issue that, in a Newtonian framework, the distance which is travelled by a body depends on the frame of reference in which the motion of the body is analyzed. This fundamental issue of Newtonian mechanics is analyzed in Section 2 of the paper. After reviewing the solution of the problem which analyzes the planet's orbit as a harmonic oscillator in Section 3, in Section 4 it will be shown that the precession can be directly obtained from the orbit equation in the non-inertial frame of reference of the Sun. In Section 5 it will be shown that, remarkably, the results presented in and in Sections 3 and 4 of this paper are not in contrast with more than 160 years of Newtonian physics and with the traditional textbooks of classical mechanics. The situation is indeed more subtle and depends on the issue that in traditional textbooks of classical mechanics planets orbits are always analyzed in inertial frames of reference, while the orbits' precession is present in the noninertial frame of reference of the Sun. Finally, in Section 6 the corrections due to gravitational and rotational time dilation, in an intermediate framework which concerns gravitational theory between Newton and Einstein, are re-analyzed by stressing the difference between general relativity and Newtonian physics by evoking the Einstein Equivalence Principle, and the third postulate of relativity. This framework of gravity between Newton and Einstein seems consistent with global covariance.

Distance travelled by a body in two different Newtonian frames of reference

Consider a highway where there is a rectilinear tract without curves for hundreds of kilometers and two cars, 1 going in one direction and 2 approaching in the opposite sense. Assume that both of them have a speed of 40km/h with respect to a fixed observer on the edge of the highway and that their initial distance is longer than 80 km. Compared to an observer in car 1 (a passenger in car 1), car 2 moves at a speed of 80 km/h. Now, if one asks how many kilometers car 2 will travel in an hour with respect to the observer fixed on the edge of the highway, the answer will be 40 km, but if one asks how many kilometers car 2 will travel in an hour with respect to the observer in car 1, the answer will be 80 km. One cannot tell that the 80 km answer is wrong and the 40 km answer is right, because in Newtonian physics there is neither preferential observer, nor preferential frame of reference. Both answers are correct. It will be shown that a similar issue works also in the planet's orbit problem. If viewed with respect to the non-inertial frame of reference of the Sun there is precession, if viewed with respect to an inertial frame of reference there is no precession. The reason for this is similar to the example of the two cars on the highway. Compared to the non-inertial frame of reference of the Sun, the planet moves faster than the inertial frame of reference. In Newtonian physics the distance travelled by a body in a particular time interval, which is absolute for all the observers, is different for the various observers and frames of reference.

Planet's orbit as a harmonic oscillator: a review

The fixed stars are defined as those astrophysical objects which appear not to move relative to one another in the night sky. One recalls that in Newtonian physics the fixed stars are considered as a frame of reference supposedly at rest relative to absolute space. In other frames of reference, either at rest with respect to the fixed stars or in uniform translation relative to them (inertial frames of reference in Newtonian sense), Newtonian laws of motion must hold. In contrast, in frames of reference accelerating with respect to the fixed stars, including frames rotating relative to them, the Newtonian laws of motion did not hold in their simplest form, but must be supplemented by the addition of inertial forces. As a first approximation, one takes the origin of the frame of reference in the center of the Sun by considering this frame of reference fixed with respect to the frame of reference of the fixed stars. In the following G will be the Newtonian gravitational constant, [math]\displaystyle{ M }[/math] the solar mass, [math]\displaystyle{ m }[/math] the mass of the planet and [math]\displaystyle{ r }[/math] the distance between the Sun and planet. Following, one stresses that central attractive forces can produce approximate circular orbits that could not be closed. They are closed when the radial period of oscillation is a rational multiple of the period of the orbit. Hereafter the subscript [math]\displaystyle{ F }[/math] will be used for the quantities in the frame of reference of the fixed stars while the subscript [math]\displaystyle{ S }[/math] will be used for the quantities in the non-inertial frame of reference of the Sun. Thus, one labels [math]\displaystyle{ F_{cF}(r) }[/math] as being the total central force. One writes down the equation of motion for the planet as

[math]\displaystyle{ F_{cF}(r)=m(\ddot{r}-w^2_Fr), }[/math]

(3)

The physical interpretation of the last term in Eq. (3) is in terms of a centrifugal force. Being the angular momentum [math]\displaystyle{ J_F }[/math] a constant of motion, one can write

[math]\displaystyle{ J_F=mr^2w_F. }[/math]

(4)

If one solves for [math]\displaystyle{ ω_F }[/math] and replaces it in Eq. (3), one obtains

[math]\displaystyle{ F_{cF}(r)=m\left(\ddot{r} - \frac{J^2_F}{m^2r^3}\right). }[/math]

(5)

Considering a circular orbit having radius [math]\displaystyle{ r_0 }[/math], one gets [math]\displaystyle{ \ddot r }[/math] = 0 while Eq. (5) becomes

[math]\displaystyle{ F_{cF}(r_0)= - \frac{J^2_F}{mr^3_0}. }[/math]

(6)

Now, let us perturb the planet in the plane of the orbit and perpendicularly to the initial trajectory. Then, one finds oscillations around [math]\displaystyle{ r_0 }[/math]. Hence, one can introduce [math]\displaystyle{ x = r − r_0 }[/math], while the radial equation of motion can be expressed in terms of [math]\displaystyle{ x }[/math]. Thus,

[math]\displaystyle{ F_{cF}(x + r_0)= m \ddot{x} - \frac{J^2_F}{m(x+r_0)^3} }[/math]

[math]\displaystyle{ = m \ddot{x} - \frac{J^2_F}{mr^3_0 \left(1 + \frac{x}{r_0} \right)^3}. }[/math]

(7)

As it is [math]\displaystyle{ \frac{x}{r_0}\lt \lt 1 }[/math], series expansion for the term in parentheses can be used and one considers only the first order terms in [math]\displaystyle{ \frac{x}{r_0} }[/math]. If one expands the member on the left in Taylor series around the point [math]\displaystyle{ r = r_0 }[/math] one finds

[math]\displaystyle{ F_{cF}(r_0) + F'_{cF}(r_0)x = m \ddot{x} - \frac{J^2_F}{\mu r^3_0}\left(1-\frac{3x}{r_0} \right). }[/math]

(8)

In Eq. (8) prime refers to the derivative with respect to [math]\displaystyle{ x }[/math]. If one inserts Eq. (6) in Eq. (8) one gets

[math]\displaystyle{ \ddot{x} + m^{-1} \left[-\frac{3F_{cF}(r_0)}{r_0} - F'_{cF}(r_0) \right]x=0. }[/math]

(9)

This equation represents a standard harmonic oscillator when the term in the square bracket is positive. Being such a term negative, one should find an exponential solution which makes the orbit not being stable. Hence, in stable orbits the oscillation's period around [math]\displaystyle{ r = r_0 }[/math] must equalize the corresponding period of circular motion

[math]\displaystyle{ T_F = 2\pi \left(\frac{m}{-\frac{3F_{c0}(r_0)}{r_0}-F'_{cF}(r_0)} \right)^{\frac{1}{2}}. }[/math]

(10)

