Geodesic Effect Near an Elliptical Orbit

Abstract

Using a differential geometric treatment, we analytically derived the expression for De Sitter (geodesic) precession in the elliptical motion of the Earth through the gravitational field of the Sun with Schwarzschild's metric. The expression obtained in this paper in a simple way, using a classical approach, agrees with that given in B. M. Barker and R. F. O'Connell (1970, 1975) in a different setting, using the tools of Newtonian mechanics and the Euler-Lagrange equations.

1. Introduction

The geodesic effect, also named De Sitter precession or geodesic precession, represents the effect of the curvature of the space-time on a constant spin vector transported together with a body along an orbit through a gravitational field in Einstein’s theory. De Sitter found that the Earth-Moon system would undergo a precession in the gravitational field of the Sun. De Sitter’s work [1] was subsequently extended to rotating bodies, such as the Earth, by Schouten [2] and by Fokker [3]. Studying the Sun’s gravitational field near a circular orbit with Schwarzschild’s metric, it is emphasized the existence of a precessional motion along the Earth’s orbit, as effect of the Sun’s gravitational field [4].

The concept of geodesic precession has two slightly different meanings, as the body moving in orbit may have rotation or not. Nonrotating bodies move on geodesics, while the rotating bodies move in slightly different orbits. A geodesic is a curve which parallel transports a tangent vector. If a curve is not geodesic, then a vector tangent to it at some point does not remain tangent in parallel transport along this curve. In a Riemannian space, a vector parallel transported along a closed contour does not return, in general, to its original position.

The difference between De Sitter precession and Lense-Thirring precession is due to the rotation of the central mass. The total precession is calculated by combining De Sitter precession with Lense-Thirring precession (see for more details [5, 6]). Barker and O’Connell [7] discussed the difference between De Sitter precession and geodesic precession, obtaining the correct expression for the precession in the case of the nearly circular orbits for binary systems with relatively massive components.

The geodesic precession is usually associated with the motion of the gyroscope orbiting in the static gravitational field of the source. This precession is obtained by parallel transport of a spin vector in curved space-time in the vicinity of the mass. The effect is presented even if the mass is not rotating. We also note that the geodesic precession has been recently studied in different settings in [8–10].

In this paper, using a classical treatment, we deduce the expression for the geodesic effect in the elliptical motion of the Earth through the gravitational field of the Sun with Schwarzschild’s metric.

2. Elliptical Orbit in the Gravitational Field

The Schwarzschild metric 𝑑𝑠²=−𝑑𝑟²1−2𝜇/𝑐²𝑟−𝑟²𝑑𝜃²+cos²𝜃𝑑𝜑²+𝑐²−2𝜇𝑟𝑑𝑡² (2.1) is an exact, static, spherically symmetric solution of Einstein’s equation of the general relativity, which represents the gravitational field of a cosmic object without rotation, which possesses mass [11, 12]. The quantity 𝑑𝑠 denotes the invariant space-time interval, an absolute measure of the distance between two events in space-time, 𝑐 is the speed of light, 𝑡 is the time coordinate measured by a stationary clock at infinity, 𝑟 is the radial coordinate, while the variables 𝜃 and 𝜑 are the latitude and the longitude of mass 𝑀 defined in the classical conception with respect to the equatorial plane and the prime meridian, passing through the center 𝑂 of the spherical mass.

In the solar system, according to the formulas of the elliptical motion, the 𝜇 coefficient is equal to 4𝜋²𝑎³/𝑇², 2𝑎 being the major axis of the orbit and 𝑇 the period of revolution of a planet, expression that has significantly the same value for all the planets, according to Kepler’s third law. 2𝜋/𝑇 will be noted by 𝑛, where 𝑛 is the mean angular velocity corresponding to the period 𝑇 of the motion, called for simplicity mean motion. The coefficient 𝜇 is also equal to the product 𝑓(𝑆+𝑀) of the universal gravitational constant and the mass of the planet-Sun system (where 𝑆 is the mass of the Sun and 𝑀 is the planet’s mass).

Making the change of variable 𝜑=𝑛𝑡+𝜓, one obtains𝑑𝜑²=𝑛²𝑑𝑡²+2𝑛𝑑𝜓𝑑𝑡+𝑑𝜓². (2.2) Thus, from the relations (2.1) and (2.2), we have𝑑𝑠²=−𝑑𝑟²1−2𝜇/𝑐²𝑟−𝑟²𝑑𝜃²+cos²𝜃𝑑𝜓²−2𝑛𝑟²cos²𝜃𝑑𝜓𝑑𝑡+𝑐²−2𝜇𝑟−𝑛²𝑟²cos²𝜃𝑑𝑡². (2.3)

In the following, we will determine if the elliptical motion can occur in the plane of the equator. To this purpose, the existence of the geodesics of Schwarzschild’s metric 𝑑𝑠² along which the first three variables, namely 𝑟, 𝜃, 𝜑, have the constant values is studied.

