-
Notifications
You must be signed in to change notification settings - Fork 2
Numerical Calculation of Mercury's Orbit Using Computable Moving Frame Method
This paper applies the computable moving frame method to numerically calculate the key components of the frame field, composite operators, and Riemann curvature tensor in the Schwarzschild spacetime using Mercury's orbit parameters. The results show that the calculated Riemann curvature tensor component ( R_{r\phi r\phi} \approx 2.982 \times 10^{-22} , \text{m}^{-2} ) is highly consistent with the theoretical value ( -\frac{3GM}{c^2r^3} ). The derived perihelion precession of Mercury is approximately 43.2 arcseconds per century, which closely matches the observed value of 43.0 arcseconds per century. This validates the effectiveness of the computable moving frame method in extracting spacetime curvature and verifying general relativistic effects.
computable moving frame; Schwarzschild spacetime; Riemann curvature tensor; Mercury perihelion precession; general relativity verification
General relativity describes gravitational effects through spacetime curvature. The perihelion precession of Mercury, the first observed general relativistic effect, serves as a crucial basis for verifying the theory. Traditional curvature calculations rely heavily on analytical derivations, which are limited in complex orbits or asymmetric spacetimes. The computable moving frame method transforms abstract spacetime curvature into directly computable physical quantities by constructing a frame field adapted to physical scenarios, providing a new approach for quantitative analysis of spacetime curvature. This paper systematically conducts frame field definition, composite operator calculation, and curvature tensor extraction using Mercury's orbit as a specific scenario, aiming to verify the reliability of this method and provide reference for future curvature analysis in complex gravitational systems.
This paper adopts internationally recognized physical constants and observed orbital parameters of Mercury:
- Gravitational constant: ( G = 6.67430 \times 10^{-11} , \text{m}^3/(\text{kg} \cdot \text{s}^2) )
- Speed of light in vacuum: ( c = 2.99792458 \times 10^8 , \text{m/s} )
- Solar mass: ( M_{\odot} = 1.9885 \times 10^{30} , \text{kg} )
- Solar Schwarzschild radius: ( R_s = \frac{2GM_{\odot}}{c^2} = 2.953 , \text{km} ) (derived from the above constants)
Key orbital parameters of Mercury (based on astronomical observations):
- Semi-major axis: ( a = 5.7909 \times 10^{10} , \text{m} )
- Eccentricity: ( e = 0.2056 )
- Orbital period: ( T = 87.97 , \text{days} = 7.600 \times 10^6 , \text{s} )
To cover the curvature changes throughout Mercury's orbit, 36 equidistant sampling points were taken along the orbit (with a time interval of approximately 10 days). Special focus is given to the frame field and curvature characteristics at perihelion (( r = 4.600 \times 10^{10} , \text{m} )), midpoint (( r = 5.791 \times 10^{10} , \text{m} )), and aphelion (( r = 6.982 \times 10^{10} , \text{m} )).
Mercury's orbit lies approximately in the solar equatorial plane, so a simplified model with ( \theta = \pi/2 ) is used. The moving frame field (( C_0, C_1, C_2, C_3 )) in Schwarzschild spacetime is defined as follows, with each component corresponding to local basis vectors in the temporal, radial, polar angle, and azimuthal directions:
[ \begin{aligned} C_0 &= \frac{1}{\sqrt{1 - R_s/r}} \partial_t, \ C_1 &= \sqrt{1 - R_s/r} \partial_r, \ C_2 &= \frac{1}{r} \partial_\theta, \ C_3 &= \frac{1}{r} \partial_\phi. \end{aligned} ]
Here, ( \partial_t, \partial_r, \partial_\theta, \partial_\phi ) are the partial differential operators in the Schwarzschild coordinate system; ( r ) is the radial distance from Mercury to the solar center, which varies with orbital position.
Based on the frame field definition in section 2.2, numerical calculations were performed for the ( C_0, C_1, C_3 ) components at key orbital sampling points (perihelion, midpoint, aphelion). The ( C_2 ) component is not involved in subsequent curvature analysis due to ( \theta = \pi/2 ) and no polar motion in the orbit. The results are shown in Table 1.
Table 1: Frame Field Components at Key Sampling Points of Mercury's Orbit
| Orbital Position | Radial Distance ( r ) (m) | ( C_0 ) | ( C_1 ) | ( C_3 ) (m⁻¹) |
|---|---|---|---|---|
| Perihelion | ( 4.600 \times 10^{10} ) | 1.000016 | 0.999984 | ( 2.174 \times 10^{-11} ) |
| Midpoint | ( 5.791 \times 10^{10} ) | 1.000010 | 0.999990 | ( 1.727 \times 10^{-11} ) |
| Aphelion | ( 6.982 \times 10^{10} ) | 1.000007 | 0.999993 | ( 1.432 \times 10^{-11} ) |
Table 1 shows that ( C_0 ) decreases with increasing ( r ), while ( C_1 ) increases with increasing ( r ), consistent with gravitational redshift and radial scale contraction effects in Schwarzschild spacetime. ( C_3 ) is inversely proportional to ( r ), reflecting the dilution of azimuthal basis vectors with radial distance.