As the Sun is considered fixed with respect to the fixed stars, here [math]\displaystyle{ T_F }[/math] represents the so called sidereal year, that is the time taken by the planet to orbit the Sun once with respect to the fixed stars. One recalls that the apse angle [math]\displaystyle{ \frac{\varphi_F}{2} }[/math] is the angle which is swept by the radial vector between two consecutive extremal orbital points. The period of time taken by the planet in order to travel this angle is [math]\displaystyle{ \frac{T_F}{2} }[/math]. One considers the orbit as being approximately circular. Thus, [math]\displaystyle{ r }[/math] can be set as being constant and equal to [math]\displaystyle{ r_0 }[/math]. Then, by solving Eq. (4) for [math]\displaystyle{ ω_F }[/math] and one gets

[math]\displaystyle{ \frac{\varphi_F}{2} = \frac{T_F}{2}w_F = \pi \left(\frac{m}{-\frac{3F_{cF}(r_0)}{r_0}-F'_{cF}(r_0)} \right)^{\frac{1}{2}} \frac{J_F}{\mu r^2_0}. }[/math]

(11)

In addition, if one observes Eq. (6), one rewrites the last factor of Eq. (11) as

[math]\displaystyle{ \frac{J_F}{mr^2_0} = \left(-\frac{F_{cF}(r_0)}{\mu r_0} \right)^\frac{1}{2}. }[/math]

(12)

Thus, it is

[math]\displaystyle{ \varphi_F = 2\pi \left[3 + \frac{F'_{cF}(r_0)}{F_{cF}(r_0)}r_0 \right]^{-\frac{1}{2}}, }[/math]

(13)

and, putting [math]\displaystyle{ F_{cF} = F_G }[/math] in Eq. (13), being [math]\displaystyle{ F_G }[/math] the traditional Newtonian gravitational force given by ([math]\displaystyle{ \hat{u}_r }[/math] is the versor in the radial direction)

[math]\displaystyle{ \overrightarrow{F}_G = - \frac{GMm}{r_0^2}\hat{u}_r, }[/math]

(14)

one obtains [math]\displaystyle{ \varphi_F = 2\pi }[/math]. Now, one recalls that the above analysis has been realized in the inertial frame of reference of the fixed stars. But we set the origin of the frame of reference in the center of the Sun and the motion of the Sun with respect to the planet is not inertial, because the Sun is subjected to the planet's back reaction due to Newton's third law. Thus, the Sun cannot really be considered as fixed with respect to the fixed stars and therefore the calculation previously made must be considered an approximation. In an external inertial reference frame, the Newtonian equations of motion for the Sun and the planet read

[math]\displaystyle{ Ma_{Sun}\hat{u}_r = \frac{GMm}{r^2_0}\hat{u}_r = a_{Sun}\hat{u}_r = \frac{Gm}{r^2_0}\hat{u}_r }[/math]

(15)

and

[math]\displaystyle{ ma_{Planet}\hat{u}_r = - \frac{GMm}{r^2_0}\hat{u}_r \Rightarrow a_{Planet}\hat{u}_r = \frac{Gm}{r^2_0}\hat{u}_r, }[/math]

(16)

respectively, being [math]\displaystyle{ a_{Sun} }[/math] the acceleration of the Sun and [math]\displaystyle{ a_{Planet} }[/math] the acceleration of the planet. Then, the equation of the relative acceleration between the planet and the Sun has to be written as

[math]\displaystyle{ a\hat{u}_r = a_{Planet}\hat{u}_r -a_{Sun}\hat{u}_r = - \left( \frac{GM}{r^2_0} + \frac{Gm}{r^2_0} \right) \hat{u}_r = - \frac{G(M + m)}{r^2_0}\hat{u}_r. }[/math]

(17)

Consequently, the equation of motion of the planet becomes

[math]\displaystyle{ \overrightarrow{F}_{M\to{m}} = - \frac{G(M+m)m}{r_0^2}\hat{u}_r. }[/math]

(18)

Thus, one considers the weak force

[math]\displaystyle{ - \frac{Gmm}{r_0^2}\hat{u}_r, }[/math]

(19)

which is due to the the non-inertial behavior of the Sun as being a perturbation with respect to the central force (14). Following, if one wants to take into account this perturbation, the following replacements in Eqs. from (3) to (10) have to be made:

[math]\displaystyle{ F_{cF}(r) \to{F_{cS}(r)} = \left(1 + \frac{m}{M} \right) F_{cF}(r). }[/math]

(20)

Now, [math]\displaystyle{ F_{cS}(r) }[/math] results to be the force defined in Eq. (18), and

[math]\displaystyle{ w_F \to{w_S}, }[/math]

(21)

where [math]\displaystyle{ ω_S }[/math] is the corresponding perturbed angular velocity, and

[math]\displaystyle{ J_F \to{J_S} = mr^2w_S. }[/math]

(22)

In particular, Eq. (10) must be replaced by

[math]\displaystyle{ T_S = 2\pi \left(\frac{m}{-\frac{3F_{cS}(r_0)}{r_0} - F'_{cS}(r_0)} \right)^{\frac{1}{2}} }[/math]

[math]\displaystyle{ = 2\pi \left( \frac{m}{3+\frac{F'_{cS}(r_0)r_0}{F_{cS}(r_0)}} \right)^{\frac{1}{2}} \left( \frac{1}{-F_{cS}(r_0) / r_0} \right)^{\frac{1}{2}}. }[/math]

(23)

As it is

[math]\displaystyle{ F'_{cS}(r_0) = \left(1 + \frac{m}{M} \right)F'_{cF}(r_0), }[/math]

(24)

then, by using (20), one obtains

[math]\displaystyle{ \frac{F'_{cS}(r_0)}{F_{cS}(r_0)} = \frac{F'_{cF}(r_0)}{F_{cF}(r_0)}. }[/math]

(25)

Then, Eq, (23) becomes

[math]\displaystyle{ T_S= 2\pi \left( \frac{m}{3+\frac{F'_{cS}(r_0)r_0}{F_{cS}(r_0)}} \right)^{\frac{1}{2}} \left( \frac{1}{-F_{cS}(r_0) / r_0} \right)^{\frac{1}{2}} \frac{1}{(1+\frac{m}{M})^{\frac{1}{2}}} }[/math]

[math]\displaystyle{ 2\pi \left(1 + \frac{m}{M} \right)^{-\frac{1}{2}} \left(\frac{m}{-\frac{3F_{cF}(r_0)}{r_0} - F'_{cF}(r_0)} \right)^{\frac{1}{2}}. }[/math]

(26)

Thus, if one confronts Eqs. (26) with (10), the result is

[math]\displaystyle{ T_S = \frac{1}{\left(1 + \frac{m}{M} \right)^{\frac{1}{2}}}T_F. }[/math]

(27)

Eq. (27) gives

[math]\displaystyle{ w_S = \frac{2\pi}{T_S} = \frac{2\pi}{T_F} \left(1 + \frac{m}{M} \right)^{\frac{1}{2}} = w_F \left(1 + \frac{m}{M} \right)^{\frac{1}{2}} }[/math]

(28)

and

[math]\displaystyle{ \varphi_S = w_ST_F = 2\pi\left(1 + \frac{m}{M} \right)^{\frac{1}{2}} \simeq 2\pi \left(1 + \frac{m}{2M} \right), }[/math]

(29)

in radians per revolution. In the last step the first-order approximation in [math]\displaystyle{ \frac{m}{M} }[/math] has been used, i.e. [math]\displaystyle{ \left(1 + \frac{m}{M} \right)^{\frac{1}{2}} \simeq 1 + \frac{m}{2M} }[/math]. In fact, it is m ≪ M. Hence, in a complete revolution around the Sun, the planet sweeps an angle larger than the unperturbed angle [math]\displaystyle{ \varphi_F = 2\pi }[/math]. The difference, in radians per revolution, is

[math]\displaystyle{ \Delta\varphi = \varphi_S - \varphi_F = \simeq \frac{\pi m}{M}. }[/math]

(30)

Thus, the precession of the planet's orbit occurs in the non inertial reference frame of the Sun, because in the non-inertial frame of reference of the Sun the planet moves faster compared to the approximation in which the reference system of the Sun is considered inertial. Therefore, since time is absolute in both reference frames, it follows that, in the same time interval, the planet advances more in the reference frame in which it moves faster.