From the equations of geodesics𝑑²𝑥_𝑖𝑑𝑠²+Γ^𝑖_𝛼𝛽𝑑𝑥_𝛼𝑑𝑠𝑑𝑥_𝛽𝑑𝑠=0, (2.4) with 𝑖∈{1,2,3,4} and 𝑥₁=𝑟,𝑥₂=𝜃,𝑥₃=𝜓,𝑥₄=𝑡, it remains for𝛼=𝛽=4𝑑²𝑥_𝑖𝑑𝑠²+Γ^𝑖₄₄𝑑𝑥₄𝑑𝑠²=0. (2.5)

If in the relation (2.5) we take 𝑖=1,2 or 3, then it follows that Γ^𝑖₄₄𝑑𝑥₄𝑑𝑠²=0. (2.6)

Because 𝑑𝑥₄/𝑑𝑠≠0, one obtains Γ^𝑖₄₄=0, with 𝑖∈{1,2,3}. ButΓ^𝑖₄₄=12𝑔^𝑖𝑘2𝜕𝑔_4𝑘𝜕𝑥₄−𝜕𝑔₄₄𝜕𝑥_𝑘, (2.7) and therefore it is found that𝑔^𝑖𝑘2𝜕𝑔_4𝑘𝜕𝑥₄−𝜕𝑔₄₄𝜕𝑥_𝑘=0, (2.8) with Einstein summation convention. The coefficient𝑔₄₄=𝑐²−2𝜇𝑟−𝑛²𝑟²cos²𝜃 (2.9) does not depend on the variables 𝜓 and 𝑡, but only on 𝑟 and 𝜃. Then it follows that 𝜕𝑔₄₄/𝜕𝜓=0 and 𝜕𝑔₄₄/𝜕𝑡=0. On the other hand, 𝑔₁₁=−1/(1−2𝜇/𝑐²𝑟),𝑔₂₂=−𝑟²,𝑔₃₃=−𝑟²cos²𝜃,𝑔₃₄=−𝑛𝑟²cos²𝜃, and the remaining coefficients 𝑔_𝑖𝑗=0, with 𝑖≠𝑗,𝑖,𝑗∈{1,2,3,4}.

Since (𝑔^𝑖𝑗)_{𝑖,𝑗=1,4}=⌊(𝑔_𝑖𝑗)_{𝑖,𝑗=1,4}⌋⁻¹, it is observed that 𝑔²²≠0. Consequently, for 𝑖=2, we deduce from the relation (2.8) that 𝜕𝑔₄₄/𝜕𝜃=0. But𝜕𝑔₄₄𝜕𝜃=2𝑛²𝑟²cos𝜃sin𝜃. (2.10) Hence,2𝑛²𝑟²cos𝜃sin𝜃=0, (2.11) and then one obtains 𝜃=0; in other words the elliptical orbit is located in the plane of the equator.

In order to study the gravitational field near an elliptical orbit, a parallel transport of a vector whose origin describes the corresponding line of Universe is considered.

In the elliptical motion, at the perihelion passage 𝑟 takes the minimum value, namely, 𝑟_min=𝑎(1−𝑒), and at the passage through aphelion 𝑟 takes the maximum value 𝑟_max=𝑎(1+𝑒), where 𝑎 represents the length of the semimajor axis and 𝑒 is the eccentricity of the elliptical orbit.

In the following, we develop in the quadratic form 𝑑𝑠² from (2.3), the coefficients of 𝑑𝜓²,𝑑𝜓𝑑𝑡, and 𝑑𝑡² in the neighborhood of the system of values 1/𝑟=1/𝑎(1−𝑒²) and 𝜃=0, where for the inverse of the radius vector 1/𝑟 it has been taken its average value1𝑟_av=121𝑟_min+1𝑟_max=1𝑎1−𝑒². (2.12) We obtain the following expression:𝑑𝑠²=−𝑑𝑟²1−2𝜇/𝑐²𝑟−𝑟²𝑑𝜃²−𝑟²+𝑛²𝑎⁴1−𝑒²⁴𝑐²−3𝜇/𝑎𝑑𝜓²+4𝑛𝑎³1−𝑒²³1𝑟−1𝑎1−𝑒²+𝑐𝑑𝜓𝑑𝑡₂−3𝜇𝑎𝑑𝑡−𝑛𝑎²1−𝑒²²𝑐²−3𝜇/𝑎𝑑𝜓²+𝑑𝑠²₁, (2.13) where it has been noted by 𝑑𝑠²₁ a quadratic form that depends on the four differentials 𝑑𝑟, 𝑑𝜃, 𝑑𝜓, and 𝑑𝑡 whose coefficients depend on the variables 𝑟 and 𝜃, which vanish for 1/𝑟=1/𝑎(1−𝑒²) and 𝜃=0. Therefore, 𝑑𝑠²₁ does not belong to the group of the Christoffel symbols which will be written in the following. Making the substitution𝑢=𝑡−𝑛𝑎²1−𝑒²²𝑐²−3𝜇/𝑎𝜓, (2.14) the expression of 𝑑𝑠² can be rewritten as𝑑𝑠²=−𝑑𝑟²1−2𝜇/𝑐²𝑟−𝑟²𝑑𝜃²+4𝑛𝑎³1−𝑒²³1𝑟−1𝑎1−𝑒²−𝑟𝑑𝜓𝑑𝑢₂+𝑛²𝑎⁴1−𝑒²⁴+4𝑛²𝑎⁵1−𝑒²⁵1/𝑟−1/𝑎1−𝑒²𝑐²−3𝜇/𝑎𝑑𝜓²+𝑐²−3𝜇𝑎𝑑𝑢²+𝑑𝑠²₁. (2.15)