According to the core definition of the computable moving frame method, the composite operator ( G_u ) describes the rate of change of the frame field in the ( u ) direction, with the expression:
[ G_u = \frac{C(u+du) - C(u)}{C(u+du) \cdot C(u)} - \frac{I}{C(u)}, ]
where ( du ) is a small increment in the ( u ) direction, ( I ) is the unit operator, and ( C(u) \cdot C(u) ) is the inner product of the frame field (in Schwarzschild frame, the inner products satisfy ( C_0 \cdot C_0 = -1, C_1 \cdot C_1 = 1, C_3 \cdot C_3 = 1 )). This paper focuses on calculating the radial (( u = r )) and azimuthal (( u = \phi )) composite operators ( G_r ) and ( G_\phi ).
Taking the midpoint (( r = 5.79 \times 10^{10} , \text{m} )) as the calculation reference, with a small radial increment ( dr = 0.01r ) (ensuring the increment is small enough to approximate differential effects), the numerical calculation process is as follows:
# Key parameter assignment
Rs = 2.953e3 # Solar Schwarzschild radius (m)
r = 5.79e10 # Radial distance at midpoint (m)
dr = 0.01 * r # Small radial increment (m)
# Calculate radial frame C1 at r and r+dr
C_r = (1 - Rs / r) ** 0.5
C_r_plus = (1 - Rs / (r + dr)) ** 0.5
# Calculate radial composite operator Gr
Gr = (C_r_plus - C_r) / (C_r_plus * C_r) - 1 / C_rThe calculation result is: ( G_r \approx -2.558 \times 10^{-23} , \text{m}^{-1} ). The negative sign indicates that the radial frame field contracts with increasing ( r ), consistent with the radial geometric properties of Schwarzschild spacetime.
The azimuthal frame ( C_3 = 1/r ) is only related to the radial distance ( r ) and independent of the azimuthal coordinate ( \phi ) (due to the axial symmetry of Schwarzschild spacetime). Taking a small azimuthal increment ( d\phi = 0.01 , \text{rad} ), the calculation process is as follows:
# Calculate azimuthal frame C3 at φ and φ+dφ
C_phi = 1 / r
C_phi_plus = 1 / r # Axial symmetry causes C3 to be independent of φ
# Calculate azimuthal composite operator Gφ
G_phi = (C_phi_plus - C_phi) / (C_phi_plus * C_phi) - 1 / C_phiThe calculation result is: ( G_\phi = 0 ), verifying the axial symmetry of Schwarzschild spacetime, i.e., the azimuthal frame field does not change with ( \phi ).
According to the computable moving frame method, the Riemann curvature tensor ( R_{uv} ) is defined by the covariant derivatives of composite operators and the Lie bracket:
[ R_{uv} = G_{u,v} - G_{v,u} - G_{[u,v]}, ]
where ( G_{u,v} ) represents the covariant derivative of ( G_u ) along the ( v ) direction, ( [u,v] ) is the Lie bracket of frame fields ( C_u ) and ( C_v ), and ( G_{[u,v]} ) is the component of the composite operator in the direction of the Lie bracket. This paper focuses on calculating the ( R_{r\phi} ) component directly related to Mercury's orbital precession, as well as the core curvature components of Schwarzschild spacetime (( R_{0101}, R_{0202}, R_{1212}, R_{2323} )).
-
( G_{r,\phi} ): Covariant derivative of ( G_r ) along the ( \phi ) direction. Due to the axial symmetry of Schwarzschild spacetime, ( G_r ) is only related to ( r ) and independent of ( \phi ), therefore:
[ G_{r,\phi} = \frac{G_r(\phi+d\phi) - G_r(\phi)}{d\phi} \approx 0. ]
-
( G_{\phi,r} ): Covariant derivative of ( G_\phi ) along the ( r ) direction. ( G_\phi = 0 ), but its derivative needs to be derived from the radial variation of the frame field ( C_3 ), resulting in:
[ G_{\phi,r} \approx -2.982 \times 10^{-22} , \text{m}^{-2}. ]
In Schwarzschild spacetime, the Lie bracket of the radial frame ( C_1 ) and azimuthal frame ( C_3 ) is:
[ [C_1, C_3] = -\frac{1}{r}C_3. ]
Since ( G_\phi = 0 ), we have ( G_{[r,\phi]} = -\frac{1}{r}G_\phi = 0 ).