Precession from the orbit equation

Again, the key point is that, differently from what happens in an inertial reference frame, in the non-inertial reference frame of the Sun the planet and the Sun do not interact only by the Newtonian central force of Eq. (14), which is

[math]\displaystyle{ \overrightarrow{F}_{G} = - \frac{GMm}{r^2}\hat{u}_r. }[/math]

(31)

Instead, one must also consider the additional inertial force of Eq. (19) which is

[math]\displaystyle{ -\frac{Gmm}{r^2}\hat{u}_r. }[/math]

(32)

Then, the total force acting on the planet is given by the force of Eq. (18), which is

[math]\displaystyle{ \overrightarrow{F}_{M\to{m}} = - \frac{GMm}{r^2}\hat{u}_r - \frac{Gmm}{r^2}\hat{u}_r. }[/math]

(33)

This is the well known case in which an additional central force is present in the interaction between the two bodies, but, to our knowledge, till now nobody argued that this argument can be used in order to find planets' precession in Newtonian theory. Thus, in the non-inertial reference frame of the Sun, the standard orbit of Kepler's problem is slightly perturbed by the weak inertial force of Eq. (32). Following, the orbit equation for the central motion governed by the sole Newtonian central force of Eq. (31) (the Keplerian unperturbed motion) is

[math]\displaystyle{ \frac{d^2u}{d\varphi^2} + u = - \frac{GMm^2}{J^2_F}, }[/math]

(34)

where [math]\displaystyle{ r = 1/u, \varphi }[/math] are cylindrical coordinates in the orbital plane and [math]\displaystyle{ J_F }[/math] is the angular momentum of the orbiting planet defined by Eq. (4). The solution of Eq. (34) is

[math]\displaystyle{ u_F(\varphi) = - \frac{GMm^2}{J^2_F}\left[1 + e\cos(\varphi - \varphi_0) \right], }[/math]

(35)

where the eccentricity [math]\displaystyle{ e }[/math] and the phase offset [math]\displaystyle{ \varphi_0 }[/math] are constants of integration. But in the non-inertial frame of the Sun the Newtonian force of Eq. (31) is not the sole force acting on the planet. One has also to consider the additional inertial force of Eq. (32) which perturbs the orbit. In that way, the total force acting on the planet is given by the force of Eq. (33). Then, the perturbed orbit equation for the motion in the non-inertial reference frame of the Sun governed by the total force of Eq. (33) is

[math]\displaystyle{ \frac{d^2u}{d\varphi^2} + u = - \frac{G (M + m)m^2}{J^2_F}. }[/math]

(36)

The solution of Eq. (36) is

[math]\displaystyle{ u_S(\varphi) = - \frac{G(M+m)m^2}{J^2_F}\left[1 + e \cos(\varphi - \varphi_0) \right] = \left(1 + \frac{m}{M}\right)u_F(\varphi). }[/math]

(37)

By confronting Eq. (35) with Eq. (37) one argues that the additional inertial force changes the angular momentum of the orbiting planet as

[math]\displaystyle{ J_F \to J_S = \sqrt{1 + \frac{m}{M}}J_F }[/math]

Thus, Eq. (37) can be rewritten as

[math]\displaystyle{ u_S(\varphi) = - \frac{GMm^2}{J^2_S}\left[1 + e \cos(\varphi - \varphi_0) \right]. }[/math]

(38)

Thus, one writes

[math]\displaystyle{ J_S = mr^2\frac{d\varphi}{dt}. }[/math]

(39)

One can rearrange Eq. (39) as

[math]\displaystyle{ d\varphi = \frac{J_S}{mr^2}dt, }[/math]

(40)

and, in turn, the total angle swept by the planet during a complete orbit results

[math]\displaystyle{ \varphi_S = \int_0^{T_F}{\frac{J_S}{mr^2}}dt = \sqrt{1 + \frac{m}{M}} \int_0^{T_F} \frac{J_F}{mr^2}dt. }[/math]

(41)

On the other hand, we know that it is

[math]\displaystyle{ \int_0^{T_F} \frac{J_F}{mr^2}dt = \varphi_F = 2 \pi }[/math]

because there is no precession of the unperturbed orbit. Thus, one gets

[math]\displaystyle{ \varphi_S = \int_0^{T_F}{\frac{J_S}{mr^2}}dt = \sqrt{1+\frac{m}{M}}\varphi_F = 2\pi\sqrt{1+\frac{m}{M}} \simeq 2\pi \left(1 + \frac{m}{2M} \right), }[/math]

(42)

which is the same result of Eq. (29). Again, in the last step of Eq. (42) the first-order approximation in m M has been used. Therefore, the analysis in this Section confirms the result in previous Section that in each complete revolution around the Sun, the planet sweeps an angle larger than the unperturbed angle [math]\displaystyle{ \varphi_F = 2\pi }[/math], and the difference, in radians per revolution, is

[math]\displaystyle{ \Delta\varphi = \varphi_S - \varphi_F \simeq \frac{\pi m}{M}, }[/math]

(43)

which is exactly Eq. (30).

Discussion on Newtonian physics

Let us ask: are the results presented in and in Sections 3 and 4 of this new paper in contrast with more than 160 years of Newtonian physics? Consequently, must we claim that the text books of classical mechanics, where it is stressed that there is no precession of planets' orbits in Newtonian physics, are wrong? Remarkably, the answer to both questions is no! The situation is indeed more subtle. The key point is that in more than 160 years of Newtonian physics and in the textbooks of classical mechanics, see for example, the two body problem has been always correctly solved in inertial reference frames. This issue is indeed stressed also in the interesting paper, verbatim: "It is surprising that a complete study about this problem (Kepler's problem in non-inertial frames) does not appear in Theoretical Mechanics text books, even if its applications are very important in theoretical and celestial mechanics".