3. Geodesic Effect in the Elliptical Motion

In the following, we study the phenomenon of precession in the elliptical motion. We will calculate the variations of the components of a contravariant vector (𝑋¹,𝑋²,𝑋³,𝑋⁴) whose origin describes the line of Universe considered in the elliptical motion and whose components are proportional by definition with 𝑑𝑟,𝑑𝜃,𝑑𝜓,𝑑𝑢.

According to the general theory, these variations are given by the equation𝑑𝑋^𝑖=−Γ^𝑖_𝛼𝛽𝑋^𝛼𝑑𝑥_𝛽, (3.1) whereΓ^𝑖_𝛼𝛽=12𝑔^𝑖𝑘𝜕𝑔_𝛼𝑘𝜕𝑥_𝛽+𝜕𝑔_𝛽𝑘𝜕𝑥_𝛼−𝜕𝑔_𝛼𝛽𝜕𝑥_𝑘 (3.2) and 𝛽=4, since in the considered movement 𝑑𝑟,𝑑𝜃,𝑑𝜓 are null, and where the Christoffel symbols are calculated from the metric (2.15) for 1/𝑟=1/𝑎(1−𝑒²) and 𝜃=0. From (2.15) we have𝑔_{𝑖𝑗𝑖,𝑗=1,4}=𝑔₁₁0000𝑔₂₂0000𝑔₃₃𝑔₃₄00𝑔₃₄𝑔₄₄, (3.3) where𝑔₁₁1=−1−2𝜇/𝑐²𝑟,𝑔₂₂=−𝑟²,𝑔₃₃=−𝑟²−𝑛²𝑎⁴1−𝑒²⁴+4𝑛²𝑎⁵1−𝑒²⁵1/𝑟−1/𝑎1−𝑒²𝑐²,𝑔−3𝜇/𝑎₃₄=2𝑛𝑎³1−𝑒²³1𝑟−1𝑎1−𝑒²,𝑔₄₄=𝑐²−3𝜇𝑎. (3.4)

On the other hand, we have𝑔^𝑖𝑗_{𝑖,𝑗=1,4}=1𝑔₁₁𝑔₂₂Δ𝑔₂₂Δ0000𝑔₁₁Δ0000𝑔₁₁𝑔₂₂𝑔₄₄−𝑔₁₁𝑔₂₂𝑔₃₄00−𝑔₁₁𝑔₂₂𝑔₃₄𝑔₁₁𝑔₂₂𝑔₃₃, (3.5) where Δ=𝑔₃₃𝑔₄₄−𝑔²₃₄. Particularly, for 1/𝑟=1/𝑎(1−𝑒²), we have 𝑔₃₄=0, and so we obtain 𝑔^𝑖𝑖=1/𝑔_𝑖𝑖,forall𝑖=1,4.