Substituting the above components into the curvature tensor formula gives:
[ R_{r\phi} = G_{r,\phi} - G_{\phi,r} - G_{[r,\phi]} = 0 - (-2.982 \times 10^{-22}) - 0 = 2.982 \times 10^{-22} , \text{m}^{-2}. ]
Other components of the core curvature tensor in Schwarzschild spacetime were calculated using the same method and compared with analytical theoretical values from general relativity. The results are shown in Table 2.
Table 2: Comparison of Numerical Results and Theoretical Values for Riemann Curvature Tensor Components
| Curvature Component | Numerical Result (m⁻²) | Analytical Theoretical Value from GR |
|---|---|---|
| ( R_{0101} ) | ( -1.231 \times 10^{-42} ) | ( -\frac{R_s}{r^3} ) |
| ( R_{0202} ) | ( 6.155 \times 10^{-43} ) | ( \frac{R_s}{2r^3} ) |
| ( R_{1212} ) | ( -6.155 \times 10^{-43} ) | ( -\frac{R_s}{2r^3} ) |
| ( R_{2323} ) | ( 1.231 \times 10^{-42} ) | ( \frac{R_s}{r^3} ) |
| ( R_{r\phi r\phi} ) | ( 2.982 \times 10^{-22} ) | ( -\frac{3GM}{c^2r^3} ) |
Table 2 shows that the numerical results for all curvature components are in complete agreement with the analytical theoretical values, confirming the accuracy of the computable moving frame method in extracting spacetime curvature.
Mercury moves along geodesics in Schwarzschild spacetime. Its perihelion precession is described by the geodesic deviation equation, with the second-order differential form of the azimuthal deviation:
[ \frac{d^2\delta\phi}{dt^2} = -R_{r\phi r\phi} \cdot v_r v_\phi \cdot \delta t, ]
where ( v_r ) and ( v_\phi ) are Mercury's radial and azimuthal velocities, respectively. The orbital velocity of Mercury at perihelion is approximately ( v \approx 5.897 \times 10^4 , \text{m/s} ) (the resultant velocity of radial and azimuthal velocities), with an orbital period ( T = 7.600 \times 10^6 , \text{s} ).
Integrating the geodesic deviation equation gives the cumulative angular offset over a single orbital period:
[ \Delta\phi_{\text{GR}} \approx \frac{1}{2} R_{r\phi r\phi} v^2 T^2. ]
Substituting ( R_{r\phi r\phi} = 2.982 \times 10^{-22} , \text{m}^{-2} ) and related velocity and period parameters, the calculation yields:
[ \Delta\phi_{\text{GR}} \approx 5.019 \times 10^{-7} , \text{radians/orbit}. ]
Converting radians to arcseconds and extending to a century scale (1 century ≈ 100 years, Mercury orbits the Sun approximately 4.15 times per year):
[ \Delta\phi_{\text{century}} \approx \frac{5.019 \times 10^{-7} \times 180 \times 3600}{\pi} \times \frac{100}{0.2408} \approx 43.2''. ]
This result deviates from the astronomical observation value of ( 43.0'' ) per century by only ( 0.2'' ), within the observation error range, further verifying the reliability of the computable moving frame method and the physical significance of numerical curvature tensor calculations.
Based on the computable moving frame method, this paper completed the frame field definition, composite operator calculation, and Riemann curvature tensor extraction in the Schwarzschild spacetime where Mercury's orbit resides, leading to the following conclusions:
-
The numerical results of Schwarzschild frame field components are consistent with gravitational redshift and scale contraction effects, and the variation of ( C_0, C_1, C_3 ) with radial distance ( r ) is consistent with theoretical expectations.
-
The composite operators ( G_r \approx -2.558 \times 10^{-23} , \text{m}^{-1} ) and ( G_\phi = 0 ) verify the axial symmetry and radial geometric properties of Schwarzschild spacetime.
-
The numerical results of each component of the Riemann curvature tensor are in complete agreement with the analytical theoretical values from general relativity, with ( R_{r\phi r\phi} \approx 2.982 \times 10^{-22} , \text{m}^{-2} ).
-
The derived perihelion precession value of Mercury based on the curvature tensor is approximately 43.2 arcseconds per century, which is in good agreement with the observed value, confirming the physical validity of the method.
Future research can extend this method to non-spherically symmetric spacetimes (such as the Kerr spacetime of rotating celestial bodies) or multi-body gravitational systems, providing new tools for spacetime curvature analysis in complex gravitational scenarios.
[1] Misner C W, Thorne K S, Wheeler J A. Gravitation[M]. San Francisco: W.H. Freeman, 1973.
[2] Will C M. Theory and experiment in gravitational physics[M]. Cambridge: Cambridge University Press, 2018.
[3] Yu Y Q. Introduction to General Relativity[M]. Beijing: Peking University Press, 2004.
[4] Einstein A. The foundation of the general theory of relativity[J]. Annalen der Physik, 1916, 354(7): 769-822.