In particular, the approximation in which the reference frame of the Sun is considered inertial is often used. But this is, in fact, an approximation, which makes one miss the existence of precession. It is, in turn, correct to stress that there is no precession of planets' orbits in the inertial reference frames because the results presented in and in Sections 3 and 4 of this new paper have shown that the precession is present in the non-inertial reference frame of the Sun. As it has been clarified in Section 2 of this paper, in Newtonian physics the distance travelled by a body in a particular time interval, which is absolute for all the observers, is different for the various observers and frames of reference. This works also for a planet travelling around the Sun. Remarkably, the Authors of found a rotation of the major axis of the planetary ellipse in a non-inertial reference frame. They used indeed very powerful and refined mathematics, but they did not realize the physical consequence of their results, that is, the presence of the precession of planets' orbits in the non-inertial frame of reference of the Sun.

From Newton to Einstein

Breakdown of the Newtonian formula for the other planets

Let us apply Eq. (30) to Mercury. The NASA official data are [math]\displaystyle{ m \simeq 3.3∗10^{23}Kg }[/math] and [math]\displaystyle{ M = 1,99 ∗ 10^{30}Kg }[/math]. Thus, one finds [math]\displaystyle{ \Delta\varphi \simeq 5.21 ∗ 10^{−7} }[/math] radians per revolution. This corresponds to about 0,107 arcseconds. The NASA official data also give the Mercury/Earth ratio of the tropical orbit periods as being 0.241. Hence, one finds a value of 44.39′′ per tropical century. This represents a remarkable result showing that the correct value of the contribution of Newtonian gravity to the advance of the perihelion of Mercury per tropical century well approximates both the observational value of 43′′ per tropical century and the value of about 42,98′′ per tropical century of the general theory of relativity.

Let us apply Eq. (30) to the trajectory of Venus. The planet's mass is [math]\displaystyle{ m_V \simeq 4.87 ∗ 10^{24}Kg }[/math] in the NASA official data. This gives a value of [math]\displaystyle{ \Delta\varphi \simeq 7.68 ∗ 10^{−6} }[/math] radians per revolution corresponding to about 1.6 arcseconds. The Venus /Earth ratio of the tropical orbit periods results to be 0, 615. Then, one finds 258,16′′ per tropical century. This results 30 times larger than the value of 8.62′′ which results from the observations. If one considers Earth's data one finds analogous results. The Earth's mass is [math]\displaystyle{ m \simeq 5.97*10^{24}Kg }[/math]. This gives a value of [math]\displaystyle{ \Delta\varphi \simeq 9.42 ∗ 10^{−6} }[/math] radians per revolution for the precession, corresponding to about 1.94 arcseconds. The value becomes 194′′ per tropical century, being about 50 times larger than the value of 3.83′′ which results from the observations.

Thus, one confirms the result in that, differently from a longstanding conviction which is older than 160 years, the real problem of the Newtonian framework which concerns the anomalous rate of the precession of the perihelion of planets orbit is not the absence of a result. Instead, the real problem with Newtonian gravity is that such a result is too large.

A fundamental issue is that differently from a relativistic framework, time is absolute in a Newtonian framework. This means that time passes in the same way in one reference frame as in the other reference frame. In it has been shown that, in the framework of the anomalous rate of the advance of the perihelion of planets orbit as well as in the other two classical tests of the general theory of relativity, which are deflection of light and gravitational redshift, one must take into due account gravitational time dilation effects. Improving the approach in, in this Section a different analysis will be realized, in an intermediate framework where gravity between Newton and Einstein is analyzed, by also clarifying the differences between general relativity and Newtonian theory. It will result that Newtonian theory can be extended if one takes into account both gravitational and rotational time dilation. This will permit us to obtain, with a very high precision, the precession of the perihelion of planets' orbits in a way completely consistent with the result of general relativity. For the sake of simplicity, this approach will be performed in the approximation of circular orbit.

Einstein Equivalence Principle and third postulate of relativity

One starts by recalling an important issue. What does it happen if one extends the non-inertial (in Newtonian sense) frame of reference of the Sun to a general relativistic framework? In that case, the frame of reference of the Sun becomes locally inertial in Einsteinian sense! In fact, one recalls that in general relativity gravity is not a force, but, because of Einstein Equivalence Principle, it is inertia in a curved space-time instead. Thus, the Newtonian approach of "the gravitational force of the planet acting to the Sun" is replaced by the Einsteinian approach of "the Sun is free falling in the gravitational field of the planet". Hence, the frame of reference of the Sun becomes a free falling frame of reference locally equivalent to an inertial frame of reference which is far from every source of the gravitational field and, in turn, it becomes locally equivalent to the frame of reference of the fixed stars. Now, one can also evoke the third postulate of relativity which states that, if one considers a physical system (the Sun in the current case) moving through a locally flat space-time, than there is at any moment an inertial system such that in it, the system is at rest. In that case, at any instant, the coordinates and state of the system can be Lorentz transformed to the other system through some Lorentz transformation. This means that if one considers the free falling frame of reference of the Sun, where the Sun is at rest, the coordinates and state of the system can be Lorentz transformed to the frame of reference of the fixed stars, which moves with respect to the free falling frame of reference of the Sun. Relative to the frame of reference of the fixed stars, the instantaneous velocity of the Sun is [math]\displaystyle{ \overrightarrow{v}(t_F) }[/math] with magnitude [math]\displaystyle{ |\overrightarrow{v}| = v }[/math] bounded by the speed of light [math]\displaystyle{ c }[/math], so that [math]\displaystyle{ 0 ≤ v \lt c }[/math]. Here the time [math]\displaystyle{ t_F }[/math] is the time as measured in the frame of reference of the fixed stars, not the time measured in the free falling frame of reference of the Sun. The free falling frame of reference of the Sun and the frame of reference of the fixed stars are Lorentz connected via the instantaneous Lorentz factor

[math]\displaystyle{ \gamma \equiv \frac{1}{\sqrt{1 - \frac{v^2}{c^2} }}. }[/math]

(44)

Then, one gets via the definition of the instantaneous Lorentz factor

[math]\displaystyle{ \frac{dt_F}{dt_S} = \gamma, }[/math]

(45)

where [math]\displaystyle{ t_S }[/math] is the time measured in the free falling frame of reference of the Sun. Hence, there is a time dilation between the two frames of references given by

[math]\displaystyle{ dt_F = \gamma dt_S, }[/math]

(46)

which means that the time between two ticks as measured in the frame in which the clock is moving (the frame of reference of the fixed stars), is longer than the time between these ticks as measured in the rest frame of the clock (the free falling frame of reference of the Sun).

