The calculation of Mercury's orbit using general relativity involves accounting for the effects of gravity, particularly the curvature of spacetime caused by massive objects like the Sun. Here's an overview of how this is done:
Equations of motion: The starting point is the set of equations that describe the motion of a planet in the presence of gravity. In the case of general relativity, these equations are derived from the theory's field equations known as the Einstein field equations.
Schwarzschild metric: The gravitational field around a spherically symmetric mass is described by the Schwarzschild metric. By applying this metric to the Sun, the curved spacetime around it is represented mathematically. The metric takes into account the mass of the Sun and the distance between the Sun and Mercury.
Geodesic equation: In general relativity, the path of a planet is determined by solving the geodesic equation, which describes the motion of a particle in curved spacetime. For Mercury's orbit, the geodesic equation is solved within the framework of the Schwarzschild metric. This involves determining the trajectory of Mercury's motion as it follows the curved spacetime around the Sun.
Perihelion precession: One of the most significant effects of general relativity on Mercury's orbit is the precession of its perihelion (the point in its orbit closest to the Sun). The curvature of spacetime causes the perihelion to advance slightly with each orbit. This effect is due to the non-uniform distribution of spacetime curvature around the Sun. The precession of Mercury's perihelion was one of the early confirmations of general relativity.
Numerical calculations: Solving the geodesic equation within the Schwarzschild metric is a complex task. To obtain accurate results, numerical methods are employed. These involve dividing the orbit into small time steps and numerically integrating the equations of motion to determine the position and velocity of Mercury at each step.
By incorporating the effects of general relativity into the calculations, the predicted orbit of Mercury matches the observations more accurately than what would be predicted by classical Newtonian physics alone. The measured perihelion precession of Mercury aligns closely with the values calculated using general relativity, providing further confirmation of the theory.