On the Problem of the Discrepancy Between Mercury's Orbit and the Law of Universal Gravitation

The problem of the discrepancy between Mercury's orbit and the law of universal gravitation was a major mystery in the scientific community until Albert Einstein's general theory of relativity was proposed. This issue is known as the "precession of the perihelion of Mercury," which refers to the fact that Mercury's orbit rotates around the Sun more slowly than expected.

Overview

1. Orbital characteristics of Mercury:

   • Mercury's orbit is elliptical and has a higher eccentricity compared to other planets. As a result, the shape of its orbit shows a significant difference between the point where it is closest to the Sun (perihelion) and the point where it is farthest (aphelion).

2. Precession of the perihelion:

   • The phenomenon where the perihelion of an orbit shifts little by little is called precession. In the case of Mercury, this precession could not be fully explained by Newton's law of universal gravitation alone. Observations showed that Mercury's perihelion shifts by about 5600 arcseconds per century, of which about 43 arcseconds remained unexplained.

Relationship with General Relativity

• Einstein's theory:

  • The general theory of relativity, proposed by Einstein in 1915, treats gravity as the curvature of spacetime. According to this theory, the mass of the Sun distorts the surrounding spacetime, and this distortion affects Mercury's orbit.

• Resolution of the problem:

  • Using Einstein's equations, the shift in Mercury's perihelion was explained, and it was found that the remaining 43-arcsecond discrepancy matched perfectly. This confirmed the validity of the general theory of relativity and had a major impact on the scientific community.

Conclusion

The problem of the precession of Mercury's perihelion was a phenomenon that could not be explained by Newtonian mechanics alone and became important evidence for the validity of the general theory of relativity. This discovery triggered a paradigm shift in physics and served as a catalyst for deepening our understanding of cosmology.

Method for calculating the 43-arcsecond discrepancy in the precession of Mercury's perihelion using general relativity

The method for calculating the 43-arcsecond discrepancy in the precession of Mercury's perihelion using general relativity is based on analyzing Mercury's orbit using Einstein's field equations. Since this calculation requires highly specialized knowledge, the basic approach is explained here.

Basic steps of the calculation

1. Use of the Schwarzschild metric:

   • In general relativity, the effect that a massive object has on the surrounding spacetime is described by the Schwarzschild solution. The Schwarzschild metric describes the geometry of spacetime around a spherically symmetric, non-rotating mass.

2. Derivation of the orbital equation:

   • Based on the Schwarzschild metric, we derive the equations of motion for a planet's orbit. Specifically, we determine the path (orbit) through spacetime taken by the planet by solving the geodesic equation.

3. Calculation of precession:

   • Considering that Mercury's orbit is elliptical, we calculate how its perihelion shifts with each orbit. Since the orbit is not a perfect ellipse but involves precession, we calculate this movement quantitatively.

4. Numerical analysis:

   • Actual calculations require advanced numerical analysis techniques to solve Einstein's field equations approximately. In particular, for small but fast-moving objects like Mercury, it is necessary to calculate even minute effects precisely.

Specific calculation formula

The specific formula for the precession of Mercury's perihelion is expressed as follows:

Δ𝜙=6𝜋𝐺𝑀/𝑐^2𝑎(1−𝑒^2)

where,

• Δ𝜙 is the angle of perihelion precession per orbit (in radians).

• 𝐺 is the gravitational constant.

• 𝑀 is the mass of the Sun.

• 𝑐 is the speed of light.

• 𝑎 is the semi-major axis of Mercury's orbit.

• 𝑒 is the eccentricity of the orbit.

This formula represents the general relativistic effect on Mercury's orbit and can accurately explain the observed 43 arcsecond discrepancy. Through this calculation, general relativity was proven to be accurate regarding the actual movement of celestial bodies.