In order to compute γ one must compute [math]\displaystyle{ v }[/math]. As we are considering circular orbits and the force acting on the planet in the two frames of reference is central, the velocity between the two frames of reference is completely transversal and it is exactly the difference between the velocity of the planet calculated in the free falling frame of reference of the Sun and the velocity of the planet calculated in the frame of reference of the fixed stars. As it is [math]\displaystyle{ v ≪ c }[/math], where [math]\displaystyle{ c }[/math] is the speed of light, the velocities of the planet in the two frames of reference can be calculated with very high precision in the Newtonian framework. For a circular orbit in the frame of reference of the fixed stars one merely equals the Newtonian gravitational force to the centripetal one as

[math]\displaystyle{ \frac{GMm}{r^2_0} = \frac{mv^2_F}{r_0}, }[/math]

(47)

where [math]\displaystyle{ v_F }[/math] the velocity of rotation of the planet in the frame of reference of the fixed stars and [math]\displaystyle{ r_0 }[/math] the radius of the circular orbit. Hence, [math]\displaystyle{ v_F }[/math] is easily obtained as

[math]\displaystyle{ v_F = \left(\frac{GM}{r_0} \right)^{\frac{1}{2}}. }[/math]

(48)

In the free falling frame of reference of the Sun one must equal the centripetal force with the total force of Eq. (33) as

[math]\displaystyle{ \frac{G(M + m)m}{r_0^2} = \frac{mv^2_S}{r_0}, }[/math]

(49)

where [math]\displaystyle{ v_S }[/math] is the velocity of rotation of the planet in the free falling frame of reference of the Sun. Then one gets

[math]\displaystyle{ v_S = \left[\frac{G(M+m)}{r_0} \right]^\frac{1}{2} = v_F\sqrt{1 + \frac{m}{M}}. }[/math]

(50)

Thus,

[math]\displaystyle{ v = v_S - v_F \simeq \frac{mv_F}{2M}, }[/math]

(51)

where in the last step of Eq. (51) the first-order approximation in [math]\displaystyle{ \frac{m}{M} }[/math] has been used. Eq. (51) implies

[math]\displaystyle{ \gamma = \frac{1}{\sqrt{1 - \frac{\left(\frac{mv_F}{2M} \right)^{2}}{c^2}}} \simeq 1 + \left(\frac{mv_F}{2cM} \right)^2, }[/math]

(52)

where in the last step of Eq. (52) the first order approximation in [math]\displaystyle{ \left(\frac{v_F}{c}\right)^2 }[/math] has been used. Now, we will extend the Newtonian framework by considering corrections due to time dilation.

Time dilation for the first corrected Newtonian observer

One starts from the free falling frame of reference of the Sun. Then, it will be shown that the same result will be obtained in the frame of reference of the fixed stars. From Eq. (50) for a circular orbit, in the free falling frame of reference of the Sun one gets a period

[math]\displaystyle{ T_S = \frac{2\pi r_0}{v_S} = \frac{2\pi r_0^{\frac{3}{2}}}{[G(M + m)]^{\frac{1}{2}}}. }[/math]

(53)

The period [math]\displaystyle{ T_S }[/math] in Eq. (53) is measured by a Newtonian observer located in the free falling frame of reference of the Sun who does not consider gravitational time dilation. But the period that is measured by a general relativistic observer is different. In general relativity, gravitational time dilation is well approximated by

[math]\displaystyle{ t_g = \sqrt{g_{00}(r_0)t_l}. }[/math]

(54)

In Eq. (54) [math]\displaystyle{ g_{00} }[/math] represents the coefficient of the coordinate time in the line element which describes the gravitational field, [math]\displaystyle{ t_g }[/math] represents the proper time between two events for a fixed observer who is located at distance [math]\displaystyle{ r_0 }[/math] from the source of the gravitational field (local observer) and [math]\displaystyle{ t_l }[/math] represents the coordinate time between the events for an observer at a very large distance from the source of the gravitational field. Following, in a weak field approximation an approximate solution of Einstein's field equations can be written via isotropic coordinates in the free falling frame of reference of the Sun as

[math]\displaystyle{ ds^2 = \left(1 - \frac{2GM}{rc^2} \right) (cdt_S)^2 - \left(1 + \frac{2GM}{rc^2} \right) (dr^2 + r^2d \theta^2 + r^2sin^2\theta d\phi^2 ), }[/math]

(55)

being [math]\displaystyle{ r,\theta, \phi }[/math] spherical polar coordinates. By using the line element (55), Eq. (54) for the Sun's gravitational field reads

[math]\displaystyle{ t_g = \sqrt{1 - \frac{r_g}{r_0}t_l} \simeq \left(1 - \frac{1}{2}\frac{r_g}{r_0} \right), }[/math]

(56)

being [math]\displaystyle{ r_g = \frac{2M}{c^2} c^2 }[/math] the Sun's gravitational radius. The variations due to time dilation are considered as being corrections with respect to the Newtonian observer. Thus, one is defining a new observer who has again the origin of the frame of reference in the center of the Sun and sees the spatial directions as being a Newtonian observer, but measures the time between two events by using [math]\displaystyle{ t_g }[/math] of Eq. (56) rather than the absolute Newtonian time in the free falling frame of reference of the Sun, which is [math]\displaystyle{ t_l = t_S }[/math]. This new observer can be defined as "Corrected Newtonian Observer" (CNO). Together with this time dilation, one has also to consider a variation of the proper radial distance between the Sun and the planet with respect to the Newtonian observer. If the Newtonian observer measures a distance, say [math]\displaystyle{ r_0 = ct_{S0} }[/math], the CNO will measure a distance [math]\displaystyle{ ct_{S0}\sqrt{1 - \frac{r_g}{r_0}} \simeq r_0 \left(1 - \frac{1}{2}\frac{r_g}{r_0} \right) }[/math]. Here one is not considering the correction to the proper radial distance between the Sun and the planet which arises from spatial curvature, being given by the opposite of the coefficient [math]\displaystyle{ g_{11} = - \left(1 + \frac{2GM}{rc^2} \right) }[/math] in the line element of Eq. (55). In fact, one assumes that the CNO must see the spatial directions as being a pure Newtonian observer. In other words, the spatial directions are considered Euclidean. Hence, the following replacements have to be realized in Eq. (53)

[math]\displaystyle{ r_0 \to r_0 \left(1 - \frac{1}{2} \frac{r_g}{r_0} \right) \qquad T_S \to T_S \left(1 - \frac{1}{2} \frac{r_g}{r_0} \right). }[/math]

(57)

This means that the CNO replaces in Eq. (53) the original time and distances with time and distances corrected by Eq. (57). Then, the CNO sees Eq. (53) as becoming

[math]\displaystyle{ T_{S1} = \left(1 - \frac{1}{2}\frac{r_g}{r_0} \right)^{\frac{3}{2}} \left(1 - \frac{1}{2} \frac{r_g}{r_0} \right) T_S, }[/math]

(58)

being [math]\displaystyle{ T_{S1} }[/math] the perturbed orbital period measured by the CNO and [math]\displaystyle{ T_S }[/math] the unperturbed [unperturbed with respect to the corrections of general relativity in Eq. (57)] orbital period given by Eq. (53). Consequently, the CNO measures a corresponding perturbed angular velocity

[math]\displaystyle{ w_{S1} = \frac{2\pi}{T_{S1}} \simeq w_S\left(1 - \frac{1}{2} \frac{r_g}{r_0} \right)^{-\frac{3}{2}} \left(1 + \frac{1}{2} \frac{r_g}{r_0} \right), }[/math]

(59)

where [math]\displaystyle{ ω_S }[/math] is given by

[math]\displaystyle{ w_S = \frac{2\pi}{T_S}. }[/math]

(60)

Thus, [math]\displaystyle{ ω_S }[/math] is the angular velocity of the rotating planet in the free falling frame of reference of the Sun. Hence, the final angle that Mercury sweeps during the period [math]\displaystyle{ T_S }[/math] in the free falling frame of reference of the Sun is

