Confirming Einstein’s Theory of General Relativity With Calculus, Part 7a: Predicting Precession I

In this series, I’m discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury’s orbit and thus confirm Einstein’s theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago.

We have shown that the motion of a planet around the Sun, expressed in polar coordinates $(r,\theta)$ with the Sun at the origin, under general relativity is

$u(\theta) = \displaystyle \frac{1 + \epsilon \cos \theta}{\alpha} + \frac{\delta}{\alpha^2} + \frac{\delta \epsilon^2}{2\alpha^2} + \frac{ \delta\epsilon}{\alpha^2} \theta \sin \theta - \frac{ \delta \epsilon^2}{6\alpha^2} \cos 2\theta - \frac{\delta(3+\epsilon^2)}{3\alpha^2} \cos \theta$ ,

where $u = \displaystyle \frac{1}{r}$ , $\displaystyle \frac{1}{\alpha} = \frac{GMm^2}{\ell^2}$ , $\delta = \displaystyle \frac{3GM}{c^2}$ , $G$ is the gravitational constant of the universe, $m$ is the mass of the planet, $M$ is the mass of the Sun, $\ell$ is the constant angular momentum of the planet, and $c$ is the speed of light.

We notice that the first term of the above solution,

$\displaystyle \frac{1 + \epsilon \cos \theta}{\alpha}$ ,

is the same as the solution found earlier under Newtonian physics, without general relativity. Therefore, the remaining terms describe the perturbation due to general relativity. All of these terms contain the small factor $\delta$ , and so these can be expected to be small adjustments to an elliptical orbit.

Of these terms, the terms

$\displaystyle \frac{\delta}{\alpha^2} + \frac{\delta \epsilon^2}{2\alpha^2}$

are constants, while the terms

$- \displaystyle \frac{ \delta \epsilon^2}{6\alpha^2} \cos 2\theta - \frac{\delta(3+\epsilon^2)}{3\alpha^2} \cos \theta$

is bounded since $-1 \le \cos \theta \le 1$ and $-1 \le \cos 2\theta \le 1$ . By contrast, the term

$\displaystyle \frac{ \delta\epsilon}{\alpha^2} \theta \sin \theta$

grows without bound. Therefore, for large values of $\theta$ , the planet’s orbit may be accurately described by only including this last perturbation:

$u(\theta) = \displaystyle \frac{1 + \epsilon \cos \theta}{\alpha} + \frac{ \delta\epsilon}{\alpha^2} \theta \sin \theta$ .

In the next post, we simplify this even further.

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Confirming Einstein’s Theory of General Relativity With Calculus, Part 7a: Predicting Precession I

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