ANU physicist shows Einstein the easy way to solve equation

Friday 2 December 2022

It’s one of Einstein’s defining moments, when his new theory of general relativity solved a longstanding puzzle, a discrepancy in Mercury’s orbit compared with Newtonian mechanics.

But Einstein calculated it the hard way, Dr Michael Hall has shown, in the American Journal of Physics.

With a simple approximation, Dr Hall, from the Department of Fundamental and Theoretical Physics has reduced the calculation of Mercury’s orbit to the same level as Newtonian calculations, that students routinely learn. 

“The resulting approximate orbit equation is very easy to solve,” said Dr Hall, whose paper was featured as a Scilight by the American Institute of Physics:

“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.”

Dr Hall’s method approximates that term, which is quadratic, with a linear term.

“I knew that Newton had developed a method for predicting precession effects due to perturbing forces, and I wanted to use this method on Einstein’s equation to simplify the calculation with a sort of ‘Newton-Einstein collaboration’,” Dr Hall said.

During the late 19th Century, astronomers measured a slight change of the orientation of Mercury’s elliptical orbit around the Sun 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.

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

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.

“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.

But does this approximation compromise the final result? No, says Dr Hall. Einstein’s methods rely on the approximation that all relevant speeds are small with respect to the speed of light. 

Hall’s method instead relies on the approximation that the distance from the Sun does not change too much over the orbit – it is this that allows replacement of the quadratic term by a linear term.

“In this sense the two methods complement each other. However, they agree precisely in the overlap of their domains, that is, when both approximations hold – as is indeed the case for Mercury orbiting the Sun, which is both low-speed and near-circular,” Dr Hall says.

Simple precession calculation for Mercury: a linearization approach, by Michael J. W. Hall, American Journal of Physics (2022)

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