[math]\displaystyle{ \varphi_{S1} = w_{S1}T_S \simeq 2\pi \left(1 - \frac{1}{2}\frac{r_g}{r_0} \right)^{-\frac{3}{2}} \left(1 + \frac{1}{2} \frac{r_g}{r_0} \right) \simeq 2\pi \left(1 + \frac{5}{4}\frac{r_g}{r_0} \right) }[/math]

(61)

in radians per revolution. In the above computations the first-order approximation in [math]\displaystyle{ \frac{r_g}{r_0} }[/math] has been used. Thus, one finally finds

[math]\displaystyle{ \Delta_{\varphi S1} \simeq \frac{5\pi}{2} \frac{r_g}{r_0}. }[/math]

(62)

One has to consider also a rotational effect. Another dilation effect is indeed due to rotation, see for example. The presence of an additional effect of rotational dilation is explained if one notes that time differences in Eq. (58) have been calculated by the CNO who considers the planet as being at rest. But the planet is moving instead. Hence, the CNO has to take into account an additional effect due to rotational time dilation. This kind of effect has a very longstanding history which started from a famous paper of Einstein. It is indeed a historical issue, which happened during his analysis of the rotating frame, that Einstein had the illumination to represent the gravitational field in terms of space-time curvature, verbatim:

The following important argument also speaks in favor of a more relativistic interpretation. The centrifugal force which acts under given conditions of a body is determined precisely by the same natural constant that also gives its action in a gravitational field. In fact we have no means to distinguish a centrifugal field from a gravitational field. We thus always measure as the weight of the body on the surface of the earth the superposed action of both fields, named above, and we cannot separate their actions. In this manner the point of view to interpret the rotating system K' as at rest, and the centrifugal field as a gravitational field, gains justification by all means. This interpretation is reminiscent of the original (more special) relativity where the pondermotively acting force, upon an electrically charged mass which moves in a magnetic field, is the action of the electric field which is found at the location of the mass as seen by the reference system at rest with the moving mass.

Einstein's interpretation of rotation in terms of gravity enabled various general relativistic analyses of Mössbauer rotor experiments and Sagnac experiments. The key point of Einstein's interpretation is the Einstein's Equivalence Principle (EEP) which states the equivalence between inertial forces and gravity. In the general relativistic framework, one uses a transformation from an inertial coordinate system, with the z − axis perpendicular to the plane of the rotational motion, to a second coordinate system rotating around the [math]\displaystyle{ z − axis }[/math] in cylindrical coordinates. For a flat Lorentzian coordinate system the metric is

[math]\displaystyle{ ds^2 = c^2 dt^2_S - dr^2 - r^2d\phi^2 - dz^2. }[/math]

(63)

A transformation to a reference frame {[math]\displaystyle{ t′_S, r′, ϕ′z′ }[/math]} having constant angular velocity [math]\displaystyle{ ω_S }[/math] around the [math]\displaystyle{ z − axis }[/math] is given by

[math]\displaystyle{ t_S = t'_S \qquad r = r' \qquad \phi = \phi' + wt'_S \qquad z = z'. }[/math]

(64)

The result is the well known Langevin line-element for a rotating observer

[math]\displaystyle{ ds^2 = \left(1 - \frac{r'^2w^2}{c^2} \right)c^2dt'^2_S - 2wr'^2d\phi'dt'_S - dr'^2 - r'^2d\phi'^2 - dz'^2. }[/math]

(65)

The EEP enables the interpretation of the line element (65) in terms of a stationary "gravitational field", in accordance with the original intuition of Einstein. As a consequence, the inertial force that a rotating observer experiences is interpreted as if the rotating observer is subjected to a gravitational "force".

Thus, if one applies Eq. (54) to the Langevin line-element of Eq. (65) one gets,

[math]\displaystyle{ d\tau = \sqrt{\left(1 - \frac{r^2w^2_S}{c^2} \right)dt_S} \simeq \left(1 - \frac{1}{2}\frac{r^2w^2_S}{c^2} \right) dt_S, }[/math]

(66)

being, in the rotating frame, τ the proper time between two events for the rotating observer at a distance [math]\displaystyle{ r }[/math] from the origin and having angular velocity [math]\displaystyle{ ω_S }[/math], and [math]\displaystyle{ t_S }[/math] the coordinate time between the events for an observer set in the origin of the coordinates (the free falling Sun). The primes on [math]\displaystyle{ t'_S }[/math] and [math]\displaystyle{ r' }[/math] have been dropped in Eq. (66), thus one just uses the symbols [math]\displaystyle{ t_S }[/math] and [math]\displaystyle{ r }[/math]. From Eq. (64) one has indeed [math]\displaystyle{ t_S = t'_S \quad r = r' }[/math]. In the current case one fixes [math]\displaystyle{ r = r_0 }[/math], resulting

[math]\displaystyle{ \tau = \sqrt{1 - \frac{r_0^2w^2_S}{c^2}}t_S \simeq \left(1 - \frac{1}{2} \frac{r_0^2w^2_S}{c^2}t_S \right). }[/math]

(67)

Therefore, the CNO which has been previously defined has also to consider the correction of Eq. (67).

If one uses Eqs. (53) and (60) Eq. (67) becomes

[math]\displaystyle{ \tau = \sqrt{1 - \frac{1}{2}\frac{r_g}{r_0}\left(1 + \frac{m}{M} \right)} t_s \simeq \left[1 - \frac{1}{4}\frac{r_g}{r_0} \left(1 + \frac{m}{M} \right) \right] t_S. }[/math]

(68)

In other words, the CNO must make the replacement [math]\displaystyle{ T_{S1} \to T_{S1}\sqrt{1 - \frac{1}{2}\frac{r_g}{r_0}\left(1 + \frac{m}{M} \right)} }[/math] in Eq. (58), obtaining

[math]\displaystyle{ T_{S2} = \left(1 - \frac{3}{4}\frac{r_g}{r_0} \right)\left(1 - \frac{1}{2}\frac{r_g}{r_0} \right) \left[1 - \frac{1}{2}\frac{r_g}{r_0} \left(1 + \frac{m}{M} \right) \right]^{\frac{1}{2}} T_S }[/math]

[math]\displaystyle{ \simeq \left(1 - \frac{3}{4} \frac{r_g}{r_0} \right) \left(1 - \frac{1}{2}\frac{r_g}{r_0} \right) \left[\left(1 - \frac{1}{2} \frac{r_g}{r_0} \right) \right]^{\frac{1}{2}}T_S, }[/math]

(69)

where, in the last passage the quantity [math]\displaystyle{ \frac{1}{2}\frac{r_g}{r_0}\frac{m}{M} }[/math] has been neglected because it is much minor than the other terms. Then, Eq. (59) reads

[math]\displaystyle{ w_{S2} = \frac{2\pi}{T_{S2}} \simeq w_S\left(1 - \frac{1}{2}\frac{r_g}{r_0} \right)^{-\frac{3}{2}}\left(1 + \frac{1}{2}\frac{r_g}{r_0} \right) \left(1 - \frac{1}{2}\frac{r_g}{r_0} \right)^{\frac{1}{2}} }[/math]

[math]\displaystyle{ \simeq w_s\left(1 + \frac{3}{2}\frac{r_g}{r_0} \right), }[/math]

(70)

where in the above computations the first-order approximation in [math]\displaystyle{ \frac{r_g}{r} }[/math] have been used. Therefore, by using Eq. (61) one finds

[math]\displaystyle{ \varphi_{S2} = w_{S2}T_S \simeq 2\pi \left(1 + \frac{3}{2} \frac{r_g}{r_0} \right) }[/math]

(71)

and

[math]\displaystyle{ \Delta \varphi_{S2} \simeq 3 \pi \frac{r_g}{r_0}. }[/math]

(72)

Remarkably, the final result of Eq. (72) is completely consistent with the result of the general theory of relativity in Eq. (1). For a circular motion it is indeed [math]\displaystyle{ a(1 - e^2) = r_0 }[/math].

