Author: Watsisname
Original post on the forum: link

Background

Often in discussions of Mercury's perihelion advance, the effect is shown greatly exaggerated.  Sometimes even the orbit is shown much more eccentric than it really is.  Before we dive in, let's take a moment to get some perspective on the true shape of the orbit and appreciate the scale of the solar system.

 
All orbits are ellipses, but most planetary orbits are close enough to circular that it can be hard to tell at a glance.  Mercury is the biggest exception with an eccentricity of 0.21.  (Or Pluto at 0.25).  What's interesting, and what this post is all about, is that Mercury's orbit is not stationary.  The ellipse slowly shifts around, it's perihelion point advancing by about 574 arcseconds (or 0.159 degrees) per century.  Much of this (531 arcseconds) can be explained by the perturbations from the other planets.  However, the remaining 43 arcseconds per century are "anomalous", unaccounted for by Newtonian mechanics.  

The puzzle of this additional observed precession was first noted by LeVerrier in 1859.  Following the successful prediction of the existence and location of Neptune from similar perturbations on Uranus' orbit, LeVerrier proposed that Mercury's orbital precession could be caused by another undiscovered planet inside of Mercury's orbit, which he named Vulcan.  Astronomers searched for this missing inner planet during solar eclipses.  Of course, Vulcan was never found... 

The mystery was not solved until Einstein began developing his general theory of relativity.  When he applied his general relativistic equations to the problem in 1915, he found that they exactly predicted the additional 43 arcseconds per century.  He considered this one of his greatest achievements.

Imagine my joy at the recognition of the feasibility of general covariance and at the result that the equations correctly yield the perihelion motion of Mercury. I was beside myself for several days in joyous excitement.

Physics

So, why does it happen? Is there an extra energy involved?  (Spoiler: Yes, in a way!)

For an elliptical orbit, we can think of the orbital motion as being made of two parts: an oscillation in azimuth (angle around the Sun), and an oscillation radially (in and out).  In Newtonian mechanics, the periods of these two oscillations are exactly equal.  That is, each time the planet completes one cycle around the Sun, it also exactly completes one cycle radially in and out.  Therefore in Newtonian mechanics there is no precession of the orbit (besides that which is caused by the influences of the other planets).

In general relativity this is no longer true.  The periods of the two oscillations are different!  To see why, it is easiest to examine the shape of the "effective potential", which you can think of as being just like the potential energy well around a spherical mass, except also taking into account the angular momentum of the orbiting body.  

If you are unfamiliar with the concept of a potential energy well, imagine a rolling landscape of hills and valleys.  If you place a ball at the bottom of a valley, it will just sit there.  It's in a stable equilibrium.  On the other hand, the top of a hill is an unstable equilibrium.  If you displace the ball slightly from the top of the hill and let it go, it will continue to roll down, exchanging gravitational potential energy for kinetic energy as it drops in height.  Finally, if you have some bowl-shaped valley and drop the ball from rest at some initial height, then it will roll down to the bottom, and (ignoring friction) keep going, rolling back up the other side until it reaches its former height, and then roll back again, oscillating back and forth indefinitely.  

Because the gravitational potential energy near Earth's surface is simply proportional to your altitude, if you make a plot of the potential energy of a ball in this landscape as a function of its position, it will be exactly the shape of the landscape.  This form of modelling of the potential is extremely useful.  In general, we can use "potential energy wells" as a way to understand motions of objects subjected to different kinds of attractive and repulsive forces, from balls rolling down hills to the vibrations of atoms bound together in molecules.

An elliptical orbit is an oscillation in an effective potential well, as well.  There is a hill to climb as you move outward due to the gravitational attraction of the star, and there is also a hill to climb as you move inward!  This is because the sideways motion is associated with angular momentum, and angular momentum acts like a repulsive centrifugal force.  By conservation of angular momentum, the sideways velocity increases as your orbit swings inward, increasing the centrifugal repulsion and making it harder to approach the center.  If the angular momentum is large enough, then the inward fall can be halted, and you swing back out again, making an orbit!  

So if we can compute the effective potential well for a planet orbiting the Sun, we can figure out some things about its motion.  The anomalous precession of an orbit can also be understood this way, by comparing the motion from the Newtonian potential with what happens in the general relativistic version of the potential.  Let's try it!

Math

The goal here is to try to understand how things work conceptually, and I will try to emphasize those concepts (especially physical concepts) as we go along.  The hard truth is that much of it will still require going through the math in order to access it, but I'll do my best to turn that math into something visualizable and comprehensible.

In Newtonian mechanics, the gravitational potential energy of a mass m a distance r around a spherical mass M is

This describes a simple curve which plummets downward indefinitely as you move to smaller radii.  As it should.  If you drop something near a massive object and give it no sideways motion (no angular momentum), then it will fall straight into it.

If we instead give the object some angular momentum L, then the potential is modified:

where μ is the reduced mass, which for a planet orbiting the Sun (m << M), reduces to approximately m.

Let's see what this looks like for Mercury:

Again the way to think about this is that the potential describes a landscape that a ball will frictionlessly roll across.  To visualize that, I added the large red dot to represent Mercury's position as a function of time.  The acceleration is determined by the slope of the potential, and I plotted the motion with 2 days per frame.  With 44 frames in all, this traces out one complete period of Mercury's orbit in the radial motion.  I also added a horizontal line to show the total energy of Mercury (its gravitational potential energy + kinetic), which is constant along the orbit.  Where this line is above the effective potential defines the range of distances from the Sun that Mercury's orbit will cover.  The vertical lines represent the extremes (Mercury's perihelion and aphelion distances).

Why is the Newtonian effective potential shaped this way?  
Because of Mercury's amount of angular momentum, the shape of the effective potential that it sees near the Sun is a valley with hills on either side.  The hill at large radii is due to the Sun's gravitational attraction, while the hill at small radii is due to centrifugal repulsion.  Even though the gravitational force grows stronger at closer distances, for an orbit the centrifugal force gets stronger more quickly, due to the conservation of angular momentum which increases your sideways speed.

A useful concept to keep in mind here is that the period of the radial oscillation depends on the curvature of the potential well.  If the well opens up more sharply, then the average acceleration is greater, and the period is shorter. 

And here's one other useful trick.  If the amplitude of the oscillation is not too large (does not reach too far away from the minimum of the well), then we can approximate the well as a parabola around the minimum.  Then for a parabolic well the motion is described very simply as a simple harmonic oscillator.  For Mercury this approximation isn't particularly good (the well is noticeably asymmetric over the region Mercury covers), but it's not terrible either.  

For a simple harmonic oscillator, the frequency is given by

which for the radial motion leads to

The frequency for the angular oscillation on the other hand is given by

Which leads to the exact same expression:

So we see in Newtonian mechanics the periods are exactly equal and there is no anomalous precession.  Now let's see how this gets modified when we move to General Relativity.

The Effective Potential in General Relativity

Mercury's Orbit in General Relativity

Going Further: The Bizarre Orbits in Strongly Curved Space-Time

Don't miss discussion of the original post on the forum: it has a lot of interesting questions and answers.