Newton-Einstein combination simplifies calculation of Mercury’s orbit

The orientation of Mercury’s elliptical orbit around the Sun changes slightly on each orbit. This precession of the perihelion is partially explained by the effects of other planets, but there is a tiny, measurable discrepancy from the predictions of classical mechanics. This remained unexplained until Einstein formulated the theory of general relativity, which affects Mercury more than the other planets due to the eccentricity of its orbit and its proximity to the Sun.

However, the math connecting Einstein’s theory to this prediction can be challenging for students learning general relativity. Textbook calculations generally use higher-order perturbation theory or require difficult integration.

Michael Hall presented a method to derive the precession of Mercury that is much simpler and requires no more math than solving Newtonian orbits. He hopes the method will be used in introductory general relativity courses.

“Einstein’s basic equation for the orbit of Mercury is not that different from Newton’s. It has one extra term, responsible for the precession,” said Hall. “I knew that Newton had developed a method for predicting precession effects due to perturbing forces and wanted to use this method on Einstein’s equation to simplify the calculation with a sort of ‘Newton-Einstein collaboration’.”

Compared to the Newtonian orbit equation, the relativistic version includes an extra term proportional to the inverse radius squared.

“The idea of linearization is to approximate this quadratic term by a linear term, which makes the equation just as easy to solve as Newton’s equation and so provides a simple derivation of the precession of the orbit,” said Hall.

Source: “Simple precession calculation for Mercury: a linearization approach,” by Michael J. W. Hall, American Journal of Physics (2022). The article can be accessed at https://doi.org/10.1119/5.0098846 .