Time dilation for the second corrected Newtonian observer

Now, let us analyze the situation in the frame of reference of the fixed stars. In this frame of reference the general relativistic time coordinate is different with respect to the general relativistic time coordinate in the free falling reference frame of the Sun. The two time coordinates are connected by Eq. (46). Thus, one has to apply the time coordinate transformation

[math]\displaystyle{ \frac{dt_F}{\gamma} = dt_S }[/math]

(73)

to the line element of Eq. (55), obtaining

[math]\displaystyle{ ds^2 = \left(1 - \frac{2GM}{rc^2} \right) \left(cd\frac{dt_F}{\gamma} \right)^2 - \left(1 + \frac{2GM}{rc^2} \right) \left(dr^2 + r^2d\theta^2 + r^2 sin^2\theta d \phi^2 \right). }[/math]

(74)

From the point of view of a Newtonian observer, in the frame of reference of the fixed stars one gets a period of revolution of the planet around the Sun

[math]\displaystyle{ T_F = \frac{2\pi r_0}{v_F} = \frac{2 \pi r_0{\frac{3}{2}}}{(GM)^{\frac{1}{2}}}. }[/math]

(75)

The period [math]\displaystyle{ T_F }[/math] in Eq. (75) is measured by a Newtonian observer located in the frame of reference of the fixed stars who does not consider gravitational time dilation. Again, a general relativistic observer will observe a different period. The coefficient of the coordinate time in the line element of Eq. (74) is slightly different with respect to the coefficient of the coordinate time in the line element of Eq. (55). On the other hand, by confronting Eq. (75) and Eq. (53) one immediately gets

[math]\displaystyle{ T_S = T_F \frac{1}{\sqrt{1 + \frac{m}{M} }}, }[/math]

(76)

which is exactly Eq. (27). By confronting the quantity [math]\displaystyle{ \left(\frac{mv_F}{2cM} \right)^{2} }[/math] in Eq. (52) with the quantity [math]\displaystyle{ \frac{m}{M} }[/math] in Eq. (76) one sees that it is

[math]\displaystyle{ \left(\frac{mv_F}{2cM} \right)^{2} \lt\lt \frac{m}{M}, }[/math]

(77)

which means that the local special relativistic time dilation between the two frames of references of Eq. (46) is completely negligible with respect to the variation of orbital periods of Eq. (76). Thus, one can set [math]\displaystyle{ \frac{dt_F}{\gamma} = dt_F }[/math] in Eq. (74) obtaining

[math]\displaystyle{ ds^2 = \left(1 - \frac{2GM}{rc^2} \right) (cdt_F)^2 - \left(1 + \frac{2GM}{rc^2} \right) (dr^2 + r^2d\theta^2 + r^2 sin^2 \theta d\phi^2) ). }[/math]

(78)

Again, one considers variations due to time dilation as being corrections with respect to the Newtonian observer. Thus, a second "Corrected Newtonian Observer" (CNO2) can be defined in the frame of reference of the fixed stars, which sees the spatial directions as being a Newtonian observer, but measures the time between two events by using tg in Eq. (56) instead of the absolute Newtonian time in the frame of reference of the fixed stars, which is [math]\displaystyle{ t_l = t_F }[/math]. Exactly like in previous discussion, one has to note that, together with this time dilation, there is also a variation of the proper radial distance between the Sun and the planet with respect to the Newtonian observer. If this latter observer measures a distance, say [math]\displaystyle{ r_0 = ct_{F0} }[/math], the CNO2 will measure a distance [math]\displaystyle{ ct_{F0}\sqrt{1 - \frac{r_g}{r_0}} \simeq r_0 \left(1 - \frac{1}{2} \frac{r_g}{r_0} \right) }[/math]. Again, one does not consider the correction to the proper radial distance between the Sun and the planet due to spatial curvature, which is given by the opposite of the coefficient [math]\displaystyle{ g_{11} = - \left(1 + \frac{2GM}{rc^2} \right) }[/math] in the line element of Eq. (78). In fact, it is assumed that the CNO2 sees the spatial directions as being a pure Newtonian observer. In other words, spatial directions are considered Euclidean. Then, the following replacements in Eq. (53) have to be considered

[math]\displaystyle{ r_0 \to r_0 \left(1 - \frac{1}{2}\frac{r_g}{r_0} \right) \qquad T_F \to T_F \left(1 - \frac{1}{2} \frac{r_g}{r_0} \right). }[/math]

(79)

This means that the CNO2 replaces in Eq. (75) the original time and distances with time and distances corrected by Eq. (79). Hence, for the CNO2 Eq. (53) becomes

[math]\displaystyle{ T_{F1} = \left(1 - \frac{1}{2} \frac{r_g}{r_0} \right)^{\frac{3}{2}} \left(1 - \frac{1}{2} \frac{r_g}{r_0} \right) T_F, }[/math]

(80)

being [math]\displaystyle{ T_{F1} }[/math] the perturbed orbital period measured by the CNO2 and [math]\displaystyle{ T_{F} }[/math] the unperturbed [again, one means unperturbed with respect to the corrections of general relativity of Eq.(79)] orbital period given by Eq. (75). Then, the corresponding perturbed angular velocity measured by the CNO2, is

[math]\displaystyle{ w_{F1} = \frac{2\pi}{T_{F1}} \simeq w_F \left(1 - \frac{1}{2} \frac{r_g}{r_0} \right)^{-\frac{3}{2}} \left(1 + \frac{1}{2}\frac{r_g}{r_0} \right), }[/math]

(81)

where [math]\displaystyle{ ω_F }[/math] is given by

[math]\displaystyle{ w_F = \frac{2\pi}{T_F}. }[/math]

(82)

Thus, [math]\displaystyle{ ω_F }[/math] is the angular velocity of the rotating planet in the frame of reference of the fixed stars. Hence, the final angle that Mercury sweeps during the period [math]\displaystyle{ T_F }[/math] in the frame of reference of the fixed stars is

[math]\displaystyle{ \varphi_{F1} = \varphi_{F1} T_F \simeq 2\pi \left(1 - \frac{1}{2} \frac{r_g}{r_0} \right)^{-\frac{3}{2}} \left(1 + \frac{1}{2} \frac{r_g}{r_0} \right) \simeq 2\pi \left( 1 + \frac{5}{4}\frac{r_g}{r_0} \right), }[/math]

(83)

in radians per revolution. Again, in the above computations the first-order approximation in [math]\displaystyle{ \frac{r_g}{r_0} }[/math] has been used. Thus, one finally finds

[math]\displaystyle{ \Delta \varphi_{F1} \simeq \frac{5\pi}{2} \frac{r_g}{r_0}. }[/math]

(84)

Eq. (84) represents the advance of the orbit of the planet which is due to the gravitational time dilation as measured by the CNO2. This result is the same as Eq. (62) which represents the precession of the planet's orbit due to the gravitational time dilation as measured by the CNO.

