Confirming Einstein’s Theory of General Relativity With Calculus, Part 6a: New Differential Equation

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 previously showed that if the motion of a planet around the Sun is expressed in polar coordinates $(r,\theta)$ , with the Sun at the origin, then under Newtonian mechanics (i.e., without general relativity) the motion of the planet follows the differential equation

$u_0''(\theta) + u_0(\theta) = \displaystyle \frac{Gm^2 M}{\ell^2}$ ,

where $u_0 = 1/r$ , $G$ is the gravitational constant of the universe, $m$ is the mass of the planet, $M$ is the mass of the Sun, and $\ell$ is the constant angular momentum of the planet. For simplicity, we wrote $\displaystyle \frac{1}{\alpha} = \displaystyle \frac{Gm^2 M}{\ell^2}$ .

We will also impose the initial condition that the planet is at perihelion (i.e., is closest to the sun), at a distance of $P$ , when $\theta = 0$ . This means that $u$ obtains its maximum value of $1/P$ when $\theta = 0$ . This leads to the two initial conditions

$u_0(0) = \displaystyle \frac{1}{P} \qquad \hbox{and} \qquad u_0'(0) = 0$ ;

the second equation arises since $u$ has a local extremum at $\theta = 0$ .

In previous posts, we showed that the solution of this initial-value problem is

$u_0(\theta) = \displaystyle \frac{1 + \epsilon \cos \theta}{\alpha}$ ,

where $\epsilon = \displaystyle \frac{\alpha - P}{P}$ . Since $r = 1/u_0$ , we see that the planet’s orbit satisfies

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

so that, as shown earlier in this series, the orbit is an ellipse with eccentricity $\epsilon$ .

At long last, we’re now ready to see what happens under general relativity. According to the theory of general relativity, the governing differential equation for the orbit of a planet should be changed ever so slightly to

$u''(\theta) + u(\theta) = \displaystyle \frac{Gm^2 M}{\ell^2} + \frac{3GM}{c^2} [u(\theta)]^2$ ,

where a second term was added to the right-hand side. (I will make no attempt here to justify the physics for this second term.) In this second term, $c$ represents the speed of light. Since $c$ is very large, this second term is very, very small. Using $\displaystyle \frac{1}{\alpha} = \displaystyle \frac{Gm^2 M}{\ell^2}$ as before and defining $\delta = \displaystyle \frac{3GM}{c^2}$ , this may be simplified to

$u''(\theta) + u(\theta) = \displaystyle \frac{1}{\alpha} + \delta [u(\theta)]^2$ ,

Finding an exact solution of this new differential equation is hopeless. While the previous differential equation was linear with constant coefficients, this new differential is decidedly nonlinear because of the new term $[u(\theta)]^2$ .

So, instead of attempting to find an exact solution, we will use the method of successive approximations, mentioned earlier in this series, to find an approximate solution that is very, very close to the exact solution. Since $\delta$ is very, very small, the solution of

$u''(\theta) + u(\theta) = \displaystyle \frac{1}{\alpha}$

will be approximately equal to the solution of the “real” differential equation under general relativity. We have already shown that the solution of this simpler differential equation is

$u_0(\theta) = \displaystyle \frac{1 + \epsilon \cos \theta}{\alpha}$ .

Therefore, the method of successive approximations suggests that a better approximation to the true orbit will be the solution of

$u''(\theta) + u(\theta) = \displaystyle \frac{1}{\alpha} + \delta [u_0(\theta)]^2$ ,

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

While significantly messier than the differential equation under Newtonian mechanics, this is now a linear differential equation that can be solved using standard techniques from differential equations. In the next few posts, we will solve this new differential equation, thus finding the predicted orbit of a planet under general relativity.

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Confirming Einstein’s Theory of General Relativity With Calculus, Part 6a: New Differential Equation

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