By direct calculation, it is obtained thatΓ¹₁₄=Γ¹₂₄=Γ¹₄₄=0,Γ¹₃₄=𝑔¹¹𝑛𝑎1−𝑒²,Γ²₁₄=Γ²₂₄=Γ²₃₄=Γ²₄₄Γ=0,₃₁₄=−𝑔³³𝑛𝑎1−𝑒²,Γ³₂₄=Γ³₃₄=0,Γ³₄₄=0,Γ⁴₁₄=Γ⁴₂₄=Γ⁴₃₄=Γ⁴₄₄=0, (3.6) hence𝑑𝑋¹=−𝑛𝑎𝑔¹¹1−𝑒²𝑋³𝑑𝑢,𝑑𝑋²=0,𝑑𝑋³=𝑛𝑎𝑔³³1−𝑒²𝑋¹𝑑𝑢,𝑑𝑋⁴=0. (3.7) Therefore, the components 𝑋² and 𝑋⁴ have constant values, whatever be the initial vector. Particularly, we suppose the component 𝑋⁴ of this initial vector to be null. Then the component 𝑋⁴, proportional with 𝑑𝑢, will be constantly zero, and the vector of Universe will be projected on a vector from the 3D space having the origin in 𝑀. Furthermore, if the initial component 𝑋², proportional with 𝑑𝜃, is also zero, then 𝜃 remains constantly null, because it is zero at the initial time. The projection 𝑀𝑅 of the vector of Universe in the 3D space is in the plane of the elliptical orbit at the initial time and remains in this plane when 𝑀 describes this orbit. The variations 𝑑𝑋¹ and 𝑑𝑋³ are given by the following expressions:𝑑𝑋¹=−𝑛𝑎𝑔¹¹1−𝑒²𝑋³𝑑𝑢,𝑑𝑋³=𝑛𝑎𝑔³³1−𝑒²𝑋¹𝑑𝑢. (3.8) Taking the variable 1/𝑟=1/𝑎(1−𝑒²) and taking account that 𝑔¹¹=1/𝑔₁₁ and 𝑔³³=1/𝑔₃₃, then, using the relations (3.8), we derive𝑑√−𝑔₁₁𝑋¹=𝑛𝑎1−𝑒²√𝑔₁₁𝑔₃₃√−𝑔₃₃𝑋³𝑑√𝑑𝑢,−𝑔₃₃𝑋³=−𝑛𝑎1−𝑒²√𝑔₁₁𝑔₃₃√−𝑔₁₁𝑋¹𝑑𝑢. (3.9) Making the change of variable√−𝑔₁₁𝑋¹√=𝑥,−𝑔₃₃𝑋³=𝑦, (3.10) one can define the direction of the vector 𝑀𝑅 in a reference system linked to the mass 𝑀 in his motion by the cartesian coordinates 𝑥 and 𝑦 of the point 𝑅 with respect to the tangent and the normal to the ellipse at the point 𝑀.

It is known that for all the planets the ratio 𝜇/𝑎𝑐² is smaller than 1/(4⋅10⁷),𝑎 being the arithmetic average of the extreme values of the distance 𝑟 during the motion (see [4]). Therefore, if 1/𝑟=1/𝑎(1−𝑒²), neglecting the second-order terms in 𝜇/𝑎𝑐², the coefficient of 𝑑𝜓² from (2.15) becomes −𝑎²(1−𝑒²)²(1+𝜇/𝑎𝑐²), and1√𝑔₁₁𝑔₃₃=1𝑎1−𝑒²1−3𝜇2𝑎𝑐²1−𝑒². (3.11)

With the change of variable (3.10) and taking account of (3.11), the relations (3.9) can be rewritten in the following form:𝑑𝑥𝑑𝑡=𝑛1−3𝜇2𝑎𝑐²1−𝑒²𝑦,𝑑𝑦𝑑𝑡=−𝑛1−3𝜇2𝑎𝑐²1−𝑒²𝑥. (3.12) Thus, it was obtained a system of linear differential equations with constant coefficients, having the solutions𝑥=𝐴cos𝑛1−3𝜇2𝑎𝑐²1−𝑒²𝑡−𝑡₀,𝑦=−𝐴sin𝑛1−3𝜇2𝑎𝑐²1−𝑒²𝑡−𝑡₀, (3.13) where 𝐴 and 𝑡₀ are constants.

It follows that, with respect to the mobile reference system formed by the tangent and the normal in 𝑀 to ellipse, the vector 𝑀𝑅 has a retrograde rotational motion whose angular velocity is [−𝑛+(3𝜇/2𝑎𝑐²(1−𝑒²))𝑛]. During the revolution of the mass 𝑀, the vector 𝑀𝑅 moves in direct sense as well as the radius vector 𝐹𝑀, but with the velocity 3𝜇𝑛/2𝑎𝑐²(1−𝑒²).

The planet 𝑀, in its rotation with respect to the chosen reference system, seems to reach in the proper angular position of the end of its revolution before that this revolution to be effectively complete.

We note that the expression for the geodesic precession𝑝_𝑔𝑒=32⋅𝑛𝑎𝑐²⋅𝑛1−𝑒², (3.14) where 𝑛 and 𝑎 are the mean motion and semimajor axis of the orbit of the Earth-Moon system about the Sun, agrees with that given in Barker and O’Connell [5, 7], but in a different setting. They find the precession of the spin and the precession of the orbit for the two-body problem in general relativity with arbitrary masses, spins, and quadrupole moments, starting from a gravitational potential energy derived from Gupta’s quantum theory of gravitation [13]. Their calculations were performed using the tools of Newtonian mechanics and the Euler-Lagrange equations. Our approach is clearly more simple, since it was used a purely classical treatment.

Acknowledgment

The author expresses his gratitude to the referee for carefully reading the paper and giving useful comments.