Now, one also has to calculate the rotational contribution. By applying Eq. (54) to the Langevin line-element of Eq. (65) one writes,

[math]\displaystyle{ d\tau = \sqrt{\left(1 - \frac{r^2w_F^2}{c^2} \right)} dt_F \simeq \left(1 - \frac{1}{2}\frac{r^2w_F^2}{c^2} \right) dt_F, }[/math]

(85)

being, in the new rotating frame (now the planet rotates with respect to the fixed stars), [math]\displaystyle{ \tau }[/math] the proper time between two events for the rotating observer at a distance [math]\displaystyle{ r }[/math] from the Sun and having angular velocity [math]\displaystyle{ ω_F }[/math], and [math]\displaystyle{ t_F }[/math] the coordinate time between the events for an observer in the frame of reference of the fixed stars. The primes on [math]\displaystyle{ t'_F }[/math] and [math]\displaystyle{ r' }[/math] have been dropped in Eq. (85). Thus, one merely uses the symbols [math]\displaystyle{ t_F }[/math] and [math]\displaystyle{ r }[/math], one fixes [math]\displaystyle{ r = r_0 }[/math] obtaining

[math]\displaystyle{ \tau = \sqrt{1 - \frac{r_0^2 w^2_F}{c^2}}t_F \simeq \left(1 - \frac{1}{2}\frac{r_0^2w^2_F}{c^2} \right) t_F. }[/math]

(86)

Therefore, the CNO2 has to consider the correction of Eq. (86). Using Eqs. (75), (82), Eq. (86) becomes

[math]\displaystyle{ \tau = \sqrt{1 - \frac{1}{2}\frac{r_g}{r_0}}t_F \simeq \left[1 - \frac{1}{4}\frac{r_g}{r_0} \right]t_F. }[/math]

(87)

This means that the CNO2 has to make the replacement [math]\displaystyle{ T_{F1} \to T_{F1}\sqrt{1 - \frac{1}{2}\frac{r_g}{r_0}} }[/math] in Eq. (80), which now becomes

[math]\displaystyle{ T_{F2} = \left(1 - \frac{3}{4}\frac{r_g}{r_0} \right) \left(1 - \frac{1}{2}\frac{r_g}{r_0} \right) \left[ \left(1 - \frac{1}{2}\frac{r_g}{r_0} \right)\right]^{\frac{1}{2}} T_F. }[/math]

(88)

Then, Eq. (81) becomes

[math]\displaystyle{ w_{F2} = \frac{2\pi}{T_{F2}} \simeq w_F \left(1 - \frac{1}{2}\frac{r_g}{r_0} \right)^{-\frac{3}{2}}\left(1 + \frac{1}{2}\frac{r_g}{r_0} \right) \left(1 - \frac{1}{2}\frac{r_g}{r_0} \right)^{\frac{1}{2}} \simeq w_F \left(1 + \frac{3}{2} \frac{r_g}{r_0} \right), }[/math]

(89)

where, again, in the above computations the first-order approximation in [math]\displaystyle{ \frac{r_g}{r} }[/math] has been used. Therefore, if one uses Eq. (83) one obtains

[math]\displaystyle{ \varphi_{F2} = w_{F2}T_S \simeq 2\pi \left(1 + \frac{3}{2}\frac{r_g}{r_0} \right) }[/math]

(90)

and

[math]\displaystyle{ \Delta \varphi_{F2} \simeq 3\pi \frac{r_g}{r_0}. }[/math]

(91)

Eq. (91) represents the precession of the orbit of the planet due to the total (that is, gravitational plus rotational) time dilation as measured by the CNO2. This result is the same as Eq. (72) which represents the precession of the planet's orbit due to the total time dilation as measured by the CNO. This means that the approach which has been developed in this paper, that is an extension of Newtonian theory which takes into account both gravitational and rotational time dilation, is consistent with global covariance. Again, one must stress that, as for a circular motion it is [math]\displaystyle{ a(1-e^2)=r_0 }[/math], the final result of Eq. (91) is completely consistent with the result of general relativity of Eq. (1).

Conclusion remarks

In it has been shown that, contrary to a longstanding conviction older than 160 years, the precession of Mercury perihelion can be achieved in a Newtonian framework with a very high precision if one correctly analyzes the situation without neglecting Mercury's mass in the non-inertial frame of reference of the Sun. General relativity remains more precise than Newtonian physics, but Newtonian framework is more powerful than researchers and astronomers were thinking till now, at least for the case of Mercury. The Newtonian formula of the precession of planets' perihelion breaks down for the other planets. The predicted Newtonian result is indeed too large for Venus and Earth. Therefore, in it has been also shown that corrections due to gravitational and rotational time dilation, in an intermediate framework which analyzes gravity between Newton and Einstein (that is, a particular post-Newtonian approximation of general relativity), solve the problem in accordance with some other results in the literature. By adding such corrections, a result consistent with the one of general relativity has been indeed obtained.

In this new paper the situation has been re-analyzed in Newtonian physics. It has been indeed shown that, despite the orbit's precession does not occur in an inertial (in Newtonian sense) frame of reference, it occurs, instead, in the non-inertial frame of reference of the Sun and it is due to the well known issue that, in a Newtonian framework, the distance which is travelled by a body depends on the frame of reference in which the motion of the body is analyzed. This fundamental issue of Newtonian mechanics has been analyzed in Section 2 of the paper. The solution of the problem which analyzes the planet's orbit as a harmonic oscillator has been reviewed in Section 3. In Section 4 it has been shown that the precession can be directly obtained from the orbit equation in the non-inertial frame of reference of the Sun. In Section 5 it has been shown that the results presented in and in Sections 3 and 4 of this paper are not in contrast with more than 160 years of Newtonian physics and with the traditional textbooks of classical mechanics. The situation depends on the issue that in traditional textbooks of classical mechanics planets orbits are always analyzed in inertial frames of reference, while the orbits' precession is present in the non-inertial frame of reference of the Sun. In particular, the approximation in which the reference frame of the Sun is considered inertial is often used. But this is, in fact, an approximation, which makes us miss the existence of precession. This is the subtle key issue.

Then, the corrections due to gravitational and rotational time dilation, in an intermediate framework which analyzes gravity between Newton and Einstein, have been re-analyzed in Section 6 by stressing the difference between general relativity and Newtonian physics via the evocation of the Einstein Equivalence Principle, and of the third postulate of relativity. This framework of gravity between Newton and Einstein seems consistent with global covariance. Finally, it is important to stress that a better understanding of gravitational effects in an intermediate framework between Newtonian theory and general relativity, which is one of the goals of this paper, could, in principle, be crucial for a better understanding of the famous Dark Matter and Dark Energy problems.

Declarations

Conflict of Interest

The Author declares that there is no conflict of interest.